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Basic Life Insurance Mathematics Ragnar Norberg Version: September 2002

Contents 1 Introduction 1.1 Banking versus insurance . . . . . . . . . 1.2 Mortality . . . . . . . . . . . . . . . . . . 1.3 Banking . . . . . . . . . . . . . . . . . . . 1.4 Insurance . . . . . . . . . . . . . . . . . . 1.5 With-profit contracts: Surplus and bonus 1.6 Unit-linked insurance . . . . . . . . . . . . 1.7 Issues for further study . . . . . . . . . . .

. . . . . . .

5 5 7 9 11 14 16 17

2 Payment streams and interest 19 2.1 Basic definitions and relationships . . . . . . . . . . . . . . . . . 19 2.2 Application to loans . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Mortality 3.1 Aggregate mortality . . 3.2 Some standard mortality 3.3 Actuarial notation . . . 3.4 Select mortality . . . . .

. . . .

28 28 33 35 37

4 Insurance of a single life 4.1 Some standard forms of insurance . . 4.2 The principle of equivalence . . . . . 4.3 Prospective reserves . . . . . . . . . 4.4 Thiele’s differential equation . . . . . 4.5 Probability distributions . . . . . . . 4.6 The stochastic process point of view

. . . . . .

39 39 43 45 52 56 57

. . . laws . . . . . .

. . . .

5 Expenses 59 5.1 A single life insurance policy . . . . . . . . . . . . . . . . . . . . 59 5.2 The general multi-state policy . . . . . . . . . . . . . . . . . . . . 62 6 Multi-life insurances 63 6.1 Insurances depending on the number of survivors . . . . . . . . . 63

CONTENTS 7 Markov chains in life insurance 7.1 The insurance policy as a stochastic process . . . . . . . . . 7.2 The time-continuous Markov chain . . . . . . . . . . . . . . 7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Selection phenomena . . . . . . . . . . . . . . . . . . . . . . 7.5 The standard multi-state contract . . . . . . . . . . . . . . 7.6 Select mortality revisited . . . . . . . . . . . . . . . . . . . 7.7 Higher order moments of present values . . . . . . . . . . . 7.8 A Markov chain interest model . . . . . . . . . . . . . . . . 7.8.1 The Markov model . . . . . . . . . . . . . . . . . . . 7.8.2 Differential equations for moments of present values 7.8.3 Complement on Markov chains . . . . . . . . . . . . 7.9 Dependent lives . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Notions of positive dependence . . . . . . . . . . . . 7.9.3 Dependencies between present values . . . . . . . . . 7.9.4 A Markov chain model for two lives . . . . . . . . . 7.10 Conditional Markov chains . . . . . . . . . . . . . . . . . . 7.10.1 Retrospective fertility analysis . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

8 Probability distributions of present values 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Calculation of probability distributions of present values mentary methods . . . . . . . . . . . . . . . . . . . . . . 8.3 The general Markov multistate policy . . . . . . . . . . 8.4 Differential equations for statewise distributions . . . . . 8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . .

. . . ele. . . . . . . . . . . .

. . by . . . . . . . .

. . . . . . . . . . . . . . . . . .

67 67 68 73 77 79 86 89 94 94 95 98 100 100 101 103 103 106 106 109 109 110 111 112 116

9 Reserves 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 General definitions of reserves and statement of some relationships between them . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Description of payment streams appearing in life and pension insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Markov chain model . . . . . . . . . . . . . . . . . . . . . . . 9.5 Reserves in the Markov chain model . . . . . . . . . . . . . . . . 9.6 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119

10 Safety loadings and bonus 10.1 General considerations . . . . . . . . . . . 10.2 First and second order bases . . . . . . . . 10.3 The technical surplus and how it emerges 10.4 Dividends and bonus . . . . . . . . . . . . 10.5 Bonus prognoses . . . . . . . . . . . . . . 10.6 Examples . . . . . . . . . . . . . . . . . . 10.7 Including expenses . . . . . . . . . . . . .

145 145 146 147 149 153 158 161

. . . . . . .

122 126 126 131 139

CONTENTS

10.8 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 11 Statistical inference in the Markov chain model 11.1 Estimating a mortality law from fully observed life lengths 11.2 Parametric inference in the Markov model . . . . . . . . . 11.3 Confidence regions . . . . . . . . . . . . . . . . . . . . . . 11.4 More on simultaneous confidence intervals . . . . . . . . . 11.5 Piecewise constant intensities . . . . . . . . . . . . . . . . 11.6 Impact of the censoring scheme . . . . . . . . . . . . . . .

. . . . . .

167 167 172 176 177 179 183

12 Heterogeneity models 185 12.1 The notion of heterogeneity – a two-stage model . . . . . . . . . 185 12.2 The proportional hazard model . . . . . . . . . . . . . . . . . . . 187 13 Group life insurance 13.1 Basic characteristics of group insurance . 13.2 A proportional hazard model for complete claim records . . . . . . . . . . . . . . . . 13.3 Experience rated net premiums . . . . . . 13.4 The fluctuation reserve . . . . . . . . . . . 13.5 Estimation of parameters . . . . . . . . .

. . . . . . individual . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . policy and . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Hattendorff and Thiele 14.1 Introduction . . . . . . . . . . . . 14.2 The general Hattendorff theorem 14.3 Application to life insurance . . . 14.4 Excerpts from martingale theory

. . . .

198 198 199 201 205

15 Financial mathematics in insurance 15.1 Finance in insurance . . . . . . . . . . . . . . . . . . 15.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . 15.3 A Markov chain financial market - Introduction . . . 15.4 The Markov chain market . . . . . . . . . . . . . . . 15.5 Arbitrage-pricing of derivatives in a complete market 15.6 Numerical procedures . . . . . . . . . . . . . . . . . 15.7 Risk minimization in incomplete markets . . . . . . 15.8 Trading with bonds: How much can be hedged? . . . 15.9 The Vandermonde matrix in finance . . . . . . . . . 15.10Two properties of the Vandermonde matrix . . . . . 15.11Applications to finance . . . . . . . . . . . . . . . . . 15.12Martingale methods . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

212 212 213 218 219 226 229 229 232 235 236 237 240

. . . .

190 190 191 194 195 197

A Calculus

B Indicator functions

C Distribution of the number of occurring events

CONTENTS

D Asymptotic results from statistics

E The G82M mortality table

F Exercises

G Solutions to exercises

Chapter 1

Introduction 1.1

Banking versus insurance

A. The bank savings contract. Upon celebrating his 55th anniversary Mr. (55) (let us call him so) decides to invest money to secure himself economically in his old age. The first idea that occurs to him is to deposit a capital of S0 = 1 (e.g. one hundred thousand pounds) on a savings account today and draw the entire amount with earned compound interest in 15 years, on his 70th birthday. The account bears interest at rate i = 0.045 (4.5%) per year. In one year the capital will increase to S1 = S0 + S0 i = S0 (1 + i), in two years it will increase to S = S1 + S1 i = S0 (1 + i)2 , and so on until in 15 years it will have accumulated to S15 = S0 (1 + i)15 = 1.04515 = 1.935 . (1.1) This simple calculation takes no account of the fact that (55) will die sooner or later, maybe sooner than 15 years. Suppose he has no heirs (or he dislikes the ones he has) so that in the event of death before 70 he would consider his savings waisted. Checking population statistics he learns that about 75% of those who are 55 will survive to 70. Thus, the relevant prospects of the contract are: – with probability 0.75 (55) survives to 70 and will then possess S15 ; – with probability 0.25 (55) dies before 70 and loses the capital. In this perspective the expected amount at (55)’s disposal after 15 years is 0.75 S15 .

(1.2)

B. A small scale mutual fund. Having thought things over, (55) seeks to make the following mutual arrangement with (55)∗ and (55)∗∗ , who are also 55 years old and are in exactly the same situation as (55). Each of the three deposits S0 = 1 on the savings account, and those who survive to 70, if any, will then share the total accumulated capital 3 S15 equally. The prospects of this scheme are given in Table 1.1, where + and − signify survival and death, respectively, L70 is the number of survivors at age 70, and 5

CHAPTER 1. INTRODUCTION

Table 1.1: Possible outcomes of a savings scheme with three participants. (55) (55)∗ + + + + − − − −

+ + − − + + − −

(55)∗∗

L70

3 S15 /L70

Probability

+ − + − + − + −

3 2 2 1 2 1 1 0

S15 1.5 S15 1.5 S15 3S15 1.5 S15 3 S15 3 S15 undefined

0.75 · 0.75 · 0.75 = 0.422 0.75 · 0.75 · 0.25 = 0.141 0.75 · 0.25 · 0.75 = 0.141 0.75 · 0.25 · 0.25 = 0.047 0.25 · 0.75 · 0.75 = 0.141 0.25 · 0.75 · 0.25 = 0.047 0.25 · 0.25 · 0.75 = 0.047 0.25 · 0.25 · 0.25 = 0.016

3 S15 /L70 is the amount at disposal per survivor (undefined if L70 = 0). There are now the following possibilities: – with probability 0.422 (55) survives to 70 together with (55)∗ and (55)∗∗ and will then possess S15 ; – with probability 2 · 0.141 = 0.282 (55) survives to 70 together with one more survivor and will then possess 1.5 S15 ; – with probability 0.047 (55) survives to 70 while both (55)∗ and (55)∗∗ die (may they rest in peace) and he will cash the total savings 3S15 ; – with probability 0.25 (55) dies before 70 and will get nothing. This scheme is superior to the one described in Paragraph A, with separate individual savings contracts: If (55) survives to 70, which is the only scenario of interest to him, he will cash no less than the amount S15 he would cash under the individual scheme, and it is likely that he will get more. As compared with (1.2), the expected amount at (55)’s disposal after 15 years is now 0.422 · S15 + 0.282 · 1.5 · S15 + 0.047 · 3 S15 = 0.985 S15 . The point is that under the present scheme the savings of those who die are bequeathed to the survivors. Thus the total savings are retained for the group so that nothing is left to others unless the unlikely thing happens that the whole group goes extinct within the term of the contract. This is essentially the kind of solidarity that unites the members of a pension fund. From the point of view of the group as a whole, the probability that all three participants will die before 70 is only 0.016, which should be compared to the probability 0.25 that (55) will die and lose everything under the individual savings program.

C. A large scale mutual scheme. Inspired by the success of the mutual fund idea already on the small scale of three participants, (55) starts to play with the idea of extending it to a large number of participants. Let us assume that a total number of L55 persons, who are in exactly the same situation as (55), agree to join a scheme similar to the one described for the three. Then the

CHAPTER 1. INTRODUCTION

total savings after 15 years amount to L55 S15 , which yields an individual share equal to L55 S15 L70

(1.3)

to each of the L70 survivors if L70 > 0. By the so-called law of large numbers, the proportion of survivors L70 /L55 tends to the individual survival probability 0.75 as the number of participants L55 tends to infinity. Therefore, as the number of participants increases, the individual share per survivor tends to 1 S15 , 0.75

(1.4)

and in the limit (55) is faced with the following situation: 1 S15 ; – with probability 0.75 he survives to 70 and gets 0.75 – with probability 0.25 he dies before 70 and gets nothing. The expected amount at (55)’s disposal after 15 years is 0.75

1 S15 = S15 , 0.75

(1.5)

the same as (1.1). Thus, the bequest mechanism of the mutual scheme has raised (55)’s expectations of future pension to what they would be with the individual savings contract if he were immortal. This is what we could expect since, in an infinitely large scheme, some will survive to 70 for sure and share the total savings. All the money will remain in the scheme and will be redistributed among its members by the lottery mechanism of death and survival. The fact that L70 /L55 tends to 0.75 as L55 increases, and that (1.3) thus stabilizes at (1.4), is precisely what is meant by saying that “insurance risk is diversifiable”. The risk can be eliminated by increasing the size of the portfolio.

1.2

Mortality

A. Life and death in the classical actuarial perspective. Insurance mathematics is widely held to be boring. Hopefully, the present text will not support that prejudice. It must be admitted, however, that actuaries use to cheer themselves up with jokes like: “What is the difference between an English and a Sicilian actuary? Well, the English actuary can predict fairly precisely how many English citizens will die next year. Likewise, the Sicilian actuary can predict how many Sicilians will die next year, but he can tell their names as well.” The English actuary is definitely the more typical representative of the actuarial profession since he takes a purely statistical view of mortality. Still he is able to analyze insurance problems adequately since what insurance is essentially about, is to average out the randomness associated with the individual risks. Contemporary life insurance is based on the paradigm of the large scheme (diversification) studied in Paragraph 1.1C. The typical insurance company

CHAPTER 1. INTRODUCTION

serves tens and some even hundreds of thousands of customers, sufficiently many to ensure that the survival rates are stable as assumed in Paragraph 1.1C. On the basis of statistical investigations the actuary constructs a so-called decrement series, which takes as it starting point a large number `0 of new-born and, for each age x = 1, 2, . . ., specifies the number of survivors, `x . Table 1.2: Excerpt from the mortality table G82M x:

`x : dx : qx : px :

100 000 58 .000579 .999421

98 083 119 .001206 .998794

91 119 617 .006774 .993226

82 339 1 275 .015484 .984516

65 024 2 345 .036069 .963931

37 167 3 111 .083711 .916289

9 783 1 845 .188617 .811383

Table 1.2 is an excerpt of the table used by Danish insurers to describe the mortality of insured Danish males. The second row in the table lists some entries of the decrement series. It shows e.g. that about 65% of all new-born will celebrate their 70th anniversary. The number of survivors decreases with age: `x ≥ `x+1 . The difference dx = `x − `x+1 is the number of deaths at age x (more precisely, between age x and age x + 1). These numbers are shown in the third row of the table. It is seen that the number of deaths peaks somewhere around age 80. From this it cannot be concluded that 80 is the “most dangerous age”. The actuary measures the mortality at any age x by the one-year mortality rate qx =

dx , `x

which tells how big proportion of those who survive to age x will die within one year. This rate, shown in the fourth row of the table, increases with the age. For instance, 8.4% of the 80 years old will die within a year, whereas 18.9% of the 90 years old will die within a year. The bottom row shows the one year survival rates `x+1 = 1 − qx . px = `x We shall present some typical forms of products that an insurance company can offer to (55) and see how they compare with the corresponding arrangements, if any, that (55) can make with his bank.

CHAPTER 1. INTRODUCTION

1.3

Banking

A. Interest. Being unable to find his perfect matches (55)∗ , (55)∗∗ ,..., our hero (55) abandons the idea of creating a mutual fund and resumes discussions with his bank. The bank operates with annual interest rate it in year t = 1, 2, .... Thus, a unit S0 = 1 deposited at time 0 will accumulate with compound interest as follows: In one year the capital increases to S1 = S0 + S0 i1 = 1 + i1 , in two years it increases to S2 = S1 + S1 i2 = (1 + i1 )(1 + i2 ), and in t years it increases to St = (1 + i1 ) · · · (1 + it ) ,

(1.6)

called the t-year accumulation factor. Accordingly, the present value at time 0 of a unit withdrawn in j years is Sj −1 =

1 , Sj

(1.7)

called the j-year discount factor since it is what the bank would pay you at time 0 if you sell to it (discount) a default-free claim of 1 at time j. Similarly, the value at time t of a unit deposited at time j < t is (1 + ij+1 ) · · · (1 + it ) =

St , Sj

called the accumulation factor over the time period from j to t, and the value at time t of a unit withdrawn at time j > t is St 1 = , (1 + it+1 ) · · · (1 + ij ) Sj the discount factor over the time period from t to j. In general, the value at time t of a unit due at time j is St Sj −1 , an accumulation factor if j < t and a discount factor if j > t (and of course 1 if j = t). From (1.6) it follows that St = St−1 (1 + it ), hence it =

St − St−1 , St−1

which expresses the interest rate in year t as the relative increase of the balance in year t. B. Saving in the bank. A general savings contract over n years specifies that at each time t = 0, ..., n (55) is to deposit an amount ct (contribution) and withdraw an amount bt (benefit). The net amount of deposit less withdrawal at time t is ct − bt . At any time t the cash balance of the account, henceforth also

CHAPTER 1. INTRODUCTION

called the retrospective reserve, is the total of past (including present) deposits less withdrawals compounded with interest, Ut = S t

t X

Sj −1 (cj − bj ) .

(1.8)

j=0

It develops in accordance with the “forward” recursive scheme Ut = Ut−1 (1 + it ) + ct − bt ,

(1.9)

t = 1, 2, . . . , n, commencing from U0 = c 0 − b 0 . Each year (55) will receive from the bank a statement of account with the calculation (1.9), showing how the current balance emerges from the previous balance, the interest earned meanwhile, and the current movement (deposit less withdrawal). The balance of a savings account must always be non-negative, Ut ≥ 0 ,

(1.10)

and at time n, when the contract terminates and the account is closed, it must be null, Un = 0 .

(1.11)

In the course of the contract the bank must maintain a so-called prospective reserve to meet its future liabilities to the customer. At any time t the adequate reserve is Vt = S t

n X

Sj −1 (bj − cj ) ,

(1.12)

j=t+1

the present value of future withdrawals less deposits . Similar to (1.9), the prospective reserve is calculated by the “backward” recursive scheme Vt = (1 + it+1 )−1 (bt+1 − ct+1 + Vt+1 ) ,

(1.13)

t = n − 1, n − 2, . . . , 0, starting from Vn = 0 . The constraint (1.11) is equivalent to n X j=0

Sj −1 cj −

n X j=0

Sj −1 bj = 0 ,

(1.14)

CHAPTER 1. INTRODUCTION

which says that the discounted value of deposits must be equal to the discounted value of the withdrawals. It implies that, at any time t, the retrospective reserve equals the prospective reserve, Ut = V t , as is easily verified. expression (1.8) with t = n into (1.11), Pn (Insert Ptthe defining Pn split the sum j=0 into j=0 + j=t+1 , and multiply with St /Sn , to arrive at Ut − Vt = 0.) C. The endowment contract. The bank proposes a savings contract according to which (55) saves a fixed amount c annually in 15 years, at ages 55,...,69, and thereafter withdraws b = 1 (one hundred thousand pounds, say) at age 70. Suppose the annual interest rate is fixed and equal to i = 0.045, so that the accumulation factor in t years is St = (1 + i)t , the discount factor in j years is Sj −1 = (1 + i)−j , and the time t value of a unit due at time j is St Sj −1 = (1 + i)t−j . For the present contract the equivalence requirement (1.14) is 14 X (1 + i)−j c − (1 + i)−15 1 = 0 , j=0

from which the bank determines (1 + i)−15 c = P14 = 0.04604 , −j j=0 (1 + i)

(1.15)

Due to interest, this amount is considerably smaller than 1/15 = 0.06667, which is what (55) would have to save per year if he should choose to tuck the money away under his mattress.

1.4

Insurance

A. The life endowment. Still, to (55) 0.04604 (four thousand six hundred and four pounds) is a considerable expense. He believes in a life before death, and it should be blessed with the joys that money can buy. He talks to an insurance agent, and is delighted to learn that, under a life annuity policy designed precisely as the savings scheme, he would have to deposit an annual amount of only 0.03743 (three thousand seven hundred and forty three pounds). The insurance agent explains: The calculations of the bank depend only on the amounts ct − bt and would apply to any customer (x) who would enter into the same contract at age x, say. Thus, to the bank the customer is really an unknown Mr. X. To the insurance company, however, he is not just Mr. X, but the significant Mr. (x) now x years old. Working under the hypothesis that (x) is one of the `x survivors at age x in the decrement series and that they all hold

CHAPTER 1. INTRODUCTION

identical contracts, the insurer offers (x) a general life annuity policy whereby each deposit or withdrawal is conditional on survival. For the entire portfolio the retrospective reserve at time t is Utp

= St

t X

Sj −1 (cj − bj ) `x+j

(1.16)

j=0

p = Ut−1 (1 + it ) + (ct − bt ) `x+t .

(1.17)

The prospective portfolio reserve at time t is Vtp

= St

n X

Sj −1 (bj − cj )`x+j

(1.18)

j=t+1 p = (1 + it+1 )−1 ((bt+1 − ct+1 )`x+t+1 + Vt+1 ).

(1.19)

In particular, for the life endowment analogue to (55)’s savings contract, the only payments are ct = c for t = 0, . . . , 14 and b15 = 1. The equivalence requirement (1.14) becomes 14 X

(1 + i)−j c `55+j − (1 + i)−15 1 `70 = 0 ,

(1.20)

j=0

from which the insurer determines (1 + i)−15 `70 c = P14 = 0.03743 . −j ` 55+j j=0 (1 + i)

(1.21)

Inspection of the expressions in (1.15) and (1.21) shows that the latter is smaller due to the fact that `x is decreasing. This phenomenon is known as mortality bequest since the savings of the deceased are bequeathed to the survivors. We shall pursue this issue in Paragraph C below. B. A life assurance contract. Suppose, contrary to the former hypothesis, that (55) has dependents whom he cares for. Then he might be concerned that, if he should die within the term of the contract, the survivors in the pension scheme will be his heirs, leaving his wife and kids with nothing. He figures that, in the event of his untimely death before the age of 70, the family would need a down payment of b = 1 (one hundred thousand pounds) to compensate the loss of their bread-winner. The bank can not help in this matter; the benefit of b would have to be raised immediately since (55) could die tomorrow, and it would be meaningless to borrow the money since full repayment of the loan would be due immediately upon death. The insurance company, however, can offer (55) a so-called term life assurance policy that provides the wanted death benefit against an affordable annual premium of c = 0.01701. The equivalence requirement (1.14) now becomes 14 X j=0

(1 + i)−j c `55+j −

15 X (1 + i)−j 1 d55+j−1 = 0 , j=1

(1.22)

CHAPTER 1. INTRODUCTION from which the insurer determines P15 −j d55+j−1 j=1 (1 + i) = 0.01701 . c = P14 −j ` 55+j j=0 (1 + i)

(1.23)

C. Individual reserves and mortality bequest. In the insurance schemes described above the contracts of deceased members are void, and the reserves of the portfolio are therefore to be shared equally between the survivors at any time. Thus, we introduce the individual retrospective and prospective reserves at time t, Ut = Utp /`x+t , Vt = Vtp /`x+t . Since we have established that Ut = Vt , we shall henceforth be referring to them as the individual reserve or just the reserve. For the general pension insurance contract in Paragraph A we get from (1.17) that the individual reserve develops as Ut

`x+t−1 (1 + it ) + (ct − bt ) `x+t dx+t−1 = Ut−1 1 + (1 + it ) + (ct − bt ) . lx+t = Ut−1

(1.24)

The bequest mechanism is clearly seen by comparing (1.24) to (1.9): the additional term Ut−1 (1 + it ) dx+t−1 /lx+t in the latter is precisely the share per survivor of the savings left over to them by those who died during the year. Virtually, the mortality bequest acts as an increase of the interest rate. Table 1.3 shows how the reserve develops for the endowment contracts offered by the bank and the insurance company, respectively. It is seen that the insurance scheme requires a smaller reserve than the bank savings scheme. Table 1.3: Reserve Ut = Vt for bank savings account and for life endowment insurance t:

Savings account: Life endowment:

0.04604 0.03743

0.25188 0.21008

0.56577 0.49812

0.95694 0.92523

For the life assurance described in Paragraph B we obtain similarly that the individual reserve develops as as shown in Table 1.4. D. Insurance risk in a finite portfolio. The perfect balance in (1.20) and (1.22) rests on the hypothesis that the decrement series `x+t follows the pattern of an infinitely large portfolio. In a finite portfolio, however, the factual numbers of survivors, Lx+t , will be subject to randomness and will be determined by

CHAPTER 1. INTRODUCTION

Table 1.4: Reserve Ut = Vt for a term life assurance of 1 against level premium in 15 years from age 55 t:

0.01701

0.04460

0.06010

0.03170

the survival probabilities px+t (some of which are) shown in Table 1.2. The difference between discounted premiums and discounted benefits, D=

14 X

(1 + i)−j 0.0374 L55+j − (1 + i)−15 L70 ,

j=0

will be a random quantity. It will have expected value 0, and its standard deviation measures how much insurance risk is left due to “imperfect diversification” in a finite portfolio. An easy exercise in probability calculus shows that the standard deviation of D/L55 is √L1 0.1685. It tends to 0 as L55 goes to infinity. 55 For the term insurance contract the corresponding quantity is √L1 0.3478, 55 indicating that term insurance is a more risky business than life endowment.

1.5

With-profit contracts: Surplus and bonus

A. With-profit contracts. Insurance policies are long term contracts, with time horizons wide enough to capture significant variations in interest and mortality. For simplicity we shall focus on interest rate uncertainty and assume that the mortality law remains unchanged over the term of the contract. We will discuss the issue of surplus and bonus in the framework of the life endowment contract considered in Paragraph 1.4.A. At time 0, when the contract is written with benefits and premiums binding to both parties, the future development of the interest rates it is uncertain, and it is impossible to foresee what premium level c will establish the required equivalence 14 X

Sj −1 c `55+j = S15 −1 1 `70 ,

(1.25)

j=0

with Sj = (1 + i1 ) · · · (1 + ij ) . If it should turn out that, due to adverse development of interest and mortality, premiums are insufficient to cover the benefit, then there is no way the insurance company can avoid a loss; it cannot reduce the benefit and it cannot increase the premiums since these were irrevocably set out in the contract at time 0. The only way the insurance company can prevent such a loss, is to charge a premium

CHAPTER 1. INTRODUCTION

’on the safe side’, high enough to be adequate under all likely scenarios. Then, if everything goes well, a surplus will accumulate. This surplus belongs to the insured and is to be repaid as so-called bonus, e.g. as increased benefits or reduced premiums. B. First order basis. The usual way of setting premiums to the safe side is to base the calculation of the premium level and the reserves on a provisional first order basis, assuming a fixed annual interest rate i∗ , which represents a worst case scenario and leads to higher premium and reserves than are likely to be needed. The corresponding accumulation factor is St∗ = (1 + i∗ )t . The individual reserve based on the prudent first order assumptions is called the first order reserve, and we denote it by Vt∗ as before. The premiums are determined so as to satisfy equivalence under the first order assumption. C. Surplus. At any time t we define the technical surplus Qt as the difference between the retrospective reserve under the factual interest development and the retrospective reserve under the first order assumption: Qt

= St

t X

Sj −1 c `55+j − St∗

j=0

= St

t−1 X

t X

Sj∗ −1 c `55+j

j=0

Sj −1 c `55+j − St∗

j=0

t−1 X

Sj∗ −1 c `55+j .

j=0

∗ Setting St = St−1 (1 + it ) and St∗ = St−1 (1 + i∗ ), writing 1 + i∗ = 1 + it − (it − i∗ ) in the latter, and rearranging a bit, we find that Qt develops as ∗ Qt = Qt−1 (1 + it ) + Vt−1 (it − i∗ ) `55+t−1 ,

(1.26)

commencing from Q0 = 0 . The contribution to the technical surplus in year t is ∗ Vt−1 (it − i∗ ) `55+t−1 ,

which is easy to interpret: it is precisely the interest earned on the reserve in excess of what has been assumed under the prudent first order assumption. The surplus is to be redistributed as bonus. Several bonus schemes are used ∗ in practice. One can repay currently the contribution Vt−1 (it − i∗ ) `55+t−1 as so-called cash bonus (a premium deductible), whereby each survivor at time t will receive ∗ Vt−1 (it − i∗ ) `55+t−1 /`55+t . Another possibility is to postpone repayment until the term of the contracts and grant so-called terminal bonus to the survivors (an added benefit), the amount

CHAPTER 1. INTRODUCTION per survivor being b+ given by S15

15 X

∗ Sj −1 Vj−1 (ij − i∗ ) `55+j = `70 b+ .

j=1

Between these two solutions there are countless other possibilities. In any case, the point is that the financial risk can be eliminated: the insurer observes the development of the factual interest and only in arrears repays the insured so as to restore equivalence on the basis of the factual interest rate development. This works well provided the first order interest rate is set on the safe side so that it ≥ i∗ for all t. There is a problem, however: Negative bonus can never be applied. Therefore the insurer will suffer a loss if the factual interest falls short of the technical interest rate. In this perspective cash bonus is the most risky solution and terminal bonus is the least risky solution. If the financial market is sufficiently rich in assets, then the interest rate guarantee that is thus inherent in the with-profit policy can be priced, and the insured can be charged an extra premium to cover it. This would ultimately eliminate the financial risk by diversifying, not only the insurance portfolio, but also the investment portfolio.

1.6

Unit-linked insurance

A quite different way of going about the financial risk is the so-called unit-linked contract. As the name indicates, the idea is to relate payments directly to the development of the investment portfolio, i.e. the interest rate. Consider the balance equation for an endowment of b against premium ct in year t = 1, . . . , 14: S15

14 X

St −1 ct `55+t − b `70 = 0 .

(1.27)

t=0

A perfect link between payments and investments performance is obtained by letting the premiums and the benefit be inflated by the index S, ct = S t c , and b15 = S15 1 . Here c is a baseline premium, which is to be determined. Then (1.27) becomes S15

14 X

St −1 St c `55+t − S15 `70 = 0 ,

t=0

which reduces to

14 X t=0

c `55+t − `70 = 0 ,

CHAPTER 1. INTRODUCTION and we find

`70 . c = P14 t=0 `55+t

Again financial risk has been perfectly eliminated and diversification of the insurance portfolio is sufficient to establish balance between premiums and benefits. Perfect linking as defined here is not common in practice. Presumably, remnants of social welfare thinking have led insurers to modify the unit-linked concept in various ways, typically by introducing a guarantee on the sum insured to the effect that it cannot be less than 1 (say). Also the premium is usually not index-linked. Under such modified variations of the unit-linked policy one cannot in general obtain balance by the simple device above. However, the problem can in principle be resolved by calculating the price of the financial claim thus introduced and to charge the insured with the needed additional premium.

1.7

Issues for further study

The simple pieces of actuarial reasoning in the previous sections involve two constituents, interest and mortality, and these are to be studied separately in the two following chapters. Next we shall escalate the discussion to more complex situations. For instance, suppose (55) wants a life insurance that is paid out only if his wife survives him, or with a sum insured that depends on the number of children that are still alive at the time of his death. Or he may demand a pension payable during disability or unemployment. We need also to study the risk associated with insurance, which is due to the uncertain developments of the insurance portfolio and the investment portfolio: the deaths in a finite insurance portfolio do not follow the mortality table (1.2) exactly, and the interest earned on the investments may differ from the assumed 4.5% per year, and neither can be predicted precisely at the outset when the policies are issued. In a scheme of the classical mutual type the problem was how to share existing money in a fair manner. A typical insurance contract of today, however, specifies that certain benefits will be paid contingent on certain events related only to the individual insured under the contract. An insurance company working with this concept in a finite portfolio, with imperfect diversification of insurance risk, faces a risk of insolvency as indicated in Paragraph 1.3.D. In addition comes the financial risk, and ways of getting around that were indicated in Sections 1.5 and 1.6. The total risk has to be controlled in some way. With these issues in mind, we now commence our studies of the theory of life insurance. The reader is advised to consult the following authoritative textbooks on the subject: [6] (a good classic – sharpen your German!), [4], [29] (lexicographic, treats virtually every variation of standard insurance products, and includes a good chapter on population theory), [45] (an excellent early text based on probabilistic models, placing emphasis on risk considerations), [11], [15] (an

CHAPTER 1. INTRODUCTION

original approach to the field – sharpen your French!), and [23] (the most recent of the mentioned texts, still classical in its orientation).

Chapter 2

Payment streams and interest 2.1

Basic definitions and relationships

A. Streams of payments. What is money? In lack of a precise definition you may add up the face values of the coins and notes you find in your purse and say that the total amount is your money. Now, if you do this each time you open your purse, you will realize that the development of the amount over time is important. In the context of insurance and finance the time aspect is essential since payments are usually regulated by a contract valid over some period of time. We will give precise mathematical content to the notion of payment streams and, referring to Appendix A, we deal only with their properties as functions of time and do not venture to discuss their possible stochastic properties for the time being. To fix ideas and terminology, consider a financial contract commencing at time 0 and terminating at a later time n (≤ ∞), say, and denote by At the total amount paid in respect of the contract during the time interval [0, t]. The payment function {At }t≥0 is assumed to be the difference of two non-decreasing, finite-valued functions representing incomes and outgoes, respectively, and is thus of finite variation (FV). Furthermore, the payment function is assumed to be right-continuous (RC). From a practical point of view this assumption is just a matter of convention, stating that the balance of the account changes at the time of any deposit or withdrawal. From a mathematical point of view it is convenient, since payment functions can then serve as integrators. In fact, we shall restrict attention to payment functions that are piece-wise differentiable (PD): Z t X At = A 0 + aτ dτ + ∆Aτ , (2.1) 0

0 t] = 1 − F (t) .

(3.2)

Fig. 3.1 shows F and F¯ for the mortality law G82M used by Danish life insurers as a basis for calculating premiums for insurances on male lives. Find the median life length and some other percentiles of this life distribution by inspection of the graphs! We assume that F is absolutely continuous and denote the density by f ; d d F (t) = − F¯ (t) . (3.3) dt dt The density of the distribution in Fig. 3.1 is depicted in Fig. 3.2. Find the mode by inspection of the graph! Can you already at this stage figure why the median and the mode of F in Fig. 3.1 appear to exceed those of the mortality law of the Danish male population? f (t) =

CHAPTER 3. MORTALITY

6 1 • .9 • .8 • .7 • .6 • .5 • .4 • .3 • .2 • .1 • 0 • 0

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Figure 3.1: The G82M mortality law: F broken line, F¯ whole line.

.04 •6 .03 • .02 • .01 • 0

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Figure 3.2: The density f for the G82M mortality law.

CHAPTER 3. MORTALITY .4 •6 .3 • .2 • .1 • 0

• 0

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• 80

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• 100

Figure 3.3: The force of mortality µ for the G82M mortality law. B. The force of mortality. The density is the derivative of −F¯ , see (3.3). When dealing with non-negative random variables representing life lengths, it is convenient to work with the derivative of − ln F¯ , d f (t) (3.4) {− ln F¯ (t)} = ¯ , dt F (t) which is well defined for all t such that F¯ (t) > 0. For small, positive dt we have µ(t) =

f (t)dt P[t < T ≤ t + dt] µ(t)dt = ¯ = P[T ≤ t + dt | T > t] . = P[T > t] F (t) (In the second equality we have neglected a term o(dt) such that o(dt)/dt → 0 as dt & 0.) Thus, for a person aged t, the probability of dying within dt years is (approximately) proportional to the length of the time interval, dt. The proportionality factor µ(t) depends on the attained age, and is called the force of mortality at age t. It is also called the mortality intensity or hazard rate at age t, the latter expression stemming from reliability theory, which is concerned with the durability of technical devices. Fig 3.3 shows the force of mortality corresponding to F in Fig. 3.1. Assess roughly the probability that a t years old person will die within one year for t = 60, 70, 80, 90! Integrating (3.4) from 0 to t and using F¯ (0) = 1, we obtain F¯ (t) = e− Relation (3.4) may be cast as

Rt 0

f (t) = F¯ (t)µ(t) = e−

(3.5)

Rt 0

µ(t) ,

(3.6)

which says that the probability f (t)dt of dying in the age interval (t, t+dt) is the product of the probability F¯ (t) of survival to t and the conditional probability µ(t)dt of then dying before age t + dt. The functions F , F¯ , f , and µ are equivalent representations of the mortality law; each of them corresponds one-to-one to any one of the others. R∞ Since F¯ (∞) = 0, we must have 0 µ = ∞. Thus, if there is a finite highest Rt attainable age ω such that F¯ (ω) = 0 and F¯ (t) > 0 for t < ω, then 0 µ % ∞ as t % ω. If, moreover, µ is non-decreasing, we must also have limt%ω µ(t) = ∞.

CHAPTER 3. MORTALITY

C. The distribution of the remaining life length. Let Tx denote the remaining life length of an individual chosen at random from the x years old members of the population. Then Tx is distributed as T − x, conditional on T > x, and has cumulative distribution function F (t|x) = P[T ≤ x + t | T > x] =

F (x + t) − F (x) 1 − F (x)

and survival function F¯ (x + t) , F¯ (t|x) = P[T > x + t | T > x] = ¯ F (x)

(3.7)

which are well defined for all x such that F¯ (x) > 0. The density of this conditional distribution is f (t|x) =

f (x + t) . F¯ (x)

(3.8)

(3.9)

Alternatively, we could insert (3.5) into (3.7) to obtain F¯ (t|x) = e−

R x+t x

µ(y) dy

= e−

Rt 0

µ(x+τ )dτ

(3.10)

which by the general relation (3.5) entails (3.9). Relation (3.9) explains why the force of mortality is particularly handy; it depends only on the attained age x + t, whereas the conditional density in (3.8) depends in general on x and t in a more complex manner. Thus, the properties of all the conditional survival distributions are summarized by one simple function of the total age only. Figs. 3.4 – 3.6 depict the functions F (t|70), F¯ (t|70), f (t|70), and µ(t|70) = µ(70 + t) derived from the life time distribution in Fig. 3.1. Observe that the first three of these functions are obtained simply by scaling up the corresponding graphs in Figs. 3.1 – 3.2 by the factor 1/F¯ (70) over the interval (70, ∞). The force of mortality remains unchanged, however.

D. Expected values in life distributions. Let T be a non-negative random variable with absolutely continuous distribution function F , and let G : R+ → R be a PD and RC function such that E[G(T )] exists and is finite. Integrating by parts in the defining expression Z ∞ E[G(T )] = G(τ ) dF (τ ) , 0

CHAPTER 3. MORTALITY

6 1 • .9 • .8 • .7 • .6 • .5 • .4 • .3 • .2 • .1 • 0 • 0

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Figure 3.4: Conditional distribution of remaining life length for the G82M mortality law: F (t|70) broken line, F¯ (t|70) whole line. .05 •6 .04 • .03 • .02 • .01 • 0

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Figure 3.5: Conditional density of remaining life length f (t|70) for the G82M mortality law. .4 •6 .3 • .2 • .1 • 0

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Figure 3.6: The force of mortality µ(t|70) = µ(70 + t), t > 0, for the G82M mortality law.

CHAPTER 3. MORTALITY we find E[G(T )] = G(0) +

∞

F¯ (τ ) dG(τ ) .

(3.11)

Taking G(t) = tk we get E[T k ] = k and, in particular,

E[T ] =

∞

tk−1 F¯ (t) dt ,

(3.12)

∞

F¯ (t)dt .

(3.13)

The expected remaining life time for an x years old person is Z ∞ F¯ (t | x) dt . e¯x =

(3.14)

From (3.10) it is seen that F¯ (t | x) is a decreasing function of x for fixed t if µ is an increasing function. Then e¯x is a decreasing function of x. One can easily construct mortality laws for which F¯ (t | x) and e¯x are not decreasing functions of x. Consider the more general function  0 ≤ t < a,  0, (t − a)k , a ≤ t < b, G(t) = ((t ∧ b) − (t ∧ a))k = (3.15)  (b − a)k , b ≤ t, that is, dG(t) = k(t − a)k−1 dt for a < t < b and 0 elsewhere. It is realized that G(T ) is the kth power of the number of years lived between age a and age b. From (3.11) we obtain Z b E[G(T )] = k (t − a)k−1 F¯ (t) dt , (3.16) a

In particular, the expected number of years lived between the ages of a and Rb b is a F¯ (t) dt, which is the area between the t-axis and the survival function in the interval from a to b. The formula can be motivated directly by noting that F¯ (t) dt is the expected number of years survived in the small time interval (t, t + dt) and using that the “expected value of the sum is the sum of the expected values”.

3.2

Some standard mortality laws

A. The exponential distribution. Suppose the force of mortality is µ(t) = λ, independent of the age. This means there are no wear-out effects; each morning when you wake up (if you wake up) life starts anew with the same prospects of longevity as for a new-born. Then the survival function (3.5) becomes F¯ (t) = e−λt ,

(3.17)

CHAPTER 3. MORTALITY

Figure 3.7: Two exponential laws with intensities λ1 and λ2 such that λ1 < λ2 ; F¯1 and F¯2 whole line, f1 and f2 broken line. and the density (3.6) becomes f (t) = λe−λt .

(3.18)

Thus, T is exponentially distributed with parameter λ. The conditional survival function (3.10) becomes F¯ (t | x) = e−λt , hence F¯ (t | x) = F¯ (t) ,

(3.19)

the same as (3.17). The exponential distribution is a suitable model for certain technical devices like bulbs and electronic components. Unfortunately, it is not so apt for description of human lives. One could arrive at the exponential distribution by specifying that (3.19) be valid for all x and t, that is, the probability of surviving another t years is independent of the age x. Then, from the general relation (3.7) we get F¯ (x + t) = F¯ (x)F¯ (t)

(3.20)

for all non-negative x and t. It follows by induction that for each pair of positive ¯ 1 m ¯ m n integers m and n, F¯ ( m n ) = F ( n ) = F (1) , hence F (t) = F¯ (1)t

(3.21)

for all positive rational t. Since F¯ is right-continuous, (3.21) must hold true for all t > 0. Putting F¯ (1) = e−λ , we arrive at (3.17). Fig. 3.7 shows the survival function and the density for two different values of λ.

CHAPTER 3. MORTALITY

B. The Weibull distribution. The intensity of this distribution is of the form µ(t) = βα−β tβ−1 ,

(3.22)

α, β > 0. The corresponding survival function is F¯ (t) = exp −( αt )β . If β > 1, then µ(t) is increasing, and if β < 1, then µ(t) is decreasing. If β = 1, the Weibull law reduces to the exponential law. Draw the graphs of F¯ and f for some different choices of α and β! We have µ(x + t) = βα−β (x + t)β−1 , and, by virtue of (3.22), F¯ (t | x) is not a Weibull law.

C. The Gompertz-Makeham distribution. This distribution is widely used as a model for survivorship of human lives, especially in the context of life insurance. Thus, as it will be frequently referred to, we shall use the acronym G-M for this law. Its mortality intensity is of the form µ(t) = α + βeγt ,

(3.23)

α, β ≥ 0. The corresponding survival function is Z t F¯ (t) = exp − (α + βeγs )ds = exp −αt − β(eγt − 1)/γ .

(3.24)

If β > 0 and γ > 0, then µ(t) is an increasing function of t. The constant term α accounts for age-independent causes of death like certain accidents and epidemic diseases, and the term βeγt accounts for all kinds of wear-out effects due to aging. We have µ(x + t) = α + βeγx eγt , and so (3.23) shows that F¯ (t|x) is also a G-M survival function with parameters α, βeγx , γ. The special case α = 0 is referred to as the (pure) Gompertz law. The G82M mortality law depicted in Fig. 3.1 is the G-M law with α = 5 · 10−4 ,

β = 7.5858 · 10−5 ,

γ = ln(1.09144) .

(3.25)

Table E.1 in Appendix E shows µ(t), F¯ (t) and f (t) for integer t.

3.3

Actuarial notation

A. Actuaries in all countries – unite! The International Association of Actuaries (IAA) has laid down a standard notation, which is generally accepted among actuaries all over the world. Familiarity with this notation is a must for anyone who wants to communicate in writing or reading with actuaries, and we shall henceforth adopt it in those simple situations where it is applicable.

CHAPTER 3. MORTALITY

B. A list of some standard symbols. According to the IAA standard, the quantities introduced so far are denoted as follows: t qx t px

µx+t

= F (t | x) , = F¯ (t | x) ,

(3.26) (3.27)

= µ(x + t) .

(3.28)

In particular, t q0 = F (t) and t p0 = F¯ (t). One-year death and survival probabilities are abbreviated as qx = 1 qx ,

px = 1 px .

(3.29)

Frequently used is also the “n-year deferred probability of death within m years”, n|m qx

= m+n qx − n qx = n px − m+n px = n px m qx+n .

The formulas in Section 3.1 are easily translated, e.g. Z t p = exp(− µx+τ dτ ), t x

(3.30)

(3.31)

f (t | x) = e¯x

t px µx+t , Z ∞

t px dt.

(3.32) (3.33)

Frequently actuaries work with expected numbers of survivors instead of probabilities. Consider a population of l0 new-born who are subject to the same law of mortality given by (3.28). The expected number of survivors at age x is lx = l 0 x p 0 .

(3.34)

The function {lx ; x > 0} is called the decrement function or, when considered only at integer values of x, the decrement series. Expressed in terms of the decrement function we find e.g. t px

= lx+t /lx ,

µx+t f (t | x)

= =

e¯x

0 −lx+t /lx+t , 0 −lx+t /lx , Z ∞

lx+t dt/lx .

(3.35) (3.36) (3.37) (3.38)

The pieces of IAA notation we have shown here are quite pleasing to the eye and also space-saving; for instance, the symbol on the left of (3.27) involves three typographical entities, whereas the one on the right involves six.

CHAPTER 3. MORTALITY

3.4

Select mortality

A. The insurance portfolio consists of selected lives. Consider an individual who purchases a life insurance at age x. In short, he will be referred to as (x) in what follows. It is quite common in actuarial practice to assume that the force of mortality of (x) depends on x and t in a more complex manner than the simple relationship (3.9), which rested on the assumption that (x) is chosen at random from the x years old individuals in the population. The fact that (x) purchases insurance adds information to the mere fact that he has attained age x; he does not represent a purely random draw from the population, but is rather selected by some mechanisms. It is easy to think of examples of such mechanisms. For instance that poor people can not afford to buy insurance and, to the extent that longevity depends on economic situation, the mortality experience for insured people would reflect that they are wealthy enough to buy insurance (’survival of the fattest’). Judging from textbooks on life insurance, e.g. [4] and [29] and many others, it seems that the underwriting standards of the insurer are generally held to be the predominant selective mechanism; before an insurance policy is issued, the insurer must be satisfied that the applicant meets certain requirements with regard to health, occupation, and other factors that are assumed to determine the prospects of longevity. Only first class lives are eligible to insurance at ordinary rates. Thus there is every reason to account for selection effects by letting the force of mortality be some more general function µx (t) or, in other words, specify that Tx follows a survival function Fx (t) that is not necessarily of the form (3.7). One then speaks of select mortality. B. More of actuarial notation. The standard actuarial notation for select mortality is τ q[x]+t τ p[x]+t

µ[x]+t

= P[Tx ≤ t + τ | Tx > t] , = P[Tx > t + τ | Tx > t] , h q[x]+t . = lim h&0 h

(3.39) (3.40) (3.41)

The idea is that the both the current age, x + t, and the age at entry, x, are directly visible in [x] + t. From a technical point of view select mortality is just as easy as aggregate mortality; we work with the distribution function t q[x] instead of t qx , and are interested in it as a function of t. For instance, Z t+τ t+τ p[x] = exp − µ ds . p = τ [x]+t [x]+s t p[x] t C. Features of select mortality. There is ample empirical evidence to support the following facts about select mortality in life insurance populations:

CHAPTER 3. MORTALITY

– For insured lives of a given age the rate of mortality usually increases with increasing duration. – The effect of selection tends to decrease with increasing duration and becomes negligible for practical purposes when the duration exceeds a certain select period. – The mortality among insured lives is generally lower than the mortality in the population. There are many possible ways of building such features into the model. For instance, one could modify the aggregate G-M intensity as µ[x]+t = α(t) + β(t) eγ (x+t) , where α and β are non-negative and non-decreasing functions bounded from above. In Section 7.6 we shall show how the selection mechanism can be explained in models that describe more aspects of the individual life histories than just survival and death.

Chapter 4

Insurance of a single life 4.1

Some standard forms of insurance

A. The single-life status. Consider a person aged x with remaining life length Tx as described in the previous section. In actuarial parlance this life is called the single-life status (x) . Referring to Appendix B, we introduce the indicator of the event of survival in t years, It = 1[Tx > t]. This is a binomial random variable with ’success’ probability t px . The indicator of the event of death within t years is 1 − It = 1[Tx ≤ t], which is a binomial variable with ’success’ probability t qx = 1 − t px . (We apologize for sometimes using technical terms where they may sound misplaced.) Note that, being 0 or 1, any indicator 1[A] satisfies 1[A]q = 1[A] for q > 0. The present section lists some standard forms of insurance that (x) can purchase, investigates some of their properties, and presents some basic actuarial methods and formulas. We assume that the investments of the insurance company yield interest at a fixed rate r so that accumulation and discounting take place in in accordance with (2.12). B. The pure endowment insurance. An n-year pure (life) endowment of 1 is a unit that is paid to (x) at the end of n years if he is then still alive. In other words, the associated payment function is an endowment of In at time n. Its present value at time 0 is P V e;n = e−rn In .

(4.1)

The expected value of P V e;n , denoted by n Ex , is n Ex

= e−rn n px .

(4.2)

For any q > 0 we have (P V e;n )q = e−qrn In (recall that Inq = In ), and so the q-th non-central moment of P V e;n can be expressed as E[(P V e;n )q ] = n Ex(qr) , 39

(4.3)

CHAPTER 4. INSURANCE OF A SINGLE LIFE

where the top-script (qr) signifies that discounting is made under a force of interest that is qr. In particular, the variance of P V e;n is V[P V e;n ] = n Ex(2r) − n Ex2 .

(4.4)

C. The life assurance. A life assurance contract specifies that a certain amount, called the sum insured, is to be paid upon the death of the insured, possibly limited to a specified period. We shall here consider only insurances payable immediately upon death, and take the sum to be 1 (just a matter of notation). First, an n-year term insurance is payable upon death within n years. The payment function is a lump sum of 1 − In at time Tx . Its present value at time 0 is P V ti;n The expected value of P V ti;n is Z A¯ 1 = xn

= e−rTx (1 − In ) .

(4.5)

e−rτ τ px µx+τ dτ ,

(4.6)

and, similar to (4.3), (qr) E[(P V ti;n )q ] = A¯ 1 .

(4.7)

In particular, (2r) V[P V ti;n ] = A¯ 1 − A¯21 xn

(4.8)

An n-year endowment insurance is payable upon death if it occurs within time n and otherwise at time n. The payment function is a lump sum of 1 at time Tx ∧ n. Its present value at time 0 is P V ei;n = e−r(Tx ∧n) .

(4.9)

The expected value of P V ei;n is Z n ¯ Ax n = e−rτ τ px µx+τ dτ + e−rn n px = A¯ 1

+ n Ex ,

(4.10)

and (qr) E(P V ei;n )q = A¯x n .

(4.11)

(2r) V[P V ei;n ] = A¯x n − A¯2x n .

(4.12)

It follows that

CHAPTER 4. INSURANCE OF A SINGLE LIFE

D. The life annuity. An n-year temporary life annuity of 1 per year is payable as long as (x) survives but limited to n years. We consider here only the continuous version. Recalling (2.30), the associated payment function is an annuity of 1 in Tx ∧ n years. Its present value at time 0 is P V a;n = a ¯Tx ∧n =

1 − e−r(Tx ∧n) . r

(4.13)

The expected value of P V a;n is Z n a ¯τ τ px µx+τ dτ + a ¯ n n px . a ¯x n = 0

A more appealing formula is a ¯x n =

e−rτ τ px dτ ,

(4.14)

displays the life annuity as a “continuum of life endowments”, a ¯x n = Rwhich n E dτ . There are several ways of proving (4.14). Using brute force, one can τ x 0 integrate by parts: Z n Z n d d a ¯ n n px = a ¯ 0 0 px + a ¯τ τ px dτ + a ¯τ τ px dτ dτ dτ 0 0 Z n Z n = e−rτ τ px dτ − a ¯τ τ px µx+τ dτ . 0

Using the brain instead, one realizes that the expected present value at time 0 of the payments in any small time interval (τ, τ + dτ ) is e−rτ dτ τ px , and summing over all time intervals one arrives at (4.14) (“the expected value of a sum is the sum of the expected values”). This kind of reasoning will be omnipresent throughout the text, and would also immediately produce formula (4.6) and (4.10). The recipe is: Find the expected present value of the payments in each small time interval and add up. We shall demonstrate below that q q X (pr) p−1 q − 1 a;n q (−1) (4.15) E[(P V ) ] = q−1 a ¯x n , r p − 1 p=1 from which we derive V[P V a;n ] =

2 (2r) a ¯x n − a ¯x n − a ¯2x n . r

(4.16)

The endowment insurance is a combined benefit consisting of an n-year term insurance and an n-year pure endowment. By (4.9) and (4.13) it is related to the life annuity by P V a;n =

1 − P V ei;n r

or P V ei;n = 1 − rP V a;n ,

(4.17)

CHAPTER 4. INSURANCE OF A SINGLE LIFE

which just reflects the more general relationship (2.31). Taking expectation in (4.17), we get ax n . A¯x n = 1 − r¯

(4.18)

Also, since P V ti;n = P V ei;n − P V e;n = 1 − rP V a;n − P V e;n , we have A¯ 1

= 1 − r¯ a x n −n Ex .

(4.19)

The formerly announced result (4.15) follows by operating with the q-th moment on the first relationship in (4.17), and then using (4.12) and (4.18) and rearranging a bit. One needs the binomial formula q X q q−p p q x y (x + y) = p p=0 Pq and the special case p=0 pq (−1)q−p = 0 (for x = −1 and y = 1). A whole-life annuity is obtained by putting n = ∞. Its expected present value is denoted simply by a ¯x and is obtained by putting n = ∞ in (4.14), that is Z ∞ e−rτ τ px dτ,

a ¯x =

(4.20)

and the same goes for the variance in (4.16) (justify the limit operations).

E. Deferred benefits. An m-year deferred n-year temporary life annuity commences only after m years, provided that (x) is then still alive, and is payable throughout n years thereafter as long as (x) survives. The present value of the benefits is PV

= P V a;m+n − P V a;m = a ¯Tx ∧(m+n) − a ¯Tx ∧m =

e−r(Tx ∧m) − e−r(Tx ∧(m+n)) r

(4.21)

The expected present value is ¯x m|n a

=a ¯x m+n − a ¯x m =

m+n

e−rt t px dt = m Ex a ¯x+m n .

(4.22)

The last expression can be obtained also by the rule of iterated expectation, and we carry through this small exercise just to illustrate the technique: E [P V ] = E E [ P V | Im ] = m px E [ P V | Im = 1] + m qx E [ P V | Im = 0] =

m px v

a ¯x+m n .

An m-year deferred whole life annuity is obtained by putting n = ∞. The expected value is denoted by m| a ¯x .

CHAPTER 4. INSURANCE OF A SINGLE LIFE

Deferred life assurances, although less common in practice, are defined likewise. For instance, an m-year deferred n-year term assurance of 1 is payable upon death in the time interval (m, m + n]. Its present value at time 0 is P V = P V ti;m+n − P V ti;m ,

(4.23)

and its expected present value is ¯ = A¯ 1

1 m|n Ax

x m+n

− A¯ 1

= m Ex A¯

x+m n

,. =

m+n

e−rτ τ px µx+τ dτ . (4.24) m

F. Computational aspects. Distribution functions of present values and many other functions of interest can be calculated easily; after all there is only one random variable in play, and finding expected values amounts just to forming integrals in one dimension. We shall, however, not pursue this approach because it will turn out that a different point of view is needed in more complex situations to be studied in the sequel. Table 4.1: Expected value (E), coefficient of variation (CV), and skewness (SK) of the present value at time 0 of a pure endowment (PE) with sum 1, a term insurance (TI) with sum 1, an endowment insurance (EI) with sum 1, and a life annuity (LA) with level intensity 1 per year, when x = 30, n = 30, µ is given by (3.25), and r = ln(1.045).

E CV SK

PE 0.2257 0.4280 −1.908

TI 0.06834 2.536 2.664

EI 0.2940 0.3140 4.451

LA 16.04 0.1308 −4.451

Anyway, by methods to be developed later, we easily compute the three first moments of the present values considered above, and find their expected values, coefficients of variation, and skewnesses shown in Table 4.1. The reader should contemplate the results, keeping in mind that the coefficient of variation may be taken as a simple measure of “riskiness”. We interpose that numerical techniques will be dominant in our context. Explicit formulas cannot be obtained even for trivial quantities like a ¯ x n under the Gompertz-Makeham law (3.23); age dependence and other forms of inhom*ogeneity of basic entities leave little room for aesthetics in actuarial science. Also relationships like (4.18) are of limited interest; they are certainly not needed for computational purposes, but may provide some general insight.

4.2

The principle of equivalence

A. A note on terminology. Like any other good or service, insurance coverage is bought at some price. And, like any other business, an insurance company

CHAPTER 4. INSURANCE OF A SINGLE LIFE

must fix prices that are sufficient to defray the costs. In one respect, however, insurance is different: for obvious reasons the customer is to pay in advance. This circ*mstance is reflected by the insurance terminology, according to which payments made by the insured are called premiums. This word has the positive connotation “prize” (reward), rather antonymous to “price” (sacrifice, due), but the etymological background is, of course, that premium means “first” (French: prime). B. The equivalence principle. The equivalence principle of insurance states that the expected present values of premiums and benefits should be equal. Then, roughly speaking, premiums and benefits will balance on the average. This idea will be made precise later. For the time being all calculations are made on an individual net basis, that is, the equivalence principle is applied to each individual policy, and without regard to expenses incurring in addition to the benefits specified by the insurance treaties. The resulting premiums are called (individual) net premiums. The premium rate depends on the premium payment scheme. In the simplest case, the full premium is paid as a single amount immediately upon the inception of the policy. The resulting net single premium is just the expected present value of the benefits, which for basic forms of insurance is given in Section 4.1. The net single premium may be a considerable amount and may easily exceed the liquid assets of the insured. Therefore, premiums are usually paid by a series of installments extending over some period of time. The most common solution is to let a fixed level amount fall due periodically, e.g. annually or monthly, from the inception of the agreement until a specified time m and contingent on the survival of the insured. Assume for the present that the premiums are paid continuously at a fixed level rate π. (This is admittedly an artificial assumption, but it can serve well as an approximate description of periodical payments, which will be treated later.) Then the premiums form an m-year temporary life annuity, payable by the insured to the insurer. Its present value is πP V a;m , with expected value π¯ ax m given by (4.14). We list formulas for the net level premium rate for a selection of basic forms of insurance: For the pure endowment (Paragraph 4.1.B) against level premium in the insurance period, π=

n Ex

a ¯x n

(4.25)

For the m-year deferred n-year temporary annuity (Paragraph 4.1.E) against level premium in the deferred period, π=

¯x m|n a a ¯x m

a ¯x m+n −1. a ¯x m

(4.26)

For the term insurance (Paragraph 4.1.C) against level premium in the insurance period, A¯ 1 π = xn . (4.27) a ¯x n

CHAPTER 4. INSURANCE OF A SINGLE LIFE

For the endowment insurance (Paragraph 4.1.C) against level premium in the insurance period, A¯x n 1 = −r, (4.28) π= a ¯x m a ¯x n the last expression following from (4.18). C. The net economic result for a policy. The random variables studied in Section 4.1 represent the uncertain future liabilities of the insurer. Now, unless single premiums are used, also the premium incomes are dependent on the insured’s life length and become a part of the insurer’s uncertainty. Therefore, the relevant random variable associated with an insurance policy is the present value of benefits less premiums, P V = P V b − πP V a;m ,

(4.29)

where P V b is the present value of the benefits, e.g. P V ei;n in the case of an n-year endowment insurance. Stated precisely, the equivalence principle lays down that E[P V ] = 0 .

(4.30)

axm| , which yields For example, with P V b = P V ei;n (4.30) becomes 0 = A¯x n −π¯ (4.28) when m = n. A measure of the uncertainty associated with the economic result of the policy is the variance V[P V ]. For example, with P V b = P V ei;n and m = n, 1 − v Tx ∧n Tx ∧n V[P V ] = V v −π = (1 + π/r)2 V[v Tx ∧n ] r (2r) ¯x n 2 a ¯x n − a −1. (4.31) = r¯ a2x n

4.3

Prospective reserves

A. The case. We shall discuss the notion of reserve in the framework of a combined insurance which comprises all standard forms of contingent payments that have been studied so far and, therefore, easily specializes to any of those. The insured is x years old upon issue of the contract, which is for a term of n years. The benefits consist of a term insurance with sum insured bt payable upon death at time t ∈ (0, n) and a pure endowment with sum bn payable upon survival at time n. The premiums consist of a lump sum π0 payable immediately upon the inception of the policy at time 0, and thereafter an annuity payable continuously at rate πt per time unit contingent on survival at time t ∈ (0, n). As before, assume that the interest rate is a deterministic function rt .

CHAPTER 4. INSURANCE OF A SINGLE LIFE

The expected present value at time 0 of total benefits less premiums under the contract can be put up directly as the sum of the expected discounted payments in each small time interval: Z n R Rn τ − π0 + e− 0 r τ px {µx+τ bτ − πτ } dτ + bn e− 0 r n px . (4.32) 0

Under the equivalence principle this is set equal to 0, a constraint on the premium function π. B. Definition of the reserve. The expected value (4.32) represents, in an average sense, an assessment of the economic prospects of the policy at the outset. At any time t > 0 in the subsequent development of the policy the assessment should be updated with regard to the information currently available. If the policy has expired by death before time t, there is nothing more to be done. If the policy is still in force, a renewed assessment must be based on the conditional distribution of the remaining life length. Insurance legislation lays down that at any time the insurance company must provide a reserve to meet future net liabilities on the contract, and this reserve should be precisely the expected present value at time t of total benefits less premiums in the future. Thus, if the policy is still in force at time t, the reserve is Z n R Rn τ (4.33) Vt = e− t r τ −t px+t {µx+τ bτ − πτ } dτ + bn e− t r n−t px+t . t

More precisely, this quantity is called the prospective reserve at time t since it “looks ahead”. Under the principle of equivalence it is usually called the net premium reserve. We will take the R τ liberty to just speak of the reserve. Upon inserting τ −t px+t = e− t µx+s ds , (4.33) assumes the form Z n R Rτ τ e− t (rs +µx+s ) ds {µx+τ bτ − πτ } dτ + e− t (rs +µx+s ) ds bn . (4.34) Vt = t

Glancing behind at (2.14), we see that, formally, the expression in (4.34) is the reserve at time t for a deterministic contract with payments given by ∆B0 = −π0 (comes from setting (4.32) equal to 0), dBt = (µx+t bt − πt ) dt, 0 < t < n and ∆Bn = bn , and with interest rate rt + µx+t . We can, therefore, reuse the relationships in Chapter 2. For instance, by (2.26) and (2.13), we have the following retrospective formula for the prospective premium reserve: Z t R Rt t Vt = e 0 (rs +µx+s ) ds π0 + e τ (rs +µx+s ) ds (πτ − µx+τ bτ ) dτ . (4.35) 0

This formula expresses Vt as the surplus of transactions in the past, accumulated at time t with the “benefit of interest and survivorship”.

CHAPTER 4. INSURANCE OF A SINGLE LIFE

6 1•

n Ex•

• n

Figure 4.1: The net reserve for an n-year pure endowment of 1 against single net premium. C. Some special cases. The net reserve is easily put up for the various forms of insurance treated in Sections 4.1 and 4.2. We assume that the interest rate is constant and that premiums are based on the equivalence principle, which can be expressed as V0 = π 0 . (4.36) First, for the pure endowment against single net premium n Ex collected at time 0, Vt = n−t Ex+t , 0 ≤ t < n. (4.37) The graph of Vt will typically look as in Fig. 4.1. At points of discontinuity a dot marks the value of the function. If premiums are payable continuously at level rate π given by (4.25) throughout the insurance period, then Vt

= =

− π¯ ax+t n−t n Ex a ¯ . n−t Ex+t − a ¯x n x+t n−t

n−t Ex+t

(4.38)

A typical graph of this function is shown in Fig. 4.2. Next, for an m-year deferred whole life annuity against level net premium π given by (4.26), ¯x+t − π¯ ax+t m−t , 0 < t < m, m−t| a Vt = a ¯x+t , t ≥ m, a ¯x − a ¯x m = a ¯x+t − a ¯x+t m−t − a ¯x+t m−t a ¯x m a ¯x = a ¯x+t − a ¯ (4.39) a ¯x m x+t m−t

CHAPTER 4. INSURANCE OF A SINGLE LIFE

6 1•

• n

Figure 4.2: The net reserve for an n-year pure endowment of 1 against level premium in the insurance period.

6 a ¯x+m •

0 0

• m

Figure 4.3: The net reserve for an m-year deferred whole life annuity against level premium in the deferred period.

CHAPTER 4. INSURANCE OF A SINGLE LIFE

• n

Figure 4.4: The net reserve for an n-year term insurance against level premium in the insurance period

6 1•

0 0

• n

Figure 4.5: The net reserve for an n-year endowment insurance with level premium payable in the insurance period.

CHAPTER 4. INSURANCE OF A SINGLE LIFE

(with the understanding that a ¯x m−t = 0 if t > m). A typical graph of this function is shown in Fig. 4.3. For the n-year term insurance against level net premium π given by (4.27), Vt

= A¯

x+t n−t

− π¯ ax+t n−t

= 1 − r¯ ax+t n−t − = 1−

n−t Ex+t

−

− (1 − n Ex )

1 − r¯ a x n − n Ex a ¯x+t n−t a ¯x n

a ¯x+t n−t . a ¯x n

(4.40)

A typical graph of this function is shown in Fig. 4.4. Finally, for the n-year endowment insurance against level net premium π given by (4.28), Vt

= A¯x+t n−t − π¯ ax+t n−t 1 − r¯ ax n = 1 − r¯ ax+t n−t − a ¯x+t n−t a ¯x n a ¯ = 1 − x+t n−t . a ¯x n

(4.41)

A typical graph of this function is shown in Fig. 4.5. The reserve in (4.41) is, of course, the sum of the reserves in (4.39) and (4.40). Note that the pure term insurance requires a much smaller reserve than the other insurance forms, with elements of savings in them. However, at old ages x (where people typically are not covered against the risk of death since death will incur soon with certainty) also the term insurance may have a Vt close to 1 in the middle of the insurance period. D. Non-negativity of the reserve. In all the examples given here the net reserve is sketched as a non-negative function. Non-negativity of Vt is not a consequence of the definition. One may easily construct premium payment schemes that lead to negative values of Vt (just let the premiums fall due after the payment of the benefits), but such payment schemes are not used in practice. The reason is that the holder of a policy with Vt < 0 is in expected debt to the insurer and would thus have an incentive to cancel the policy and thereby get rid of the debt. (The agreement obliges the policy-holder only to pay the premiums, and the contract can be terminated at any time the policy-holder wishes.) Therefore, it is in practice required that Vt ≥ 0, t ≥ 0.

(4.42)

E. The reserve considered as a function of time. We will now take a closer look at the prospective reserve as a function of time, bearing in mind that it should be non-negative. The building blocks are the expected present values n−t Ex+t , a ¯x+t:n−t , A¯x+t n−t , and A¯ 1 appearing in the formulas in Section 4.3.

x+t n−t

CHAPTER 4. INSURANCE OF A SINGLE LIFE First, n−t Ex+t

= e−

Rn t

(r+µx+s ) ds

is seen to be an increasing function of t no matter what are the interest rate and the mortality rate. The derivative is d n−t Ex+t = n−t Ex+t (r + µx+t ) . dt We interpose here that nothing is changed if r depends on time. The expressions above show that, for this pure survival benefit, r and µ play identical parts in the expected present value. Thus, mortality bequest acts as an increase of the interest rate. Next consider Z n R τ e− t (r+µx+s ) ds dτ . a ¯x+t:n−t = t

The following inequalities are obvious: a ¯x+t:n−t ≤

1 r + inf s≥t µx+s

≤

1 . r

The last expression is just the present value of a perpetuity, (2.32). If µ is an increasing function, then a ¯x+t:n−t ≤

1 . r + µx+t

We find the derivative d a ¯ = (r + µx+t )¯ ax+t:n−t − 1 . dt x+t:n−t It follows that a ¯x+t:n−t is a decreasing function of t if µ is increasing, which is quite natural. You can easily invent an example where a ¯ x+t:n−t is not decreasing. From the identity A¯x+t n−t = 1 − r¯ ax+t:n−t we conclude that A¯x+t n−t is an increasing function of t if µ is increasing. For A¯ 1 = 1 − r¯ ax+t:n−t − n−t Ex+t x+t n−t

no general statement can be made as to whether it is decreasing or increasing. Looking back at the formulas derived in Paragraph C above, we can conclude that the reserve for the pure life endowment against single premium, (4.37), is always increasing. Assume henceforth that µ is increasing, as is usually the case at ages when people are insured and certainly holds for the GompertzMakeham law. Then also the reserve (4.38) for the pure life endowment against level premium during the term of the contract is increasing, and the same is the case for the reserve (4.41) of the endowment insurance. It is left to the diligent

CHAPTER 4. INSURANCE OF A SINGLE LIFE

reader to show that the reserve in (4.39) is increasing throughout the deferred period and thereafter turns decreasing. This is best done by examining (4.33) and (4.35) in the cases t ≤ m and t ≤ m, respectively. It follows in particular that the reserve is non-negative. The same trick serves also to show that (4.40) is first increasing and thereafter decreasing.

4.4

Thiele’s differential equation

A. The differential equation. We turn back to the general case with the reserve given by (4.33) or (4.33), the latter being the more convenient since we can draw on the results in Chapter 2. The differential form (2.19) translates to the celebrated Thiele’s differentila equation, d Vt = πt − bt µx+t + (r + µx+t ) Vt , (4.43) dt valid at each t where b, π, and µ are continuous. The right hand side expression in (4.43) shows how the fund per surviving policy-holder changes per time unit at time t. It is increased by the excess of premiums over benefits (which may be negative, of course), by the interest earned, rVt , and by the fund inherited from those who die, µx+t Vt . When combined with the boundary condition Vn−

= bn ,

(4.44)

the differential equation (4.43) determines Vt for fixed b and π. If the principle of equivalence is exercised, then we must add the condition (4.36). This represents a constraint on the contractual payments b and π; typically, one first specifies the benefit b and then determines the premium rate for a given premium plan (shape of π). Thiele’s differential equation is a so-called backward differential equation. This term indicates that we take our stand at the beginning of the time interval we are interested in and also that the differential equation is to be solved by a backward scheme starting from the ultimo condition (2.21). The differential equation may be put up by the direct backward construction which goes as follows. Suppose the policy is in force at time t ∈ (0, n). Use the rule of iterated expectation, conditioning on what happens in the small time interval (t, t + dt]: with probability µx+t dt + o(dt) the insured dies, and the conditional expected value is then just bt ; with probability 1 − µx+t dt + o(dt) the insured survives, and the conditional expected value is then −πt dt + e−r dt Vt+dt . We gather Vt = bt µx+t dt − πt dt + (1 − µx+t dt)e−r dt Vt+dt + o(dt).

(4.45)

Subtract Vt+dt on both sides, divide by dt and let dt tend to 0. Observing that (e−rdt − 1)/dt → −r as dt → 0, one arrives at (4.43)

CHAPTER 4. INSURANCE OF A SINGLE LIFE

B. Savings premium and risk premium. Suppose the equivalence principle is in use. Rearrange (4.43) as πt =

d Vt − rVt + (b − Vt )µx+t . dt

(4.46)

This form of the differential equation shows how the premium at any time decomposes into a savings premium, πts =

d Vt − rVt , dt

(4.47)

and a risk premium, πtr = (bt − Vt )µx+t .

(4.48)

The savings premium provides the amount needed in excess of the earned interest to maintain the reserve. The risk premium provides the amount needed in excess of the available reserve to cover an insurance claim. C. Uses of the differential equation. In the examples given above, Thiele’s differential equation was useful primarily as a means of investigating the development of the reserve. It was not required in the construction of the premium and the reserve, which could be put up by direct prospective reasoning. In the final example to be given Thiele’s differential equation is needed as a constructive tool. Assume that the pension treaty studied above is modified so that the reserve is paid back at the moment of death in case the insured dies during the contract period, the philosophy being that “the savings belong to the insured”. Then the scheme is supplied by an (n + m)-year temporary term insurance with sum bt = Vt at any time t ∈ (0, m + n). The solution to (4.43) is easily obtained as π¯ st , 0 < t < m, Vt = b¯ am+n−t , m < t < m + n, Rt where s¯t = 0 (1 + i)t−τ dτ . The reserve develops just as for ordinary savings contracts offered by banks. D. Dependence of the reserve on the contract elements. A small collection of results due to Lidstone (1905) and, in the time-continuous set-up, Norberg (1985), deal with the dependence of the reserve on the contract elements, in particular mortality and interest. The starting point in the time-continuous case is Thiele’s differential equation. For the sake of concreteness, we adopt the model assumptions and the contract described in Section 4.4 and will refer to this as the standard contract. For ease of reference we fetch Thiele’s differential from (4.43): d Vt = πt − µx+t bt + (rt + µx+t ) Vt . dt

(4.49)

CHAPTER 4. INSURANCE OF A SINGLE LIFE The boundary condition following from the very definition of the reserve is Vn− = bn .

(4.50)

With premiums determined by the principle of equivalence, we also have V0 = π 0 ,

(4.51)

where π0 is the lump sum premium payment collected upon the inception of the policy (it may be 0, of course). Now consider a different model with interest rt∗ and mortality µ∗x+t and a different contract with benefits b∗t and premiums πt∗ . This will be referred to as the special contract. The reserve function Vt∗ under this contract satisfies d ∗ V = πt∗ − µ∗x+t b∗t + (rt∗ + µ∗x+t ) Vt∗ , dt t

(4.52)

∗ Vn− = b∗n ,

(4.53)

V0∗

(4.54)

π0∗

Assume that π0∗ = π0 ,

b∗n = bn .

(4.55)

Vt∗

We are interested in the difference − Vt , and a few words are in order to motivate this: The reserve is accounted as a liability on the part of the insurance company. To be on the safe side, the company should, at any time, provide a reserve in excess of what seems likely to be needed. This is usually obtained by using ’technical’ elements rt∗ and µ∗x+t that are different from the ’realistic’ elements rt and µx+t , and that produce a reserve Vt∗ bigger than the ’realistic’ Vt . Subtract (4.49) from (4.52) to get d (V ∗ − Vt ) = ηt + (rt∗ + µ∗x+t ) (Vt∗ − Vt ) , dt t

(4.56)

where ηt = (πt∗ − πt ) + (µx+t bt − µ∗x+t b∗t ) + (rt∗ − rt + µ∗x+t − µx+t ) Vt . Integrate (4.56) from 0 to t, using V0 = V0∗ , to obtain Vt∗ − Vt =

t R t

(r ∗ +µ∗ )

ηs ds .

∗ Similarly, integrate from t to n, using Vn− = Vn− , to obtain Z n R s ∗ ∗ Vt∗ − Vt = − e− t (r +µ ) ηs ds . t

(4.57)

CHAPTER 4. INSURANCE OF A SINGLE LIFE

From these relations conclude: If there exists a t0 ∈ [0, n] such that ηt

≤ < 0 for t t , ≥ > 0

(4.58)

then Vt∗ ≤ Vt for all t. In particular, this is the case if ηt is non-decreasing. The result remains valid if all inequalities are made strict. We can now prove the following: (1) For a contract with level premium intensity throughout the insurance period, and with non-decreasing reserve, a uniform increase of the interest rate results in a decrease of the reserve. Proof: Now rt∗ − rt = ∆r is a positive constant, π ∗ and π are both constants, all other elements are unchanged, and Vt increasing. Then ηt = (π ∗ − π) + ∆r Vt is increasing. (2) Consider an endowment insurance with fixed sum insured and level premium rate throughout the insurance period. Prove that a change of mortality from µ to µ∗ such that µ∗t − µt is positive and non-increasing, leads to a decrease of the reserve. Proof: Now (µ∗x+t − µx+t ) is positive and decreasing (non-increasing), π ∗ and π are constants, bt = b∗t = b constant, Vt increasing (this is the case for the endowment insurance if µ in increasing). Then, since Vt ≤ b, we have ηt = (π ∗ − π) − (µ∗x+t − µx+t )(b − Vt ) is increasing. (3) Consider a policy with no down premium payment at time 0 and no life endowment at time n. Let the special contract be the same as the standard one, except that the special contract charges so-called natural premium, πt∗ = bt µx+t . Then Vt∗ = 0 for all t, and (4.58) can be used to check whether the reserve Vt is non-negative (as it should be). Proof: Putting πt∗ = µx+t bt , means premiums covers current expected benefits, so there is no accumulation of reserve; Vt∗ = 0. Now ηt = −πt + µx+t bt , so if this is increasing, then 0 = Vt∗ ≤ Vt . This is the case e.g. if π and b are constants and µt is increasing. The reason why the impact on the reserve of a change in valuation and/or contract elements is a bit involved is that, under the equivalence principle, the premium is also affected by the change. However, if we require that the premium be constant as function of t, then (π ∗ − π) appearing in the expression for ηt is constant and does not affect the monotonicity properties of ηt . Note also that, since Vt∗ − Vt starts and ends at 0, ηt cannot be strictly positive in some part of (0, n) without being strictly negative in some other part.

CHAPTER 4. INSURANCE OF A SINGLE LIFE

4.5

Probability distributions

A. Motivation. The basic paradigm being the principle of equivalence, life insurance mathematics centers on expected present values. The key tool is Thiele’s differential equation, which describes the development of such expected values and forms a basis for computing them by recursive methods. In Chapter 7 we shall obtain analogous differential equations for higher order moments, which will enable us to compute the variance, skewness, kurtosis, and so on of the present value of payments under a fairly general insurance contract. We shall give an example of how to determine the probability distribution of a present value, which is at the base of the moments and of any other expected values of interest. Knowledge of this distribution, and in particular its upper tail, gives insight into the riskiness of the contract beyond what is provided by the mean and some higher order moments. The task is easy for an insurance on a single life since then the model involves only one random variable (the life length of the insured). De Pril [14] and Dhaene [16] offer a number of examples. In principle the task is simple also for insurances involving more than one life or, more generally, a finite number of random variables. In such situations the distributions of present values (and any other functions of the random variables) can be obtained by integrating the finite-dimensional distribution. B. A simple example. Consider the single life status (x) with remaining life time Tx distributed as described in Chapter 3. Suppose (x) buys an n year term insurance with fixed sum b and premiums payable continuously at level rate π per year as long as the contract is in force (see Paragraphs 4.1.C-D). The present value of benefits less premiums on the contract is U (Tx ) = be−rTx 1[0 < Tx < n] − π¯ aTx ∧n , Rt where a ¯t = 0 e−rτ dτ = (1 − e−rt )/r is the present value of an annuity certain payable continuously at level rate 1 per year for t years. The function U is non-increasing in Tx , and we easily find the probability distribution  , u < −π¯ an ,   0   P[Tx > n] , −π¯ a ≤ u < b e−rn − π¯ an , n h i (4.59) P[U ≤ u] = 1 br+π −rn P T > , be − π¯ a ≤ u < b, ln  x n r ur+π    1 , u ≥ b.

The jump at −π¯ an is due to the positive probability of survival to time n. Similar effects are to be anticipated also for other insurance products with a finite insurance period since, in general, there is a positive probability that the policy will remain in the current state until the contract terminates. The probability distribution in (4.59) is depicted in Fig. 4.6 for the G82M case with r = ln(1.045) and µ(t|x) = 0.0005 + 10−4.12+0.038(x+t) when x = 30, n = 30, b = 1, and π = 0.0042608 (the equivalence premium).

CHAPTER 4. INSURANCE OF A SINGLE LIFE

6 1•

• −π¯ an|

• 0

• b

Figure 4.6: The probability distribution of the present value of a term insurance against level premium.

4.6

The stochastic process point of view

A. The processes indicating survival and death. In Paragraph A of Section 4.1 we introduced the indicator of the event of survival to time t, It = 1[Tx > t], and the indicator of the complementary event of death within time t, Nt = 1 − It = 1[Tx ≤ t]. Viewed as functions of t, they are stochastic processes. The latter counts the number of deaths of the insured as time is progresses and is thus a simple example of a counting process as defined in Paragraph D of Appendix A. This motivates the notation Nt . By their very definitions, It and Nt are RC. In the present context, where everything is governed by just one single random variable, Tx , the process point of view is not important for practical purposes. For didactic purposes, however, it is worthwhile taking it already here as a rehearsal for more complicated situations where stochastic processes cannot be dispensed with. The payment functions of the benefits considered in Section 4.1 can be recast in terms of the processes It and Nt . In differential form they are dBte;n

= It dεn (t) ,

dBtti;n dBta;n dBtei;n

= 1(0,n] (t) dNt , = It 1(0,n) (t) dt , = dBtti;n + dBte;n .

Their present values at time 0 are P V e;n P V ti;n

= e − 0 r In , Z n R τ = e− 0 r dNτ , 0

CHAPTER 4. INSURANCE OF A SINGLE LIFE P V a;n P V ei;n

e− 0 ti;n

= V

Rτ 0

Iτ dτ ,

+ V e;n .

The expressions in (4.14) and (4.10) are obtained directly by taking expectation under the integral sign, using the obvious relations E [Iτ ] = E [dNτ ] =

τ px , τ px µx+τ

dτ .

The relationship (4.19) re-emerges in its more basic form upon integrating by parts to obtain Z n R Z n R Rn τ τ e− 0 r dIτ , e− 0 r (−rτ )Iτ dτ + e− 0 r In = 1 + 0

and setting dIt = −dNt in the last integral.

Chapter 5

Expenses 5.1

A single life insurance policy

A. Three categories of expenses. Any firm has to defray expenditures in addition to the net production costs of the commodities or services it offers, and these expenses must be taken account of in the prices paid by the customers. Thus, the rate of premium charged for a given insurance contract must not merely cover the contractual net benefits, but also be sufficient to provide for all items of expenditure connected with the operations of the insurance company. For the sake of concreteness, and also of loyalty to standard actuarial notation, we shall introduce the issue of expenses in the framework of the simple single life policy encountered in Chapter 4. To get a case that involves all main types of payments, let us consider a life (x) who purchases an n-year endowment insurance with a fixed sum insured, b, and premium payable continuously at level rate as long as the policy is in force. We recall that the net premium rate determined by the principle of equivalence is 1 A¯x n =b −r , (5.1) π=b a ¯x n a ¯x n and that the corresponding net premium reserve to be provided if the insured is alive at time t, is a ¯x+t n−t ¯ Vt = bAx+t n−t − π¯ ax+t n−t| = b 1 − . (5.2) a ¯x n The term net means “net of administration expenses”. When expenses are included in the accounts, one will have to charge the policy with a gross premium rate π 0 , which obviously must be greater than the net premium rate, and the gross premium reserve Vt0 to be provided if the policy is in force at time t will also in general differ from the net premium reserve. The precise definitions of these quantities can only be made after we 59

CHAPTER 5. EXPENSES

have made specific assumptions about the structure of the expenses, which we now turn to. The expenses are usually divided into three categories. In the first place there are the so-called α-expenses that incur in connection with the establishment of the contract. They comprise sales costs, including advertising and agent’s commission, and costs connected with health examination, issue of the policy, entering the details of the contract into the data files, etc. It is assumed that these expenses incur immediately at time 0 and that they are of the form α0 + α00 b .

(5.3)

In the second place there are the so-called β-expenses that incur in connection with collection and accounting of premiums. They are assumed to incur continuously at constant rate β 0 + β 00 π 0

(5.4)

throughout the premium-paying period. Finally, in the third place there are the so-called γ-expenses that comprise all expenditures not included in the former two categories, such as wages to employees, rent, taxes, fees, and maintenance of the business operations in general. These expenses are assumed to incur continuously at rate γ 0 + γ 00 b + γ 000 Vt0

(5.5)

at time t if the policy is then in force. The constant terms α0 , β 0 , and γ 0 represent costs that are the same for all policies. The terms α00 b, β 00 π 0 , and γ 00 b represent costs that are proportional to the size of the contract as measured by the amounts specified in the policy. Typically this is the case for the agent’s commission, which may be a considerable portion of the α-expenses on individual insurances sold in an open competitive market, and also for the debt collector’s or solicitor’s commission, which in former days made up the major part of the β-expenses. The term γ 000 Vt0 represents expenses in connection with management of the investment portfolio, which can reasonably be divided between the policy-holders in proportion to their current reserves. B. The gross premium and the gross premium reserve. Upon exercising the equivalence principle in the presence of expenses, one will determine the gross premium rate π 0 and the corresponding gross premium reserve function Vt0 . When expenses depend on the reserve, as specified in (5.5), we have to resort to the Thiele technique to construct π 0 and Vt0 . We can immediately put up the following differential equation by adding the expenses to the benefits in the set-up of Section 4.4: d 0 V = π 0 − β 0 − β 00 π 0 − γ 0 − γ 00 b − γ 000 Vt0 − µx+t b + rVt0 + µx+t Vt0 . dt t

(5.6)

CHAPTER 5. EXPENSES The appropriate side conditions are 0 Vn− = b,

(5.7)

V00 = −(α0 + α00 b) .

(5.8)

and As before, (5.7) is a matter of definition and relies only on the fact that the endowment benefit falls due upon survival at time n, and (5.8) is the equivalence requirement, which determines π 0 for given benefits and expense factors. Gathering terms involving Vt0 on the left of (5.6) and multiplying on both Rn 000 sides with e t (r−γ +µ) gives Rn 000 d R n (r−γ 000 +µ) 0 et Vt = e t (r−γ +µ) {(1 − β 00 )π 0 − β 0 − γ 0 − (γ 00 + µx+t )b} . dt (5.9)

Now integrate (5.9) between t and n, using (5.7), and rearrange a bit to obtain Z n R τ 000 Vt0 = e− t (r−γ +µ) {β 0 + γ 0 + (γ 00 + µx+τ )b − (1 − β 00 )π 0 } dτ t

+ e−

Rn t

(r−γ 000 +µ)

(5.10)

Upon inserting t = 0 into (5.10) and using (5.8), we find Rn Rn Rτ 000 000 α0 + α00 b + 0 e− 0 (r−γ +µ) {β 0 + γ 0 + (γ 00 + µx+τ )b}dτ + e− 0 (r−γ +µ) b 0 π = . Rn Rτ 000 (1 − β 00 ) 0 e− 0 (r−γ +µ) dτ

(5.11)

In the special case where γ 000 = 0 we could determine π 0 and Vt0 directly from the defining relations without using the differential equation. That goes, in fact, also for the general case with γ 000 6= 0 by the following consideration: By inspection of the differential equation (5.6) and the side conditions, it is realized that, formally, the problem amounts to determining the “net premium rate” (1 − β 00 )π 0 and “net premium reserve” Vt0 for a policy with (admittedly unrealistic) benefits consisting of a lump sum payment of α0 + α00 b at time 0, a continuous level life annuity of β 0 + γ 0 + γ 00 b per year, and an endowment insurance of b, when the interest rate is r − γ 000 . Easy calculations show that, when γ 000 = 0, the gross and net quantities are related by α0 + α00 b 1 0 0 00 0 +β +γ +γ b , (5.12) π + π = 1 − β 00 a ¯x n and Vt0 = Vt −

a ¯x+t n−t 0 (α + α00 b). a ¯x n

(5.13)

CHAPTER 5. EXPENSES

It is seen that π 0 > π, as was anticipated at the outset. Furthermore, Vt0 < Vt for 0 ≤ t < n, which may be less obvious. The relationship (5.13) can be explained as follows: All expenses that incur at a constant rate throughout the term of the contract are compensated by an equal component in the “effective” gross premium rate (1 − β 00 )π 0 , see (5.12). Thus, the only expense factor that appears in the gross reserve is the non-amortized initial α-cost, which is the last term on the right of (5.13). It represents a debt on the part of the insured and is therefore to be subtracted from the net reserve. In Paragraph 4.3.D we have advocated non-negativity of the reserve. Now, already from (5.8) it is clear that the gross premium sets out negative at the time of issue of the contract and it will remain negative for some time thereafter until a sufficient amount of premium has been collected. The only way to get around this problem would be to charge an initial lump sum premium no less than the initial expense, but this is usually not done in practice (presumably) because a substantial down payment might keep customers with liquidity problems from buying insurance.

5.2

The general multi-state policy

A. General treatment of expenses. Consider now the general multi-state insurance policy treated in Chapter 7. Expenses are easily accommodated in the theory of that chapter since they can be treated as additional benefits of annuity and assurance type. Thus, from a technical point of view expenses do not create any additional difficulties, and we can therefore suitably end this chapter here. We round off by saying that expenses are still of conceptual and great practical importance. Assumptions about the various forms of expenses are part of the technical basis, which must be verified by the insurer and is subject to approval of the supervisory authority. Thus, just as statistical and economic analysis is required as a basis for assumptions about mortality and interest, thorough cost analysis are required as a basis for assumptions about the expense factors.

Chapter 6

Multi-life insurances 6.1

Insurances depending on the number of survivors

A. The single-life status reinterpreted. In the treatment of the single life status (x) in Chapters 3–4 we were having in mind the remaining life time T of an x year old person. From a mathematical point of view this interpretation is not essential. All that matters is that T is a non-negative random variable with an absolutely continuous distribution function, so that the survival function is of the form R − 0t µx+τ dτ . (6.1) t px = e The footscript x serves merely to indicate what mortality law is in play. Regardless of the nature of the status (x) and the notion of lifetime represented by T , the previous results remain valid. In particular, all formulas for expected present values of payments depending on T are preserved, the basic ones being the endowment, n Ex

= v n n px ,

(6.2)

the life annuity, a ¯x n =

n t

v t px dt = 0

n t Ex dt ,

(6.3)

the endowment insurance, ax n , A¯x n = 1 − r¯

(6.4)

A¯1x n = A¯x n − n Ex .

(6.5)

and the term insurance,

These formulas demonstrate that present values of all main types of payments in life insurance — endowments, life annuities, and assurances — can be traced 63

CHAPTER 6. MULTI-LIFE INSURANCES

back to the present value t Ex of an endowment and, as far as the mortality law is concerned, to the survival function t px . Once we have determined t px , all other functions of interest are obtained by integration, possibly by some numerical method, and elementary algebraic operations. B. Multi-dimensional survival functions. Consider a body of r individuals, the j-th of which is called (xj ) and has remaining lifetime Tj , j = 1, . . . , r. For the time being we shall confine ourselves to the case with independent lives. Thus, assume that the Tj are stochastically independent, and that each Tj possesses an intensity denoted by µxj +t and, hence, has survival function t p xj

= e−

Rt 0

µxj +τ dτ

(6.6)

(The function µ need not be the same for all j as the notation suggests; we have dropped an extra index j just to save notation.) The simultaneous distribution of T1 , . . . , Tr is given by the multi-dimensional survival function r r Y P ∩j=1 {Tj > tj } =

tj p x j

= e−

j=1

R tj 0

µxj +τ dτ

or, equivalently, by the density r Y

tj pxj µxj +tj

(6.7)

j=1

C. The joint-life status. The joint life status (x1 . . . xr ) is defined by having remaining lifetime Tx1 ...xr = min{T1 , . . . , Tr } . (6.8) Thus, the r lives are looked upon as a single entity, which continues to exist as long as all members survive, and terminates upon the first death. The survival function of the joint-life is denoted by t px1 ...xr and is t px1 ...xr

R t Pr = P ∩rj=1 {Tj > t} = e− 0 j=1 µxj +τ dτ .

(6.9)

From this survival function we form the present values of an endowment n Ex1 ...xr , a life annuity a ¯x1 ...xr n , an endowment insurance A¯x1 ...xr n , and a term insur1 ¯ ance, Ax1 ...xr n , by just putting (6.9) in the role of the survival function in (6.2) – (6.5). By inspection of (6.9), the mortality intensity of the joint-life status is simply the sum of the component mortality intensities, µx1 ...xr (t) =

r X j=1

µxj +t .

(6.10)

CHAPTER 6. MULTI-LIFE INSURANCES

In particular, if the component lives are subject to G-M mortality laws with a common value of the parameter c, µxj +t = αj + βj eγ (xj +t) ,

(6.11)

then (6.10) becomes µx1 ...xr (t) = α0 + β 0 eγ t

(6.12)

with α0 =

r X

αj , β 0 =

j=1

r X

βj e γ x j ,

(6.13)

j=1

again a G-M law with the same γ as in the component laws. D. The last-survivor status. The last survivor status x1 . . . xr is defined by having remaining lifetime Tx1 ...xr = max{T1 , . . . , Tr } .

(6.14)

Now the r lives are looked upon as an entity that continues to exist as long as at least one member survives, and terminates upon the last death. The survival function of this status is denoted by t px1 ...xr . By the general addition rule for probabilities (Appendix C), = P ∪rj=1 {Tj > t} t px1 ...xr X X r−1 = t p xj − t pxj1 xj2 + . . . + (−1) t px1 ...xr . (6.15) j

j1 t] = P[S > s] P[T > t] for all s and t. In particular, stochastic independence implies that C(g(S), h(T )) = 0 for all functions g and h such that the covariance is well defined. (We let C and V denote covariance and variance, respectively.) Mortality statistics suggest that life lengths of husband and wife are dependent and, moreover, that they are positively correlated. It is easy to think of possible explanations to this empirical fact. For instance, that people who marry

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

101

do so because they have something in common (’birds of a feather fly together’), or that married people share lifestyle and living conditions and therefore also hazards of diseases and accidents, or that death of the spouse impairs the living conditions for the survivor (’a grief effect’, or maybe the husband just does not know where the kitchen is and starves to death shortly after the loss of the spouse). Correlation is a rather special measure of dependence – essentially it measures linear dependence between random variables – and it is not sufficiently refined for our purposes.

7.9.2

Notions of positive dependence

There are various notions of positive dependence between pairs of random variables, and we will introduce three of them here. A comprehensive reference text is [5]. Definition PQD: S and T are positively quadrant dependent, written PQD(S, T ), if P[S > s , T > t] ≥ P[S > s] P[T > t] for all s and t.

(7.92)

This definition is symmetric in the two variables, so PQD(S, T ) is the same as PQD(T, S). The defining inequality (7.92) is equivalent to P[S > s | T > t] ≥ P[S > s] ,

(7.93)

which is easy to interpret: knowing e.g. that the wife will survive at least s years improves the survival prospects of the husband. Definition AS: S and T are associated, written AS(S, T ), if C(g(S, T ), h(S, T )) ≥ 0

(7.94)

for all real-valued functions g and h that are increasing in both arguments (and for which the covariance exists). Also the definition of AS is symmetric in the two variables, so AS(S, T ) is the same as AS(T, S). Definition RTI: S is right tail increasing in T , written RTI(S|T ), if P[S > s | T > t] is an increasing function of t for each fixed s.

(7.95)

The definition of RTI is not symmetric in the two variables. To each notion of positive dependence there is a corresponding notion of negative dependence. We can reasonably say that S and T are negatively quadrant dependent if the inequality (7.92) is reversed. This is the same as PQD(−S, T ), see Exercise 21. We can say that S and T are negatively associated (’dissociated’

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

102

does not have the right connotation) if the inequality (7.94) is reversed. This is the same as AS(−S, T ), see Exercise 21. We say that S is right tail decreasing in T , written RTD(S|T ) if P[S > s | T > t] is a decreasing function of t for each fixed s. This is the same as RTD(−S|T ), see Exercise 21. Since results on positive dependence thus translate into results on negative dependence, we will henceforth focus on the former. Theorem 1: RTI(S|T ) ⇒ AS(S, T ) ⇒ PQD(S, T ). Proof (incomplete): The first implication, RTI(S|T ) ⇒ AS(S, T ), is the hard part. The proof is long and technical and can be found in [20]. The second implication AS(S, T ) ⇒ PQD(S, T ) is easy. For g(S, T ) = 1(s,∞) (S) = 1[S > s] and h(S, T ) = 1(t,∞) (T ) = 1[T > t], (7.94) reduces to C(1[S > s], 1[T > t]) ≥ 0 ,

(7.96)

which is just a reformulation of the defining inequality (7.92). As a partial compensation for the absence of proof of the first implication, let us prove the shortcut implication RTI(S|T ) ⇒ PQD(S, T ): if RTI(S|T ), then P[S > s | T > t] ≥ P[S > s | T > 0] for t > 0, which is the same as (7.93). The following result is a partial converse to the second implication in Theorem 1. It could be formulated by saying that positive quadrant dependence is equivalent to “marginal association”. Lemma 1: PQD(S, T ) ⇔ C(g(S), h(T )) ≥ 0 for increasing functions g and h. Proof : By (7.96) the result holds for increasing indicator functions. Then it P holds P for increasing simple functions, g(S) = g0 + m i=1 gi 1[S > si ] and h(T ) = n h0 + j=1 hj 1[T > tj ] (with constant coefficients gi and hj and gi > 0, i = 1, . . . , m and hj > 0, j = 1, . . . , n), as is seen from C(g(S), h(T )) =

m X n X

gi hj C (1[S > si ], 1[T > tj ]) ≥ 0 .

i=1 j=1

Then it holds for all increasing functions g(S) and h(T ) since since any increasing function from R to R can be written as the limit of a sequence of increasing simple functions (monotone convergence). In the definitions of PQD, AS, and RTI we could equally reasonably have entered the events S ≥ s and T ≥ t. Due to the continuity property of probability measures, it does not matter which inequalities we use, > or ≥. See Exercise 21. for A1 ⊆ A2 ⊆ . . . lim P[An ] = P[∪n An ] for A1 ⊇ A2 ⊇ . . . lim P[An ] = P[∩n An ]

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

7.9.3

103

Dependencies between present values

The following table lists formulas for the life lengths and the survival functions of the four statuses husband, wife, their joint life, and their last survivor. Status (z) Husband (x) Wife (y) Joint life (x, y) Last survivor x, y

Life length U

Survival function P[U > τ ]

S T S ∧T S ∨T

P[S > τ ] P[T > τ ] P[S > τ , T > τ ] P[S > τ ] + P[T > τ ] − P[S > τ , T > τ ]

The next table recapitulate the formulas for present values and their expected values for the most basic insurance benefits to a status (z) with remaining life length U . Payment scheme

Present value

Expected present value

Pure endowment Life annuity Term insurance

−rn R n e −rτ 1[U > n] 1[U > τ ] dτ 0 e e−rU 1[U < n]

−rn P[U > n] n Ez R= e n −rτ a ¯z n = 0 e P[U > τ ] dτ A¯ 1 = 1 − n Ez − a ¯z n xn

The life lengths of the statuses listed in the first table are increasing functions of both S and T . From the second table we see that, for a general status with remaining life length U , the present value of a pure survival benefit (life endowment or life annuity) is an increasing functions of U , whereas the present value of the pure death benefit is a decreasing function of U (we assume the interest rate r is positive.) Combining these observations and Theorem 1, we can infer the following (and many other things): If PQD(S, T ), then pure survival benefits on any two statuses are positively dependent, pure death benefits on any two statuses are positively dependent, and any pure survival benefit and any death benefit are negatively dependent. We can also draw conclusions about the bias introduced in equivalence premiums by erroneously adopting the independence hypothesis. For instance, if PQD(S, T ) and we work under the independence hypothesis, then the present value of a survival benefit on the joint life will be underestimated, whereas the present value of a survival benefit on the last survivor will be overestimated. For the death benefit it is the other way around. Combining these things we may conclude e.g. that, for a death benefit on the joint life against level premium during joint survival, the equivalence premium will be overestimated.

7.9.4

A Markov chain model for two lives

It is not easy to create a given form of dependence between the life lengths S and T by direct specification of their joint distribution. However, the process point

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

104

of view, which is a powerful one, quite naturally allows us to express various ideas about dependencies between life lengths of a couple. A suitable framework is the Markov model sketched in Figure F.4. The following formulas are obvious: p00 (s, t) p01 (s, t) p02 (s, t)

= e− s µ+ν , Z t R Rt 0 τ = e− s µ+ν µτ e− τ ν dτ , s Z t R Rt 0 τ = e− s µ+ν ντ e− τ µ dτ . s

The joint survival function of S and T is p00 (0, t) + p00 (0, s)p01 (s, t) , s ≤ t , P[S > s , T > t] = p00 (0, s) + p00 (0, t)p02 (s, t) , s > t , ( Rt 0 Rt R t Rτ e− R0 µ+ν + s e− R0 µ+ν µτ e− Rτ ν , s ≤ t , Rs = (7.97) s τ s 0 e− 0 µ+ν + t e− 0 µ+ν ντ e− τ µ , s > t . The marginal survival function of T is (put s = 0 in (7.97))

P[T > t] = p00 (0, t) + p01 (0, t) Z t R Rt 0 R τ − 0t µ+ν e− 0 µ+ν µτ e− τ ν , t ≥ 0 . + = e

(7.98)

It is intuitively obvious that S and T are independent if µ0τ = µτ and ντ0 = ντ for all τ , and this will follow from Theorem 2 below (see also Exercise 1). It is also intuitively clear that S and T will become dependent if we let the mortality rates depend on marital status. Let us see what happens if the mortality rate increases upon the loss of the spouse. Theorem 2: If µ0τ ≥ µτ and ντ0 ≥ ντ for all τ , then S and T are positively dependent in the sense RTI(S|T ) (hence AS(S, T ) and PQD(S, T )). If µ0τ ≤ µτ and ντ0 ≤ ντ for all τ , then S and T are negatively dependent in the sense RTD(S|T ) (hence AS(−S, T ) and PQD(−S, T )). If µ0τ = µτ and ντ0 = ντ for all τ , then S and T are independent. Proof : Consider first the case s ≤ t. From (7.97) and (7.98) we get Rt Rt 0 R t Rτ e− 0 µ+ν + s e− 0 µ+ν µτ e− τ ν dτ Rt Rt P[S > s | T > t] = R t Rτ 0 e− 0 µ+ν + 0 e− 0 µ+ν µτ e− τ ν dτ R R s − τ µ+ν−ν 0 e 0 µτ dτ 0 Rt . = 1− R t − R τ µ+ν−ν 0 − 0 µ+ν−ν 0 0 e + 0e µτ dτ

Now we need only to study the denominator in the second term as a function of t. Its derivative is Rs 0 e− 0 µ+ν−ν (νt0 − νt ) .

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

105

It follows that P[S > s | T > t] is an increasing function function of t if νt0 ≥ νt and a decreasing function function of t if νt0 ≤ νt . Next consider the case s > t. This is a bit more complicated. From (7.97) and (7.98) we get Rs Rs 0 R s Rτ e− 0 µ+ν + t e− 0 µ+ν ντ e− τ µ dτ Rt Rt . P[S > s | T > t] = Rt Rτ 0 e− 0 µ+ν + 0 e− 0 µ+ν µτ e− τ ν dτ

∂ By the rule d(u/v) = (v du − u dv)/v 2 , the sign of ∂t P[S > s | T > t] is the same as that of R Z t R R s 0 R Rt − 0τ µ+ν − τt ν 0 − 0t µ+ν e µτ e dτ −e− 0 µ+ν νt e− t µ e + 0 R Z s R R s − 0 µ+ν − 0τ µ+ν − τs µ0 − e + e ντ e dτ × t R Z t R R R − 0t µ+ν − 0t µ+ν − 0τ µ+ν − τt ν 0 0 e (−µt − νt ) + e µt + e µτ e dτ (−νt ) . 0

In this expression two terms cancel in the last parenthesis. Further, to get rid Rt Rs of some common factors, let us multiply with e 0 µ+ν e 0 µ+ν , which preserves the sign and turns the expression into R Z t R t s 0 0 µ+ν−ν − 1+ eτ µτ dτ e t µ−µ +ν νt 0 Z s R Z t R s t 0 0 + 1+ e τ µ−µ +ν ντ dτ νt + e τ µ+ν−ν µτ dτ νt0 . 0

Substituting Z s R s 0 e τ µ−µ +ν ντ dτ

s R s

Rs t

µ−µ0 +ν

s R s

= e

(µτ − µ0τ + ντ ) dτ

µ−µ0 +ν

(µ0τ − µτ ) dτ Z s R s 0 −1+ e τ µ−µ +ν (µ0τ − µτ ) dτ

and rearranging a bit, we arrive at Z Z t R t 0 µ+ν−ν 0 τ νt + ν t e µτ dτ 0

t R s

µ+ν−ν

s R s

µ−µ0 +ν

(µ0τ − µτ ) dτ

µτ dτ (νt0 − νt ) .

It follows that P[S > s | T > t] is an increasing function function of t if µ0t ≥ µt and νt0 ≥ νt and a decreasing function function of t if µ0t ≤ µt and νt0 ≤ νt .

106

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

7.10

Conditional Markov chains

7.10.1

Retrospective fertility analysis

In connection with a pension insurance scheme there is an additional benefit which is a sum insured payable to possible dependent children less than 18 years old at the time of death of the insured. In the technical basis we therefore need to make assumptions about births. We have to distinguish by sex, and in the following we consider female insured only. The Figure below shows a flowchart for possible life histories with death and births (at most J). To keep things simple, we assume that the insured enters the scheme in state 0 at age 0 and that the process is Markov: for a t year old who has given birth to j children, the mortality rate µj (t) and the fertility rate φj (t) are functions of t and j only.

φ0 -

Alive 0 births

···

φj−1 j Alive j births

µ0 ?

φj -

···

φJ−1 J Alive J births

µj ?

µJ ?

d. Dead

Assume now that the past history of births and death is observed only upon death of the insured, when the additional benefit to the possible dependents is due. Suppose that the statistical data comprise only those who are dead at the time of consideration and that for each of those there is a complete record of the times of possible births and of death. In these data the observed life history of a woman, who entered the scheme u years ago, is governed by a Markov process as described above, but with intensities µ∗j (t) φ∗j (t)

1 , pjd (t, u) pj+1,d (t, u) . = φj (t) pjd (t, u) = µj (t)

(7.99) (7.100)

We see that µ∗j (t) ≥ µj (t), which is easy to explain (we are looking at the mortality, given death). We are going to prove a more interesting result: If mortality increases with the number of births, that is, µj (t) ≤ µj+1 (t) , j = 0, . . . , J − 1, t > 0 ,

(7.101)

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

107

then φ∗j (t) ≥ φj (t) , j = 0, . . . , J − 1 , t > 0 .

(7.102)

We need to prove that pj+1,d (t, u) ≥ pjd (t, u), j = 1, . . . , J − 1. It is convenient to work with pj (t, u) = 1 − pjd (t, u) =

J X

pjk (t, u) ,

(7.103)

k=j

the probability that a t year old with j births will survive to age u, and to prove the hypothesis Hj : pk (t, u) ≤ pj (t, u) ,

k = j + 1, . . . , J,

(7.104)

for j = 0, . . . , J − 1. The proof goes by induction ’downwards’, proving that Hj+1 implies Hj . Thus assume Hj+1 is true. By direct reasoning (or an easy calculation) the mortality intensity at age u (> t) associated with the survival function (F.49) is P k≥j pjk (t, u)µk (u) P , (7.105) µj (t, u) = k≥j pjk (t, u)

hence

pj (t, u) = e−

Ru t

µj (t,s)ds

Two more expressions for pj (t, u), both obvious, are X pj (t, u) = pjk (t, τ ) pk (τ, u) ,

(7.106)

(7.107)

k≥j

t ≤ τ ≤ u, and pj (t, u) = e−

Ru t

(φj +µj )

e− t

Rτ t

(φj +µj )

φj (τ ) pj+1 (τ, u) dτ .

(7.108)

By (F.47) and (7.105) we have µj (u) ≤ µj+1 (t, u) , hence e− Therefore, from (F.53) we get

Ru t

µj

(7.109)

≥ pj+1 (t, u) .

pj (t, u) ≥ e− t φj pj+1 (t, u) Z u R τ + e− t φj φj (t, τ ) pj+1 (t, τ ) pj+1 (τ, u) dτ . t

(7.110)

CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE

108

Focusing on the last two factors under the integral, use in succession (F.49), the induction hypothesis (7.104), and (F.50), to deduce pj+1 (t, τ )pj+1 (τ, u) =

J X

pj+1,k (t, τ )pj+1 (τ, u)

k=j+1

≥

J X

pj+1,k (t, τ )pk (τ, u)

k=j+1

= pj+1 (t, u) . Putting this into (7.110), we obtain R Z u R − tτ φj − tu φj e φj (t, τ ) dτ pj+1 (t, u) = pj+1 (t, u) . pj (t, u) ≥ e + t

It follows that Hj is true. Since HJ−1 is obviously true, we are done. Comment: The inequality (F.48) means that the fertility rates will be overestimated if one uses the estimators for the φ∗j based on diseased participants in the scheme. If the inequalities (F.47) are reversed, then also the inequality (F.48) will be reversed, and the estimators the φ∗j will underestimate the fertility. In particular it follows that, under the hypothesis of non-differential mortality, the fertility rates will be unbiasedly estimated from the selected material of diseased participants.

Chapter 8

Probability distributions of present values Abstract: A system of integral equations is obtained for the statewise probability distributions of the present value of future payments on a multistate life insurance policy under Markov assumptions. They are brought on a differential form convenient for computation and applied to some cases. Key words: Multistate life insurance, Markov counting process, optional sampling, stochastic interest.

8.1

Introduction

A. Background and motive of the present study. By tradition, life insurance mathematics centers on conditional expected values of discounted cashflows. A key tool of the theory are Thiele’s differential equations, which describe the development of such expected values for a multistate policy driven by a Markov process. In two recent papers one of the authors (Norberg, 1994, 1995) obtains differential equations for higher order moments of present values and offers examples of their potential uses in solvency assessments and in construction of untraditional insurance products. In continuance of those results, we undertake to determine the probability distributions that are at the base of the moments and of any other expected values of interest. Knowledge of the distribution of the present value, and in particular its upper tail, gives insight into the riskiness of the contract beyond what is provided by the mean and the higher order moments. B. Contents of the paper. By way of introduction, Section 2 deals briefly with models involving only a finite number of random variables. In such situations the distributions of present values (and any other functions of the random variables) can be obtained by 109

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES110 integrating the finite-dimensional distribution. This approach comes down overagainst more complex situations where stochastic processes have to be employed. In Section 3 we consider a general multistate insurance policy with payments of assurance and annuity types and with state-dependent force of interest. Assuming that the state process is Markov, we derive in Section 4 a set of integral equations for the conditional probability distribution function of the present value of future benefits less premiums, given the state at the time of consideration. The equations are converted to a differential form that forms the basis of a computational scheme. Examples of applications, including numerical illustrations, are given in Section 5.

8.2

Calculation of probability distributions of present values by elementary methods

A. A simple example involving only one life length. De Pril (1989) and Dhaene (1990) compile lists of distributions of present values of standard single-life insurance benefits ; pure endowment, term assurance, endowment assurance, and life annuity. To offer an example along this line, consider a life insurance policy specifying that the sum assured b is to be paid out immediately upon the (possible) death of the insured within n years after the date of issue of the policy and that premiums are payable continuously at level rate c per year as long as the contract is in force. Suppose interest accumulates with constant intensity δ so that the τ years discount factor is v τ = e−δτ . Denoting the remaining life time of the insured by T , the present value of benefits less premiums on the contract is U (T ) = bv T 1(0,n] (T ) − c¯ aT ∧n| , Rt τ −δt where a ¯t| = 0 v dτ = (1 − e )/δ is the present value of an annuity certain payable continuously at level rate 1 per year for t years. The function U is nonincreasing in T and, letting T be a random variable, we easily find the probability distribution (4.59). The jump at −c¯ an| is due to the positive probability of survival to time n. Similar effects are to be anticipated also for more complex finite term insurance products since, in general, there is a positive probability that the policy will remain in the current state until the contract terminates. Fig. 1 shows the graph of the function in (4.59) for the case where δ = Rt ln(1.045) = 0.044017, P[T > t] = e− 0 µ(τ )dτ with µ(t) = 0.0005 + 0.000075858 · 100.038(30+t) , n = 30, b = 1, and c = 0.0042608; interest and mortality are according to the first order technical basis currently used by most Danish insurers, supposing the insured is a male who is 30 years old at the time when the policy is issued, and the given value of c is the net premium rate. This contract will henceforth be referred to as the term insurance policy. Figure 1 about here. Fig. 1: Probability distribution of the present value at time 0 of the term in-

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES111 surance policy. B. Models involving a finite number of random variables. The analysis above was straightforward because the present value is a function of only one random variable. The approach works whenever only a finite number of random variables, T1 , . . . , Tr , are involved, e.g. for a multilife insurance depending on r lifetimes. The probability distribution of a present value U (T1 , . . . , Tr ), say, is obtained by integrating the joint probability function of the Ti -s over the sets {(t1 , . . . , tr ); U (t1 , . . . , tr ) ≤ u} for (in principle) all u. When applicable at all, this procedure would usually be far more complicated than the one we are going to demonstrate. Thus, with no further ado, we now turn to the principal message of the paper.

8.3

The general Markov multistate policy

A. Payments, interest, and present values. Consider a stream of payments in respect of a contract issued at time 0 and terminating at time n, say, and denote by B(t) the total amount paid in the time interval [0, t]. The payment function {B(t)}t≥0 is assumed to be rightcontinuous and of bounded variation. Money is currently invested in (or borrowed from) a fund that at any time t yields δ(t) in return per unit of time and unit amount on deposit, that is, δ(t) is the force of interest R τ at time t. Then the discounted value at time t of a unit due at time τ is e− t δ , and so the present value at time t of the future payments under the contract is Z n R τ (8.1) e− t δ dB(τ ). t

R R The short-hand exemplified by δ = δ(s)ds will be in frequent use throughR Rb R out. By convention, a means (a,b] if b < ∞ and (a,∞) if b = ∞.

B. The multi-state insurance policy. We adopt the standard set-up of life insurance mathematics as presented in Norberg (1994). There is a set of states, J = {1, . . . , J}, such that at any time t ∈ [0, n] the policy is in one an only one state. Denote by X(t) the state of the policy at time t. Considered as a function of t, {X}t≥0 is taken to be rightcontinuous, and X(0) = 1 implying (as a convention) that the policy commences in state 1. Introduce Ij (t) = 1[X(t) = j], the indicator of the event that the policy is in state j at time t, and Njk (t) = ]{τ ; τ ∈ (0, t], Xτ − = j, X(τ ) = k}, the total number of transitions of X from state j to state k (6= j) by time t. The payment function B is assumed to be of the form (7.41), where each Bj is a deterministic payment function specifying payments due during sojourns in state j (a general life annuity) and each bjk is a deterministic function specifying payments due upon transitions from state j to state k (a general life assurance). The left-limit in Ij (t−) means that the state j annuity is effective at time t if

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES112 the policy is in state j just prior to (but not necessarily at) time t. Consistently we define Ij (0−) = 1. We shall allow the force of interest to depend on the current state, that as in (7.74. C. The time-continuous Markov model. It is assumed that {X(t)}t≥0 is a (continuous-time) Markov chain. Denote the transition probabilities by pjk (t, u) = P[X(u) = k|X(t) = j]. The transition intensities µjk (t) = lim pjk (t, t + h) h↓0

are assumed to exist for P all j, k ∈ J , j 6= k. The total intensity of transition out of state j is µj· (t) = k;k6=j µjk (t). The probability of staying uninterruptedly Ru in state j during the time interval from t to u is e− t µj· .

8.4

Differential equations for statewise distributions

A. The statewise probability distributions. The problem is to determine the conditional probability distribution of the liability in (8.1), given the information available at time t. Since the Markov assumption implies conditional independence between past and future for fixed present state of the policy, the relevant functions are the statewise probability distributions defined by

Pj (t, u) = P

−

Rτ t

t ∈ [0, n], u ∈ R, j ∈ J .

dB(τ ) ≤ u Ij (t) = 1 ,

(8.1)

B. A system of integral equations. A simple heuristic argument will establish that the probabilities in (8.1) satisfy the integral equations

Pj (t, u) =

X Z

k;k6=j

e− t

Rs t

µj·

µjk (s) ds

Z s eδj (s−τ ) dBj (τ ) − bjk (s) ·Pk s, eδj (s−t) u − t Z n R − tn µj· −δj (τ −t) +e 1 e dBj (τ ) ≤ u . t

(8.2)

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES113 Rs

In the first terms on the right here the factor e− t µj· µjk (s)ds is the probability that the policy stays in state until time s (< n) and then makes a transfer to state k (6= j) in the small time interval [s, s + ds). In this case the annuity Bj is in force during the time interval (t, s], the lump sum bjk (s) falls due at time s, and the interest rate during this time interval is δj , and so the event Z

e− t

Rτ t

dB(τ ) ≤ u

(8.3)

takes place if Z

e−δj (τ −t) dBj (τ ) + e−δj (s−t) bjk (s) + e−δj (s−t) t

e− s

Rτ s

dB(τ ) ≤ u

or, equivalently, Z

e− s

Rτ s

Z dB(τ ) ≤ eδj (s−t) u −

s t

e−δj (τ −t) dBj (τ ) − bjk (s).

Thus, the corresponding conditional probability is Z s δj (s−t) δj (s−τ ) u− e dBj (τ ) − bjk (s) . Pk s, e t

Summing over all times s and states k, we obtain the first terms on the right of (8.2), which thus is the part of the total probability that pertains to exit from state j before time n. Likewise it is realized that the last term on the right of (8.2) is the remaining part of the probability, pertaining to the case of no transition out of state j before time n. This heuristic argument is made rigorous by applying Doob’s optional sampling theorem to the martingale generated by the indicator of the event in (8.3) and the stopping time defined as the the minimum of n and the time of the first transition after t from the current state j. C. A system of differential equations. Already (8.2) might serve as a basis for computation of the statewise probability functions, but is not convenient since the integrand on the right depends on t. Introduce the auxiliary functions Qj defined by

Qj (t, u) = Pj or

Z t e−δj τ dBj (τ ) t, eδj t u − 0

(8.4)

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES114

Pj (t, u) = Qj

t, e

−δj t

−δj τ

dBj (τ ) .

(8.5)

Rt Rt Multiply by e− 0 µj· in (8.2), insert eδj t u − 0 e−δj τ dBj (τ ) in the place of u, and rearrange a bit to obtain e−

Rt 0

µj·

Qj (t, u) =

e− t

Rs 0

·Qk s, e −e

µj·

(δj −δk )s

+ e−

Rn 0

µjk (s) ds

k;k6=j

µj·

−δj τ

e−δk τ dBk (τ ) 0

dBj (τ ) − e

−δk s

bjk (s)

e−δj τ dBj (τ ) ≤ u .

(8.6)

Now the integrand on the right does not contain t, and we are allowed to differentiate along t on the right hand side by simply substituting t forR s in minus the t integrand. Performing this and cancelling the common factor e− 0 µj· , we arrive at the following main result, where the side condition (7.76) comes directly out of (8.6) by letting t ↑ n:

Theorem. The functions Qj in (8.4) are the unique solutions to the differential equations dt Qj (t, u) = µj· (t)dt Qj (t, u) −

µjk (t) dt

k;k6=j

Z t e−δk τ dBk (τ ) ·Qk t, e(δj −δk )t u + 0

−e(δj −δk )t

e−δj τ dBj (τ ) − e−δk t bjk (t) ,

(8.7)

0 ≤ t ≤ n, subject to the conditions

Qj (n, u) = 1

e 0

−δj τ

dBj (τ ) ≤ u .

Having determined the auxiliary Qj , we obtain the Pj from (8.5).

(8.8)

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES115 Remark: The differentiation is in the Stieltje’s sense for functions of bounded variation and does not require differentiability or any other smoothness properties of the functions involved. D. Computational scheme. A simple numerical procedure consists in approximating the functions Qj by the functions Q∗j obtained from the finite difference version of (8.7): Q∗j (t − h, u) = (1 − µj· (t)h)Q∗j (t, u) + h

µjk (t)

k;k6=j

·Q∗k −e

t, e

(δj −δk )t

−δj τ

e−δk τ dBk (τ ) 0

dBj (τ ) − e

−δk t

bjk (t) .

(8.9)

Starting from (8.8) (with Q∗j in the place of Qj ), one calculates first the functions Q∗j (n − h, ·) by (8.9) and continues recursively until the Q∗j (0, ·) have been calculated in the final step. The Q∗j (t, u) are defined for t ∈ {0, h, 2h, . . . , n} and u ∈ {a, a + h0 , a + 0 2h , . . . , b}, say, where the steplengths h and h0 must be sufficiently small and a and b must be chosen such that the supports of the Qj (t, ·) are sufficiently well covered by [a, b]. What is “sufficient” must be decided on in each individual case by judgement and by trial and error. If two states j and k are intercommunicating, then Qj (t, ·) and Qk (t, ·) have the same support. The supports are finite and usually easy to determine in situations where the number of assurance payments has a non-random upper bound. The u-arguments in the functions Q∗k on the right of (8.9) must, of course, be rounded to the nearest point in {a, a + h0 , a + 2h0 , . . . , b}. E. Comments on the method. Before turning to applications, we pause in this paragraph to offer some motivation and discussion of our approach. It is not needed in the sequel and may be skipped on the first reading. The equation (8.7) is just the differential form of the integral equation (8.6). ∂ It does not require that the derivatives ∂t Qj (t, u) exist, which they do not in general for the obvious reason that the statewise annuity functions on the right of (8.7) may have jumps. If one should attempt to construct differential equations along the lines of Norberg (1994), the starting point would be the martingale M defined by

M (t) = P

e− 0

Rτ 0

dB(τ ) ≤ u | Ft ,

where Ft = σ{X(τ ); τ ≤ t} is the information generated by the state process

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES116 up to time t. Using the Markov property of conditional independence between past and future, given the present, we find

M (t) =

Ij (t) Pj

t, e

Rt 0

u−

e 0

−

Rτ 0

dB(τ )

Now, the recipe would be to apply the change of variable formula to the expression on the right and then to identify the martingale component that is predictable (and of bounded variation) and hence constant. Accomplishing this without caring about justification, would lead to the first order partial differential equations ∂ ∂ Pj (t, u) + Pj (t, u)(δj u − bj (t)) ∂t ∂u X µjk (t) (Pk (t, u − bjk (t)) − Pj (t, u)) = 0, + k;k6=j

valid between jumps of the contractual annuity functions Bj , and Pj (t−, u) = Pj (t, u − ∆Bj (t)), valid at jumps of the Bj , and subject to the condition that the Pj (n, u) are 0 and 1 according as u < 0 or u ≥ 0. The approach requires that the functions Pj possess first order derivatives in both directions. As we have seen already in the introductory example of Section 2 they generally do not. This difficulty might possibly be circumvented by conditioning on the ultimate state at time n, but proving differentiability of the conditional probabilities would require additional assumptions and would not be straightforward. These remarks serve to show that the method developed in Section 4 is not just one among several candidate approaches to the problem; it is the only mathematically sound solution we are able to offer. One general conclusion we can extract is that there is no single general technique for solving the bulk of problems of the kind considered here; the method will have to be designed for each individual problem at hand and will depend on the model assumptions and the functional of interest.

8.5

Applications

A. The Poisson distribution. In continuance of the example in Paragraph 6B of Norberg (1994), consider the special case with two states, J = {1, 2}, no interest, δ1 = δ2 = 0, and the only payments being an assurance of 1 payable upon each transition, b12 = b21 = 1.

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES117 Then, taking n = 1, the present value in (8.1) is just the number of transition in the time interval (t, 1], N12 (1) + N21 (1) − N12 (t) − N21 (t). Furthermore, take µ12 = µ21 = µ, a constant (> 0). Then it is seen from the defining relation (8.4) that the functions Pj and Qj are all the same. Denoting this function by P , we can work with (8.2), which becomes

P (t, u) =

e−µ(s−t) µP (s, u − 1)ds + e−µ(1−t) 1[0 ≤ u].

(8.1)

Using that P (t, u) = 0 for u < 0, we readily obtain from (8.1) that P (t, u) = e−µ(1−t) for 0 ≤ u < 1. Then, for 1 ≤ u < 2, it follows from (8.1) that P (t, u) = µ(1 − t)e−µ(1−t) + e−µ(1−t) . Proceeding by induction we obtain, for each u ≥ 0, that

P (t, u) =

[u] X (µ(1 − t))i i=0

e−µ(1−t) ,

which is the Poisson distribution with parameter (1 − t)µ, of course. As a check on the accuracy of the numerical method, we list the computed values of P (0, u), u = 0, 1, . . . , 8, for t = 0 and µ = 1 together with the exact values of the Poisson probabilities (in parantheses): 0.3670 (0.3679), 0.7358 (0.7358), 0.9202 (0.9197), 0.9813 (0.9810), 0.9965 (0.9864), 0.9994 (0.9994), 0.9999 (0.9999), 0.9999 (0.9999), 1.0000 (1.0000). These results were obtained with h = 1/200, h0 = 1/100, a = −0.5, and (truncating the infinite support) b = 9.5. B. The term insurance policy. To analyse the term insurance policy in Paragraph 2A, take J = {1, 2}, n = 30, µ12 (t) = 0.0005 + 0.000075858 · 100.038(30+t) , δ1 = ln(1.045), b1 = −0.0042608, b12 = 1, and all other intensities and payments null. Again, as a check on the accuracy of the numerical method, we list the computed values of P (0, u) together with exact values (in parantheses): P (0, u) = 0 for u < −0.0705 (0 for u < −0.0709), P (0, u) = 0.8453 for u ∈ [−0.0705, 0.1965) (0.8452 for u ∈ [−0.0709, 0.1961)), P (0, 0.2) = 0.8491 (0.8490), P (0, 0.4) = 0.9465 (0.9467), P (0, 0.6) = 0.9774 (0.9776), P (0, 0.8) = 0.9918 (0.9919), P (0, 1) = 1.0000 (1.0000). These results are based on h = 1/1000, h0 = 1/2000, a = −0.1, and b = 1.1. C. A combined insurance policy. In our final numerical example we consider what will be referred to as the combined policy, which is the same as the one in Paragraph B, but with a disability pension added. More specifically, 1 is payable upon death, an annuity with level intensity 0.5 is payable during disability, and premium is payable with level intensity in active state. The relevant model entities are J = {1, 2, 3},

CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES118 n = 30, b13 = b23 = 1, b1 = −0.013108 (net premium when the intensities are as specified below), b2 = 0.5, and, adoping the standard Danish technical basis except for the recovery intensity, δ = ln(1.045) (independent of state), and µ13 (t) = µ23 (t) = 0.0005 + 0.000075858 · 100.038x , µ12 (t) = 0.0004 + 0.0000034674 · 100.06(30+t) , µ21 (t) = 0.005, all other payments and intensities being null. Figure 2 about here Fig. 2: Probability distribution of the present value of the combined insurance policy in (a) state 1 and (b) state 2. This example is a follow-up of Paragraphs 4B-C in Norberg (1994), where the first three moments of the present value are calculated for the combined policy. D. Numerical evaluation of multiple integrals. Numerical integration in higher dimensions is in general complicated, and there exists no technique held to be universally superior. The technique developed here can be used to evaluate integrals that, possibly after a reinterpretation, can be recognized as a probability related to a present value for a suitably specified policy. Just to illustrate the idea, suppose T1 and T2 are independent positive random variables with cumulative distribution functions F1 and F2 with densities f1 and f2 , respectively, and that we seek P[(T2 ∧ 1) − (T1 ∧ 1) ≤ u]. It is realized that this probability is found as P (0, u) for the policy with J = {1, 2, 3, 4}, µ12 (t) = µ34 (t) = f1 (t)/(1 − F1 (t)), µ13 (t) = µ24 (t) = f2 (t)/(1 − F2 (t)), b1 (t) = −1, b2 (t) = 1, n = 1, and no interest. Countless examples of this kind can be constructed.

Chapter 9

Reserves Prospective and retrospective reserves are defined as conditional expected values, given some information available at the time of consideration. Each specification of the information invoked gives rise to a corresponding pair of reserves. Relationships between reserves are established in the general set-up. For the prospective reserve the present definition conforms with, and generalizes, the traditional one. For the retrospective reserve it appears to be novel. Special attention is given to the continuous time Markov chain model frequently used in the context of life and pension insurance. Thiele’s differential equation for the prospective reserve is shown to have a retrospective counterpart. It is pointed out that the prospective and retrospective differential equations have, respectively, the Kolmogorov backward and forward differential equations as special cases. Practical uses of the differential equations are demonstrated by examples.

9.1

Introduction

A. Sketch of the idea. The concept of prospective reserve is no matter of dispute in life insurance mathematics. It is defined as the conditional expected present value of future benefits less premiums on the policy, given its present state. A straightforward generalization is obtained by conditioning on some other piece of information, e.g. on the policy’s staying in some subset of the state space. It is proposed here to define the retrospective reserve analogously as the conditional expected present value of past premiums less benefits. B. An example: insurance of a single life. A person aged x buys a life insurance policy specifying that the sum assured, b, is payable immediately upon death before age x + n and that premiums are to be contributed continuously with level intensity c throughout the insurance period. Let Tx denote the person’s remaining life length after the policy is issued at time 0, say. R Assume that t the survival function t px = P {Tx > t} is of the form t px = e− 0 µx+s ds , with 119

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continuous force of mortality, µ. Finally, assume that interest is earned with a constant, nonrandom intensity δ so that v = e−δ is the annual discount factor and i = eδ − 1 is the annual rate of interest (1 + i = v −1 is the annual interest factor). At any time t ∈ [0, n] the policy is either in state 0 = ”alive” or in state 1 = ”dead”. The prospective reserves in the two states, indicated by subscripts 0 and 1, are

V0+ (t) =

v τ −t t

τ −t px+t {bµx+τ

− c} dτ

(9.1)

(by the usual heuristic argument, the sum of expected discounted benefits minus premiums in small time intervals (τ, τ + dτ ), 0 < τ < t, and, of course, V1+ (t) = 0.

(9.2)

The statewise retrospective reserves as defined above are

V0− (t)

(1 + i)t−τ dτ

(9.3)

(trivial) and

V1− (t) =

1 1 − t px

(1 + i)t−τ {c (τ px − t px ) − b τ px µx+τ } dτ

(9.4)

(use the same kind of argument as in (9.1) noting that, conditional on death within time t, the probability of survival to τ is (τ px − t px )/(1 − t px ) and the probability of death in (τ, τ + dτ ) is τ px µx+τ dτ /(1 − t px ), 0 < τ < t) . The state at time t is X(t) = 1[Tx ≤ t], the ”number of deaths” of the person within time t. This is the information on which the reserves in (9.1) – (9.4) are based. Now, suppose the complete prehistory of the policy is currently recorded, so that it is known at any time if the person is alive or dead and, in the latter case, when he died. The information available at time t is the pair (X(t), min(Tx, t)). Denote the reserves correspondingly by a double subscript. The reserves in state ± 0 remain as above, V0,t (t) = V0± (t), and so does the prospective reserve in state + 1, of course, V1,Tx (t) = V1+ (t) = 0. Only the retrospective reserve in state 1 is affected by the additional information on the exact time of death. It now becomes simply the value at time t of past premiums less the benefit payment,

− (t) V1,T x

Tx 0

(1 + i)t−τ dτ − b(1 + i)t−Tx .

(9.5)

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The quantity in (9.1) is what traditionally is referred to as the prospective reserve. The notion of retrospective reserve launched here differs from the traditional one, which in the present example is

−

1 t px

V0 (t) =

(1 + i)t−τ τ px (c − bµx+τ )dτ.

(9.6)

This quantity emerges from the principle of equivalence, which requires that benefits and premiums should balance in the mean at the outset: Z Splitting

Rn 0

into

Rt 0

v τ τ px (bµx+τ − c)dτ = 0.

(9.7)

Rn t

in (9.7) and substituting from (9.1) and (9.6), yields −

V0 (t) = V0+ (t) .

(9.8)

Thus, the traditional concept of ”retrospective reserve” is rather a retrospective formula for the prospective reserve, valid for b and c satisfying the equivalence principle. For general b and c the quantity − V0 (t) is not an expected value, and it has no probabilistic interpretation in the present model involving one single policy. In an extended (artificial) model, with m independent replicates of the policy issued at time 0, it may be interpreted as the almost sure limit of the total accumulated surplus per survivor by time t as m tends to infinity. As compared with (9.8), the retrospective reserve introduced here is related to the prospective reserve under the equivalence principle by the identity − t px V0 (t)

+ (1 − t px )V1− (t) = t px V0+ (t) .

(9.9)

C. Outline of the paper. In Section 2 the present notions of reserves are defined for quite general stochastic payment streams and discounting rules, and certain relationships between them are established. No particular reference to the insurance context is made at this stage. In Sections 3 and 4 the framework of the further discussions is presented: payments of the life annuity and life insurance types in a continuous time Markov chain model. A useful auxiliary result is that a Markov process behaves like a composition of mutually independent Markov processes in disjoint intervals when its values at the dividing points between the intervals are fixed. This together with standard results for Markov chains is used in Section 5 to investigate the properties of reserves in the Markov chain case. The prospective reserve, given the state at the time of consideration, is the traditional one, which satisfies the well-known generalized Thiele’s differential equations (see e.g. Hoem, 1969a), here also referred to as the prospective differential equations. The statewise retrospective reserves turn out to satisfy a set of retrospective differential equations, different from the prospective ones. It

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is pointed out the differential equations for the reserves have the Kolmogorov differential equations for the transition probabilities as special cases. Surprisingly, maybe, it is the retrospective equations that generalize the Kolmogorov forward equations, while the prospective equations generalize the Kolmogorov backward equations. In Section 6 some more examples are supplied.

9.2

General definitions of reserves and statement of some relationships between them

A. Payment streams and their discounted values. First some basic definitions and results are quoted from Norberg (1990). Consider a stream of payments commencing at time 0. It is defined by a finite-valued payment function A, which for each time t ≥ 0 specifies the total amount A(t) paid in [0, t]. Negative payments are allowed for; it is only required that A be of bounded variation in finite intervals and, by convention, rightcontinuous. This means that A = B − C, where B and C are non-negative, nondecreasing, finite-valued, and right-continuous functions representing the outgoes and incomes, respectively, of some business. In the context of insurance B represents benefits and C represents contributed premiums on an insurance policy (or a portfolio of policies). The payment function extends in a unique way to a payment measure on the Borel sets, which is also denoted by A. Assuming that payments are valuated by a piecewise monotone and continuous discount function v, the present value of A at time t is

V (t, A) =

1 v(t)

v(τ ) dA(τ )

(9.1)

[0,∞)

(the sum of all payments in small intervals discounted at time 0, multiplied by the interest factor 1/v(t) for the interval from 0 to t). Just to obtain transparent formulas, it will be assumed throughout that v is of the form v(t) = e−

Rt 0

(9.2)

with piecewise continuous interest intensity δ. (The shorthand exemplified by Rt Rt δ = 0 δ(τ ) dτ will be in frequent use throughout.) 0

B. Definitions of retrospective and prospective reserves. The restriction of A to a (measurable) time set T is the measure AT counting only those A-payments that fall due in T ; AT {S} = A{S ∩ T }. At any time t ≥ 0 the payment stream splits into payments after time t and payments up to and including time t; A = A(t,∞) + A[0,t] . The present value in (9.1) splits correspondingly into

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V (t, A) = V + (t, A) − V − (t, A),

(9.3)

where V + (t, A) = V (t, A(t,∞) ) =

1 v(t)

v(τ ) dA(τ )

(9.4)

(t,∞)

is the discounted value of future net outgoes (in the insurance context benefits less premiums), and V − (t, A) = V (t, −A[0,t] ) =

1 v(t)

v(τ ) d(−A)(τ )

(9.5)

[0,t]

is the value, with accumulation of interest, of past net incomes (in the insurance context premiums less benefits). The quantity V − (t, A) is observable by time t and can suitable be called the individual retrospective reserve of the policy at time t. If the future development of (v, A) were known, then V + (t, A) would be the appropriate amount to set aside to cover future excess of benefits over premiums on the individual policy. However, if the future course of (v, A) is uncertain, it is not possible to provide V + (t, A) as a prospective reserve on an individual basis. Assume now that A and, possibly, also v are stochastic processes on some probability space (Ω, F, P ). An operational definition of the prospective reserve must depend solely on information that is at hand at the moment when the reserve is to be provided. Let F = {Ft }t≥0 be a family of sub-sigmaalgebras of F, Ft representing some piece of information available at time t. The family F may be increasing, that is, Fs ⊂ Ft , s < t, but this is not required in general. Reserves are defined as conditional expected values, given the information provided by F. At time t the prospective F-reserve is VF+ (t, A) = EFt V + (t, A),

(9.6)

and the retrospective F-reserve is VF− (t, A) = EFt V − (t, A),

(9.7)

where the subscript on the expectation sign signifies conditioning. The prospective F-reserve meets the operationality requirement formulated above as it is determined by the current information. Even though the retrospective individual reserve in (9.5) is observable by time t, it may be judged relevant to calculate retrospective reserves with respect to some more summary information F. For a given realization of {v(τ )}0≤τ ≤t , Ft may be thought of as a classification of

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the policies, whereby all policies with the same characteristics as specified by Ft are grouped together. Forming the mean, conditional on Ft , means averaging over all policies in the same group, roughly speaking. The reserves are conditional means of the present values V ± (t, A). Other features of the conditional distributions of these random variables may be of interest. In particular, as measures of variability, introduce the variances ±(2)

(t, A) = V arFt V ± (t, A).

(9.8)

C. Relationships between reserves. When only one payment stream is considered, notation can be saved by dropping the symbol A from V (t, A). Thus, abbreviations like V (t) and VF± (t) will be frequently used in the sequel. By (9.1) – (9.3), the value of A at time 0 is related to its value at any time t ≥ 0 by V (0) = v(t)V (t) = v(t){V + (t) − V − (t)}. Taking expectation gives E V (0) = E {v(t)V (t)} +

(9.9) −

= E {v(t)(V (t) − V (t))}.

(9.10)

The equivalence principle of insurance states that premiums and benefits should balance on the average as seen at the outset, that is, E V (0) = 0.

(9.11)

It does not imply EV (t) = 0 for t > 0 unless v is a deterministic function, confer (9.9). Taking iterated expectations in (9.10), the equivalence requirement can be cast as E {v(t)VF− (t)} = E {v(t)VF+ (t)}

(9.12)

if v(t) is determined by Ft , and E VF− (t) = E VF+ (t)

(9.13)

if v is deterministic. Relation (9.9) is a special case of (9.13). Let F0 = {Ft0 }t≥0 be some sub-sigmaalgebra representing more summary information than F = {Ft }t≥0 in the sense that Ft0 ⊂ Ft , t ≥ 0. The rule of iterated expextations yields the following relationship between reserves on different levels of information:

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CHAPTER 9. RESERVES

VF±0 (t) = EFt0 VF± (t).

(9.14)

By the general rule V arX = E V arY X + V ar EY X, variances denoted as in (9.8) are related by ±(2)

VF 0

±(2)

(t) = EFt0 {VF

(t) + (VF± (t))2 } − (VF±0 (t))2 .

(9.15)

There exist also useful relationships between reserves at different times for a fixed source of information, F. The discounted values of future net outgoes at two different points of time, t < u, are related by v(t)V + (t) =

v(τ ) dA(τ ) + v(u)V + (u). (t,u]

Similarly, for s < t, v(t)V − (t) = v(s)V − (s) +

v(τ ) d(−A)(τ ). (s,t]

Taking expectations in these two identities, yields a pair of basic relationships under the assumption that v is deterministic. If {A(τ )}τ ≥u depends stochastically only on Fu for given Ft and Fu , then v(t)VF+ (t)

(t,u]

v(τ ) d EFt A(τ ) + v(u)EFt VF+ (u).

(9.16)

Likewise, if {A(τ )}τ ≤s depends stochastically only on Fs for given Fs and Ft , then v(t)VF− (t) = v(s)EFt VF− (s) +

(s,t]

v(τ ) d(−EFt A)(τ ).

(9.17)

D. Right-continuity of the reserve processes. As defined by (9.6) and (9.7) the reserves are right-continuous stochastic processes. They could alternatively be made left-continuous by letting the integrals in (9.4) and (9.5) extend over [t, ∞) and [0, t), respectively. This would be in keeping with tradition, but the rightcontinuous versions are chosen here since they fit into the general apparatus of stochastic integrals and differential equations and thus are the more convenient quantities to deal with in anticipated applications of the theory to complex models. Anyway, the right-continuous and the left-continuous versions differ only at points of time where non-null amounts fall due with positive probability.

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CHAPTER 9. RESERVES

9.3

Description of payment streams appearing in life and pension insurance

A. Specification of the insurance treaty terms. Having insurances of persons in mind, consider insurance treaties specifying terms of the following general form. There is a set J = {0, . . . , J} of possible states of the policy. The policy is issued at time 0, say. At any time t ≥ 0 it is in one and only one of the states in J , commencing in state 0. Payments are of two kinds: general life annuities by which the amount A◦g (t) − A◦g (s) is paid during a sojourn in state g throughout the time interval (s, t], and general life insurances by which an amount a◦gh (t) is paid immediately upon a transition from state g to state h at time t. They comprise benefits, administration expenses of all kinds, and premiums (negative). The contractual functions A◦g and a◦gh are, respectively, a payment function and a finite-valued, right-continuous function. B. The form of the payment function. Let X(t) be the state of the policy at time t. The development of the policy is given by {X(t)}t≥0 . This process, regarded as a function from [0, ∞) to J , is assumed to be right-continuous, with a finite number of jumps in any finite time interval. Let Ngh be the process counting the transitions from state g to state h, that is, Ngh (t) = #{τ ∈ [0, t]; X(τ −) = g, X(τ ) = h}. The stream of net payments is of the form A=

X g

{Ag +

Agh },

h;h6=g

with dAg (t) = 1[X(t) = g] dA◦g (t), dAgh (t) = a◦gh (t) dNgh (t).

(9.1) (9.2)

The behaviour of the retrospective and prospective reserves is now to be studied for some specifications of F that are of relevance in insurance. The model framework will be the traditional Markov chain, which yields lucid results.

9.4

The Markov chain model

A. Model assumptions and basic relationships. The process {X(t)} t≥0 is assumed to be a continuous time Markov chain on the state space J . The transition probabilities are denoted pjk (t, u) = P {X(u) = k | X(t) = j}. The transition intensities

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CHAPTER 9. RESERVES

µjk (t) = lim u↓t

pjk (t, u) u−t

are assumed to exist for all t and j 6= k. To simplify matters, the functions µjk are furthermore assumed to be piecewise continuous. The total transition intensity from state j is µj =

µjk .

k; k6=j

From the Chapman-Kolmogorov equations, pjk (t, u) =

pjg (t, τ ) pgk (τ, u),

g∈J

valid for t ≤ τ ≤ u, one obtains Kolmogorov’s differential equations, the forward, X ∂ pij (s, t) = pig (s, t) µgj (t) − pij (s, t)µj (t), ∂t

(9.1)

g; g6=j

and the backward, X ∂ pjk (t, u) = µj (t)pjk (t, u) − µjg (t)pgk (t, u). ∂t

(9.2)

g; g6=j

Together with the initial conditions pjk (t, t) = δjk , j, k ∈ J , (the Kronecker delta) they determine the transition probabilities uniquely. The Kolmogorov equations are the major tools for constructing the transition probabilities from the intensities, which are the basic entities in the system; they are functions of one argument only and, being readily interpretable, they form the natural starting point for specification of the model. The conditional probability of staying uninterruptedly in state j throughout the time interval [t, u], given that X(t) = j, is (solve a simple differential equation) pjj (t, u) = e−

Ru t

µj

(9.3)

B. Moments of present values. Throughout the balance of the paper it will be assumed that the interest intensity is nonstochastic. Consider the general annuities and insurances defined by (9.1) and (9.2). The present values at time t of their contributions in some time interval T are

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CHAPTER 9. RESERVES

V (t, AgT ) = V (t, AghT ) =

Z 1 v(τ )1[X(τ ) = g]dA◦g (τ ), v(t) T Z 1 v(τ )a◦gh (τ ) dNgh (τ ). v(t) T

Here follows a list of formulas for the first and second order moments of such present values, conditional on X(s), with T ⊂ [s, ∞). The mean values are

EX(s)=i V (t, AgT )

EX(s)=i V (t, AghT )

Z 1 v(τ )pig (s, τ ) dA◦g (τ ), v(t) T Z 1 v(τ )a◦gh (τ )pig (s, τ )µgh (τ ) dτ. v(t) T

(9.4) (9.5)

The noncentral second order moments, for intervals S, T ⊂ [s, ∞), are EX(s)=i {V (t, AeS )V (t, AgT )} ZZ 1 v(ϑ)v(τ ){1[ϑ ≤ τ ]pie (s, ϑ)peg (ϑ, τ ) = 2 v (t) S×T +1[ϑ > τ ]pig (s, τ )pge (τ, ϑ)} dA◦e (ϑ) dA◦g (τ ),

(9.6)

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CHAPTER 9. RESERVES

EX(s)=i {V (t, Aef S )V (t, AghT )} ZZ 1 = 2 [ v(ϑ)v(τ ){1[ϑ < τ ]pie (s, ϑ)pf g (ϑ, τ ) v (t) S×T +1[ϑ > τ ]pig (s, τ )phe (τ, ϑ)}µef (ϑ)µgh (τ )a◦ef (ϑ)a◦gh (τ ) dϑ dτ Z v 2 (τ )pig (s, τ )µgh (τ )a◦gh2 (τ ) dτ ], (9.7) +δef,gh S∩T

EX(s)=i {V (t, AeS )V (t, AghT )} ZZ 1 = 2 v(ϑ)v(τ ){1[ϑ ≤ τ ]pie (s, ϑ)peg (ϑ, τ ) v (t) S×T +1[ϑ > τ ]pig (s, τ )phe (τ, ϑ)}dA◦e (ϑ) µgh (τ )a◦gh (τ ) dτ.

(9.8)

Variances and covariances are now formed by the rule Cov(X, Y ) = E(XY ) − EXEY . The formulas are easily established by a heuristic type of argument. To motivate e.g. (9.8), put Z

v(τ )a◦gh (τ ) dNgh (τ ) } EX(s)=i { v(ϑ)1[X(ϑ) = T S ZZ = v(ϑ)v(τ )EX(s)=i {1[X(ϑ) = e] dNgh (τ )}a◦gh (τ ) dA◦e (ϑ), e] dA◦e (ϑ)

S×T

and calculate the expected value in the integrand for ϑ ≤ τ and for ϑ > τ . Rigorous proofs will not be given here. Some of the results in (9.4) – (9.8) are proved for absolutely continuous A◦g by Hoem (1969a) and Hoem & Aalen (1978). C. Conditional Markov chains. Viewing reserves as conditional expected values, one may be concerned with various sub-sigmaalgebras of the basic sigmaalgebra F. They will typically be of the form FT = σ{X(τ ); τ ∈ T }, the sigmaalgebra representing all information provided by the process X in the time set T . Of particular interest are F[0,t] and F{t0 ,...,tq } . The Markov property means precisely that for B ∈ F(t,∞) , P {B | F[0,t] } = P {B | F{t} },

(9.9)

that is, the future development of the process depends on its past and present states only through the present. An equivalent formulation is that, for A ∈ F(0,t) and B ∈ F(t,∞) ,

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CHAPTER 9. RESERVES

P {A ∩ B | F{t} } = P {A | F{t} }P {B | F{t} },

(9.10)

which says that for a fixed present state of the process its future and past are conditionally independent. The equivalence of (9.9) and (9.10) is easily established in the present situation with finite state space. By induction (9.10) extends to the following: for 0 = t0 < t1 < . . . < tq < tq+1 = ∞ and Ap ∈ F(tp−1 ,tp ) , p = 1, . . . , q + 1,

P {∩q+1 p=1 Ap | F{t0 ,...,tq } } =

q+1 Y

p=1

P {Ap | F{tp−1 ,tp } },

where F{tq ,∞} = F{tq } . Thus, conditional on X(0), X(t1 ), . . . , X(tq ), the developments of the process in the intervals (tp−1 , tp ) are mutually independent. So, to study the conditional process, given its values at fixed points, interest can be focused on the behaviour of the segment {X(τ )}s 0. In particular, V0− (0) = −A◦0 (0), of course, whereas Vj− (0) is not defined for j 6= 0. Formula (9.3) is easy to interpret by direct heuristic reasoning: conditionally, given X(0) = 0 and X(t) = j, the probability of staying in state g at time τ is p0g (0, τ )pgj (τ, t)/p0j (0, t), and the probability of a transfer from state g to state h in the time interval (τ, τ + dτ ) is p0g (0, τ )µgh (τ )dτ phj (τ, t)/p0j (0, t). The conditional variances, ±(2)

(t) = V arX(t)=j V ± (t),

are composed from (9.6) – (9.8) upon inserting the relevant conditional transition probabilities and intensities. Finally, note that the future and the past developments of the process are independent, hence uncorrelated, for fixed present X(t) = j. C. More general reserves. Referring to Paragraph 2C, let F be as in Paragraph B and let F0 = {Ft0 }t≥0 be the more summary information with Ft0 generated by the events X(t) ∈ Jl , l = 1, . . . , m, where {J1 , . . . , Jm } is some partitioning of J . Thus, F0 -reserves at time t are now conditional on X(t) ∈ Jl , l ∈ {1, . . . , m}. For any K ⊂ J and t ≤ u put pjK (t, u) = P {X(u) ∈ K | X(t) = j} =

pjk (t, u).

k∈K

Applying (9.14) and (9.15), one obtains (with obvious notation) VJ±l (t) =

X 1 p0j (0, t)Vj± (t) p0Jl (0, t) j∈Jl

and ±(2)

VJ l

(t) =

X 1 ±(2) p0j (0, t){Vj (t) + Vj± (t)2 } − VJ±l (t)2 , p0Jl (0, t) j∈Jl

hence F0 -reserves and F0 -variances can be composed from the corresponding single-state quantities.

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A variety of reserves based on different choices of F can be analysed by the results in Section 4. For instance, taking Ft = F{t1 ,t2 ,...}∩[0,t] means that reserves are formed by averaging over policies that are known to have visited certain states at given epochs in the past. The reserves and variances are found for each of the intervals and, by conditional independence, simply added to give the required total reserve and variance. D. Differential equations for prospective reserves in single states. In the present Markov chain set-up (9.16) specializes to Wj+ (t)

v(τ ) (t,u]

pjg (t, τ ){dA◦g (τ ) +

a◦gh (τ )µgh (τ )dτ }

h;h6=g

pjg (t, u)Wg+ (u),

(9.4)

where, for each j ∈ J , Wj+ (t) = v(t)Vj+ (t).

(9.5)

Consider the case where all payments are restricted to a finite interval [0, n]. Assume first that the functions A◦g are absolutely continuous between 0 and n, that is, dA◦g (t) = a◦g (t) dt, t ∈ (0, n). Then all Wg+ are continuous in (0,n). Assume furthermore that all the functions a◦g , a◦gh , µgh , and δ are continuous in (0, n). Then, for u = t + dt (9.4) becomes Wj+ (t)

= v(t){a◦j (t) +

µjg (t)a◦jg (t)} dt

g; g6=j

+ (1 − µj (t) dt)Wj+ (t + dt) +

µjg (t)dtWg+ (t + dt) + o(dt),

g; g6=j

where o(dt)/dt → 0 as dt → 0. Now, subtract Wj+ (t + dt) and divide by dt on both sides, and let dt ↓ 0 to obtain the prospective differential equations, which are the multistate generalization of Thiele’s classical equation: X ∂ + µjg (t)a◦jg (t)} Wj (t) = − v(t){a◦j (t) + ∂t g; g6=j X + µjg (t)Wg+ (t). + µj (t)Wj (t) −

(9.6)

g; g6=j

For fixed contractual functions the Vj+ are uniquely determined by the equations (9.6) together with the conditions

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CHAPTER 9. RESERVES

Vj+ (n−) = A◦j (n) − A◦j (n−).

(9.7)

If the principle of equivalence (9.11) is invoked, then the additional condition V0+ (0) = −A◦0 (0)

(9.8)

must be imposed — a constraint on the contractual functions. The procedure is to be modified only slightly if A◦j has jumps at points tj1 , . . . , tj,qj −1 in (0, n). Put tj0 = 0 and tjqj = n. The solution of (9.6) has to be obtained in each of the intervals (tj,p−1 , tjp ), p = 1, . . . , qj , and the piecewise solutions in adjacent intervals must be hooked together by the conditions Vj+ (tjp −) = Vj+ (tjp ) + A◦j (tjp ) − A◦j (tjp −),

(9.9)

p = 1, . . . , qj , which comprise (9.7). A comment is needed also on possible discontinuities of the functions a◦jg , µjg , and δ. At such points and at the points tgp (if any) where A◦g jumps and ∂ Wj+ does not exist. The Wg+ is discontinuous for some g 6= j, the derivative ∂t left and right derivatives exist, however, and if the discontinuities of the kind mentioned are finite in number, they will cause no technical problem as they will not affect the integrations that have to be performed to obtain the solution of the differential equations. Pursuing the previous remark, switch the term µj (t)Wj+ (t) appearing on the Rt

right of (9.6) over to the left, multiply by the integrating factor e− 0 µj to form R ∂ − 0t µj the complete differential ∂t {e Wj+ (t)} on the left, and finally integrate over an interval (t, u) containing no jumps of A◦j to obtain an integral equation. Insert (9.5) and solve (recall (9.2) and (9.3))

Vj+ (t)

1 [ v(t)

u t

v(τ )pjj (t, τ ){a◦j (τ ) +

+ v(u)pjj (t, u)Vj+ (u−) ] .

µjg (τ )(a◦jg (τ ) + Vg+ (τ ))}dτ

g; g6=j

(9.10)

This expression is easy to interpret: it decomposes the future payments into those that fall due before and those that fall due after time u or the time of the first transition out of the current state, whichever occurs first. By fixed contractual functions the integral equations in conjunction with the conditions (9.9) determine the Vj+ . E. Differential equations for retrospective reserves in single states. The starting point is (9.17), which now specializes to

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CHAPTER 9. RESERVES

v(t)Vj− (t) = v(s)

X p0g (0, s)pgj (s, t) p0j (0, t)

−

v(τ )

(s,t]

Vg− (s)

X p0g (0, τ ) {dA◦g (τ )pgj (τ, t) p (0, t) 0j g

a◦gh (τ )µgh (τ )phj (τ, t) dτ }.

(9.11)

h; h6=g

Upon multiplying by p0j (0, t) and introducing Wj− (t) = v(t)p0j (0, t)Vj− (t),

(9.12)

j ∈ J , (9.11) can be reshaped as Wj− (t)

Wg− (s)pgj (s, t)

−

v(τ ) (s,t]

p0g (0, τ ){dA◦g (τ )pgj (τ, t)

a◦gh (τ )µgh (τ )phj (τ, t)dτ }.

(9.13)

h; h6=g

Again, consider the case where all payments are restricted to a finite interval [0, n], with dA◦g (t) = a◦g (t) dt, t ∈ (0, n), and a◦g , a◦gh , µgh , and δ all continuous in (0, n). With t and t + dt in the places of s and t, (9.13) becomes Wj− (t + dt)

Wg− (t)µgj (t)dt + Wj− (t)(1 − µj (t)dt)

g;g6=j

−v(t)

p0g (0, t){a◦g (t)dt pgj (t, t + dt)

a◦gh (t)µgh (t)phj (t, t + dt)dt} + o(dt).

h; h6=g

As pij (t, t + dt) is µij (t)dt + o(dt) for i 6= j and 1 − µj (t)dt + o(dt) for i = j, all terms pij (t, t + dt) on the right can be replaced by δij (the difference is absorbed in o(dt)). Then proceed in the same manner as in the previous paragraph to obtain the retrospective differential equations ∂ W − (t) ∂t j

Wg− (t)µgj (t) − Wj− (t)µj (t)

g;g6=j

− v(t){p0j (0, t)a◦j (t) +

g; g6=j

p0g (0, t)a◦gj (t)µgj (t)}. (9.14)

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CHAPTER 9. RESERVES

For fixed contractual functions the retrospective reserves are determined uniquely by (9.14) together with the conditions Wj− (0) = −δ0j A◦0 (0).

(9.15)

The equivalence condition (9.11) can be cast in terms of the retrospective reserves as X

p0j (0, n)Vj− (n) = 0.

(9.16)

j∈J

Possible discontinuities are accounted for in the same manner as for the prospective reserves, the retrospective counterpart of (9.9) being Vj− (tjp ) = Vj− (tjp −) − (A◦j (tjp ) − A◦j (tjp −)),

(9.17)

p = 1, . . . , qj . Copying essentially the steps leading to (9.10), one obtains from (9.14) the integral equation Vj− (t−) =

1 [ v(s)p0j (0, s)pjj (s, t)Vj− (s) v(t)p0j (0, t) Z t − v(τ ){p0j (0, τ )pjj (τ, t)a◦j (τ ) s X + p0g (0, τ )µgj (τ )pjj (τ, t)(a◦gj (τ ) − Vg+ (τ ))}dτ ], g; g6=j

(9.18)

valid in intervals (s, t) containing no jumps of A◦j . It decomposes the past payments into those that fall due before and those that fall due after time s or the time of the last transition into the current state, whichever is the latter. F. The prospective and retrospective differential equations are generalizations of Kolmogorov’s backward and forward differential equations, respectively. This result is established by considering a simple endowment in the special case where v(t) ≡ 1 (no interest). First, assume that the only payment provided is a unit benefit payable at time u contingent upon X(u) = k, that is, A◦k = εu , the measure with a unit mass at u, and all other A◦g and all a◦gh are null. Then, for t < u, Wj+ (t) in (9.5) reduces to pjk (t, u), and the prospective differential equations in (9.6) specialize to the Kolmogorov backward equations in (9.2). Second, the Kolmogorov forward differential equations in (9.1) are obtained as a specialization of the retrospective differential equations in (9.14) by letting the

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CHAPTER 9. RESERVES

only payment be a unit premium at time s, contingent upon X(s) = i, whereby Wj− (t) in (9.12) reduces to p0i (0, s)pij (s, t) for t > s. Likewise, integral equations for the transition probabilities come out as special cases. From (9.10) one gets

pjk (t, u) =

u t

pjj (t, τ )

µjg (τ )pgk (τ, u)dτ + δjk pjj (t, u),

(9.19)

g; g6=j

and from (9.18)

pij (s, t) = δij pii (s, t) +

X Z

g; g6=j

t s

pig (s, τ )µgj (τ )pjj (τ, t)dτ.

(9.20)

These equations are easy to interpret. In a similar manner one may also find differential and integral equations for expected sojourn times. For v(t) ≡ 1, dA◦k (t) = dt, and all other contractual functions null, Vj+ (t) and Vj− (t) are just the expected future and past total sojourn times in state k, conditional on X(t) = j. G. Uses of the differential equations. In the case where the contractual functions do not depend on the reserves, the defining relations (9.2) and (9.3) are explicit expressions for the reserves. The differential equations (9.6) and (9.14) are not needed for constructive purposes — they serve only to give insight into the dynamics of the policy. The situation is entirely different if the contractual functions are allowed to depend on the reserves in some way or other. The most typical examples are repayment of a part of the reserve upon withdrawal (a state ”withdrawn” must then be included in the state space J ) and expenses depending partly on the reserve. Also the primary insurance benefits may in some cases be specified as functions of the reserve. In such situations the differential equations are indispensable tools in the construction of the reserves and determinaton of the equivalence premium. The simplest case is when the payments are linear functions of the reserve, with coefficients possibly depending on time. Then the operations leading to (9.10) and (9.18) can basically be reproduced after collecting all terms involving the reserve in state j on the left hand side and multiplying by the appropriate integrating factor. Such techniques are standard and frequently used for the traditional prospective reserve. They carry over to the retrospective reserve, as will be illustrated by examples in the final section. Apart from some very simple situations, like the one encountered in Paragraph 1B, the computation of the reserves will usually require numerical solution of the set of differential equations (or the equivalent integral equations). There is, however, an important class of situations which allow for more direct computation by iterated numerical integrations, and a comment shall be rendered on those. When returns to formerly visited states are impossible, µjg = 0 for

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CHAPTER 9. RESERVES

g < j, say, the equations (9.10) form the basis of a recursive computational procedure. The sum over g on the right of (9.10) extends only over g > j (void if j = J), and so one can suitably start by determining VJ+ , which is easy since the equation for VJ+ involves no other Vg+ (typically the policy is no longer in force in state J and VJ+ is identically 0). Then proceed downwards through the state + space: having determined Vj+1 , . . . , VJ+ , use (9.10) with u = tjp and (9.9) in each interval (tj,p−1 , tjp ), starting from time tjqj = n. Similarly, the equations (9.18) are solved recursively starting with the equation for V0− , which involves + , use (9.18) with s = tj,p−1 and no other Vg− : having determined V0+ , . . . , Vj−1 (9.17) in each interval (tj,p−1 , tjp ), starting from time tj0 = 0. Again, the examples in the following section are referred to. H. Behaviour of the retrospective reserves in the vicinity of 0. In establishing the retrospective differential equations, the auxiliary functions Wj− defined in (9.12) are more convenient to work with than the reserves themselves. The Vj− are defined only for those t where p0j (0, t) > 0, whereas the Wj− are well defined in all of [0, ∞). The Wj− are right-continuous, and their values in 0, given by (9.15), are simple initial conditions for the differential equations. The reserves Vj− are also right-continuous, but in 0 only V0− is defined (−A◦0 (0)). For j 6= 0 the limit Vj− (0+) must be obtained by letting t ↓ 0 in (9.3). Leaving details aside, only the intuitively appealing result is reported. A state j 6= 0 is said to be immediately accessible from state 0 in q steps via the directed path g = (g1 g2 . . . gq−1 ) if µg (0) = µ0g1 (0)µg1 g2 (0) . . . µgq−1 j (0) > 0. Let qj be the minimum of all q for which this property holds for some path, and denote the set of such minimal paths by Pj . Then

Vj− (0+)

−A◦0 (0)

−

g∈Pj

µg (0){a◦0g1 (0) + a◦g1 g2 (0) + . . . + a◦gq −1 j (0)} j P . µ (0) g∈Pj g

I. A different notion of retrospective reserve. In a recent paper Wolthuis & Hoem (1990) have launched a notion of retrospective reserve quite different from the one introduced here. Working in the Markov chain model, they require that the statewise retrospective reserves should satisfy EX(0)=i V (0) = v(t)

pij (0, t){Vj+ (t) − Vj− (t)}

(9.21)

j∈J

for i = 0, which conforms with (9.10). Then, imagining that the policy might start from any state different from 0, they require that (9.21) be valid for all i, whereby some hypothetical values must be chosen for the EX(0)=i V (0), i 6= 0. There exists no such choice that can produce the retrospective reserves (9.3), and so the approach is incompatible with the one taken here. The same is the case for the approach proposed by Hoem (1988), where (9.21) is required for i = 0 and Vj− (t) is put equal to Vj+ (t) for j 6= 0.

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CHAPTER 9. RESERVES

9.6

Some examples

A. Life insurance of a single life (continued from Paragraph 1B). In this simple situation the prospective differential equations (9.6) are ∂ + W (t) = v t (c − µx+t b) + µx+t W0+ (t) − µx+t W1+ (t), ∂t 0 ∂ + W (t) = 0. ∂t 1

(9.1) (9.2)

With the conditions W0+ (n−) = W1+ (n−) = 0

(9.3)

they lead to (9.1) and (9.2), which could be put up by direct prospective reasoning. The equivalence premium is

c=b

Rn τ v p µ dτ 0 R τ x x+τ . n τ v p dτ τ x 0

Suppose the treaty is modified so that the prospective reserve is paid out as an additional benefit upon death before time n; a◦01 (t) = b + V0+ (t). Then the reserve and the equivalence premium cannot be determined directly by prospective reasoning, and it is necessary to employ the differential equations (9.6). Now, instead of (9.1) one gets ∂ + W (t) = v t {c − µx+t (b + V0+ (t))} + µx+t W0+ (t) − µx+t W1+ (t) ∂t 0 = v t {c − µx+t b} − µx+t W1+ (t). (9.4) Equation (F.38) and the conditions (F.39) remain unchanged. One finds V1+ (t) = 0 as before, and V0+ (t) =

v τ −t (µx+τ b − c) dτ.

(9.5)

The equivalence premium is determined upon inserting t = 0 in (9.5) and equating to 0: Rn τ v µx+τ dτ c = b 0Rn τ . v dτ 0

(9.6)

These techniques and results are classical, and are referred here for the sake of comparison with what now follows.

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CHAPTER 9. RESERVES

For the standard contract, with no repayment of the reserve, the differential equations (9.14) become ∂ − W (t) = −W0− (t)µx+t + v t t px c, ∂t 0 ∂ − W (t) = W0− (t)µx+t − v t t px µx+t b. ∂t 1 With the conditions W0− (0) = W1− (0) = 0

(9.7) (9.8)

(9.9)

they lead to (9.3) and (9.4), which could be put up directly. Suppose now that the retrospective reserve is to be paid out as an additional benefit upon death; a◦01 (t) = b + V0− (t). Then the retrospective differential equations must be employed. Instead of (9.8) one gets ∂ − W (t) = W0− (t)µx+t − v t t px µx+t (b + V0− (t)) ∂t 1 = −v t t px µx+t b, whereas equation (9.7) and the conditions (9.9) remain unchanged. One arrives at the same expression for V0− as before, of course, and V1− (t)

= −

b 1 − t px

(1 + i)t−τ τ px µx+τ dτ. 0

The equivalence premium c is determined by (9.16):

c=b

Rn 0

v τ τ px µx+τ dτ Rn . τ n px 0 v dτ

B. Widow’s pension. A married couple buys a widow’s pension policy specifying that premiums are to be paid with intensity c as long as both husband and wife are alive, and pensions are to be paid with intensity b as long as the wife is widowed. The policy terminates at time n or upon the death of the wife, whichever occurs first. The relevant Markov model is sketched in Fig. reffig:twolives. Expressions for the transition probabilities are easily obtained by direct reasoning or by use of (9.19). The reserves can be picked directly from (9.2) and (9.3): Z n 1 v(τ ){b p01 (t, τ ) − c p00 (t, τ )}dτ, v(t) t Z n b v(τ )p11 (t, τ )dτ, v(t) t

V0+ (t)

V1+ (t)

V2+ (t)

= V3+ (t) = 0,

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CHAPTER 9. RESERVES

V0− (t)

V1− (t) =

V2− (t) = V3− (t) =

c v(t)

v(τ ) dτ,

(9.10)

v(τ ) {c p00 (0, τ )p01 (τ, t) v(t) p01 (0, t) 0 − b p01 (0, τ )p11 (τ, t)}dτ, Z t c v(τ ) p00 (0, τ )p02 (τ, t) dτ, v(t) p02 (0, t) 0 Z t 1 v(τ ) {c p00 (0, τ )p03 (τ, t) v(t) p03 (0, t) 0 − b p01 (0, τ )p13 (τ, t)}dτ.

(9.11) (9.12)

(9.13)

The differential equations are not needed to construct these formulas. The retrospective ones are listed for ease of reference: ∂ − W (t) ∂t 0 ∂ − W (t) ∂t 1 ∂ − W (t) ∂t 2 ∂ − W (t) ∂t 3

= − W0− (t)(µ01 (t) + µ02 (t)) + v(t)p00 (0, t) c,

(9.14)

= W0− (t)µ01 (t) − W1− (t)µ13 (t) − v(t)p01 (0, t) b,

(9.15)

= W0− (t)µ02 (t) − W2− (t)µ23 (t),

(9.16)

= W1− (t)µ13 (t) + W2− (t)µ23 (t).

(9.17)

Consider a modified policy, by which the retrospective reserve is to be paid back to the husband in case he is widowered before time n, the philosophy being that couples receiving no pensions should have their savings back. Now the retrospective differential equations are needed. The equations above remain unchanged except that the term v(t)p00 (0, t)V0− (t)µ02 (t) = W0− (t)µ02 (t) must be subtracted on the right of (9.16), which then changes to ∂ − W (t) = −W2− (t)µ23 (t). ∂t 2

(9.18)

Together with the conditions Wj− (0) = 0, j = 0, 1, 2, 3, these equations are easily solved. Obviously, the expressions for V0− (t) and V1− (t) remain the same as in (9.10) and (9.11). From (9.18) follows V2− (t) = 0, which is also obvious. Finally, (9.17) gives

V3− (t)

v(τ ) {c p00 (0, τ )p013 (τ, −, t) v(t) p03 (0, t) 0 − b p01 (0, τ )p13 (τ, t)} dτ,

(9.19)

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CHAPTER 9. RESERVES where

p013 (τ, −, t) =

p01 (τ, ϑ)µ13 (ϑ)dϑ τ

is the probability of passing from state 0 to state 3 via state 1 in the time interval [τ, t], given that X(τ ) = 0. As a final example the widow’s pension shall be analysed in the presence of administration expenses that depend partly on the reserve. Consider again the policy terms described in the introduction of this paragraph, but assume that administration expenses incur as follows. At time t expenses fall due with intensity e00 (t) + e000 (t)c in state 0 (the latter term represents encashment commision) and with intensity e01 (t) in state 1. In addition, expenses related to maintenance of the reserve fall due with intensity e(t)×(current retrospective reserve) throughout the entire period [0, n]. Instead of (9.14) – (9.17) one now gets

143

CHAPTER 9. RESERVES

∂ − W (t) ∂t 0

= −W0− (t)(µ01 (t) + µ02 (t)) − v(t)p00 (0, t){e00 (t) + e000 (t)c + e(t)V0− (t) − c},

∂ − W (t) ∂t 1

(9.20)

= W0− (t)µ01 (t) − W1− (t)µ13 (t) − v(t)p01 (0, t){b + e01 (t) + e(t)V1− (t)},

∂ − W (t) ∂t 2

(9.21)

= W0− (t)µ02 (t) − W2− (t)µ23 (t) −v(t)p02 (0, t)e(t)V2− (t),

∂ − W (t) ∂t 3

(9.22)

= W1− (t)µ13 (t) + W2− (t)µ23 (t) −v(t)p03 (0, t)e(t)V3− (t),

(9.23)

Wj− (0)

and the conditions (9.15) become = 0, j = 0, 1, 2, 3. Now a small trick. ∂ Wj− (t), there appears In each of the equations (9.20) – (9.23), say the one for ∂t − − a term −v(t)p0j (0, t)e(t)Vj (t) = −e(t)Wj (t) on the right hand side. Switch Rt

this over to theR left and multiply on both sides by e 0 e . Form a complete t ∂ differential ∂t {e 0 e Wj− (t)} on the left hand side and absorb everywhere the Rt

factor e 0 e into v(t) = e− 0 δ (remember v(t) is a factor in Wj− (t)). What remains are retrospective differential equations for the the same situation as the original one, modified to the effect that the administration costs related to the reserve have vanished and the interest intensity δ has been decreased by e. Thus, one can apply the explicit expression (9.3) for this modified case, and Rt arrive at the following formulas, where v ∗ (t) = e− 0 (δ−e) : 1 ∗ v (t)

V0− (t) = V1− (t) =

V2− (t) = V3− (t) =

t 0

v ∗ (τ ){c(1 − e000 (τ )) − e00 (τ )} dτ,

Z t 1 v ∗ (τ ) [ p00 (0, τ )p01 (τ, t){c(1 − e000 (τ )) − e00 (τ )} v ∗ (t) p01 (0, t) 0 − p01 (0, τ )p11 (τ, t)(b + e01 (τ )) ] dτ, Z t 1 v ∗ (τ )p00 (0, τ )p02 (τ, t){c(1 − e000 (τ )) − e00 (τ )}dτ, v ∗ (t) p02 (0, t) 0 Z t 1 v ∗ (τ ) [ p00 (0, τ )p03 (τ, t){c(1 − e000 (τ )) − e00 (τ )} v ∗ (t) p03 (0, t) 0 − p01 (0, τ )p13 (τ, t)(b + e01 (τ )) ] dτ.

Using the equivalence principle in the form (9.16), one obtains the equivalence premium

Rn 0

v ∗ (τ ){p00 (0, τ )e00 (τ ) + p01 (0, τ )(b + e01 (τ ))} dτ Rn . v ∗ (τ )p00 (0, τ )(1 − e000 (τ )) dτ 0

(9.24)

CHAPTER 9. RESERVES

144

If the cost element e(t)Vj− (t) is replaced by e(t)Vj+ (t), the prospective differential equations must be used. The procedure above can essentially be repeated, and just as for the retrospective reserves it turns out that the prospective reserves are those correspending to e(t) = 0 and discount function v ∗ . The equivalence premium remains as in (9.24).

Chapter 10

Safety loadings and bonus 10.1

General considerations

A. Bonus – what it is. The word bonus is Latin and means ’good’. In insurance terminology it denotes various forms of repayments to the policyholders of that part of the company’s surplus that stems from good performance of the insurance portfolio, a sub-portfolio, or the individual policy. We shall here concentrate on the special form it takes in traditional life insurance. The issue of bonus presents itself in connection with every standard life insurance contract, characteristic of which is its specification of nominal contingent payments that are binding to both parties throughout the term of the contract. All contracts discussed so far are of this type, and a concrete example is the combined policy described in 7.4: upon inception of the contract the parties agree on a death benefit of 1 and a disability benefit of 0.5 per year against a level premium of 0.013108 per year, regardless of future developments of the intensities of mortality, disability, and interest. Now, life insurance policies like this one are typically long term contracts, with time horizons wide enough to capture significant variations in intensities, expenses, and other relevant economicdemographic conditions. The uncertain development of such conditions subjects every supplier of standard insurance products to a risk that is non-diversifiable, that is, independent of the size of the portfolio; an adverse development can not be countered by raising premiums or reducing benefits, and also not by cancelling contracts (the right of withdrawal remains one-sidedly with the insured). The only way the insurer can safeguard against this kind of risk is to build into the contractual premium a safety loading that makes it cover, on the average in the portfolio, the contractual benefits under any likely economic-demographic development. Such a safety loading will typically create a systematic surplus, which by statute is the property of the insured and has to be repaid in the form of bonus.

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CHAPTER 10. SAFETY LOADINGS AND BONUS

146

B. Sketch of the usual technique. The approach commonly used in practice is the following. At the outset the contractual benefits are valuated, and the premium is set accordingly, on a first order (technical) basis, which is a set of hypothetical assumptions about interest, intensities of transition between policy-states, costs, and possibly other relevant technical elements. The first order model is a means of prudent calculation of premiums and reserves, and its elements are therefore placed to the safe side in a sense that will be made precise later. As time passes reality reveals true elements that ultimately set the realistic scenario for the entire term of the policy and constitute what is called the second order (experience) basis. Upon comparing elements of first and second order, one can identify the safety loadings built into those of first order and design schemes for repayment of the systematic surplus they have created. We will now make these things precise. To save notation, we disregard administration expenses for the time being and discuss them separately in Section 10.7 below.

10.2

First and second order bases

A. The second order model. The policy-state process Z is assumed to be a time-continuous Markov chain as described in Section 7.2. In the present context we need to equip the indicator processes and counting processes related to the Z . The probability measure process Z with a topscript, calling them IjZ and Njk and expectation operator induced by the transition intensities are denoted by P and E, respectively. The investment portfolio of the insurance company bears interest with intensity r(t) at time t. The intensities r and µjk constitute the experience basis, also called the second order basis, representing the true mechanisms governing the insurance business. At any time its past history is known, whereas its future is unknown. We extend the set-up by viewing the second order basis as stochastic, whereby the uncertainty associated with it becomes quantifiable in probabilistic terms. In particular, prediction of its future development becomes a matter of modelbased forecasting. Thus, let us consider the set-up above as the conditional model, given the second order basis, and place a distribution on the latter, whereby r and the µjk become stochastic processes. Let Gt denote their complete history up to, and including, time t and, accordingly, let E[ · | Gt ] denote conditional expectation, given this information. For the time being we will work only in the conditional model and need not specify any particular marginal distribution of the second order elements. B. The first order model. We let the first order model be of the same type as the conditional model of second order. Thus, the first order basis is viewed as deterministic, and we denote its elements by r ∗ and µ∗jk and the corresponding probability measure and expectation operator by P∗ and E∗ , respectively. The first order basis represents a prudent initial assessment of the development of

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the second order basis, and its elements are placed on the safe side in a sense that will be made precise later. By statute, the insurer must currently provide a reserve to meet future liabilities in respect of the contract, and these liabilities are to be valuated on the first order basis. The first order reserve at time t, given that the policy is then in state j, is Z n R − tτ r ∗ ∗ ∗ e dB(τ ) Z(t) = j Vj (t) = E t   Z n R X X τ ∗ = e− t r p∗jg (t, τ ) dBg (τ ) + bgh (τ )µ∗gh (τ ) dτ  .(10.1) t

h;h6=g

We need Thiele’s differential equations

dVj∗ (t) = r∗ (t)Vj∗ (t) dt − dBj (t) −

∗ Rjk (t) µ∗jk (t) dt ,

(10.2)

k; k6=j

where ∗ Rjk (t) = bjk (t) + Vk∗ (t) − Vj∗ (t)

(10.3)

is the sum at risk associated with a possible transition from state j to state k at time t. The premiums are based on the principle of equivalence exercised on the first order valuation basis, Z n Rτ ∗ E∗ (10.4) e− 0 r dB(τ ) = 0 , 0−

or, equivalently, V0∗ (0) = −∆B0 (0) .

10.3

(10.5)

The technical surplus and how it emerges

A. Definition of the mean portfolio surplus. With premiums determined by the principle of equivalence (10.4) based on prudent first order assumptions, the portfolio will create a systematic technical surplus if everything goes well. Quite naturally, the surplus is some average of past net incomes valuated on the factual second order basis less future net outgoes valuated on the conservative first order basis. The portfolio-wide mean surplus thus construed is Z t R X t r p0j (0, t) Vj∗ (t) S(t) = E e τ d(−B)(τ ) Gt − 0−

= −e −

Rt 0

X j

e−

0−

Rτ 0

X j

p0j (0, t) Vj∗ (t) .



p0j (0, τ ) dBj (τ ) +

k;k6=j



bjk (τ )µjk (τ ) dτ 

(10.6)

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The definition conforms with basic principles of insurance accountancy; at any time the balance is the difference between, on the debit, the factual income in the past and, on the credit, the reserve that by statute is to be provided in respect of future liabilities. In particular, due to (10.5), S(0) = 0

(10.7)

and, due to Vj∗ (n) = 0, S(n) = E

e 0−

Rn τ

d(−B)(τ ) Gn ,

(10.8)

as it ought to be. Note that the expression in (10.6) involves only the past history of the second order basis, which is currently known. B. The contributions to the surplus. Differentiating (10.6), applying the Kolmogorov forward equation (7.20) and the Thiele backward equation (10.2) to the last term on the right, leads to   Z t Rt Rτ X X p0j (0, τ ) dBj (τ ) + bjk (τ )µjk (τ ) dτ  dS(t) = − e 0 r r(t) dt e− 0 r 0−

−

X j

−

X j

−

X j

p0j (0, t) dBj (t) +

 

g; g6=j

k;k6=j



k;k6=j



bjk (t)µjk (t) dt



p0g (0, t) µgj (t) dt − p0j (0, t) µj· (t) dt . Vj∗ (t) 

p0j (0, t) r∗ (t)Vj∗ (t) dt − dBj (t) −

k; k6=j



∗ Rjk (t) µ∗jk (t) dt .

Reusing the relation (10.6) in the first line here and gathering terms, we obtain X dS(t) = r(t) dt S(t) + p0j (0, t)cj (t) dt , j

with cj (t) = {r(t) − r ∗ (t)} Vj∗ (t) +

∗ Rjk (t){µ∗jk (t) − µjk (t)} .

(10.9)

k; k6=j

Finally, integrating up and using (10.7), we arrive at Z t R X t e τr S(t) = p0j (0, τ )cj (τ ) dτ , 0

(10.10)

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149

which expresses the technical surplus at any time as the sum of past contributions compounded with second order interest. One may arrive at the definition of the contributions (10.9) by another route, starting from the individual surplus defined, quite naturally, as Z t Rt Rτ X IjZ (t)Vj∗ (t). (10.11) Sind (t) = e 0 r e− 0 r d(−B)(τ ) − 0−

Upon differentiating this expression, and proceeding along the same lines as above, one finds that SRind (t) consists of a purely erratic term and a systematic Rt t P term. The latter is 0 e τ r j IjZ (τ )cj (τ ) dτ , which is the individual counterpart of (10.10), showing how the contributions emerge at the level of the individual policy. They form a random payment function C defined by X IjZ (t) cj (t) dt . (10.12) dC(t) = j

With this definition, we can recast (10.10) as Z t R t r τ e dC(τ ) Gt . S(t) = E

(10.13)

C. Safety margins. The expression on the right of (10.9) displays how the contributions arise from safety margins in the first order force of interest (the first term) and in the transition intensities (the second term). The purpose of the first order basis is to create a non-negative technical surplus. This is certainly fulfilled if r(t) ≥ r∗ (t) (10.14) (assuming that all Vj∗ (t) are non-negative as they should be) and ∗ sign {µ∗jk (t) − µjk (t)} = sign Rjk (t) .

10.4

(10.15)

Dividends and bonus

A. The dividend process. Legislation lays down that the technical surplus belongs to the insured and has to be repaid in its entirety. Therefore, to the contractual payments B there must be added dividends, henceforth denoted by D. The dividends are currently adapted to the development of the second order basis and, as explained in Paragraph 10.1.A, they can not be negative. The purpose of the dividends is to establish, ultimately, equivalence on the true second order basis: Z n Rτ (10.16) E e− 0 r d{B + D}(τ ) Gn = 0 . 0−

CHAPTER 10. SAFETY LOADINGS AND BONUS We can state (10.16) equivalently as Z n R n r E e τ d{B + D}(τ ) Gn = 0 .

150

(10.17)

0−

The value at time t of past individual contributions less dividends, compounded with interest, is Z t R t U d (t) = e τ r d{C − D}(τ ) . (10.18) 0−

This amount is an outstanding account of the insured against the insurer, and we shall call it the dividend reserve at time t. By virtue of (10.8) and (10.13) we can recast the equivalence requirement (10.17) in the appealing form E[U d (n) | Gn ] = 0 .

(10.19)

From a solvency point of view it would make sense to strengthen (10.19) by requiring that compounded dividends must never exceed compounded contributions: E[U d (t) | Gt ] ≥ 0 , (10.20) t ∈ [0, n]. At this point some explanation is in order. Although the ultimate balance requirement is enforced by law, the dividends do not represent a contractual obligation on the part of the insurer; the dividends must be adapted to the second order development up to time n and can, therefore, not be stipulated in the terms of the contract at time 0. On the other hand, at any time, dividends allotted in the past have irrevocably been credited to the insured’s account. These regulatory facts are reflected in (10.20). If we adopt the view that “the technical surplus belongs to those who created it”, we should sharpen (10.19) by imposing the stronger requirement U d (n) = 0 .

(10.21)

This means that no transfer of redistributions across policies is allowed. The solvency requirement conforming with this point of view, and sharpening (10.20), is U d (t) ≥ 0 , (10.22) t ∈ [0, n]. The constraints imposed on D in this paragraph are of a general nature and leave a certain latitude for various designs of dividend schemes. We shall list some possibilities motivated by practice. B. Special dividend schemes. The so-called contribution scheme is defined by D = C, that is, all contributions are currently and immediately credited to the account of the insured. No dividend reserve will accrue and, consequently,

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151

the only instrument on the part of the insurer in case of adverse second order experience is to cease crediting dividends. In some countries the contribution principle is enforced by law. This means that insurers are compelled to operate with minimal protection against adverse second order developments. By terminal dividend is meant that all contributions are currently invested and their compounded total is credited to the insured as a lump sum dividend payment only upon the termination of the contract at some time T after which no more contributions are generated. Typically T would be the time of transition to an absorbing state (death or withdrawal), truncated at n. If compounding is at second order rate of interest, then D(t) = 1[t ≥ T ]

e 0

RT τ

dC(τ ) .

Contribution dividends and terminal dividends represent opposite extremes in the set of conceivable dividend schemes, which are countless. One class of intermediate solutions are those that yield dividends only at certain times T1 < · · · < TK ≤ n, e.g. annually or at times of transition between certain R Ti R T i r states. At each time Ti the amount ∆D(Ti ) = Ti−1 e τ dC(τ ) (with T0 = 0) is entered to the insured’s credit. C. Allocation of dividends; bonus. Once they have been allotted, dividends belong to the insured. They may, however, be disposed of in various ways and need not be paid out currently as they fall due. The actual payouts of dividends are termed bonus in the sequel, and the corresponding payment function is denoted by B b . The compounded value of credited dividends less paid bonuses at time t is Z t R t (10.23) e τ r d{D − B b }(τ ) . U b (t) = 0

This is a debt owed by the insurer to the insured, and we shall call it the bonus reserve at time t. Bonuses may not be advanced, so B b must satisfy U b (t) ≥ 0

(10.24)

for all t ∈ [0, n]. In particular, since D(0) = 0, one has B b (0) = 0. Moreover, since all dividends must eventually be paid out, we must have U b (n) = 0 .

(10.25)

We have introduced three notions of reserves that all appear on the debit side of the insurer’s balance sheet. First, the premium reserve V ∗ is provided to meet net outgoes in respect of future events; second, the dividend reserve U d is provided to settle the excess of past contributions over past dividends; third, the bonus reserve U b is provided to settle the unpaid part of dividends credited in the past. The premium reserve is of prospective type and is a predicted

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152

amount, whereas the dividend and bonus reserves are of retrospective type and are indeed known amounts summing up to Z t R t d b U (t) + U (t) = e τ r d{C − B b }(τ ) , (10.26) 0

the compounded total of past contributions not yet paid back to the insured. D. Some commonly used bonus schemes. The term cash bonus is, quite naturally, used for the scheme B b = D. Under this scheme the bonus reserve is always null, of course. By terminal bonus, also called reversionary bonus, is meant that all dividends, with accumulation of interest, are paid out as a lump sum upon the termination of the contract at some time T , that is, Z T R T e τ r dD(τ ) . B b (t) = 1[t ≥ T ] 0

Here we could replace the integrator D by C since terminal bonus obviously does not depend on the dividend scheme; all contributions are to be repaid with accumulation of interest. Assume now, what is common in practice, that dividends are currently used to purchase additional insurance coverage of the same type as in the primary policy. It seems natural to let the additional benefits be proportional to those stipulated in the primary policy since they represent the desired profile of the product. Thus, the dividends dD(s) in any time interval [s, s + ds) are used as a single premium for an insurance with payment function of the form dQ(s){B + (τ ) − B + (s)} , τ ∈ (s, n], where the topscript ”+ ” signifies, in an obvious sense, that only positive payments (benefits) are counted. Supposing that additional insurances are written on first order basis, the proportionality factor dQ(s) is determined by ∗+ dD(s) = dQ(s)VZ(s) (s),

where ∗+ (s) VZ(s)

∗

e s

−

Rτ s

r∗

(10.27)

dB (τ ) Z(s) +

is the single premium at time s for the future benefits under the policy. Now the bonus payments B b are of the form dB b (t) = Q(t)dB + (t) .

(10.28)

Being written on first order basis, also the additional insurances create technical surplus. The total contributions under this scheme develop as dC(t) + Q(t)dC + (t) ,

(10.29)

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153

where the first term on the right stems from the primary policy and the second term stems from the Q(t) units of additional insurances purchased in the past, each of which payment function B + producing contributions C + of the form P has + Z dC (t) = j Ij (t) c+ j (t) dt, with X ∗+ ∗+ ∗ c+ Rjk (t){µ∗jk (t) − µjk (t)} , j (t) = {r(t) − r (t)}Vj (t) + k; k6=j

∗+ Rjk (t)

b+ jk (t)

+ Vk∗+ (t) − Vj∗+ (t) .

The present situation is more involved than those encountered previously since, not only are dividends driven by the contractual payments, but it is also the other way around. To keep things relatively simple, suppose that the contribution principle is adopted so that the dividends in (10.27) are set equal to the contributions in (10.29). Then the system is governed by the dynamics ∗+ dC(t) + Q(t)dC + (t) = dQ(t)VZ(t) (t) ∗+ or, realizing that VZ(t) (t) is strictly positive whenever dC(t) and dC + (t) are,

dQ(t) − Q(t) dG(t) = dH(t) ,

(10.30)

where G and H are defined by dG(t)

dH(t) =

1 ∗+ VZ(t) (t)

dC + (t) ,

(10.31)

dC(t) .

(10.32)

Multiplying with exp(−G(t)) to form a complete differential on the left and then integrating from 0 to t, using Q(0) = 0, we obtain Z t Q(t) = eG(t)−G(τ ) dH(τ ) . (10.33) 0

10.5

Bonus prognoses

A. A Markov chain environment. We shall adopt a simple Markov chain description of the uncertainty associated with the development of the second order basis. Let Y (t), 0 ≤ t ≤ n, be a time-continuous Markov chain with finite state space Y = {1, . . . , q} and constant intensities of transition, λef . Denote the associated indicator processes by IeY . The process Y represents the “economic-demographic environment”, and we let the second order elements depend on the current Y -state: X r(t) = IeY (t) re = rY (t) , e

µjk (t)

X e

IeY (t) µe;jk (t) = µY (t);jk (t) .

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154

The re are constants and the µe;jk (t) are intensity functions, all deterministic. With this specification of the full two-stage model it is realized that the pair X = (Y, Z) is a Markov chain on the state space X = Y × Z, and its intensities of transition, which we denote by κej,f k (t) for (e, j), (f, k) ∈ X , (e, j) 6= (f, k), are κej,f j (t) = λef ,

e 6= f,

(10.34)

κej,ek (t) = µe;jk (t) ,

j 6= k,

(10.35)

and null for all other transitions. In this extended set-up the contributions, whose dependence on the second order elements was not visualized earlier, can appropriately be represented as X dC(t) = c(t) dt = IeY (t)IjZ (t)cej (t) dt, e,j

where cej (t) = {re − r∗ (t)} Vj∗ (t) +

∗ Rjk (t){µ∗jk (t) − µe;jk (t)} .

(10.36)

k; k6=j

Under the scheme of additional benefits described in Paragraph 10.4.D a similar convention goes for C + and c+ and, accordingly, (10.31) and (10.32) become X IeY (t)IjZ (t)gej (t) dt , (10.37) dG(t) = g(t) dt = e,j

gej (t)

c+ ej (t) Vj∗+ (t)

dH(t) = h(t) dt =

(10.38) X

IeY (t)IjZ (t)hej (t) dt ,

(10.39)

e,j

hej (t)

cej (t) . Vj∗+ (t)

(10.40)

B. Preparatory remarks on the issue of bonus prognoses. There is no single functional of the future bonus stream that presents itself as the relevant quantity to prognosticate. One could e.g. take the total bonuses discounted by some suitable inflation rate, or the undiscounted total bonuses, or the rate at which bonus will be paid at certain times, and one could apply any of these possibilities to the random development of the policy or to some representative fixed development. We shall focus on the expected value, and in the simplest cases also higher order moments, of the future bonuses discounted by the stochastic second order interest. From this we can easily deduce predictors for a number of other relevant quantities. We turn now to the analysis of some of the schemes described in Section 10.4.

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155

C. Contribution dividends and cash bonus. This case, where B b = C = D, is particularly simple since the bonus payments at any time depend only on the current state of the process. We can then employ the appropriate version of Thiele’s differential equation to calculate the state-wise expected discounted future bonuses (= contributions), Z n R − tτ r e c(τ ) dτ X(t) = (e, j) . Wej (t) = E t

They are determined by the appropriate version of Thiele’s differential equation, d Wej (t) dt

= re Wej (t) − cej (t) −

λef (Wf j (t) − Wej (t))

f ;f 6=e

−

µe;jk (t) (Wek (t) − Wej (t)) ,

(10.41)

k;k6=j

subject to ∀e, j .

Wej (n−) = 0 ,

(10.42)

D. Terminal dividend and/or bonus. Under the terminal bonus scheme dividends and bonuses are the same, of course. The problem of predicting the total bonus payments discounted with respect to second order interest is basically the same as in the previous paragraph since it amounts to adding the total amount of compounded past contributions, which is known, and the state-wise predictor of discounted future contributions. Suppose instead that at time t, the policy still being in force, it is decided to predict the undiscounted value of the terminal bonus amount, W =

e 0

RT τ

c(τ ) dτ =

t R t

c(τ ) dτ W 0 (t) + W 00 (t) ,

where RT

W 0 (t)

= e Z =

W 00 (t)

e t

RT τ

c(τ ) dτ .

We need the state-wise expected values We0 (t) = E[W 0 (t) | Y (t) = e], 00 Wej (t) = E[W 00 (t) | X(t) = (e, j)] , to find the state-wise predictors of W in (10.43), Wej (t) =

t R t

00 c(τ ) dτ We0 (t) + Wej (t) .

(10.43)

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CHAPTER 10. SAFETY LOADINGS AND BONUS

We shall find these functions by the backward construction, starting from W 0 (t) = er dt W 0 (t + dt), W 00 (t) = c(t) dt W 0 (t) + W 00 (t + dt) . Conditioning on what happens in the small time interval (t, t + dt], we get   X We0 (t) = ere dt (1 − λe· dt) We0 (t + dt) + λef (t) dt Wf0 (t + dt) , f ; f 6=e

and

00 00 Wej (t) = cej (t) dt We0 (t) + (1 − (λe· + µe;j· (t)) dt) Wej (t + dt) X 00 + λef (t) dt Wf j (t + dt) f ; f 6=e

00 µe;jk (t) dt Wek (t + dt) .

k; k6=j

From these relationships we easily obtain the differential equations d 0 W (t) dt e d 00 W (t) dt ej

= −re We0 (t) −

f ; f 6=e

λef Wf0 (t) − We0 (t) ,

= − cej (t) We0 (t) −

00 λef Wf00j (t) − Wej (t)

f ; f 6=e

−

k; k6=j

00 00 µe;jk (t) Wek (t) − Wej (t) ,

(10.44)

(10.45)

which are to be solved subject to We0 (n−) = 1 ,

00 (n−) = 0 , Wej

∀e, j .

(10.46)

E. Additional benefits. Suppose we want to predict the total future bonuses discounted with respect to second order interest, Z n R τ W (t) = e− t r Q(τ ) dB + (τ ) , t

with Q defined by (10.33). Recalling (10.37)–(10.40), we reshape W (t) as Z n R Z τ R τ τ W (t) = e− t r e r g h(r) dr dB + (τ ) t 0 Z n R Z t R Z τ R Rτ τ t τ = e− t r e r g h(r) dr e t g + e r g h(r) dr dB + (τ ) 0

t R t

h(r) dr W (t) + W (t) ,

(10.47)

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CHAPTER 10. SAFETY LOADINGS AND BONUS with W 0 (t) 00

W (t)

n R τ

Zt n

e−

(g−r)

Rτ t

dB + (τ ),

W 0 (τ ) h(τ ) dτ .

Thus, we need the state-wise expected values 0 Wej (t) = E[W 0 (t) | X(t) = (e, j)] , 00 Wej (t) = E[W 00 (t) | X(t) = (e, j)] ,

in order to find the state-wise predictors of W (t) in (10.47), Wej (t) =

t R t

0 00 h(r) dr Wej (t) + Wej (t) .

The backward equations start from W 0 (t)

= dB + (t) + e(g(t)−r(t)) dt W 0 (t + dt) ,

W 00 (t)

= W 0 (t) h(t) dt + e−r(t) dt W 00 (t + dt) ,

from which we proceed in the same way as in the previous paragraph to obtain 0 dWej (t)

0 = − dBj+ (t) + (re − gej (t)) dt Wej (t) X 0 0 − λef dt Wf j (t) − Wej (t) f ; f 6=e

− 00 dWej (t)

0 0 µe;jk (t) dt b+ jk (t) + Wek (t) − Wej (t) ,

00 λef dt Wf00j (t) − Wej (t)

k;k6=j 0 − Wej (t)hej (t) dt

−

00 + re dt Wej (t)

f ; f 6=e

−

k; k6=j

(10.48)

00 00 µe;jk (t) dt Wek (t) − Wej (t) .

(10.49)

The appropriate side conditions are 0 Wej (n−) = ∆Bj+ (n) ,

00 Wej (n−) = 0 ,

∀e, j .

(10.50)

F. Predicting undiscounted amounts. If the undiscounted total contributions or additional benefits is what one wants to predict, one can just apply the formulas with all re replaced by 0.

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158

G. Predicting bonuses for a given policy path. Yet another form of prognosis, which may be considered more informative than the two mentioned above, would be to predict bonus payments for some possible fixed pursuits of a policy instead of averaging over all possibilities. Such prognoses are obtained from those described above upon keeping the realized path Z(τ ) for τ ∈ [0, t], where t is the time of consideration, and putting Z(τ ) = z(τ ) for τ ∈ (t, n], where z(·) is some fixed path with z(t) = Z(t). The relevant predictors then become essentially functions only of the current Y -state and are simple special cases of the results above. As an example of an even simpler type of prognosis for a policy in state j at time t, the insurer could present the expected bonus payment per time unit at a future time s, given that the policy is then in state i, and do this for some representative selections of s and i. If Y (t) = e, then the relevant prediction is X pYef (t, s)cf i (s) . E[cY (s)i (s) | Y (t) = e] = f

10.6

Examples

A. The case. For our purpose, which is to illustrate the role of the stochastic environment in model-based prognoses, it suffices to consider simple insurance products for which the relevant policy states are Z = {a, d} (’alive’ and ’dead’). We will consider a single life insured at age 30 for a period of n = 30 years, and let the first order elements be those of the Danish technical basis G82M for males: r∗ µ∗ad (t)

= ln(1.045) , = µ∗ (t) = 0.0005 + 0.000075858 · 100.038(30+t) .

Three different forms of insurance benefits will be considered, and in each case we assume that premiums are payable continuously at level rate as long as the policy is in force. First, a term insurance (TI) of 1 = bad (t) with first order premium rate 0.0042608 = −ba (t). Second, a pure endowment (PE) of 1 = ∆Ba (30) with first order premium rate 0.0140690 = −ba (t). Third, an endowment insurance (EI), which is just the combination of the former two; 1 = bad (t) = ∆Ba (30), 0.0183298 = −ba (t). Just as an illustration, let the second order model be the simple one where interest and mortality are governed by independent time-continuous Markov chains and, more specifically, that r switches with a constant intensity λi between the first order rate r ∗ and a better rate i r∗ (i > 1) and, similarly, µ switches with a constant intensity λm between the first order rate µ∗ and a better rate m µ∗ (m < 1). (We choose to express ourselves this way although (10.15) shows that, for insurance forms with negative sum at risk, e.g. pure endowment insurance, it is actually a higher second order mortality that is “better” in the sense of creating positive contributions.)

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CHAPTER 10. SAFETY LOADINGS AND BONUS

bb,a

r∗ , µ∗ , alive

λi

gb,a i r∗ , µ∗ , alive

λi 6

µ∗ (t) ? ··,d

λm

dead

m µ∗ (t)

µ∗ (t)

λm

6 ∗ m µ (t)

? bg,a

r∗ , m µ∗ , alive

λi

gg,a i r∗ , m µ∗ , alive

λi

Figure 10.1: The Markov process X = (Y, Z) for a single life insurance in an environment with two interest states and two mortality states. The situation fits into the framework of Paragraph 10.5.A; Y has states Y = {bb, gb, bg, gg} representing all combinations of “bad” (b) and “good” (g) interest and mortality, and the non-null intensities are λbb,gb = λgb,bb = λbg,gg = λgg,bg = λi , λbb,bg = λbg,bb = λgb,gg = λgg,gb = λm . The first order basis is just the worst-scenario bb. Adopting the device (10.34)–(10.35), we consider the Markov chain X = (Y, Z) with states (bb, a), (gb, a), etc. It is realized that all death states can be merged into one, so it suffices to work with the simple Markov model with five states sketched in Figure 10.1.

B. Results. We shall report some numerical results for the case where i = 1.25, m = 0.75, and λi = λm = 0.1. Prognoses are made at the time of issue of the policy. Computations were performed by the fourth order Runge-Kutta method, which turns out to work with high precision in the present class of situations.

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160

Table 10.1 displays, for each of the three policies, the state-wise expected values of discounted contributions obtained by solving (10.41)–(10.42). We shall be content here to point out two features: First, for the term insurance the mortality margin is far more important than the interest margin, whereas for the pure endowment it is the other way around (the latter has the larger reserve). Note that the sum at risk is negative for the pure endowment, so that the first order assumption of excess mortality is really not to the safe side, see (10.15). Second, high interest produces large contributions, but, since high initial interest also induces severe discounting, it is not necessarily true that good initial interest will produce a high value of the expected discounted contributions, see the two last entries in the row TI. The latter remark suggests the use of a discounting function different from the one based on the second order interest, e.g. some exogenous deflator reflecting the likely development of the price index or the discounting function corresponding to first order interest. In particular, one can simply drop discounting and prognosticate the total amounts paid. We shall do this in the following, noting that the expected value of bonuses discounted by second order interest must in fact be the same for all bonus schemes, and are already shown in Table 10.1. Table 10.2 shows state-wise expected values of undiscounted bonuses for three different schemes; contribution dividends and cash bonus (C, the same as total undiscounted contributions), terminal bonus (T B), and additional benefits (AB). We first note that, now, any improvement of initial second order conditions helps to increase prospective contributions and bonuses. Furthermore, expected bonuses are generally smaller for C than for T B and AB since bonuses under C are paid earlier. Differences between T B and AB must be due to a similar effect. Thus, we can infer that AB must on the average fall due earlier than T B, except for the pure endowment policy, of course. One might expect that the bonuses for the term insurance and the pure endowment policies add up to the bonuses for the combined endowment insurance policy, as is the case for C and T B. However, for AB it is seen that the sum of the bonuses for the two component policies is generally smaller than the bonuses for the combined policy. The explanation must be that additional death benefits and additional survival benefits are not purchased in the same proportions under the two policy strategies. The observed difference indicates that, on the average, the additional benefits fall due later under the combined policy, which therefore must have the smaller proportion of additional death benefits. C. Assessment of prognostication error. Bonus prognoses based on the present model may be equipped with quantitative measures of the prognostication error. By the technique of proof shown in Section 10.5 we may derive differential equations for higher order moments of any of the predictands considered and calculate e.g. the coefficient of variation, the skewness, and the kurtosis.

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Table 10.1: Conditional expected present value at time 0 of total contributions for term insurance policy (TI), pure endowment policy (PE), and endowment insurance policy (EI), given initial second order states of interest and mortality (b or g). bb

10.7

TI :

.00851 .00854 .01061 .01059

PE :

.01613 .01823 .01595 .01807

EI :

.02463 .02677 .02656 .02865

Including expenses

A. The form of the expenses. Expenses are assumed to incur in accordance with a non-decreasing payment function A of the same type as the contractual payments, that is, X X Z ajk (t) dNjk (t) . (10.51) IjZ (t) dAj (t) + dA(t) = j

j6=k

It is common in practice to assume, furthermore, that expenses of annuity type incur with a lump sum of initial costs at time 0 and thereafter continuously at a rate that depends on the current state, that is, ∆A0 (0) > 0 and dAj (t) = aj (t) dt for t > 0. The transition costs ajk (t) are not explicitly taken into account in practice, but we include them here since they add realism without adding mathematical complexity. B. First order assumptions. The elements ∆A0 (0), aj (t), and ajk (t) will in general depend on the second order development, and the first order basis must, therefore, specify prudent estimates ∆A∗0 (0), a∗j (t), and a∗jk (t). Denote the corresponding payment function by A∗ . C. Surplus and contributions in the presence of expenses. The introduction of expenses adds a new feature to the previous set-up in that also the payments become dependent on the second order development. However, the essential parts of the analyses in the previous sections carry over with merely notational modifications; all it takes is to replace everywhere the contractual payment function B with A + B in the past and A∗ + B in the future. One finds, in particular, that the first order equivalence relation (10.5) now turns into V0∗ (0) = −∆A∗0 (0) − ∆B0 (0) , (10.52) the surplus at time 0 becomes S(0) = ∆A∗0 (0) − ∆A0 (0) ,

(10.53)

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CHAPTER 10. SAFETY LOADINGS AND BONUS

Table 10.2: Conditional expected value (E) of undiscounted total contributions (C), terminal bonus (T B), and total additional benefits (AB) for term insurance policy (TI), pure endowment policy (PE), and endowment insurance policy (EI), given initial second order states of interest and mortality (b or g). bb

E C: E TB : E AB :

.02153 .03693 .02949

.02222 .03916 .03096

.02436 .04600 .03545

.02505 .04847 .03706

PE: E C: E TB : E AB :

.04342 .07337 .07337

.04818 .08687 .08687

.04314 .07264 .07264

.04791 .08615 .08615

EI:

.06495 .11030 .10723

.07040 .12603 .12199

.06750 .11864 .11501

.07296 .13462 .13003

TI:

E C: E TB : E AB :

and the contributions consist of a jump ∆C(0) = ∆A∗0 (0) − ∆A0 (0)

(10.54)

at time 0 and thereafter a continuous part, which is defined upon replacing (10.9) with cj (t)

= {r(t) − r ∗ (t)} Vj∗ (t) + {a∗j (t) − aj (t)} X + {a∗jk (t) − ajk (t)}µjk (t) k; k6=j

∗ Rjk (t){µ∗jk (t) − µjk (t)} ,

(10.55)

k; k6=j

where now ∗ Rjk (t) = a∗jk (t) + bjk (t) + Vk∗ (t) − Vj∗ (t) .

(10.56)

Referring to the discussion in Paragraph 10.3.C, we see that the contributions emerge from safety margins in all first order elements, r ∗ , µ∗jk , and A∗ . D. Prediction in the presence of expenses. The complexity of the prediction problem depends heavily on the assumptions made about the second order expenses, and at this point some new problems may arise. Just to get started, suppose first that the expense elements ∆A0 (0), aj (t), and ajk (t) are deterministic. Then the methods in Section 10.5 carry over with only trivial modifications. Presumably, this simplistic model is at the base of the frequently encountered claim that “administration expenses can be regarded

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163

as additional benefits”. Unfortunately, real life expenses are of a different, and typically less pleasant, nature. An exhaustive discussion of this issue could easily exhaust the reader, so we shall be content with just outlining some tentative ideas. The problem is that expenses are made up of wages, commissions, rent, taxes and other items that are governed by the economic development. In the framework of the Markov model in Paragraph 10.5.A, one simple way of accounting for such effects is to make the second order expenses dependent on the current state of Y , that is, X ∆A0 (0) = IeY (t) ∆Ae0 (0) , e

aj (t) =

IeY (t) aej (t) ,

ajk (t) =

IeY (t) aejk (t) ,

with deterministic ∆Ae0 , aej , and aejk . By enriching sufficiently the state space of Y , one can in principle create a fairly realistic model. Perhaps the most reasonable point of view is that expenses are R t inflated by some time-dependent rate γ(t) so that we should put aj (t) = e 0 γ a0j (t) and Rt ajk (t) = e 0 γ a0jk (t) with a0j and a0jk deterministic. One possibility is to put the second order force of interest r in the role of γ. More realistically one should let γ be something else, but still related to r through joint dependence on a suitably specified Y . We shall not pursue this idea any further here, but note, by way of warning, that prognostication in this kind of inflation model will present problems in addition to those solved in Section 10.5.

10.8

Discussions

A. The principle of equivalence. This principle, as formulated in (10.4), is basic in life insurance. The expected value represents averaging over a large (really infinite) portfolio of policies, the philosophy being that, even if the individual policy creates a (possibly large) loss or gain, there will be balance on the average between outgoes and incomes in the portfolio as a whole if the premiums are set by equivalence. The deviation from perfect balance, which is inevitable in a finite world with finite portfolios, represents profit or loss on the part of the insurer and has to be settled by an adjustment of the equity capital. (The possibility of loss, about as likely and about as large as the possible profit, might seem unacceptable to an industry that needs to attract investors, but it should be kept in mind that salaries to employees and dividends to owners are accounted as part of the expenses discussed in Section 10.7.) B. On the notion of second order basis. The definition of the second order basis as the true one is slightly at variance with practical usage (which is

CHAPTER 10. SAFETY LOADINGS AND BONUS

164

not uniform anyway). The various amendments made to our idealized definition in practice are due to administrative and procedural bottlenecks: The factual development of interest, mortality, etc. has to be verified by the insurer and then approved by the supervisory authority. Since this can not be a continuous operation, any regulatory definition of the second order basis must to some extent involve realistic, still typically conservative, short term forecasts of the future development. However, our definition can certainly be agreed upon as the intended one. C. Model deliberations. The Markov chain model is mathematically tractable since state-wise expected values are determined by solving (in most cases simple) systems of first order ordinary differential equations. At the same time, when equipped with a sufficiently rich state space and appropriate intensities of transition, it is able to picture virtually any conceivable notion of the real object of the model. The Markov chain model is particularly apt to describe the development of life insurance policy since the paths of Z are of the same kind as the true ones. When used to describe the development of the second order basis, however, the approximative nature of the Markov chain is obvious, and it will surface immediately as e.g. the experienced force of interest takes values outside of the finite set allowed by the model. This is not a serious objection, however, and the next paragraph explains why. D. The role of the stochastic environment model. A paramount concern is that of establishing equivalence conditional on the factual second order history in the sense of (10.16). Now, in this conditional expectation the marginal distribution of the second order elements does not appear and is, in this respect, irrelevant. Also the contributions and, hence, the dividends are functions only of the realized experience basis and do not involve the distribution of its elements. Then, what remains the purpose of placing a distribution on the second order elements is to form a basis for prognostication of bonus. Subsidiary as it is, this role is still an important part of the play; although a prognosis does not commit the insurer to pay the forecasted amounts, it should as much as possible be a reliable piece of information to the insured. Therefore, the distribution placed on the second order elements should set a reasonable scenario for the course of events, but it need not be perfectly true. This is comforting since any view of the mechanisms governing the economic-demographic development is to some extent guess-work. When the accounts are eventually made up, every speculative element must be absent, and that is precisely what the principle (10.16) lays down. E. A digression: Which is more important, interest or mortality? Actuarial wisdom says it is interest. This is, of course, an empirical statement based on the fact that, in the era of contemporary insurance, mortality rates

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165

have been smaller and more stable than interest rates. Our model can add some other kind of insight. We shall again be content with a simple illustration related to the single life described in Section 10.6. Table 10.3 displays expected values and standard deviations of the present values at time 0 of a term life insurance and a life annuity under various scenarios with fixed interest and mortality, that is, conditional on fixed Y -state throughout the term of the policy. The impact of interest variation is seen by reading column-wise, and the impact of mortality variation is seen by reading row-wise. The overall impression is that mortality is the more important element by term insurance, whereas interest is the (by far) more important by life annuity insurance.

166

CHAPTER 10. SAFETY LOADINGS AND BONUS

Table 10.3: Expected value (E) and standard deviation (SD) of present values of a term life insurance (TI) with sum 1 and a life annuity (LA) with level intensity 1 per year, with interest r = i r∗ and mortality µ = m µ∗ for various choices of i and m . m :

1.5

TI 1.0

0.5

1.5

LA 1.0

0.5

i : 0.5

E : .14636 .10119 .05250 SD : .27902 .24104 .18041

20.545 20.996 21.467 03.691 03.101 02.257

1.0

E : .09927 .06834 .03531 SD : .20245 .17330 .12857

15.750 16.039 16.340 02.505 02.097 01.521

1.5

E : .06976 .04782 .02460 SD : .15858 .13437 .09868

12.466 12.655 12.852 01.759 01.468 01.061

Chapter 11

Statistical inference in the Markov chain model Think of the insurance company as a car: At the steering wheel sits the managing director trying to keep the vehicle steadily on the road. In the front passenger seat sits the sales manager pushing the speed pedal. At the rear sits the actuary peeping out the back window and giving the directions.

11.1

Estimating a mortality law from fully observed life lengths

A. Completely observed life lengths. Let the life length of an individual be represented by a non-negative random variable T with cumulative distribution function of the form F (t; θ) = 1 − e−

0 µ(s;θ) ds

(11.1)

The mortality intensity µ is assumed to be continuous (at least piece-wise) so that the density f (t) = µ(t)(1 − F (t))

(11.2)

exists.

A. Right-censored life times.

Let the total life length of an individual be represented by a non-negative random variable T with cumulative distribution function of the form Rt F (t) = 1 − e− 0 µ(s) ds . (11.3) The mortality intensity µ is assumed to be continuous (at least piece-wise) so that the density f (t) = µ(t)(1 − F (t)) ,

(11.4)

exists. Suppose that the individual is observed continually in z years from its birth so that only the truncated life length W = T ∧z is observed. A technical term for this kind of observational plan is right-censoring at time z. The cumulative distribution of W is F (t), 0 < t < z , P[W ≤ t] = 1, t ≥ z.

167

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL168 and the density (with respect to Lebesgue measure on (0, z) and the unit mass at z) is (recall (11.4)) µ(t)(1 − F (t)), 0 < t < z , g(t) = 1 − F (z), t =z. Introduce d(t) = 1(0,z) (t) = to obtain the closed expression

1, 0,

0 < t < z, t ≥ z,

g(t) = µ(t)d(t) (1 − F (t)) , 0 < t ≤ z .

(11.5)

(11.6)

Denote the indicator function of survival to age u > 0 by I(u) = 1[U > u]. The indicator of death before age z is D = d(T ) = 1 − I(z−) = 1 − I(z) , (11.7)

where the last equality holds with probability 1. We will need the following formulas, valid whenever the displayed moments exist: E[D k ] k

E[T ]

F (z) , k = 1, 2, . . . , Z z tk−1 (1 − F (t))dt , k = 1, 2, . . . , k

(11.8) (11.9)

E[DT ]

E[T ] − z(1 − F (z)) . (11.10) Rz Rz To verify (11.10), use (11.7) and T = 0 I(t)dt to write DT = 0 I(t)dt − zI(z), and take expectation.

B. The truncated exponential distribution.

We set out by analyzing the simple case with constant mortality intensity, partly as a motivating example, but also because the techniques are at the base of an important class of procedures in actuarial life history analysis. Thus, assume that U is exponentially distributed with cumulative distribution function F (u; µ) = 1 − e−µu , u > 0 , that is, µ is constant, independent of age. The expected life length is Z ∞ 1 ν= (1 − F (u; µ))du = . µ 0

(11.11)

(11.12)

Using (11.8) – (11.11) one easily calculates E[D]

V[D]

1 − e−µz ,

e−µz (1 − e−µz ) ,

(11.13) (11.14)

E[T ]

1 − e−µz , µ

(11.15)

V[T ]

1 − 2µze−µz − e−2µz , µ2

(11.16)

E[D − µT ]

V[D − µT ]

(11.17)

1 − e−µz .

(11.18)

C. Maximum likelihood estimators based on censored exponential variates. Let Ui , i = 1, 2, . . ., be independent replicates of U . Consider the problem of estimating µ from a sample of n censored life lengths, Ti = Ui ∧ zi , i = 1, . . . , n. The interpretation is that a mortality study is carried out in a population during a certain period of time terminating at time t¯, say, the sample being n individuals born during the period at times t¯ − zi , i = 1, . . . , n.

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL169 Referring to (11.7), put Ni = 1[Ti < zi ] , i = 1, . . . , n. By (11.6) and (11.11), the likelihood of the observables is Ni −µTi Λ = Πn e = µN e−µW = eln µ N −µW , i=1 µ

(11.19)

where N

n X

i=1 n X

Ni , the total number of deaths occurred, Ti , the total time exposed to risk of death.

i=1

Clearly, (N, W ) is a sufficient statistic. Take the logarithm, ln Λ = ln µ N − µW , and form the derivatives

N ∂ ln Λ = −W , ∂µ µ

(11.20)

N ∂2 ln Λ = − 2 . (11.21) ∂µ2 µ Putting the expression in (11.20) equal to 0 and noting that the second derivative is nonpositive, we find that the maximum likelihood estimator (MLE) of µ is the so-called occurrenceexposure rate, (OE rate) N µ ˆ= , (11.22) W the number of deaths occurred per unit of time exposed to risk of death in the sample. It is the empirical counterpart of the mortality intensity, which is the expected number of deaths per time unit, roughly speaking. The MLE of the expected life length in (11.12) is νˆ =

W , N

(11.23)

defined as +∞ when N = 0. This estimator has no mean (and no higher order moments). The expressions for the likelihood in (11.19) and the MLE in (11.22) do not appear to depend on the censoring mechanoism. The censoring is, however, decisive of the probability distribution of µ ˆ and, hence, of its performance as an estimator of µ. Unfortunately, this probability distribution is not easy to calculate in general, and we shall therefore have to add assumptions about the censoring mechanism, ranging from the special case of no censoring, where everything is simple and a lot of powerful results can be proved, to weak conditions under which only certain asymptotic properties are in reach.

D. The special case with no censoring. Suppose now that the n lives are completely observed without censoring, that is, zi = ∞ and Ti = Ui , i = 1, . . . , n. Then all Ni P are 1, N = n, W = n i=1 Ti , and the likelihood in (11.19) becomes Λ = eln µ n−µW .

(11.24)

In this simple situation it is easy to investigate the small sample properties of the estimators. The sum of the life lengths, W , is now a sufficient statistic. It has a gamma distribution with shape parameter n and scale parameter ν = 1/µ, whose density is µn n−1 −µw w e , w > 0. Γ(n) One finds (perform the easy calculations) for k > −n that E[W k ] =

Γ(n + k) , Γ(n)µk

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL170 hence

nµ , n > 1, n−1 n2 µ2 V[ˆ µ] = , n > 2, (n − 1)2 (n − 2) E[ˆ µ] =

and

E[ˆ ν] = ν , n ≥ 1 ,

ν2 , n ≥ 1. n The estimator µ ˆ is biased and, on the average, overestimates µ by µ/(n − 1), which is negligible for large n. An unbiased estimator of µ is µ ˇ = (n − 1)/W . Its variance is µ 2 /(n − 2). The estimator νˆ is now just the observed average life length, the empirical counterpart of ν. It is unbiased, of course. In fact, µ ˇ and νˆ are UMVUE (uniformly minimum variance unbiased estimators) since they are based on W , which is a sufficient and complete statistic: by (11.24), the distribution belongs to an exponential family with canonical parameter µ varying in the open set (0, ∞). V[ˆ ν] =

E. Asymptotic results by uniform censoring. say. Writing µ ˆ=

1 Pn i=1 Ni n 1 Pn i=1 Ti n

Suppose all zi are equal to z,

and noting that E[Ni ] = µETi by (11.13) and (11.15), it follows by the strong law of large numbers that the estimator is strongly consistent, a.s.

µ ˆ → µ. To investigate its asymptotic distribution, look at Pn √1 √ i=1 (Ni − µTi ) n . n(ˆ µ − µ) = 1 Pn i=1 Ti n

The denominator of this fraction converges a.s. to E[Ti ] given by (11.15). By the central limit theorem, the limiting distribution of the numerator is normal with mean 0 (recall (11.17)) and variance given by (11.18). It follows that µ2 µ ˆ ∼as N µ, . (11.25) n(1 − e−µz ) Copying the arguments above (or using (D.6) in Appendix D), it can also be concluded that νˆ defined by (11.23) is strongly consistent and that 1 νˆ ∼as N ν, . (11.26) 2 −µz nµ (1 − e ) No strong conclusions as to optimality can be drawn in parallel to those in the previous paragraph. The reason is seen from (11.19): the distribution belongs to a general exponential family with canonical parameter (ln µ, µ), which does not vary in an open (two-dimensional) set. Therefore, the sufficient statistic (N, W ) cannot be proved to be complete (not the usual way at least), and standard theory for inference in regular exponential families of distributions cannot be employed.

F. Asymptotic results by fairly general censoring.

Consider now the general situation in Paragraph C with censoring varying among the individuals. A bit more effort must now be put into the study of the asymptotic properties of the MLE. It turns out that a sufficient condition for consistency and asymptotic normality is that the expected exposure grows to infinity in the sense that n X i=1

E[Ti ] → ∞ ,

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL171 which by (11.15) is equivalent to n X i=1

(1 − e−µzi ) → ∞ ,

(11.27)

that is, the expected number of deaths grows to infinity. Thus assume that (11.27) is satisfied. In the following the relationships (11.13) – (11.18) will be used frequently without explicit mentioning. First, to prove consistency, use (11.15) and (11.18) to write Pn i=1 (Ni − µTi ) Pn µ ˆ−µ = . i=1 Ti Pn Pn −1 Ti (Ni − µTi ) . (11.28) Pni=1 = Pni=1 µ i=1 E[Ti ] i=1 V[Ni − µTi ] The first factor in (11.28) has expected value 0 and variance

1 1 , = Pn V[N − µT ] (1 − e−µzi ) i i i=1 i=1

which tends to 0 as n increases. Therefore, this Pn Pnfactor tends to 0 in probability. The second factor in (11.28) is the inverse of i=1 Ti /µ i=1 E[Ti ], which has expected value 1/µ and variance equal to 1/µ2 times Pn Pn −µzi − e−2µzi ) i=1 (1 − 2µzi e i=1 V[Ti ] 2 = Pn Pn −µz −µzi ) 2 i ) i=1 (1 − e i=1 (1 − e Pn −µzi ) 1 i=1 a(µzi )(1 − e = Pn Pn , −µz i) (1 − e (1 − e−µzi ) i=1 i=1 (11.29)

where a is defined as

1 − 2te−t − e−2t , t ≥ 0. 1 − e−t The function a is bounded since it is continuous and tends to 0 as t & 0 (use l’Hospital’s rule) and to 1 as t % ∞. The first factor in (11.29) is bounded since it is a weighted average of values of a, and the second factor tends to 0 by assumption. It follows that the expression in (11.29) tends to 0 and, consequently, that the second factor in (11.28) converges in probability to µ. It can be concluded that the expression in (11.28) converges in probability to 0, so that µ ˆ is weakly consistent; p µ ˆ → µ. a(t) =

Next, to prove asymptotic normality, look at v u n Pn −1 Pn uX Ti (Ni − µTi ) t (1 − e−µzi )(ˆ Pni=1 . µ − µ) = pPi=1 n −µzi ) µ i=1 E[Ti ] i=1 (1 − e i=1

In the presence of (11.27) the first factor on the right converges in distribution to a standard normal variate. (Verify that the Lindeberg condition is satisfied, see Appendix D.) The second factor has just been proved to converge in probability to µ. It follows that µ2 µ ˆ ∼as N µ, Pn . (11.30) −µzi ) i=1 (1 − e Likewise, it also holds that νˆ is weakly consistent, and 1 νˆ ∼as N ν, 2 Pn . −µzi ) µ i=1 (1 − e

(11.31)

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL172 G. Random censoring.

In Paragraph E the censoring time was assumed to be the same for all individuals. Thereby the pairs (Ni , Ti ) became stochastic replicates, and we could invoke simple asymptotic theory for i.i.d. variates to prove strong consistency and asymptotic normality of MLE-s. In Paragraph F the censoring was allowed to vary among the individuals, but it turned out that the asymptotic results essentially remained true, although only weak consistency could be achieved. All we required was (11.27), which says that the censoring must not turn too severe so that information deteriorates in the end: there must be a certain stability in the censoring pattern so that individuals with sufficient exposure time enter the study sufficiently frequently in the long run. One way of securing such stability is to regard the censoring times as outcomes of i.i.d. random variables. Such an assumption seems particularly apt in a non-experimental context like insurance. The censoring is not subject to planning, and the censoring times are just as random in their nature as anything else observed about the individuals. Thus, we henceforth work with an augmented model where the assumptions in Paragraph F constitute the conditional model for given censoring times Zi = zi , i = 1, 2, . . ., and the Zi are independent selections from some distribution function H with (generalized) density h independent of µ. This way the triplets (Ni , Ti , Zi ), i = 1, 2, . . ., become stochastic replicates, and the i.i.d. situation is restored with all its powers. The likelihood of the observations now becomes Ni −µTi Λ = Πn e h(Zi ) = eln µ N −µW Πn i=1 µ i=1 h(Zi ) .

(11.32)

Maximizing (11.32) with respect to µ is equivalent to maximizing the likelihood (11.19) in the conditional model for fixed censoring, hence the MLE remains the same as before. Its distribution is affected by the structure now added to the model, however. It is easy to prove that the results in Paragraph E carry over to the present case, only that the expression 1 − e−µz is everywhere to be replaced by 1 − E e−µZ , where Z ∼ H.

11.2

Parametric inference in the Markov model

A. The likelihood of a time-continuous Markov process. Consider now the general set-up, whereby the development of an insurance policy is represented by a continuous time Markov process Z on a finite state space Z = {0, 1, . . . , J}. As usual, let I g (t) and Ngh (t) denote, respectively, the indicator of the event that the process is staying in state g at time t ≥ 0, and the number of transitions from state g to state h in the time interval (0, t]. The transition intensities µgh are assumed to exist, and to be piecewise continuous. Suppose the policy is observed continuously throughout the time period [t, t¯], commencing in state g0 at time t. One then speaks of left-censoring and right-censoring at times t and t¯, respectively, and the triplet z = (t, t¯, g0 ) will be referred to as observational design or censoring scheme of the policy. Consider a specific realization of the observed part of the process:  g0 , t < τ < t1 ,    , t1 + dt1 < τ < t2 ,  g1 ... X(τ ) =     gq−2 , tq−2 + dtq−2 < τ < tq−1 , gq−1 , tq−1 + dtq−1 < τ < t¯.

¯ By the given P censoring, the probability of this realization is as follows, where t 0 = t , tq = t, and µg = h6=g µgh denotes the total intensity of transition out of state g: Z t Z t 1 2 exp − µg0 µg0 g1 (t1 )dt1 exp − µg1 µg1 g2 (t2 )dt2 . . . t0

exp

−

tq−1 tq−2 q−1 Y

p=1

µgq−2

µgq−2 gq−1 (tq−1 )dtq−1 exp

µgp−1 gp (tp )dtp exp

−

q Z X 1

−

tp tp−1

µgp−1

tq tq−1

µgq−1

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL173



exp 

q−1 X

p=1

ln µgp−1 gp (tp ) −

q Z X 1



tp tp−1

µgp−1  dt1 . . . dtq−1 .

It follows that the likelihood of the observables is  XZ X Z t¯ Λ = exp  ln µgh (τ )dNgh (τ ) − g6=h



exp 

g6=h

t¯



t¯

µg (τ )Ig (τ )dτ  

{ln µgh (τ )dNgh (τ ) − µgh (τ )Ig (τ )dτ } .

(11.33)

B. ML estimation of parametric intensities. Now consider a parametric model where the intensities are of the form µgh (t, θ), with θ = (θ1 , . . . , θs )0 varying in an open set in the s-dimensional euclidean space, s < ∞. We assume they are twice continuously differentiable functions of θ. Suppose that inference is to be made about the intensities or, equivalently, the parameter θ on the basis of data from a sample of n similar policies. Equip all quantities related to the mth policy by topscript (m). The processes X (m) are assumed to be stochastically independent replicates of the process Z described above, but their censoring schemes z (m) may be different. By independence, the likelihood of the whole data set is the product of the individual Q (m) . Thus, by (11.33), likelihoods: Λ = n m=1 Λ XZ ln Λ = ln µgh (τ, θ)dNgh (τ ) − µgh (τ, θ)Ig (τ )dτ , (11.34) g6=h

with Ngh =

n X

(m)

Ngh , Ig =

m=1

n X

(m)

m=1

The censoring schemes are not visualized in (11.34), and they need not be if, as a matter of (m) (m) definition, dNgh (t) and Ig (t) are taken as 0 for t ∈ / [t(m) , t¯(m) ]. Likewise, introduce (m)

(t) = p

(m)

(t(m) , t)1 [t(m) ,t¯(m) ] (t) ,

the probability that the censored process Z (m) stays in g at time t, by definition taken as 0 for t ∈ / [t(m) , t¯(m) ]. In the MLE construction we need the derivatives of (11.34), of first order (an s-vector), XZ ∂ ∂ ln Λ = ln µgh (τ, θ){dNgh (τ ) − µgh (τ, θ)Ig (τ )dτ } , (11.35) ∂θ ∂θ g6=h and of second order (an s × s matrix), X Z ∂2 ∂2 ln Λ = ln µgh (τ, θ){dNgh (τ ) − µgh (τ, θ)Ig (τ )dτ } ∂θ∂θ 0 ∂θ∂θ 0 g6=h ∂ ∂ − ln µgh (τ, θ) 0 µgh (τ, θ)Ig (τ ) dτ ∂θ ∂θ By (11.35) the MLE θˆ is the solution to XZ ∂ ln µgh (τ, θ){dNgh (τ ) − µgh (τ, θ)Ig (τ )dτ } |θ=θˆ= 0sx1 . ∂θ g6=h

(11.36)

(11.37)

A comment on the form of the likelihood (9.31): For each type of transition g → h (1)

)

introduce Ngh , the number of transitions of that type, and (if Ngh > 0) Tgh , . . . , Tgh gh , the

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL174 times when such transitions occurred. In terms of these quantities the log likelihood in (9.32) assumes the form gh XN X

g6=h j=1

(j)

ˆ − ln µgh (Tgh , θ)

ˆ Ig (τ )dτ , µgh (τ, θ)

g6=h

and the ML equations (9.35) become (j) ˆ ∂ µ (T , θ) ∂θi gh gh (j) ˆ µgh (Tgh , θ) g6=h j=1 N

gh XX

g6=h

∂ ˆ Ig (τ )dτ , µgh (τ, θ) ∂θi

(11.38)

i = 1, . . . , s. The form (11.38) is explicit and is, of course, the one we will work with when it comes to numerical computation of the MLE: The good thing about the form (9.32) is that it, by use of the counting processes, writes the sum on the left as a sum of contributions from all small time intervals. This is particularly useful in the derivation of the statistical properties of the MLE. (A similar remark could be made about the benefit of using the counting processes to define the payment stream for a general insurance policy in Section 7.5.) Referring to Appendix D, the large sample distribution properties of the MLE are given by θˆ ∼as N(θ, Σ(θ)) , where Σ(θ) is given by its inverse, the so-called information matrix, ∂ Σ(θ)−1 = −E ln Λ . ∂θ∂θ 0

(11.39)

(11.40)

Taking expectation in (11.36), noting that the terms dNgh (τ ) − µgh (τ, θ)Ig (τ )dτ have zero means, we obtain X n XZ 1 ∂ ∂ (m) Σ(θ)−1 = µgh (τ, θ) 0 µgh (τ, θ) pg (τ, θ)dτ . µgh (τ, θ) ∂θ ∂θ m=1 g6=h (11.41) The expression in parentheses under the integral sign is an s×s matrix and all other quantities are scalar. It is seen that the information matrix tends to infinity, hence the variance matrix of Pn (m) the MLE tends to 0, if the terms (τ, θ) grow to infinity as n increases, roughly m=1 pg speaking, which means that the expected number of individuals exposed to risk in different states gets unlimited.

C. Estimating the parameters of a Gompertz-Makeham mortality law. The actuarial office in a life insurance company is to estimate the mortality law governing the company’s portfolio of term insurance policies. (The mortality law for the portfolio of life annuities may be different since people who (believe they) are in good health would probably find a pure survival benefit more useful and profitable than a pure death benefit. Thus, it seems appropriate to perform a separate mortality investigation for each line of life insurance business. Moreover, since mortality also depends on sex, the study would typically include only males or only females.) Suppose the statistical data comprises n individuals who have been insured under the scheme during a certain period of time. For each individual No. m (= 1, . . . , n) there is a policy record with the following pieces of information: – xm , the age on entry into the study; – ym , the age on exit from the study; – Nm , the number of deaths during the study (0 or 1).

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL175 Here xm would typically be the age at issue of the policy. If Nm = 1, then ym is the age at death, and if Nm = 0, then ym is the age at the time of censoring, either at the term of the contract or upon termination of the study, whichever occurred first. In any case y m − xm is the time spent under observation as alive during the study. With these definitions xm takes the role of t(m) in the general set-up and, for Nm = 0, ym takes the role of t¯(m) . The state space is now just Z = {0, 1} (”alive” and ”dead”). Assume the mortality law is Gompertz-Makeham so that the mortality intensity at age t is µ(τ, θ) = α + βeγτ , with

 α  θ = β . γ We need the derivatives of the intensity w.r.t. all three parameters, ∂ µ(τ, θ) = 1 , ∂α ∂ µ(τ, θ) = eγτ , ∂β ∂ µ(τ, θ) = β eγτ τ , ∂γ and their integrals with respect to time, Z y ∂ µ(τ, θ) dτ = y − x , x ∂α Z y eγy − eγx ∂ µ(τ, θ) dτ = , γ x ∂β γy Z y ∂ e y − eγx x eγy − eγx µ(τ, θ) dτ = β − . γ γ2 x ∂γ The MLE equations (9.35) specialize to X X 1 = (ym − xm ) , ˆ γˆ ym α ˆ + βe 

m; Nm =1

eγˆym ˆ γˆ ym α ˆ + βe

eγˆ ym ym ˆ γˆ ym α ˆ + βe

X eγˆ ym − eγˆxm , γ ˆ m X eγym ym − eγxm xm eγym − eγxm − . γ γ2 m

To find the information matrix (9.38) we tives,  ∂ ∂ µ(τ, θ) 0 µ(τ, θ) =  ∂θ ∂θ (m)

need the matrix with the products of the deriva1 · ·

(symmetric) and the probabilities p0 (m) p0 (τ, θ)

eγτ e2γτ ·

 βeγτ τ βe2γτ τ  β 2 e2γτ τ 2

(τ, θ), which are Z τ exp − (α + βeγ s ) ds

(11.42)

eγτ − eγxm , (11.43) γ for τ ∈ [xm , ym ] (the survival probability) and 0 otherwise. (There is only one kind of transition, from 0 to 1, and the summation over g, h in the information matrix (9.38) can be dropped.) We see that all ingredients in the asymptotic variance matrix are given by explicit formulas, and it remains only to perform a numerical integration to find its value for given θ. =

exp −α(τ − xm ) − β

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL176

11.3

Confidence regions

A. An asymptotic confidence ellipsoid. MLE it follows that

From the asymptotic normality of the

(θˆ − θ)0 Σ−1 (θ)(θˆ − θ) ∼as χ2s ,

(11.44)

ˆ 0 Σ−1 (θ)(θ − θ) ˆ ≤ χ2 (θ − θ) s, 1−ε .

(11.45)

the chi-squared distribution with s degrees of freedom. Therefore, denoting the (1 − ε)-fractile of this distribution by χ2s, 1−ε , an asymptotic 1−ε confidence region is the set of all θ satisfying

The expression on the left here will typically be a complicated function of θ, and it is in general not easy to find the values of θ that satisfy the inequality and constitute a confidence ˆ and that the region. Now, suppose Σ(θ) can be estimated by some function of the data, Σ, estimator is consistent in the sense that ˆ −1 (θ) → I . ΣΣ

(11.46)

Then it is easy to show that also the relation ˆ 0Σ ˆ ≤ χ2 ˆ −1 (θ − θ) (θ − θ) s, 1−ε

(11.47)

determines an asymptotic 1 − ε confidence region. The relation (11.47) defines an ellipsoid, which is a fairly simple geometric figure and, as we shall see in the following paragraph, a convenient basis for deriving other confidence statements of interest. ˆ would be to replace θ in Σ(θ) by the consistent A straightforward way of constructing Σ ˆ that is, put estimator θ, ˆ . ˆ = Σ(θ) Σ This works well if the entries in Σ(θ) are closed expressions in θ. Unfortunately, this is the case only in certain simple situations, typically when the state space Z is small and the pattern of transitions is hierarchical. One example is the mortality study with parametric mortality law, e.g. of G-M type. In more complex situations we cannot in general find closed formulas for the (m) probabilities pg (τ ) involved in Σ, even if the intensities themselves are simple parametric (m)

functions. Then a different construction is required. A simple device is to replace the p g by their empirical counterparts

(m) Ig (τ ) n X

m=1

(τ )

and put (m)

(τ ) ≈ Ig (τ ) .

(11.48)

B. Simultaneous confidence intervals. The confidence ellipsoid (11.47) can be resolved in simultaneous confidence intervals for all linear functions of θ in the following way. The Schwarz inequality says that for all vectors a and x in Rs , √ |a0 x| ≤ a0 a x0 x , with equality for a = cx. Thus, noting that the quadratic form on the left of (11.47) is ˆ −1/2 (θ − θ)) ˆ 0 (Σ ˆ −1/2 (θ − θ)), ˆ the confidence statement can be cast equivalently as (Σ q ˆ ≤ a 0 a χ2 ˆ −1/2 (θ − θ)| (11.49) |a0 Σ s, 1−ε , ∀a .

ˆ is of full rank, the vector Σ ˆ −1/2 a ranges through all of Rs as a ranges in Rs . Thus, Since Σ ˆ Σ ˆ −1/2 a), (11.49) is equivalent to ˆ −1/2 a)0 Σ( writing a0 a = (Σ q ˆ ≤ a0 Σa ˆ χ2 |a0 (θ − θ)| s, 1−ε , ∀a , that is,

a0 θ ∈ [a0 θˆ −

ˆ , a0 θˆ + χ2s, 1−ε a0 Σa

q ˆ , ∀a . χ2s, 1−ε a0 Σa]

(11.50)

The intervals in (11.50) are (asymptotic) simultaneous confidence intervals for all linear functions of θ in the sense that the probability is at least 1 − ε that they all hold true.

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL177 C. Confidence band for the G-M mortality intensity. Returning to the mortality study example in Paragraph 11.2.C, let c be taken as known so that the mortality intensity is a linear function of the unknown parameter θ = (α, β)0 . The MLE is obtained by solving the equations (11.42) and (11.42), and the appropriate variance matrix Σ is obtained by inverting the upper left 2 × 2 block in the information matrix defined by (11.41), (11.42), and (11.43). From (11.50) we obtain simultaneous confidence intervals for all µ(τ ) = α + βe γτ , constituting a confidence band in the space of mortality intensity functions; q q ˆ γτ − χ2 ˆ γτ − χ2 µ(τ ) ∈ [α ˆ + βe ˆτ , α ˆ + βe ˆτ ] , ∀τ > 0 , (11.51) 2, 1−ε σ 2, 1−ε σ where

ˆ σ ˆτ = (1, eγτ )Σ

11.4

1 eγτ

More on simultaneous confidence intervals

A. Simultaneous confidence intervals based on a confidence region. Let X ∈ X be some observation(s) with distribution Pθ , θ ∈ Θ, some s-dimensional set. A 1 − ε confidence region for θ is a function C from X to the set of subsets of Θ such that the random set C = C(X) satisfies Pθ {θ ∈ C} = 1 − ε, ∀θ ∈ Θ . From a confidence region we readily obtain confidence intervals for the values g(θ) of all real-valued functions g : Θ → R as follows. For a fixed g, define the random variables g = inf θ∈C g(θ) and g = supθ∈C g(θ). Clearly, θ ∈ C implies g ≤ g(θ) ≤ g for all g, hence Pθ {g ≤ g(θ) ≤ g , ∀g} ≥ 1 − ε , ∀θ ∈ Θ . Thus, by stating that g(θ) ∈ [ g, g ] , ∀g ,

(11.52)

all statements hold true simultaneously with a probability no less than 1 − ε. In this sense the intervals in (11.52) are simultaneous confidence intervals with (simultaneous) confidence level 1 − ε. A practical consequence is that we are allowed to snoop around in the data set, looking for possible interesting effects (within the model), and still keep control of the probability of making false statements. B. Confidence ellipsoid for a normal mean; Scheff´ e intervals. Let θˆ is an estimator of an s-dimensional parameter vector θ and assume that θˆ ∼ N (θ, Σ) , with Σ known. A 1 − ε confidence region of θ is the ellipsoid defined by (3.2): ˆ 0 Σ−1 (θ − θ) ˆ ≤ χ2 C = {θ ; (θ − θ) s, 1−ε } .

To construct the confidence interval (11.52) for a specific function g(θ), we are left with the mathematical problem of finding the extrema of g over the ellipsoid, which may be a difficult task. For linear functions it is simple, however, and it goes by the the technique in Paragraph 3B, P which is due to Scheff´e. The simultaneous confidence intervals for linear functions a0 θ = sp=1 ap θp are, with a bit sloppy notation, a0 θ ∈ a0 θˆ ± σa

where

q χ2s, 1−ε , ∀a ∈ Rs ,

σa2 = a0 Σ a ˆ is the variance of the point estimator a0 θ.

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL178 C. Narrowing the confidence intervals. Generally speaking, in the presence of uncertainty, the price we have to pay for making many safe statements is that each individual statement has to be vague. In our situation this general truth takes a very manifest form: for a fixed confidence level 1−ε the lengths of the confidence intervals increase with the dimension of θ since χ2s, 1−ε is an increasing function of s (why?). We can gain precision in terms of lengths of the intervals by reducing the number of statements we want to make. Suppose we are only interested in drawing inferences about linear combinations of r linearly independent functions b0j θ, j = 1, . . . , r, r < s. Thus, putting B = (b1 , . . . , br ), an s×r matrix, we are only interested in linear functions a0 θ with a = Bc for some r-vector c, that is, a ∈ R(B), the r-dimensional linear space spanned by the b j . Then, start from B 0 θˆ ∼ N (B 0 θ, B 0 ΣB) , and apply the results above to obtain that simultaneous confidence intervals for all linear functions of B 0 θ are given by q (11.53) c0 B 0 θ ∈ c0 B 0 θˆ ± c0 B 0 ΣBc χ2r, 1−ε , ∀ c ∈ Rr ,

or, equivalently,

a0 θ ∈ a0 θˆ ± σa

q χ2r, 1−ε , ∀a ∈ R(B) .

It is seen that, by reducing the ”dimension of our statements” q from s to r, we have gained a χ2r, 1−ε /χ2s, 1−ε .

reduction of the lengths of the confidence intervals by a factor

B. Bonferroni intervals for a finite number of parameter functions. Let gj (θ), j = 1, . . . , r, be a finite number of real-valued parameter functions, and assume that for each gj (θ) we have constructed an individual confidence interval [ g j , g j ] with level 1 − εj . Thus, denoting the event gj (θ) ∈ [gj , g j ] by Aj , we have Pθ (Aj ) ≥ 1 − εj for each j. The probability that all Aj hold true at the same time is Pθ (∩j Aj ) = 1 − Pθ ∪j Acj ≥

1−

≥

1−

r X

Pθ (Acj )

j=1 r X

εj .

j=1

It follows that the intervals taken together are simultaneous confidence intervals with simulP taneous confidence level no less than 1 − ε, where ε = rj=1 εj . This simple device, due to Bonferroni, represents an attractive alternative to the approach in Paragraphs A – C in situations where we take interest only in a finite number of parameter functions. It turns out that the Bonferroni intervals often are shorter than the Scheff´e intervals, which aim at an infinite number of parameter functions. Bonferroni intervals with simultaneous confidence level 1 − ε for q linear functions a0j θ, j = 1, . . . , q, are q (11.54) a0j θ ∈ a0j θˆ ± σaj χ21, 1−ε/q , q (Note that χ21, 1−ε/q is just the (1 − ε/2q)-fractile of the standard normal distribution, so

we recognize (11.54) as the traditional individual 1 − ε/q confidence interval for a normal mean.) Let r (≤ s) be the dimension of the space spanned by the coefficient vectors a j . The corresponding Scheff´e intervals based on (11.53) are q a0j θ ∈ a0j θˆ ± σaj χ2r, 1−ε . q The ratio of the length of the intervals by the two constructions is B/S(q, r, ε) = χ21, 1−ε/q /χ2r, 1−ε .

Clearly, the ratio decreases with r. It increases with q, and (for r > 1) it starts from q = 1

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL179 with a value smaller than 1 and tends to infinity as q grows. There will be some value q(r, ε) such that the ratio is ≤ 1 for q ≤ q(r, ε). Inspection of a table of the chi-square fractiles shows e.g. that q(2, 0.025) = 5, q(4, 0.1) = 20, q(6, 0.25) = 50 (approximately). E. Confidence intervals based on consistent and asymptotically normal point estimators. The results and considerations in the previous paragraphs carry over to the situation in Section 3, where (3.4) formed the basis for simultaneous inference. Suppose we are interested in functions of a set of parameter functions f j (θ), j = 1, . . . , r, r < s. Put f = (f1 , . . . , fr )0 . If f is continuously differentiable, first order Taylor expansion gives ˆ ∼as N f (θ), Σf (θ)) , f (θ)

with

Σf (θ) = Df (θ)0 Σ(θ) Df (θ) ,

and

∂ f (θ) , ∂θ an s × r matrix. We obtain the asymptotic confidence ellipsoid Df (θ) =

ˆ 0Σ ˆ ≤ χ2 ˆ −1 (f − f (θ)) Cf = {f ; (f − f (θ)) r, 1−ε } , f

ˆ Asympˆ f is some consistent estimator of Σf in the sense of (3.3), e.g. Σ ˆ f = Σf (θ). where Σ totic Scheff´e intervals for all functions g(θ) = h(f (θ)), with h real-valued and continuously differentiable, are q ˆ ±σ g(θ) ∈ g(θ) ˆ g χ2r, 1−ε , where

σ ˆg =

∂ ∂ g(θ) Σ(θ) g(θ) |θ=θˆ . ∂θ 0 ∂θ

Asymptotic Bonferroni intervals for a finite collection of functions g j , j = 1, . . . , q, are obtained upon replacing r and ε with 1 and ε/q. F. The G-M mortality intensity revisited. The confidence intervals (3.8) are infinitely many, so Bonferroni ideas cannot help here. If also c is to be estimated, we obtain asymptotic confidence intervals by the device above. The same goes for any function of actuarial relevance, like the reserve of a life insurance or a portfolio of insurances. Think of examples. Returning to the mortality study example in Paragraph 2C, let γ be taken as known so that the mortality intensity is a linear function of the unknown parameter θ = (α, β) 0 . The MLE is obtained by solving the eqnarrays (11.42) and (11.42), and the appropriate variance matrix Σ is obtained by inverting the upper left 2 × 2 block in the information matrix defined by (11.41), (11.42), and (11.43). From (11.50) we obtain simultaneous confidence intervals for all µ(τ ) = α + βe γτ , constituting a confidence band in the space of mortality intensity functions; q q ˆ γτ − χ2 ˆ γτ − χ2 µ(τ ) ∈ [α ˆ + βe ˆτ , α ˆ + βe ˆτ ] , ∀τ > 0 , 2, 1−ε σ 2, 1−ε σ where

ˆ σ ˆτ = (1, eγτ )Σ

11.5

1 eγτ

Piecewise constant intensities

A. Piecewise constant intensities.

Let 0 = t0 < t1 < · · · < · · · < tr = t¯ be some finite partition of the time interval [0, t¯], and assume that the intensities are step functions of

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL180 the form µgh (τ )

= =

µgh,q , τ ∈ [tq−1 , tq ) , q = 1, . . . , r , r X

1[tq−1 ,tq ) (τ )µgh,q ,

(11.55)

q=1

where the µgh,q take values in (0, ∞), with no relationships between them. The situation fits into the general framework with θ = (. . . , µgh,q , . . .)0 , a vector of (typically high) dimension J × J × r.

B. The MLE estimators are O-E rates.

The log likelihood in (11.34) now

becomes ln Λ =

r XX

g6=h q=1

{ln µgh,q Ngh,q − µgh,q Wg,q } ,

(11.56)

where Ngh,q

Wg,q

Z Z

tq tq−1

dNgh (τ ) ,

(11.57)

Ig (τ )dτ ,

(11.58)

tq tq−1

are, respectively, the total number of transitions from state g to state h and the total time spent in state g during the age interval [tq−1 , tq ). Since the µgh,q are functionally unrelated, the log likelihood decomposes into terms that depend on one and only one of the basic parameters, and finding maximum amounts to maximizing each term. The derivatives involved in the ML construction now become particularly simple: 1 ∂ ln Λ = Ngh,q − Wg,q , (11.59) ∂µgh,q µgh,q 1 ∂2 ln Λ = −δghq,g 0 h0 q 0 2 Ngh,q . ∂µgh,q ∂µg 0 h0 ,q 0 µgh,q

(11.60)

It follows from (11.59) that the MLE is µ ˆgh,q =

Ngh,q , Wg,q

(11.61)

an O-E rate of the same kind as in the simple model of 11.2.B. Noting that, by (11.57), Z tq X n (m) E[Ngh,q ] = µgh,q pg (τ )dτ , tq−1 m=1

we obtain from (11.60) that the asymptotic variance matrix becomes   µgh,q  Σ(θ) = Diag . . . , R t , . . . , Pn (m) q (τ )dτ tq−1 m=1 pg

(11.62)

a diagonal matrix, implying that the estimators of the µgh,q are asymptotically independent. An estimator of Σ is obtained upon replacing the parameter functions appearing on the right of (11.62) by their straightforward estimators: put µgh,q ≈ µ ˆgh,q defined by (11.61) and, by the device (11.48), Z tq Z tq X n (m) Ig (τ )dτ = Wg,q , pg (τ )dτ ≈ tq−1

tq−1 m=1

to obtain

ˆ = Diag Σ

...,

Ngh,q ,... 2 Wg,q

(11.63)

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL181 C. Smoothing O-E rates.

The MLE of the intensity function is obtained upon inserting the estimators (11.61) in (11.55). The resulting function will typically have a ragged appearance due to the estimation error in a finite sample. This is unsatisfactory since the intensities are expected to be smooth functions: for instance, there are a priori reasons to assume that the mortality intensity is a continuous and non-decreasing function of the age. Now, the very assumption of piecewise constant intensities is artificial, of course, and the estimates obtained under this assumption cannot serve as an ultimate answer in practice. In fact, they represent only the first step in a two-stage procedure, where the second step is to fit some smooth functions to the raw estimates delivered by the O-E rates. The functions used for fitting constitute the model we have in mind. It may be objected that the two-stage procedure is a detour since, if the intensities are assumed to be functions of a smaller set of parameters, one could follow the prescription in Section 11.2 and maximize the likelihood directly. There are two reasons why the two-stage procedure never the less merits special treatment: in the first place, the O-E rates and their asymptotic variance matrix are easy to construct; in the second place, a comparative plot of the fitted functions and the O-E rates makes it possible to detect systematic deviations between model assumptions and facts. A commonly used fitting technique is the so-called generalized least squares method, which amounts to minimizing a positive definite quadratic form in the deviations between the raw estimates and the fitting functions. In the following brief outline of the procedure we focus on one given intensity and drop the subscripts g, h. For each interval [tq−1 , tq ) choose a ”representative” point τq , e.g. the interval midpoint. Put µ ˆ = (. . . , µ ˆ q , . . .)0 , the vector of O-E rates, and (with a bit sloppy notation) µ(θ) = (. . . , µ(τq , θ) , . . .)0 , the vector of true values. Let A = (apq ) be some positive definite matrix of order r × r. Estimate θ by θ ∗ minimizing X (µ(θ) − µ ˆ)0 A(µ(θ) − µ ˆ) = apq (µ(τp , θ) − µ ˆp )(µ(τq , θ) − µ ˆq ) . (11.64) pq

If the intensity is a linear function of θ (like in the G-M study with known γ), µ(θ) = Y (τ )θ ,

(11.65)

then

θˆ = (Y 0 AY )−1 Y 0 Aˆ µ. (11.66) ˆ The asymptotic variance of θ is (Y 0 AY )−1 Y 0 AΣ(θ)AY (Y 0 AY )−1 . By the Gauss-Markov theorem it is minimized by taking A = Σ(θ)−1 , and the minimum is (Y 0 Σ(θ)−1 Y )−1 . Thus, ˆ −1 , where Σ ˆ is some estimate of Σ satisfying (11.46). asymptotically the best choice of A is Σ Write Σ = Σ(θ). The symmetric, pd matrix Σ has a symmetric pd square root W such that Σ = W 2. ∆

= = =

(Y 0 AY )−1 Y 0 AΣAY (Y 0 AY )−1 − (Y 0 Σ−1 Y )−1 (Y 0 AY )−1 Y 0 AW I − W Y ((W Y )0 (W Y ))−1 (W Y )0 W AY (Y 0 AY )−1 (Y 0 AY )−1 Y 0 AW H W AY (Y 0 AY )−1 .

where The matrix H = I − H = H 0 H and

P (P 0 P )−1 P 0

H = I − P (P 0 P )−1 P 0 . is symmetric, H = H 0 , and idempotent, H 2 = H, Thus,

∆ = HW AY (Y 0 AY )−1

HW AY (Y 0 AY )−1

which is indeed pd. More on analytic graduation - the G-M example: Let us focus on one intensity that is to be graduated and, to fix ideas, assume it is the mortality intensity in the simple model with two states ’alive’ and ’dead’.. The first step is to assume that the intensity is piece-wise constant: µ(t) = µq ,

q − 1 ≤ t < q , j = 1, 2, . . .

The log likelihood (9.53) is ln Λ =

X q

(ln µq Nq − µq Wq ) ,

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL182 where Nq and Wq are, respectively, the number of deaths and the total time spent alive in the age interval [q − 1, q). Each µq is a parameter which is functionally unrelated to all the others, so there are many parameters in this model! For instance, if we are interested in mortality up to age 100 and have data in the age range from 0 to 100, there are 100 parameters, which is quite a lot. Remember, however, that this model is just a first step in a two-stage procedure where the second step is to graduate (smooth) the ML estimators resulting from the present naive model with piece-wise constant intensity. The ML estimators are the occurrence-exposure rates, µ ˆq =

Nq , Wq

which are well defined for all q such that Wq > 0 (i.e. in age intervals where there were survivors exposed to risk of death). The µ ˆ q are asymptotically (as n increases) normally distributed, mutually independent, unbiased, and with variances given by µq σq2 = as.V[ˆ µq ] = , (11.67) E[Wq ] where the expected exposure is E[Wq ] =

n Z X

m=1

p(m) (τ ) dτ ,

q−1

p(m) (τ ) being the probability that individual No. m is alive and under observation at time τ . The variance σq2 is inversely proportional to the corresponding expected exposure. In the present simple model, with only one intensity of transition from the state ’alive’ to the absorbing state ’dead’, we find explicit expressions for the expected exposure. For instance, suppose we have observed each individual life from birth until death or until attained age 100, whichever occurs first (i.e. censoring at age 100). Then, for τ ∈ [q − 1, q) with q = 1, . . . , 100, we have Z τ p(m) (τ ) = exp − µ(s) ds 0   q−1 X  = exp − (11.68) µp − (τ − (q − 1)) µq  , p=1

hence

E[Wq ]

= and σq2 =

q q−1



q−1 X



exp −



n exp −

p=1

µp 



µp − (τ − (q − 1)) µq  dτ 1 − exp (−µq ) , µq

1 µq . n exp − Pq−1 µp (1 − exp (−µq )) p=1

(11.69)

You should look at other censoring schemes and discuss the impact of censoring on the variance. Take e.g. the case where person No m enters at age zm and is observed until death or age 100, whichever occurs first (all zm less than 100). Estimators σ ˆ q2 of the variances are obtained upon replacing the µj in (G.55) by the estimators µ ˆ j . Simpler estimators are obtained by just replacing µq and E[Wq ] in (G.53) with their straightforward empirical counterparts: σ ˆ q2 = µ ˆ q /Wq = Nq /Wq2 . Now to graduation. The occurrence-exposure rates will usually have a ragged appearance. Assuming that the real underlying mortality intensity is a smooth function, we will therefore

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL183 fit a suitable function to the occurrence-exposure rates. Suppose we assume that the true mortality rate is a Gompertz Makeham function, µ(t) = α + βeγt . Then, take some representative age τq (typically τq = q − 0.5) in each age interval and fit the parameters α, β, γ by minimizing a weighted sum of squared errors X Q= aq (ˆ µq − α − βeγτq )2 . q

This is a matter of non-linear regression. The optimal weights aq are the inverse of the variances, but since these are unknown, we plug in the estimators and use a q = 1/ˆ σq2 . The minimizing values α∗ , β ∗ , and γ ∗ are obtained by differentiating Q with respect to each of the three parameters and setting the derivatives equal to 0. The derivatives are: ∂ Q ∂α

aq 2 (ˆ µq − α − βeγτq ) (−1) ,

∂ Q ∂β

aq 2 (ˆ µq − α − βeγτq ) (−eγτq ) ,

∂ Q ∂γ

aq 2 (ˆ µq − α − βeγτq ) (−βeγτq τq ) .

Thus α∗ , β ∗ , and γ ∗ are the solution to the equations X ∗ aq µ ˆ q − α ∗ − β ∗ e γ τq

∗ ∗ aq µ ˆ q − α ∗ − β ∗ e γ τq e γ τq

∗ ∗ aq µ ˆ q − α ∗ − β ∗ e γ τq e γ τq τ q

X q

This is in general a set of non-linear equations that does not allow of an explicit solution. Actually these equations are just as involved as the maximum likelihood equations in Paragraph 9.2C above, which is disappointing since the two-stage procedure considered here was supposed to be simpler. (Occurrence-exposure rates are easy to find and they are asymptotically independent, which makes it easy to find their asymptotic variances. The graduation is, however, messy.) Suppose now that γ is taken to be known. Then only the two first equations above are relevant and they reduce to X X X aq α ∗ + aq eγτq β ∗ = aq µ ˆq , q

X q

aq e

γ ∗ τq

∗

α −

X q

aq e

2γ ∗ τq

∗

aq µ ˆ q eγ

∗

τq

This is a linear system of equations with an explicit solution, which the industrious reader certainly will find.

11.6

Impact of the censoring scheme

A. The precision of the estimation. The precision of the MLE depends on the amount of information provided by the censoring scheme of the study. Asymptotically it is the variance matrix Σ(θ) that determines everything, and in Section 11.2 it was pointed out that the size of this matrix depends on the censoring scheme only through the functions Pn (m) (τ ), g = 0, . . . , J, the expected numbers of individuals staying in each state g at m=1 pg time τ . (It depends also on the parametric structure of the intensities, of course.) We shall look at two censoring schemes frequently encountered in practice.

CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL184 B. Longitudinal observation (cohort studies).

The term cohort stems from Latin and originally signified a unit division in an ancient Roman legion. In demography it means a class of individuals born in a particular year or more general period of time (a ”generation”), and a cohort study is one where a cohort is observed over a certain period, possibly until it is extinct. This was the situation in Paragraph 11.2.C. Thus, let the n Markov processes in the general set-up be stochastic replicates, all commencing in state 0 at time 0 and thereafter observed continuously throughout the time interval [0, t¯]. In this case n X (m) (1) pg (τ ) = n pg (τ ) , g = 0, . . . , J, m=1

and



1 X Σ(θ) =  n g6=h

t¯ 0

−1 1 ∂ ∂ (1) µ(τ, θ) 0 µ(τ, θ)pg (τ )dτ  . µ(τ, θ) ∂θ ∂θ

This matrix tends to 0 as n increases if the inverse matrix indicated exists.

C. Cross-sectional observation.

In a cross-sectional study a population is observed over a certain period of time. As an example, suppose the G-M mortality study in Paragraph 11.2.C is conducted cross-sectionally throughout a calendar period of duration t¯, and that it comprises n individuals at ages t(m) , m = 1, . . . , n, at the beginning of the study. In this case the factor depending on the design in the information matrix is ! (m) n n X X (eγτ − 1) eγt 1 [ t(m) ,t(m) +t¯ ] (τ ) exp −α(τ − t(m) ) − β p(m) (τ ) = . γ m=1 m=1

Chapter 12

Heterogeneity models 12.1

The notion of heterogeneity – a two-stage model

The life length T of an individual depends on a number of factors like biological inheritance (some are strong and healthy, others are weak and sickly), occupation (mining and forestry have higher accident rates than office work), leisure activities (mountain climbing is more wholesome but also more dangerous than philately), nutrition (see the weekly magazines for current wisdom), smoking and drinking habits. The list might be extended endlessly. Let all such individual characteristics be represented by a parameter θ, which may be quite complex, possibly of large dimension comprising numbers and strings of letters. The dependence of T on these characteristics is accounted for by letting the probability law of T depend on θ. Thus, write Fθ (t) for the probability that a person with characteristics θ dies within age t and F¯θ (t) for the corresponding survival probability. Assuming that Fθ possesses a density R fθ , the force of mortality is µθ (t) = fθ (t)/F¯θ (t), and F¯θ (t) = exp − 0t µθ (s) ds , F¯θ (t | x) = R exp − xx+t µθ (s) ds , and so on.

Apparently the aggregate mortality law treated in Section *** conflicts with the myopic viewpoint taken here, but this is really not so: the aggregate law describes the mortality pattern in the population as a whole and represents the prospects of longevity for a person when nothing is known as to his/her personal characteristics. To make this precise we let the individual characteristics of a randomly chosen newly born be a random element Θ with a distribution G, and the conditional distribution of the life length for fixed Θ = θ is Fθ . If the value of Θ is observed to be θ, then Fθ is the relevant basis for making predictions about T . If Θ is not observed, then predictions about T can only be based on the unconditional distribution function Z F (t) = E[FΘ (t)] = Fθ (t) dG(θ) (12.1) or any of the equivalent functions in (3.2) – (3.4), which now assume the forms Z F¯ (t) = F¯θ (t) dG(θ), Z f (t) = fθ (t) dG(θ), R R fθ (t) dG(θ) µθ (t)F¯θ (t) dG(θ) R µ(t) = f (t)/F¯ (t) = R ¯ = Fθ (t) dG(θ) F¯θ (t) dG(θ) =

E[µΘ (t) | T > t] .

185

(12.2) (12.3) (12.4) (12.5)

186

CHAPTER 12. HETEROGENEITY MODELS Formula ( 12.5 ) calls for a comment. The joint distribution of T and Θ is given by Z Z Z P [T ∈ A, Θ ∈ B] = fθ (t) dt dG(θ) = Fθ [A] dG(θ). B A

The marginal distribution of T is given by P[T ∈ A] = F [A] =

Fθ [A] dG(θ) ,

where the integral sign without indication of the area signifies integration over the entire range of the variable. The conditional distribution of Θ, given T ∈ A, is R Fθ [A] dG(θ) (12.6) G[B | T ∈ A] = RB Fθ [A] dG(θ)

or, in terms of the generalized density,

dG(θ | T ∈ A) = R

Fθ [A] dG(θ) . Fθ 0 [A] dG(θ 0 )

(12.7)

In particular, the conditional distribution of Θ, given T > t, is given by F¯θ (t) dG(θ) , dG(θ | T > t) = R ¯ Fθ 0 (t) dG(θ 0 )

(12.8)

and the last expression in (12.4) is recognized as the expected value of µ Θ (t) with respect to this distribution. Inserting A = [t, t + dt] and Fθ [A] = fθ dt in (12.7) gives (somewhat informally) fθ dG(θ) . (12.9) dG(θ | T = t) = R fθ 0 (t) dG(θ 0 )

The probabilities in (12.7) have a straightforward interpretation as proportions in a cohort of individuals born at the same time. They cannot in general be interpreted as proportions in a given population, and the reason for this is that no assumptions have been made as to the birth rates that would govern the development of the age composition of the population over time. It is certainly true that (12.8) is the distribution of Θ amongst those who, at a given moment, are exactly t years old in the population. It is not the distribution of Θ amongst those who, at a given moment, are t years or older (as the conditioning on T > t might suggest). Likewise, (12.9) is the distribution of Θ amongst those who are known to have died at age t in the past. The conditional probability P[T ∈ A0 | T ∈ A] may be expressed as F [A0 | A]

= = = =

F [A0 ∩ A] F [A] R Fθ [A0 ∩ S] dG(θ) F [A] Z Fθ [A0 ∈ A] Fθ [A] dG(θ) Fθ [A] F [A] Z Fθ [A0 | A] dG(θ | T ∈ A).

(12.10)

Formula ( 12.10 ) says that, when T ∈ A is given, the probability of T ∈ A0 is to be formed in the usual way, by taking the average of the probability of T ∈ A0 for fixed Θ over the distribution of Θ, all distributions being conditional on T ∈ A. In particular, Z F¯ (t | x) = F¯θ (t | x) dG(θ | T > x), (12.11) which resembles (12.2), only that the distributions of T and Θ are updated in regard of the information that the person has survived to age x. Those who adhere to a deterministic world picture would presumably fancy the idea that T could be exactly determined if only the individual and its surroundings could be sufficiently

CHAPTER 12. HETEROGENEITY MODELS

187

accurately described – down to the atoms. Then T would be just a function T (Θ) of Θ, and Fθ would reduce to a one-point distribution in T (θ) which need no longer be explicated in the model since the mortality law would simply be GT −1 . The model formulation (12.1) with Fθ a non-degenerate distribution, expresses the point of view that something remains unexplained beyond Θ. Two interpretations are possible. Either that there exists such a thing as pure randomness in the world, or that not all explanatory factors are included in Θ. Now, leaving such speculations to the philosophers, let us pursue the chosen approach.

12.2

The proportional hazard model

The perhaps simplest means of describing mortality variations is the so-called proportional hazard model, which specifies that Θ is a positive random variable, and µθ (t) = θµ◦ (t),

(12.12)

µ◦ (t)

where is some ”baseline” force of mortality. According to this assumption, the mortality pattern is basically the same for all people, only that some die ”faster” than others at all ages. It does not allow for the possibility that e.g. some people have mortality above the average in the youth and below the average in the old age. Introduce the accumulated baseline intensity at age t, Z x W (x) = µ◦ (s) ds. (12.13) 0

Under the assumption (12.11), the conditional survival function by fixed Θ = θ is

and the conditional density is

F¯θ (t) = e−θW (t) ,

(12.14)

fθ (t) = θµ◦ (t) e−θW (t) .

(12.15)

The functions in (12.2) – (12.5) become Z F¯ (t) = e−θW (t) dG(θ), Z f (t) = µ◦ (t) θ e−θW (t) dG(θ), R θ e−θW (t) dG(θ) = µ◦ (t) E(Θ | T > t). µ(t) = µ◦ (t) R −θW (t) e dG(θ)

(12.16) (12.17) (12.18)

Although the unconditional distribution F is the only thing that matters when Θ is unobservable, the present two-stage model is of interest also in such circ*mstances. In the first place it is of theoretical interest due to its explanatory import, and in the second place it is of practical value since it produces candidates for suitable aggregate laws. Assume now that G is Gγ,δ , the gamma distribution with shape parameter γ and scale parameter δ −1 , whose density is δ γ γ−1 −θδ gγ,δ (θ) = θ e , θ > 0, (12.19) Γ(γ) and γ, δ > 0. Here Γ is the gamma function Z ∞ Γ(γ) = tγ−1 e−t dt , 0

which satisfies Γ(γ) = (γ − 1) Γ(γ − 1), γ > 1,

(easy to prove by integration by parts) and Γ(1) = 1. It is immediately seen that δγ Γ(r + γ) E Θr e−Θs = , r > −γ, s > −δ, Γ(γ) (s + δ)r+γ =

δ γ (r + γ − 1)(r) , r = 0, 1, · · · , s > −δ. (s + δ)r+γ

(12.20)

188

CHAPTER 12. HETEROGENEITY MODELS In particular, all moments of Θ exist, and EΘ

γ/δ,

(12.21)

V Θ = γ/δ 2 .

(12.22) θ0

The role of the scale parameter becomes clear by substituting = θδ in the integral Z θ Z θδ γ 0 δ 1 Gγ,δ (θ) = θ γ−1 e−θδ dθ = θ 0−γ−1 e−θ dθ 0 = Gγ,1 (θδ), (12.23) Γ(γ) 0 Γ(γ) 0 which is an increasing function of δ. By use of ( 12.20 ), it is readily seen that the present case (12.16)—(12.18) specialize to γ δ , (12.24) F¯ (t) = W (t) + δ γ δ γ f (t) = µ◦ (t) , (12.25) (W (t) + δ)γ+1 γ µ(t) = µ◦ (t) . (12.26) W (t) + δ Observe that P [W (t) > w]

F¯ (W −1 (w)) =

δ w+δ

that is, W (t) + δ is Pareto-distributed with parameters (δ, γ). Observe also that µ(t) is µ◦ (t) times a factor that decreases with t. Notice that µ◦ is not a force of mortality in the unconditional law. The survival function for an x year old is obtained from ( 12.24 ) as γ W (x) + δ , F¯ (t | x) = F¯ (x + t)/F¯ (x) = W (x + t) + δ (12.27)

which can be written as F¯ (t | x) = with W (t | x)

δ(x)

δ(x) W (t | x) + δ(x)

W (t + x) − W (x) = W (x) + δ.

(12.28)

µ◦ (x + τ ) dτ,

Thus, the survival distribution of an x-year old is of the same form as that of a newly born, see ( 12.24 ), but with updated value of the scale parameter. Despite what has been said about Θ and G as elements of a mere explanatory background of the ”surface” entities T and F , it may be of interest to study the conditional distribution ( 12.6 ) for some given aspect T ∈ A of the life length. As an example, take the general proportional hazard model and focus attention at (see ( 12.6 ) R ∞ −θ 0 W (t) e dG(θ 0 ) ¯ | T > t) = P [Θ > θ | T > t] = Rθ G(θ . ∞ −θ 0 W (t) dG(θ 0 ) 0 e To study the dependence of this function on t, differentiate:

∂ ¯ G(θ | T > t) = ∂t R ∞ −θ 0 W (t) R R R 0 0 0 e dG(θ 0 ) θ∞ e−θ W (t) (−θ 0 µ◦ (t)) dG(θ 0 ) − 0∞ e−θ W (t) (−θ 0 µ◦ (t)) dG(θ 0 ) θ∞ e−θ W (t) dG(θ 0 ) 0 (12.29) R ∞ 2 e−θ 0 W (t) dG(θ 0 ) 0

CHAPTER 12. HETEROGENEITY MODELS

189

By introduction of the probability measure H given by

( 12.29 ) can be cast as

e−θW (x) dG(θ) , dH(θ) = R ∞ −θ 0 W (t) e dG(θ 0 ) 0

µ◦ (t) {−EH (Θ 1[Θ > θ]) + EH Θ EH 1[Θ > θ]} = −µ◦ (t)CovH (Θ, 1[Θ > θ]),

(12.30)

where subscript H signifies that the operators are formed by the distribution H. Now, Θ and 1[Θ > θ] are both increasing functions of Θ, hence associated and positively correlated. It follows that the expression in ( 12.30 ) is negative, hence that the probability G(θ | T > t) is a decreasing function of t and so RT D(Θ | T ). Thus, old people are likely to have a small value of θ. Specialize now to the gamma case ( 12.19 ) again. By ( 12.8), the conditional density of Θ, given X > x, is gγ,δ (θ | T > t) = cθ γ−1 e−θ(W (t)+δ) , (12.31) where c does not depend on θ. The constant c need not be calculated since, by inspection of (12.19) and ( 12.31 ) it follows that gγ,δ (θ | T > t) = gγ,W (t)+δ (θ) .

(12.32)

Formula ( 12.28 ) is also obtained from ( 12.11 ) upon inserting Fθ (t | x) = exp[−θ{W (x + t) − W (x)}] and, from ( 12.32 ), dG(θ | X > x) = gγ,W (x)+δ (θ) dθ and using ( 12.20 ). From a computational point of view this is a detour, but it uncovers the role of the hidden Θ in the play.

Chapter 13

Group life insurance 13.1

Basic characteristics of group insurance

. A group insurance treaty is an arrangement whereby a group of persons is covered by a single contract with an insurer. In its broadest context, group insurance would include a variety of coverages, including life insurance, accident insurance, health insurance, annuities, civil property insurance, and others. The major types of groups eligible for group insurance are individual employer groups (the employees of a firm are covered by a contract between an insurer and the employer, typically as a part of a labour–management negotiated employee welfare and security plan), multiple employer groups (the same as individual employer groups, except that the individual firm is extended to two or more firms/employers, e.g. a trade association or an entire industry), labour union groups (the members of a labour union are covered by a contract issued directly to the union), and creditor – debtor groups (life and/or health insurance is provided for debtors through a contract issued to the creditor, e.g. a commercial bank; if the borrower dies or is disabled, benefits are paid to the lender to cancel the insured part of the debt). A great variety of groups beyond the foregoing classifications are covered by group insurance. Among such miscellaneous groups are associations of public and private employees, professional organizations, fraternal societies, and many others. When group insurance is contrasted with individual insurance, a number of characteristic features are evident, In the first place, the coverage is offered to all members of the group, usually without medical examination or other evidence of individual insurability. Thus, the criteriae by which individuals are recruited to a group are considered to provide a sufficient guarantee against adverse selection of high-risk individuals, so-called antiselection against the insurer. For instance, it is to be expected that the staff of an engineering workshop or publishing house or the membership of an association of lawyers or teachers has only a small infusion of impaired lives. To further preclude the possibility of antiselection, there are usually some requirements pertaining to the minimum number of persons needed to constitute a group and to the minimum proportion covered in the entire group. Another feature characteristic of group insurance is that the persons insured under a contract are not parties to the contract, since legally the contract is between the insurer and the policyholder (usually an employer or an organization). A third characteristic of group insurance is that it is essentially low-cost, mass protection. Marketing and administration costs are far below the level typical of individual insurance. As a fourth characteristic, it should be pointed out that group insurance contracts are of a continuing nature, in that the contract and the plan may last long beyond the lifetime, or membership in the plan, of any one individual. New persons are added to the group from time to time, and others terminate their coverage. The contract is renewed regularly, typically annually. Therefore, both contract terms and premium rate can be currently adjusted in

190

15.2. HAZARD MODEL FOR POLICY AND RECORDS

191

accordance with the observed development of costs, risk conditions and other circ*mstances influencing the economic result. This feature sets a difference of great principal and practical importance between group contracts and individual contracts in life insurance. An individual life insurance policy usually specifies premiums and benefits that remain unalterned throughout the contract period, which may extend over several decades. Therefore, a substantial safety loading is usually built into the individual premium to meet possible unfavourable future developments of mortality and expenses. In group life insurance there is no need for this kind of safety loading since the premium rate can be currently adjusted in accordance with the experiences. The last remark points directly to the final feature of group insurance to be mentioned here. To save expenses, usually only very summary characteristics of the groups are observed and used as a basis for the rating of premiums at the outset. Those risk characteristics that are not observed may vary considerably between the groups and give rise to substantial risk differentials between them, despite that they “appear to be similar”. As an example, in group life insurance one may choose to observe only the number of persons insured under the plan of a group and leave other characteristics such as occupation and age composition unobserved. If the groups differ considerably with respect to these unobserved risk characteristics, they will have different “true underlying risk premiums”. These differentials will be reflected by the risk experiences of the individual groups as time passes and claims statistics accrues. Thus, the individual claims record of a group provides some information on its “risk profile”, which could be taken into account in the current adjustment of the premium. When the premium is regulated this way for each group in regard of its claims experience, one speaks of experience rating. There is yet another point of difference between individual and group insurance which ought to be mentioned because it explains why experience rating is widely used in group life insurance. If you should ask holders of individual life insurance policies if they find the premiums reasonable, the answers would typically be “I guess so” or “I don’t know”. They don’t know and don’t haggle over the price, simply because they have no access to statistics from which they could judge the fairness of the premiums. In group life insurance this is different. Each master contract is managed by a policyholder who can compare premium payments with received benefits in the long run. Those policy holders who find that premiums exceed by far the benefits, will sooner or later call for a discount (the others will remain silent). Therefore, a competitive market will tend to enforce experience rating of groups life contracts.

13.2

A proportional hazard model for complete individual policy and claim records

Consider a group life portfolio for which statistical records have been maintained during the period (τ 0 , τ 00 ), where τ 00 is the present moment. The portfolio comprises I master contracts, labeled by i = 1, . . . , I. Let (τi0 , τi00 ) be the period during which contract i has been in force (τi00 < τ 00 if the contract has been terminated in its entirety). Let Ji be the number of persons currently or formerly insured under the plan of contract i. They are labeled by (i, j), j = 1, . . . , Ji . For each individual (i, j) introduce the following quantities, which are observable by time τ 00 : 0 , τij xij , Tij , Kij ,

Mij ,

the time of entry into the group, the age at enty, the time exposed to risk as insured before time τi00 the number of times the coverage has been terminated on an individual basis before time τi00 , the number of deaths as insured before time τi00 .

The pairs (Kij , Mij ) can only assume the values (0,0), (1,0), (0,1) (implying that participation in the group will not be resumed once it has been terminated). The hidden mortality

15.2. HAZARD MODEL FOR POLICY AND RECORDS

192

characteristics of group i are represented by a latent quantity Θi . The Tij , Kij , Mij , and Θi are viewed as random variables, and the following assumptions are made. (i) Variables belonging to different groups are stochastically independent and Θ 1 , . . . , ΘI are iid (independent and identically distributed). (ii) Variables belonging to different persons within one and the same group i are conditionally independent, given Θi . (iii) All persons in one and the same group follow the same pattern of mortality and termination. More specifically, it is assumed that the Θi are positive and that, conditional on Θi = θi , a person who entered group i at age x and is still a member of the group at age x + t (x, t > 0), then has a force of termination κi (x, t)

(13.1)

θi µ(x, t).

(13.2)

and a force of mortality of the form

Assumption (i) corresponds to the idea that the groups are independent random selections from a population of groups that are comparable, but not entirely similar. It is this assumption, in conjunction with (15.2), that establishes a relationship between the groups and forms the rationale of combining portfoliowide mortality experience with the mortality experience of a given group in an assessment of the mortality in that group. The “proportional hazard” assumption (15.2) represents, perhaps, the simplest possible way of modelling mortality variations between groups. It states that the risk characteristics specific of a group act on the force of mortality only through a multiplicative factor, implying that the mortality pattern is basically the same for all groups. Such an assumption is not apt for describing more complex mortality differences, e.g. that a group may have a mortality below the average at early ages and above the average towards the end of life. For example, it is thinkable that such hazardous occupations as blast furnace operation and mining attract only physically fit and healthy applicants and that those who are employed quickly get worn out by the severe working conditions. The statistical data presently available from group i are the individual entrance times, 0 , entrance ages, x , and histories as insurees, τij ij Oi = {(Kij , Mij , Tij ); j = 1, . . . , Ji }. The conditional distribution of (Kij , Mij , Tij ), when Θi = θi , is given by P {Kij = 0, Mij = 1, tij < Tij tij + dtij | Θi = θi } Z t ij {κi (xij , t) + θi µ(xij , t)} dt , = θi µ(xij , tij ) dtij exp − 0

0 , 0 < tij < τi00 − τij

P {Kij = 1, Mij = 0, tij < Tij tij + dtij | Θi = θi } Z t ij = κi (xij , tij ) dtij exp − {κi (xij , t) + θi µ(xij , t)} dt , 0

0 , 0 < tij < τi00 − τij

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193

P {Kij = Mij = 0, Tij = tij | Θi = θi } Z t ij {κi (xij , t) + θi µ(xij , t)} dt , = exp − 0

0 . tij = τi00 − τij

From these expressions we gather the following formula for the conditional likelihood of O i : Ji Y

j=1

(κi (xij , Tij )Kij {θi µ(xij , Tij )}Mij exp[−

Tij 0

{κi (xij , t) + θi µ(xij , t)} dt]),

(13.3)

0 or (K , M ) = (0, 0) and T 00 (Kij , Mij ) ∈ {(0, 1), (1, 0)} and 0 < Tij < τi00 − τij ij ij ij = τi − 0 τij , j = 1, . . . , Ji . For each person (i, j) introduce the cumulative basic force of mortality

Wij =

Tij

µ(xij , t) dt.

(13.4)

It is seen from (15.3) that a set of sufficient statistics for group i are Mi Wi

= =

P Ji

j=1

PJ i

j=1

Mij ,

the total number of deaths,

(13.5)

Wij ,

the sum of cumulative basic intensities,

(13.6)

and that the conditional likelihood, considered as a function of θi , is proportional to Mi

θi

e−θi Wi .

(13.7)

The expression in (15.7) is gamma shaped. Motivated by the convenience of the analysis in the previous chapter, it is assumed that the common distribution of the latent Θ i is the gamma distribution with density gγ,δ (θi ) =

δγ θ γ−1 e−θi δ , θi > 0. Γ(γ) i

(13.8)

The conditional density of Θi , given O, is proportional to the product of the expressions in (15.7) and (15.8), hence gγ,δ (θi | Oi ) = gMi +γ,Wi +δ (θi ).

(13.9)

By use of (14.25)–(14.27), it follows that E(Θi | Oi )

V ar(Θi | Oi )

E(e−wΘi | Oi ) =

Wi + δ w + Wi + δ

Mi + γ , Wi + δ Mi + γ , (Wi + δ)2 Mi +γ

, w > −(Wi + δ).

(13.10) (13.11)

(13.12)

˜ i (say) of Θi with respect to squared The conditional mean in (15.10) is the Bayes estimator Θ loss. It can be cast as

15.3. EXPERIENCE RATED NET PREMIUMS

194

˜ i = ζi Θ ˆ i + (1 − ζi )γ/δ, Θ

(13.13)

ˆ i = Mi /Wi Θ

(13.14)

where

is the maximum likelihood estimator of θi in the conditional model, given Θi = θi , and ζi = Wi /(Wi + δ).

(13.15)

ˆ i and the unconditional The expression in (15.13) is a weighted mean of the sample estimator Θ mean, EΘi = γ/δ. The weight ζi attached to the experience of the group, is an increasing function of the exposure times Tij , confer (15.4).

13.3

Experience rated net premiums

The set of master contracts in force at the present moment is I = {i : τi00 = τ 00 }, and for each group i ∈ I the set of persons presently covered under the plan of the group is 0 Ji = {j : τij + Tij = τ 00 }. 00 and S 00 denote, respectively, the number of For each person (i, j) presently insured let Mij ij deaths and the sum payable by death in the next year, (τ 00 , τ 00 + 1). To prevent technicalities 00 will remain from obscuring the main points, disregard interest and assume that all the S ij constant throughout the year. For a group i ∈ I the net annual premium based on the available information O i is X 00 00 PiA = Sij E(Mij | Oi ). (13.16) j∈Ji

The expected values appearing in (15.16) are 00 E(Mij | Oi )

= =

00 E{E(Mij | Θi , Oi ) | O i } Z 1 µ(xij , Tij + t) dt | Oi ], E[1 − exp −Θi 0

which can be calculated by formula (15.12). Defining Z 1 00 wij = µ(xij , Tij + t) dt

(13.17)

and Qi,w = 1 −

Wi + δ w + Wi + δ

Mi +γ

(13.18)

one finds 00 E(Mij | Oi ) = Qi,w00 . ij

Substituting (15.19) into (15.16) yields

(13.19)

15.4. THE FLUCTUATION RESERVE X

PiA =

195

00 Sij Qi,w00 ,

(13.20)

j∈Ji

with Qi,w00 defined by (15.17) and (15.18). ij

As an alternative to the premium (15.20), which is exact on an annual basis, one could use the “instantaneous net premium” per time unit at time τ 00 ,    X 00 00 I (13.21) Sij Mij (∆τ ) | Oi /∆τ, Pi = lim E   ∆τ ↓0 j∈Ji

where Now,

00 (∆τ ) Mij

is the number of deaths of person (i, j) in the time interval (τ 00 , τ 00 + ∆τ ).

00 E{Mij (∆τ ) | Oi }

00 E[E{Mij (∆τ )) | Θi , Oi } | Oi ]

= =

E{Θi µ(xij , Tij )∆τ + o(∆t) | Mi , Wi } ˜ i + o(∆t), µ(xij , Tij )∆tΘ

(13.22)

the last passage being a consequence of (15.10) and (15.13). Combine (15.21) and (15.22), to obtain X 00 ˜ i. Sij µ(xij , Tij )Θ (13.23) PiI = j∈Ji

To see that PiI is an approximation to PiA , apply the first order Taylor expansion (1 + x)−α ≈ 00 in (15.17) by 1 − αx to the second term on the right of (15.18) and then approximate w ij µ(xij , Tij ), which gives Qi,w00

00 1 − {1 + wij /(Wi + δ)}−(Mi +γ)

≈

00 1 − {1 − (Mi + γ)wij /(Wi + δ)}

00 ˜ wij Θi

≈

˜ i. µ(xij , Tij )Θ

PiA

(13.24) 00 wij

PiI .

Using (15.25) in (15.20), yields ≈ The approximation is good if the are 0. By Lemma 1 below (in Paragraph D), the predictable process f (t, t, U (t), j) has no jumps coinciding with jumps of the counting processes, and so the sum in (14.16) is 0. By Lemma 3 below, the term in (14.17) is predictable (or, more presicely, the increment of a predictable process). It follows that Ij (t)dWj (t) is predictable. Finally, by Lemma ˜◦j (t) is predictable since dA ˜◦j (t) is. Comparison of the predictable 2 below, Ij (t)dA (bounded variation) parts of (3.4) and (3.11) leads to Theorems 2 and 3.

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With the present definition of the statewise reserves it makes no difference if we replace Wk (t−) with Wk (t) in (3.12) (and Vk (t) with Vk (t−) in (3.14)). This observation settles the apparent disagreement with the result (3.5) in Møller (1993). He considers the case where the contractual payment functions, the transition intensities and, hence, the reserve depend only on t and S(t) or, equivalently, on t and t − S(t) (he uses U (t) to denote the latter).

C. Construction of the reserve Usually Thiele’s differential equations are mobilized when the reserve cannot be put up by a direct prospective argument, typically when the payments depend on the current reserve. However, as long as the payments and the transition intensities do not depend on the past, the statewise reserves remain functions of t only and can, therefore, be determined by a set of simple differential equations. Real difficulties arise only when payments and transition intensities are allowed to depend on the past in a more or less complex manner; then the reserves will be functions of several variables and it is rather obvious that they cannot be determined by just a set of first order ordinary differential equations. Let us look briefly at the case mentioned at the balance of the previous paragraph, where payments, intensities, and hence the reserve depend on S(t) (and t). We seek the two-dimensional statewise discounted reserve-functions Wj (s, t) (say), 0 ≤ s ≤ t ≤ n, j ∈ J , where n is the time of expiry of the contract. Conditioning on whether or not there is a transition out of state j within time n and, in case there is, the time and the direction of the first such transition, we obtain the integral equation Wj (s, t)

e−

Rτ

t µj· (s,u)du

µjk (s, τ )dτ

k;k6=j

˜◦j (s, t) + a ˜ ◦jk (s, τ ) + Wk (τ, τ )) ·(A˜◦j (s, τ ) − A +e

R − tn µj· (s,u)du

(14.18)

˜◦j (s, n) − A ˜◦j (s, t)). (A

Again, we see that Wk (τ, τ ) can be replaced with Wk (τ −, τ −) since the integration with respect to dτ annihilates the at most countable number of differences between them. To determine the reserve functions, solve first the Wj (t, t) from the integral equations with s = t, and then solve the Wj (s, t) from the general equations. To obtain a differential form of the integral equation (14.18), add A˜◦j (s, t) and Rt multiply by e− 0 µj· (s,u)du on bothR sides, then differentiate with respect to t, and finally t divide by the common factor e− 0 µj· (s,u)du , which gives the following equivalent of (3.15): ˜◦j (s, t)) = − dt (Wj (s, t) + A

k;k6=j

µjk (s, t)dt(˜ a◦jk (s, t) + Wk (t, t) − Wj (s, t)). (14.19)

D. Some auxiliary results Lemma 1. If H is a predictable process, then Z t Z t H(τ )dNjk (τ ). H(τ −)dNjk (τ ) = 0

(14.20)

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211

Consequently, H has no jumps in common with the counting processes. Proof: Let Mjk be the martingale defined by dMjk (t) = dNjk (t) − λjk (t)dt. The difference between the two integrals in (14.20) is Z

(H(τ ) − H(τ −))dNjk (τ )

(H(τ ) − H(τ −))λjk (τ )dτ Z t + (H(τ ) − H(τ −))dMjk (τ ). 0

The first term on the right is zero because the integrand differs from 0 at most at a countable set of points. Likewise, the second term is also 0 because its variance is Z t E (H(τ ) − H(τ −))2 λjk (τ )dτ. 0

The last assertion is an obvious implication of the argument above.

Lemma 2. If A is a predictable semimartingale, then so is also the process H defined by dH(t) = Ij (t)dA(t). Proof: By the rule of integration by parts, we have the identities dH(t) = dIj (t)A(t−) + Ij (t)dA(t) = dIj (t)A(t) + Ij (t−)dA(t). Forming the difference between the last two expressions and using (3.1), gives Ij (t)dA(t) = Ij (t−)dA(t) +

k;k6=j

(A(t) − A(t−))(dNkj (t) − dNjk (t)).

The first term on the right is predictable and the second term is 0 by Lemma 1.

Lemma 3. If G is a semimartingale, then the process H defined by dH(t) = I j (t−)Ij (t)dG(t) is predictable. Proof: Inserting the Doob-Meyer decomposition X dG(t) = dG0 (t) + Gik (t)(dNik (t) − λik (t)dt), i6=k

with G0 (t) and the Gik (t) predictable, and noting that Ij (t−)Ij (t)dNik (t) = 0, we get X dH(t) = Ij (t−)Ij (t)dG0 (t) − Gik (t)λik (t)dt. i6=k

Here the first term is predictable by Lemma 2, and the second term is certainly predictable as it is continuous.

Chapter 15

Financial mathematics in insurance 15.1

Finance in insurance

Finance was always an essential part of insurance. Trivially, one might say, because any business has to attend to its money affairs. However, for at least two reasons, insurance is not just any business. In the first place insurance products are not physical goods or services, but financial contracts with obligations related to uncertain future events. Therefore, pricing is not just a piece of accountancy involving the four basic arithmetical operations, but requires assessment and management of risk by sophisticated mathematical models and methods. In the second place, insurance contracts are more or less long term (in life insurance for up to several decades), and they are typically paid in advance (hence the term ’premium’ for ’price’ derived from French ’prime’ for ’first’). Therefore, the insurance industry is a major accumulator of capital, and insurance companies especially pension funds are major institutional investors in today’s society. It follows that the financial operations (investment strategy) of an insurance company may be as decisive of its revenues as its insurance operations (design of products, risk management, premium rating, procedures of claims assessment, and the pure randomness in the claims process). Accordingly one speaks of assets risk or financial risk and liability risk or insurance risk. We anticipate here that financial risk may well be the more severe: Insurance risk created by random deviations of individual claims from their expected is diversifiable in the sense that, by the law of large numbers, it can be eliminated in a sufficiently large insurance portfolio. This notion of diversification does not apply to catastrophe coverage and it does not account for the risk associated with long term contracts insurance ic insurance risk Financial risk created random economic events are by booms, recessions and, rare but disastrous, crashes in the market as a whole is not diversifiable; it is the uncertain events are part of our the world history nonreplicable averaged out in any meaningful sense. Financial risk created by the day-today rises and falls of individual stocks, is not diversifiable

212

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

213

on large random ups and downs is held to be indiversifiable since the entire portfolio is affected by the development of the economy. On this background one may ask why insurance mathematics traditionally centers on measurement and control of the insurance risk. The answer may partly be found in institutional circ*mstances: The insurance industry used to be heavily regulated, solvency being the primary concern of the regulatory authority. Possible adverse developments of economic factors (e.g. inflation, weak returns on investment, low interest rates, etc.) would be safeguarded against by placing premiums on the safe side. The comfortable surpluses, which would typically accumulate under this regime, were redistributed as bonuses (dividends) to the policyholders only in arrears, after interest and other financial parameters had been observed. Furthermore, the insurance industry used to be separated from other forms of business and protected from competition within itself, and severe restrictions were placed on its investment operations. In these circ*mstances financial matters appeared to be something the traditional actuary did not need to worry about. Another reason why insurance mathematics used to be void of financial considerations was, of course, the absence of a well developed theory for description and control of financial risk. All this has changed. National and institutional borders have been downsized or eliminated and regulations have been liberalized: Mergers between insurance companies and banks are now commonplace, new insurance products are being created and put on the market virtually every day, by insurance companies and other financial institutions as well, and without prior licencing by the supervisory authority. The insurance companies of today find themselves placed on a fiercely competitive market. Many new products are directly linked to economic indices, like unit-linked life insurance and catastrophe derivatives. By so-called securitization also insurance risk can be put on the market and thus open new possibilities of inviting investors from outside to participate in risk that previously had to be shared solely between the participants in the insurance insurance schemes. These developments in practical insurance coincide with the advent of modern financial mathematics, which has equipped the actuaries with a well developed theory within which financial risk and insurance risk can be analyzed, quantified and controlled. A new order of the day is thus set for the actuarial profession. The purpose of this chapter is to give a glimpse into some basic ideas and results in modern financial mathematics and to indicate by examples how they may be applied to actuarial problems involving management of financial risk.

15.2

Prerequisites

A. Probability and expectation. Taking basic measure theoretic probability as a prerequisite, we represent the relevant part of the world and its uncertainties by a probability space (Ω, F, P). Here Ω is the set of possible outcomes ω, F is a sigmaalgebra of subsets of Ω representing the events to which we want to assign probabilities, and P : F 7→ [0, 1] is a probability measure. A set A ∈ F such that P[A] = 0 is called a nullset, and a property that takes place in all of Ω, except possibly on a nullset, is said to hold almost surely (a.s.). If more than one probability measure are in play, we write “nullset (P)” and “a.s. ˜ and P are said to (P)” whenever emphasis is needed. Two probability measures P ˜ be equivalent, written P ∼ P, if they are defined on the same F and have the same nullsets.

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Let G be some sub-sigmaalgebra of F. We denote the restriction of P to G by PG ; PG [A] = P[A], A ∈ G. Note that also (Ω, G, PG ) is a probability space. A G-measurable random variable (r.v.) is a function X : Ω 7→ R such that X −1 (B) ∈ G for all B ∈ R, the Borel sets in R. We write X ∈ G in short. value of a r.v. X is the probability-weighted average E[X] = R The expected R X dP = Ω X(ω) dP(ω), provided this integral is well defined. The conditional expected value of X, given G, is the r.v. E[X|G] ∈ G satisfying E{E[X|G] Y } = E[XY ]

(15.1)

for each Y ∈ G such that the expected value on the right exists. It is unique up to nullsets (P). To motivate (15.1), consider the special case when G = σ{B1 , B2 , . . .}, the sigma-algebra generated by the F-measurable sets B1 , B2 , .P . ., which form a partition of Ω. Being G-measurable, E[X|G] must be of the form k bk 1[Bk ]. Putting this together with Y = 1[Bj ] into the relationship (15.1) we arrive at R X dP X B , E[X|G] = 1[Bj ] j P[Bj ] j as it ought to be. In particular, taking X = 1[A], we find the conditional probability P[A|B] = P[A ∩ B]/P[B]. One easily verifies the rule of iterated expectations, which states that, for H ⊂ G ⊂ F, E { E[X|G]| H} = E[X|H] .

(15.2)

F. Change of measure. If L is a r.v. such that L ≥ 0 a.s. (P) and E[L] = 1, we can define a probability ˜ on F by measure P Z ˜ P[A] = L dP = E[1[A]L] . (15.3) A

˜ ∼ P. If L > 0 a.s. (P), then P ˜ is The expected value of X w.r.t. P ˜ E[X] = E[XL]

(15.4)

if this integral exists; by the definition (15.3), the relation (15.4) is true for indicators, hence for simple functions R and, by R passing to limits, it holds for measurable functions. ˜ = XL dP suggests the notation dP ˜ = L dP or Spelling out (15.4) as X dP ˜ dP = L. dP

(15.5)

˜ w.r.t. P. The function L is called the Radon-Nikodym derivative of P ˜ is formed by the rule Conditional expectation under P E[XL|G] ˜ E[X|G] = . E[L|G]

(15.6)

To see this, observe that, by definition, ˜ E[X|G] ˜ ˜ E{ Y } = E[XY ]

(15.7)

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215

for all Y ∈ G. The expression on the left of (15.7) can be reshaped as ˜ ˜ E{E[X|G] Y L} = E{E[X|G] E[L|G] Y } . The expression on the right of (15.7) is E[XY L] = E{E[XL|G] Y } . It follows that (15.7) is true for all Y ∈ G if and only if ˜ E[X|G] E[L|G] = E[XL|G] , which is the same as (15.6). For X ∈ G we have ˜ G [X] = E[X] ˜ E = E[XL] = E {X E[L|G]} = EG {X E[L|G]} ,

(15.8)

showing that ˜G dP = E[L|G] . dPG

(15.9)

B. Stochastic processes. To describe the evolution of random phenomena over some time interval [0, T ], we introduce a family F = {Ft }0≤t≤T of sub-sigmaalgebras of F, where Ft represents the information available at time t. More precisely, Ft is the set of events whose occurrence or non-occurrence can be ascertained by time t. If no information is ever sacrificed, we have Fs ⊂ Ft for s < t. We then say that F is a filtration, and (Ω, F, F, P) is called a filtered probability space. A stochastic process is a family of r.v.-s, {Xt }0≤t≤T . It is said to be adapted to the filtration F if Xt ∈ Ft for each t ∈ [0, T ], that is, at any time the current state (and also the past history) of the process is fully known if we are currently provided with the information F. An adapted process is said to be predictable if its value at any time is entirely determined by its history in the strict past, loosely speaking. For our purposes it is sufficient to think of predictable processes as being either left-continuous or deterministic. C. Martingales. An adapted process X with finite expectation is a martingale if E[Xt |Fs ] = Xs for s < t. The martingale property depends both on the filtration and on the probability measure, and when these need emphasis, we shall say that X is martingale (F, P). The definition says that, “on the average”, a martingale is always expected to remain on its current level. One easily verifies that, conditional on the present information, a martingale has uncorrelated future increments. Any integrable r.v. Y induces a martingale {Xt }t≥0 defined by Xt = E[Y |Ft ], a consequence of (15.2). Abbreviate Pt = PFt , introduce Lt =

˜t dP , dPt

216

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE and put L = LT . By (15.9) we have Lt = E[L|Ft ] ,

(15.10)

which is a martingale (F, P). D. Counting processes. As the name suggests, a counting process is a stochastic process N = {Nt }0≤t≤T that commences from zero (N0 = 0) and thereafter increases by isolated jumps of size 1 only. The natural filtration of N is FN = {FtN }0≤t≤T , where FtN = σ{Ns ; s ≤ t} is the history of N by time t. This is the smallest filtration to which N is adapted. The N strict past history of N at time t is denoted by Ft− . An FN -predictable process {Λt }0≤t≤T is called a compensator of N if the process M defined by Mt = N t − Λ t

(15.11)

is a zero mean FN -martingale. If Λ is absolutely continuous, that is, of the form Z t Λt = λs ds , 0

then the process λ is called the intensity of N . We may also define the intensity informally by λt dt = P [dNt = 1 | Ft− ] = E [dNt | Ft− ] ,

and we sometimes write the associated martingale (15.11) in differential form, dMt = dNt − λt dt .

(15.12)

A stochastic integral w.r.t. the martingale M is an FN -adapted process H of the form Z t hs dMs , (15.13) Ht = H 0 + 0

F0N -measurable

where H0 is and h is an FN -predictable process such that H is integrable. The stochastic integral is also a martingale. A fundamental representation result states that every FN martingale is a stochastic integral w.r.t. M . It follows that every integrable FtN measurable r.v. is of the form (15.13). R t (1) R t (2) (1) (1) (2) (2) If Ht = H0 + 0 hs dMs and Ht = H0 + 0 hs dMs are stochastic integrals with finite variance, then an easy heuristic calculation shows that Z T (2) (1) (2) λ ds | F h (15.14) h(1) Cov[HT , HT |Ft ] = E s t , s s t

and, in particular, Var[HT |Ft ] = E

h2s λs

ds | Ft .

H (1) and H (2) are said to be orthogonal if they have conditionally uncorrelated increments, that is, the covariance in (15.14) is null. This is equivalent to saying that H (1) H (2) is a martingale.

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The intensity is also called the infinitesimal characteristic if the counting process since it entirely determines it probabilistic properties. If λt is deterministic, then Nt is a Poisson process. If λ depends only on Nt− , then Nt is a Markov process. A comprehensive textbook on counting processes in life history analysis is [3]. E. The Girsanov transform. Girsanov’s theorem is a celebrated one in stochastics, and it is basic in mathematical finance. We formulate and prove the counting process variation: Theorem (Girsanov). Let Nt be a counting process with (F, P)-intensity λt , and let ˜ t be a given non-negative F-adapted process such that λ ˜ t = 0 if and only if λt = 0. λ ˜ such that P ˜ ∼ P and N has (F, P)-intensity ˜ Then there exists a probability measure P ˜ t . The likelihood process (15.10) is λ Z t Z t ˜ s − ln λs ) dNs + ˜ s ) ds . (ln λ (λs − λ Lt = exp 0

Proof: We shall give a constructive proof, starting from a guessed L in (15.5). Since L must be strictly positive a.e. (P), a candidate would be L = LT , where Z t Z t Lt = exp φs dNs + ψs ds 0

with φ predictable and ψ adapted. In the first place, Lt should be a martingale (F, P). By Itˆ o’s formula, dLt

= =

Lt ψt dt + Lt− (eφt − 1) dNt Lt ψt + eφt − 1 λt dt + Lt− eφt − 1 dMt .

The representation result (15.13) tells us that to make L a martingale, we must make the drift term vanish, that is, ψt = 1 − e φ t λ t , (15.15) whereby

dLt = Lt− eφt − 1 dMt ,

˜ given by In the second place, we want to determine φt such that the process M ˜ t dt ˜ t = dNt − λ dM ˜ Thus, we should have E[ ˜M ˜ t |Fs ] = M ˜ s or, by (15.6), is a martingale (F, P). i h ˜ t L|Fs E M ˜s . =M E [L|Fs ] Using the martingale property (15.10) of Lt , this is the same as i h ˜ t Lt |Fs = M ˜ s Ls E M

(15.16)

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˜ t Lt should be a martingale (F, P). Since i.e. M ˜ t Lt ) d(M

˜ t dt)Lt + M ˜ t (eφt − 1)Lt (−λt dt) (−λ ˜ t− + 1)Lt− eφt − M ˜ t− Lt− ) dNt + (M ˜ t + e φ t λt + ( M ˜ t− + 1)Lt− eφt − M ˜ t− Lt− ) dMt . Lt dt −λ

˜ t − ln λt . we conclude that the martingale property is obtained by choosing φt = ln λ The multivariate case goes in the same way; just replace by vector-valued processes.

15.3

A Markov chain financial market - Introduction

A. Motivation. The theory of diffusion processes, with its wealth of powerful theorems and model variations, is an indispensable toolkit in modern financial mathematics. The seminal papers of Black and Scholes [10] and Merton [32] were crafted with Brownian motion, and so were most of the almost countless papers on arbitrage pricing theory and its bifurcations that followed over the past quarter of a century. A main course of current research, initiated by the martingale approach to arbitrage pricing ([24] and [25]), aims at generalization and unification. Today the core of the matter is well understood in a general semimartingale setting, see e.g. [13]. Another course of research investigates special models, in particular various Levy motion alternatives to the Brownian driving process, see e.g. [18] and [40]. Pure jump processes have been widely used in finance, ranging from plain Poisson processes introduced in [33] to quite general marked point processes, see e.g. [8]. And, as a pedagogical exercise, the market driven by a binomial process has been intensively studied since it was launched in [12]. The present paper undertakes to study a financial market driven by a continuous time hom*ogeneous Markov chain. The idea was launched in [39] and reappeared in [19], the context being limited to modelling of the spot rate of interest. The purpose of the present study is two-fold: In the first place, it is instructive to see how well established theory turns out in the framework of a general Markov chain market. In the second place, it is worthwhile investigating the feasibility of the model from a theoretical as well as from a practical point of view. Poisson driven markets are accommodated as special cases. B. Preliminaries: Notation and some useful results. Vectors and matrices are denoted by in bold letters, lower and upper case, respectively. They may be equipped with topscripts indicating dimensions, e.g. An×m has n rows and m columns. We may write A = (ajk )k∈K j∈J to emphasize the ranges of the row index j and the column index k. The transpose of A is denoted by A0 . Vectors are invariably taken to be of column type, hence row vectors appear as transposed. The identity matrix is denoted by I, the vector with all entries equal to 1 is denoted by 1, and the vector with all entries equal to 0 is denoted by 0. By Dj=1,...,n (aj ), or just D(a), is meant the diagonal matrix with the entries of a = (a1 , . . . , an )0 down the principal diagonal. The n-dimensional Euclidean space is denoted by Rn , and the linear subspace spanned by the columns of An×m is denoted by R(A).

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219

A diagonalizable square matrix An×n can be represented as A = Φ Dj=1,...,n (ρj ) Φ−1 =

n X

ρj φj ψ 0j ,

(15.1)

j=1

where the φj are the columns of Φn×n and the ψ 0j are the rows of Φ−1 . The ρj are the eigenvalues of A, and φj and ψ 0j are the corresponding right and left eigenvectors, respectively. Eigenvectors (right or left) corresponding to eigenvalues that are distinguishable and non-null are mutually orthogonal. These results can be looked up in e.g. [30]. The exponential function of An×n is the n × n matrix defined by exp(A) =

∞ n X X 1 p eρj φj ψ 0j , A = Φ Dj=1,...,n (eρj ) Φ−1 = p! p=0 j=1

(15.2)

where the last two expressions follow from (15.1). The matrix exp(A) has full rank. If Λn×n is positive definite symmetric, then hζ 1 , ζ 2 iΛ = ζ 01 Λζ 2 defines an inner product on Rn . The corresponding norm is given by kζkΛ = hζ, ζi1/2 . If Fn×m has Λ full rank m (≤ n), then the h · , · iΛ -projection of ζ onto R(F) is ζ F = PF ζ ,

(15.3)

where the projection matrix (or projector) PF is PF = F(F0 ΛF)−1 F0 Λ .

(15.4)

The projection of ζ onto the orthogonal complement R(F)⊥ is ζ F⊥ = ζ − ζ F = (I − PF )ζ . Its squared length, which is the squared h · , · iΛ -distance from ζ to R(F), is kζ F⊥ k2Λ = kζk2Λ − kζ F k2Λ = ζ 0 Λ(I − PF )ζ .

(15.5)

The cardinality of a set Y is denoted by |Y|. For a finite set it is just its number of elements.

15.4

The Markov chain market

A. The continuous time Markov chain. At the base of everything (although slumbering in the background) is some probability space (Ω, F, P). Let {Yt }t≥0 be a continuous time Markov chain with finite state space Y = {1, . . . , n}. We assume that it is time hom*ogeneous so that the transition probabilities pjk t = P[Ys+t = k | Ys = j] depend only on the length of the transition period. This implies that the transition intensities λjk = lim

t&0

pjk t , t

(15.1)

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j 6= k, exist and are constant. To avoid repetitious reminders of the type “j, k ∈ Y”, we reserve the indices j and k for states in Y throughout. We will frequently refer to Y j = {k; λjk > 0} , the set of states that are directly accessible from state j, and denote the number of such states by nj = |Y j | .

Put

λjj = −λj· = −

λjk

k;k∈Y j

(minus the total intensity of transition out of state j). We assume that all states j intercommunicate so that pjk t > 0 for all j, k (and t > 0). This implies that n > 0 for all j (no absorbing states). The matrix of transition probabilities, Pt = (pjk t ), and the infinitesimal matrix,

Λ = (λjk ) ,

are related by (F.41), which in matrix form reads Λ = limt&0 1t (Pt − I), and by the backward and forward Kolmogorov differential equations, d Pt = Pt Λ = ΛPt . dt

(15.2)

Under the side condition P0 = I, (15.2) integrates to Pt = exp(Λt) .

(15.3)

In the representation (15.2), Pt = Φ Dj=1,...,n (eρj t ) Φ−1 =

n X

eρj t φj ψ 0j ,

(15.4)

j=1

the first (say) eigenvalue is ρ1 = 0, and corresponding eigenvectors are φ1 = 1 and jn ψ 01 = (p1 , . . . , pn ) = limt%∞ (pj1 t , . . . , pt ), the stationary distribution of Y . The remaining eigenvalues, ρ2 , . . . , ρn , are all strictly negative so that, by (15.4), the transition probabilities converge exponentially to the stationary distribution as t increases. Introduce Itj = 1[Yt = j] ,

(15.5)

the indicator of the event that Y is in state j at time t, and Ntjk = |{s; 0 < s ≤ t, Ys− = j , Ys = k}| ,

(15.6)

the number of direct transitions of Y from state j to state k ∈ Y j in the time interval (0, t]. For k ∈ / Y j we define Ntjk ≡ 0. Taking Y to be right-continuous, the same goes for the indicator processes I j and the counting processes N jk . As is seen from (15.5), (15.6), and the obvious relationships X X j (Ntkj − Ntjk ) , Yt = jIt , Itj = I0j + j

k;k6=j

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the state process, the indicator processes, and the counting processes carry the same information, which at any time t is represented by the sigma-algebra FtY = σ{Ys ; 0 ≤ s ≤ t}. The corresponding filtration, denoted by FY = {FtY }t≥0 , is taken to satisfy the usual conditions of right-continuity (Ft = ∩u>t Fu ) and completeness (F0 contains all subsets of P-nullsets), and F0 is assumed to be the trivial (∅, Ω). This means, essentially, that Y is right-continuous (hence the same goes for the I j and the N jk ) and that Y0 deterministic. The compensated counting processes M jk , j 6= k, defined by dMtjk = dNtjk − Itj λjk dt

(15.7)

and M0jk = 0, are zero mean, square integrable, mutually orthogonal martingales w.r.t. (FY , P). We now turn to the subject matter of our study and, referring to introductory texts like [9] and [43], take basic notions and results from arbitrage pricing theory as prerequisites. B. The continuous time Markov chain market. We consider a financial market driven by the Markov chain described above. Thus, Yt represents the state of the economy at time t, FtY represents the information available about the economic history by time t, and FY represents the flow of such information over time. In the market there are m + 1 basic assets, which can be traded freely and frictionlessly (short sales are allowed, and there are no transaction costs). A special role is played by asset No. 0, which is a “locally risk-free” bank account with state-dependent interest rate X j j rt = r Yt = It r , j

where the state-wise interest rates r , j = 1, . . . , n, are constants. Thus, its price process is ! Z t X jZ t j Bt = exp rs ds = exp r Is ds , 0

with dynamics

dBt = Bt rt dt = Bt

rj Itj dt .

(Setting B0 = 1 a just a matter of convention.) The remaining m assets, henceforth referred to as stocks, are risky, with price processes of the form   X X ijk jk X ij Z t j i β Nt  , (15.8) α Is ds + St = exp  j

k∈Y j

i = 1, . . . , m, where the αij and β ijk are constants and, for each i, at least one of the β ijk is non-null. Thus, in addition to yielding state-dependent returns of the same form as the bank account, stock No. i makes a price jump of relative size γ ijk = exp β ijk − 1

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upon any transition of the economy from state j to state k. By the general Itˆ o’s formula, its dynamics is given by   X X ijk X ij j jk  i i  α It dt + . (15.9) γ dNt dSt = St− j

k∈Y j

Taking the bank account as numeraire, we introduce the discounted stock prices ˜t ≡ 1, which S˜ti = Sti /Bt , i = 0, . . . , m. (The discounted price of the bank account is B is certainly a martingale under any measure). The discounted stock prices are   Z t X ij X X ijk jk i j j S˜t = exp  (α − r ) β Nt  , (15.10) Is ds + 0

k∈Y j

with dynamics



i  dS˜ti = S˜t−

X j

(αij − rj )Itj dt +

X X j

k∈Y j



γ ijk dNtjk  ,

(15.11)

i = 1, . . . , m. We stress that the theory we are going to develop does not aim at explaining how the prices of the basic assets emerge from supply and demand, business cycles, investment climate, or whatever; they are exogenously given basic entities. (And God said “let there be light”, and there was light, and he said “let there also be these prices”.) The purpose of the theory is to derive principles for consistent pricing of financial contracts, derivatives, or claims in a given market. C. Portfolios. A dynamic portfolio or investment strategy is an m + 1-dimensional stochastic process θ 0t = (ηt , ξ 0t ) , where ηt represents the number of units of the bank account held at time t, and the i-th entry in ξt = (ξt1 , . . . , ξtm )0 represents the number of units of stock No. i held at time t. As it will turn out, the bank account and the stocks will appear to play different parts in the show, the latter being the more visible. It is, therefore, convenient to costume the two types of assets and their corresponding portfolio entries accordingly. To save notation, however, it is useful also to work with double notation θt = (θt0 , . . . , θtm )0 , with θt0 = ηt , θti = ξti , i = 1, . . . , m, and work with S∗t = (St0 , . . . , Stm )0 ,

St0 = Bt .

The portfolio θ is adapted to FY (the investor cannot see into the future), and the shares of stocks, ξ, must also be FY -predictable (the investor cannot, e.g. upon a sudden crash of the stock market, escape losses by selling stocks at prices quoted just before and hurry the money over to the locally risk-free bank account.)

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The value of the portfolio at time t is Vtθ = ηt Bt +

m X

ξti Sti = ηt Bt + ξ0t St = θ 0t S∗t

i=0

Henceforth we will mainly work with discounted prices and values and, in accordance with (15.10), equip their symbols with a tilde. The discounted value of the portfolio at time t is ˜ t = θ 0t S ˜ ∗t . V˜tθ = ηt + ξ0t S The strategy θ is self-financing (SF) if ˜ ∗t = dV˜tθ = θ 0t dS

dVtθ m X

θ 0t

(15.12) dS∗t

or, equivalently,

ξti dS˜ti .

(15.13)

i=1

We explain the last step: Put Yt = Bt−1 , a continuous process. The dynamics of the ˜ ∗t = Yt S∗t is then dS ˜ ∗t = dYt S∗t + Yt dS∗t . Thus, for V˜tθ = Yt Vtθ , we discounted prices S have ˜ ∗t , dV˜tθ = dYt Vtθ + Yt dVtθ = dYt θ 0t S∗t + Yt θ0t dS∗t = θ 0t (dYt S∗t + Yt dS∗t ) = θ 0t dS hence the property of being self-financing is preserved under discounting. The SF property says that, after the initial investment of V0θ , no further investment inflow or dividend outflow is allowed. In integral form: Z t Z t ˜s . ˜ ∗s = V˜0θ + ξ0s dS (15.14) θ 0s dS V˜tθ = V˜0θ + 0

˜ ∗t , Obviously, a constant portfolio θ is SF; its discounted value process is V˜tθ = θ0 S hence (15.13) is satisfied. More generally, for a continuous portfolio θ we would have ˜ ∗t + θ0t dS ˜ ∗t , and the self-financing condition would be equivalent to the a dV˜t (θ) = dθ 0t S 0 ˜∗ budget constraint dθ t St = 0, which says that any purchase of assets must be financed by a sale of some other assets. We urge to say that we shall typically be dealing with portfolios that are not continuous and, in fact, not even right-continuous so that “dθ t ” is meaningless (integrals with respect to the process θ are not well defined). D. Absence of arbitrage. An SF portfolio θ is called an arbitrage if, for some t > 0, V0θ < 0 and Vtθ ≥ 0 a.s. P , or, equivalently,

V˜0θ < 0 and V˜tθ ≥ 0 a.s. P .

A basic requirement on a well-functioning market is the absence of arbitrage. The assumption of no arbitrage, which appears very modest, has surprisingly far-reaching consequences as we shall see. ˜ that is equivalent to P and A martingale measure is any probability measure P ˜ ∗t are martingales (F, P). ˜ such that the discounted asset prices S The fundamental theorem of arbitrage pricing says: If there exists a martingale measure, then there is

224

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

no arbitrage. This result follows from easy calculations starting from (15.14): Forming ˜ under P ˜ and using the martingale property of S ˜ we find ˜ ∗ under P, expectation E Z t ˜ s ] = V˜0θ ξ0s dS E[V˜tθ ] = V˜0θ + E[ 0

(the stochastic integral is a martingale). It is seen that arbitrage is impossible. We return now to our special Markov chain driven market. Let ˜ jk ) ˜ = (λ Λ ˜ jk = 0 if and only be an infinitesimal matrix that is equivalent to Λ in the sense that λ jk ˜ equivalent to P, under if λ = 0. By Girsanov’s theorem, there exists a measure P, ˜ Consequently, the processes which Y is a Markov chain with infinitesimal matrix Λ. ˜ jk , j = 1, . . . , n, k ∈ Y j , defined by M ˜ tjk = dNtjk − Itj λ ˜ jk dt , dM

(15.15)

˜ jk M 0

˜ Rewrite and = 0, are zero mean, mutually orthogonal martingales w.r.t. (F , P). (15.11) as     X X X X i  ˜ jk  Itj dt + ˜ tjk  , (15.16) αij − rj + dS˜ti = S˜t− γ ijk dM γ ijk λ Y

k∈Y j

˜ if and only if i = 1, . . . , m. The discounted stock prices are martingales w.r.t. (FY , P) the drift terms on the right vanish, that is, X ijk jk ˜ = 0, γ λ (15.17) αij − rj + k∈Y j

j = 1, . . . , n, i = 1, . . . , m. From general theory it is known that the existence of such ˜ implies absence of arbitrage. an equivalent martingale measure P The relation (15.17) can be cast in matrix form as j

˜ , rj 1 − αj = Γj λ

(15.18)

j = 1, . . . , n, where 1 is m × 1 and αj = αij

i=1,...,m

k∈Y j Γj = γ ijk

i=1,...,m

˜j = λ ˜ jk λ

k∈Y j

The existence of an equivalent martingale measure is equivalent to the existence of a ˜ j to (15.18) with all entries strictly positive. Thus, the market is arbitragesolution λ free if (and we can show only if) for each j, r j 1 − αj is in the interior of the convex cone of the columns of Γj . Assume henceforth that the market is arbitrage-free so that (15.16) reduces to X X ijk i ˜ tjk , γ dM (15.19) dS˜ti = S˜t− j

k∈Y j

˜ for some measure P ˜ ˜ jk defined by (15.15) are martingales w.r.t. (FY , P) where the M that is equivalent to P.

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Inserting (15.19) into (15.13), we find that θ is SF if and only if dV˜tθ =

m X X X j

i ˜ tjk , ξti S˜t− γ ijk dM

(15.20)

k∈Y j i=1

˜ and, in particular, implying that V˜ θ is a martingale w.r.t. (FY , P) ˜ V˜Tθ | Ft ] . V˜tθ = E[

(15.21)

˜ denotes expectation under P. ˜ (Note that the tilde, which in the first place was Here E introduced to distinguish discounted values from the nominal ones, is also attached to the equivalent martingale measure and certain related entities. This usage is motivated by the fact that the martingale measure arises from the discounted basic price processes, roughly speaking.) E. Attainability. A T -claim is a contractual payment due at time T . Formally, it is an FTY -measurable random variable H with finite expected value. The claim is attainable if it can be perfectly duplicated by some SF portfolio θ, that is, ˜. V˜Tθ = H

(15.22)

If an attainable claim should be traded in the market, then its price must at any time be equal to the value of the duplicating portfolio in order to avoid arbitrage. Thus, denoting the price process by πt and, recalling (15.21) and (15.22), we have ˜H ˜ | Ft ] , π ˜t = V˜tθ = E[

(15.23)

i h R ˜ e− tT r H Ft . πt = E

(15.24)

By (15.23) and (15.20), the dynamics of the discounted price process of an attainable claim is d˜ πt =

m X X X j

i ˜ tjk . ξti S˜t− γ ijk dM

(15.25)

k∈Y j i=1

F. Completeness. Any T -claim H as defined above can be represented as Z TX X ˜ H] ˜ = E[ ˜ + ˜ tjk , H ζtjk dM 0

(15.26)

k∈Y j

where the ζtjk are FY -predictable and integrable processes. Conversely, any random variable of the form (15.26) is, of course, a T -claim. By virtue of (15.22), and (15.20), attainability of H means that Z T ˜ = V˜0θ + H dV˜tθ 0

V˜0θ

X X X j

k∈Y j

i ˜ tjk . γ ijk dM ξti S˜t−

(15.27)

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Comparing (15.26) and (15.27), we see that H is attainable iff there exist predictable processes ξt1 , . . . , ξtm such that m X

i ξti S˜t− γ ijk = ζtjk ,

i=1

for all j and k ∈ Y . This means that the nj -vector ζ jt = (ζtjk )k∈Y j 0

is in R(Γj ). The market is complete if every T -claim is attainable, that is, if every nj -vector is 0 in R(Γj ). This is the case if and only if rank(Γj ) = nj , which can be fulfilled for each j only if m ≥ maxj nj .

15.5

Arbitrage-pricing of derivatives in a complete market

A. Differential equations for the arbitrage-free price. Assume that the market is arbitrage-free and complete so that prices of T -claims are uniquely given by (15.23) or (15.24). Let us for the time being consider a T -claim of the form H = h(YT , ST` ) .

(15.1) (ST`

Examples are a European call option on stock No. ` defined by H = − K)+ , a + YT + caplet defined by H = (rT − g) = (r − g) , and a zero coupon T -bond defined by H = 1. For any claim of the form (15.1) the relevant state variables involved in the conditional expectation (15.24) are t, Yt , St` , hence πt is of the form πt =

n X

Itj f j (t, St` ) ,

(15.2)

j=1

where the i h RT ˜ e− t r H Yt = j, St` = s f j (t, s) = E

(15.3)

are the state-wise price functions. ˜ The discounted price (15.23) is a martingale w.r.t. (FY , P). Assume that the functions f j (t, s) are continuously diferentiable. Using Itˆ o on π ˜ t = e−

0 r

n X

Itj f j (t, St` ) ,

(15.4)

j=1

we find d˜ πt

∂ j ∂ j f (t, St` ) + f (t, St` )St` α`j ∂t ∂s j Rt X X ` ` ) dNtjk +e− 0 r (1 + γ `jk )) − f j (t, St− f k (t, St−

e−

0 r

Itj

−rj f j (t, St` ) +

k∈Y j

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE =

227

∂ ∂ j Itj −rj f j (t, St` ) + f j (t, St` ) + f (t, St` )St` α`j ∂t ∂s j X ` ` ˜ jk dt + {f k (t, St− (1 + γ `jk )) − f j (t, St− )}λ

e−

0 r

k∈Y j R − 0t r

X X j

k∈Y j

` ` ˜ tjk . f k (t, St− (1 + γ `jk )) − f j (t, St− ) dM

(15.5)

By the martingale property, the drift term must vanish, and we arrive at the nonstochastic partial differential equations ∂ j ∂ f (t, s)sα`j −rj f j (t, s) + f j (t, s) + ∂t ∂s X k ˜ jk = 0 , f (t, s(1 + γ `jk )) − f j (t, s) λ +

(15.6)

k∈Y j

j = 1, . . . , n, which are to be solved subject to the side conditions f j (T, s) = h(j, s) ,

(15.7)

j = 1, . . . , n. In matrix form, with R = Dj=1,...,n (rj ) ,

A` = Dj=1,...,n (α`j ) ,

and other symbols (hopefully) self-explaining, the differential equations and the side conditions are − Rf (t, s) +

∂ ∂ ˜ (t, s(1 + γ)) = 0 , f (t, s) + sA` f (t, s) + Λf ∂t ∂s f (T, s) = h(s) .

(15.8) (15.9)

B. Identifying the strategy. Once we have determined the solution f j (t, s), j = 1, . . . , n, the price process is known and given by (15.2). The duplicating SF strategy can be obtained as follows. Setting the drift term to 0 in (15.5), we find the dynamics of the discounted price; Rt X X ` ` ˜ tjk . ) dM (15.10) d˜ πt = e− 0 r (1 + γ `jk )) − f j (t, St− f k (t, St− j

k∈Y j

Identifying the coefficients in (15.10) with those in (15.25), we obtain, for each state j, the equations m X i=1

` ` i ), (1 + γ `jk )) − f j (t, St− γ ijk = f k (t, St− ξti St−

(15.11)

k ∈ Y j . The solution ξ jt = (ξti,j )0i=1,...,m (say) certainly exists since rank(Γj ) ≤ m, and it is unique iff rank(Γj ) = m. Furthermore, it is a function of t and St− and is thus predictable. This simplistic argument works on the open intervals between the jumps ˜ jk dt. For the dynamics (15.10) and (15.25) to ˜ tjk = −Itj λ of the process Y , where dM

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228

be the same also at jump times, the coefficients must clearly be left-continuous. We conclude that n X j It− ξt , ξt = j=1

which is predictable. Finally, η is determined upon combining (15.12), (15.23), and (15.4): ! n m R X X j j ` j i,j i − 0t r ξ t St . It f (t, St ) − It− ηt = e j=1

i=1

C. The Asian option. As an example of a path-dependent claim, let us consider an Asian option, which R + T essentially is a T -claim of the form H = 0 Sτ` dτ − K , where K ≥ 0. The price process is " # Z T + RT − r ` Y ˜ e t πt = E Sτ dτ − K Ft 0 Z n t X j j It f t, St` , = Sτ` dτ , 0

j=1

where j

f (t, s, u)

" Z R − tT r ˜ E e

Sτ`

The discounted price process is π ˜ t = e−

0 r

n X

Itj f j

j=1

# + ` +u−K Yt = j, St = s . Z t, St` ,

Ss`

We obtain partial differential equations in three variables. The special case K = 0 is simpler, with only two state variables. D. Interest rate derivatives. A particularly simple, but still important, class of claims are those of the form H = h(YT ). Interest rate derivatives of the form H = h(rT ) are included since rT = rYT . For such claims the only relevant state variables are t and Yt , so that the function in (15.3) depends only on t and j. The equation (15.6) reduces to X k d j ˜ jk , f = rj ftj − (ft − ftj )λ dt t j

(15.12)

k∈Y

and the side condition is (put h(j) = hj ) fTj = hj . In matrix form, d ˜ − Λ)f ˜ t, ft = ( R dt

(15.13)

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subject to fT = h . The solution is ˜ − R)(T − t)}h . ft = exp{(Λ

(15.14)

It depends on t and T only through T − t. In particular, the zero coupon bond with maturity T corresponds to h = 1. We will henceforth refer to it as the T -bond in short and denote its price process by p(t, T ) and its state-wise price functions by p(t, T ) = (pj (t, T ))j=1,...,n ; ˜ − R)(T − t)}1 . p(t, T ) = exp{(Λ

(15.15)

For a call option on a U -bond, exercised at time T (< U ) with price K, h has entries hj = (pj (T, U ) − K)+ . In (15.14) – (15.15) it may be useful to employ the representation shown in (15.2), ˜ − R)(T − t)} = Φ ˜ Dj=1,...,n (eρ˜j (T −t) ) Φ ˜ −1 , exp{(Λ

(15.16)

say.

15.6

Numerical procedures

A. Simulation. The hom*ogeneous Markov process {Yt }t∈[0,T ] is simulated as follows: Let K be the number of transitions between states in [0, T ], and let T1 , . . . , TK be the successive times of transition. The sequence {(Tn , YTn )}n=0,...,K is generated recursively, starting from the initial state Y0 at time T0 = 0, as follows. Having arrived at Tn and YTn , generate the next waiting time Tn+1 − Tn as an exponential variate with parameter λYn · (e.g. − ln(Un )/λYn · , where Un has a uniform distribution over [0, 1]), and let the new state YTn+1 be k with probability λYn k /λYn · . Continue in this manner K + 1 times until TK < T ≤ TK+1 . B. Numerical solution of differential equations. Alternatively, the differential equations must be solved numerically. For interest rate derivatives, which involve only ordinary first order differential equations, a Runge Kutta will do. For stock derivatives, which involve partial first order differential equations, one must employ a suitable finite difference method, see e.g. [46].

15.7

Risk minimization in incomplete markets

A. Incompleteness. The notion of incompleteness pertains to situations where a contingent claim cannot be duplicated by an SF portfolio and, consequently, does not receive a unique price from the no arbitrage postulate alone. In Paragraph 15.4F we were dealing implicitly with incompleteness arising from a scarcity of traded assets, that is, the discounted basic price processes are incapable of ˜ and, in particular, reproducing spanning the space of all martingales w.r.t. (FY , P) the value (15.26) of every financial derivative (function of the basic asset prices).

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Incompleteness also arises when the contingent claim is not a purely financial derivative, that is, its value depends also on circ*mstances external to the financial market. We have in mind insurance claims that are caused by events like death or fire and whose claim amounts are e.g. inflation adjusted or linked to the value of some investment portfolio. In the latter case we need to work in an extended model specifying a basic probability space with a filtration F = {Ft }t≥0 containing FY and satisfying the usual conditions. Typically it will be the natural filtration of Y and some other process that generates the insurance events. The definitions and conditions laid down in Paragraphs 15.4C-E are modified accordingly, so that adaptedness of η and predictability of ξ are taken to be w.r.t. (F, P) (keeping the symbol P for the basic probability measure), a T -claim H is FT measurable, etc. B. Risk minimization. Throughout the remainder of the paper we will mainly be working with discounted prices and values without any other mention than the notational tilde. The reason is that the theory of risk minimization rests on certain martingale representation results that apply to discounted prices under a martingale measure. We will be content to give just a sketchy review of some main concepts and results from the seminal paper of F¨ ollmer and Sondermann [21]. ˜ be a T -claim that is not attainable. This means that an admissible portfolio Let H θ satisfying ˜ V˜Tθ = H ˜tθ , of the portfolio by time t is defined as that part of the cannot be SF. The cost, C value that has not been gained from trading: Z t ˜tθ = V˜tθ − ˜τ . C ξ0τ dS 0

The risk at time t is defined as the mean squared outstanding cost, i h ˜ (C ˜t = E ˜Tθ − C ˜tθ )2 Ft . R

(15.1)

By definition, the risk of an admissible portfolio θ is Z T 0 ˜ 2 θ θ ˜ ˜ ˜ ˜ ξ τ d S τ ) Ft , Rt = E ( H − Vt − t

which is a measure of how well the current value of the portfolio plus future trading gains approximates the claim. The theory of risk minimization takes this entity as its object function and proves the existence of an optimal admissible portfolio that minimizes the risk (15.1) for all t ∈ [0, T ]. The proof is constructive and provides a recipe for how to actually determine the optimal portfolio. ˜ at time t as One sets out by defining the intrinsic value of H i h ˜ H ˜ | Ft . V˜tH = E

Thus, the intrinsic value process is the martingale that represents the natural current forecast of the claim under the chosen martingale measure. By the Galchouk-KunitaWatanabe representation, it decomposes uniquely as Z t 0 H ˜ ˜ H] ˜ + ξH V˜tH = E[ t dS t + L t , 0

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˜ which is orthogonal to S. ˜ The portfolio θ H dewhere LH is a martingale w.r.t. (F, P) fined by this decomposition minimizes the risk process among all admissible strategies. The minimum risk is Z T ˜ ˜ tH = E dhLH iτ Ft . R t

C. Unit-linked insurance. As the name suggests, a life insurance product is said to be unit-linked if the benefit is a certain predetermined number of units of an asset (or portfolio) into which the premiums are currently invested. If the contract stipulates a minimum value of the benefit, disconnected from the asset price, then one speaks of unit-linked insurance with guarantee. A risk minimization approach to pricing and hedging of unit-linked insurance claims was first taken by Møller [34], who worked with the Black-ScholesMerton financial market. We will here sketch how the analysis goes in our Markov chain market, which conforms well with the life history process in that they both are intensity-driven. Let Tx be the remaining life time of an x years old who purchases an insurance at time 0, say. The conditional probability of survival to age x + u, given survival to age x + t (0 ≤ t < u), is u−t px+t

= P[Tx > u | Tx > t] = e−

t µx+s ds

(15.2)

where µy is the mortality intensity at age y. We have d u−t px+t = u−t px+t µx+t dt .

(15.3)

Introduce the indicator of survival to age x + t, It = 1[Tx > t] , and the indicator of death before time t, Nt = 1[Tx ≤ t] = 1 − It . The process Nt is a (very simple) counting process with intensity It µx+t , that is, M given by dMt = dNt − It µx+t dt

(15.4)

is a martingale w.r.t. (F, P). Assume that the life time Tx is independent of the ˜ obtained by replacing the economy Y . We will work with the martingale measure P ˜ and leaving the rest of the model intensity matrix Λ of Y with the martingalizing Λ unaltered. Consider a unit-linked pure endowment benefit payable at a fixed time T , contingent on survival of the insured, with sum insured equal to one unit of stock No. `, but guaranteed no less than a fixed amount g. This benefit is a contingent T -claim, H = (ST` ∨ g) IT . The single premium payable as a lump sum at time 0 is to be determined. Let us assume that the financial market is complete so that every purely financial derivative has a unique price process. Then the intrinsic value of H at time t is V˜tH = π ˜t It T −t px+t ,

232

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE where π ˜t is the discounted price process of the derivative ST` ∨ g. Using Itˆ o and inserting (15.4), we find dV˜tH

T −t px+t

µx+t dt + (0 − π ˜ t− T −t px+t ) dNt

d˜ πt It−

T −t px+t

+π ˜t It−

d˜ πt It−

T −t px+t

−π ˜t− T −t px+t dMt .

It is seen that the optimal trading strategy is that of the price process of the sum insured multiplied with the conditional probability that the sum will be paid out, and that dLH ˜t− dMt . t = −T −t px+t π Consequently,

˜ tH R

15.8

2 T −s px+s

T −t px+t

2 ˜ π E ˜s Ft s−t px+t µx+s ds

2 ˜ π E ˜s Ft T −s px+s µx+s ds .

(15.5)

Trading with bonds: How much can be hedged?

A. A finite zero coupon bond market. Suppose an agent faces a contingent T -claim and is allowed to invest only in the bank account and a finite number m of zero coupon bonds with maturities Ti , i = 1, . . . , m, all post time T . For instance, regulatory constraints may be imposed on the investment strategies of an insurance company. The question is, to what extent can the claim be hedged by self-financed trading in these available assets? An allowed SF portfolio has discounted value process V˜tθ of the form dV˜tθ =

m X i=1

ξti

X X j

k∈Y j

˜ tjk = (˜ pk (t, Ti ) − p˜j (t, Ti ))dM

˜ jt )0 Fjt ξ , d(M t

˜ tjk )k∈Y j is the nj -dimensional row vector comprising ˜ jt = (M where ξ is predictable, M ˜ t = (M ˜ tjk ), and the non-null entries in the j-th row of M Fjt = Yj Ft where i=1,...,m ˜ (t, Tm )) , Ft = (˜ pj (t, Ti ))j=1,...,n = (˜ p(t, T1 ), · · · , p

(15.1)

and Yj is the nj × n matrix which maps Ft to (˜ pk (t, Ti ) − p˜j (t, Ti ))i=1,...,m . If e.g. k∈Y j n n p×p p×(n−p−1) Y = {1, . . . , p}, then Y = (I ,0 , −1p×1 ). The sub-market consisting of the bank account and the m zero coupon bonds is complete in respect of T -claims iff the discounted bond prices span the space of all ˜ over the time interval [0, T ]. This is the case iff, for each j, martingales w.r.t. (FY , P) rank(Fjt ) = nj . Now, since Yj obviously has full rank nj , the rank of Fjt is determined by the rank of Ft in (15.1). We will argue that, typically, Ft has full rank. Thus, suppose c = (c1 , . . . , cm )0 is such that Ft c = 0n×1 .

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Recalling (15.15), this is the same as m X

˜ − R)Ti }1 = 0 , ci exp{(Λ

i=1

˜ has full rank, or, by (15.16) and since Φ Dj=1,...,n (

m X

ci eρ˜

˜ −1 1 = 0 . )Φ

(15.2)

i=1

˜ −1 has full rank, the entries of the vector Φ ˜ −1 1 cannot be all null. Typically Since Φ all entries are non-null, and we assume this is the case. Then (15.2) is equivalent to m X

ci eρ˜

= 0,

j = 1, . . . , n.

(15.3)

i=1

Using the fact that the generalized Vandermonde matrix has full rank, we know that (15.3) has a non-null solution c if and only if the number of distinct eigenvalues ρ˜j is less than m, see Section 15.9 below. In the case where rank(Fjt ) < nj for some j we would like to determine the Galchouk-Kunita-Watanabe decomposition for a given FTY -claim. The intrinsic value process has dynamics X X X jk ˜ jt )0 ζ j . ˜ tjk = ˜t = ζt d M (15.4) dH d(M t j

k∈Y j

We seek a decomposition of the form X X jk X i ˜ tjk p(t, Ti ) + ψt d M dV˜t = ξt d˜ j

X X X j

j∈Y j

ξti

˜ tjk + (˜ p (t, Ti ) − p˜j (t, Ti )) dM

˜ jt )0 Fjt ξj + d(M t

k∈Y j

˜ jt )0 ψ j , d(M t

X X j

˜ tjk ψtjk dM

k∈Y j

such that the two martingales on the right hand side are orthogonal, that is, X j X j j 0 j j ˜ ψ = 0, It− (Ft ξt ) Λ t j

k∈Y j

˜ j = D(λ ˜ j ). This means that, for each j, the vector ζ jt in (15.4) is to be where Λ j decomposed into its h , iΛ ˜ j projections onto R(Ft ) and its orthocomplement. From (15.3) and (15.4) we obtain Fjt ξjt = Pjt ζ jt , where

0 j j −1 j 0 j ˜ , ˜ Ft ) Ft Λ Pjt = Fjt (Fjt Λ

hence 0 j j −1 j 0 j j ˜ ζ . ˜ Ft ) Ft Λ ξjt = (Fjt Λ t

(15.5)

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Furthermore, ψ jt = (I − Pjt )ζ jt ,

(15.6)

and the risk is Z

tj pYs−t

λjk (ψsjk )2 ds .

(15.7)

k∈Y j

The computation goes as follows: The coefficients ζ jk involved in the intrinsic value process (15.4) and the state-wise prices pj (t, Ti ) of the Ti -bonds are obtained by simultaneously solving (15.6) and (15.12), starting from (15.9) and (15.12), respectively, and at each step computing the optimal trading strategy ξ by (15.5) and the ψ from (15.6), and adding the step-wise contribution to the variance (15.7) (the step-length times the current value of the integrand). B. First example: The floorlet. For a simple example, consider a ’floorlet’ H = (r ∗ − rT )+ , where T < mini Ti . The motivation could be that at time T the insurance company will ascribe interest to the insured’s account at current interest rate, but not less than a prefixed guaranteed rate r∗ . Then H is the amount that must be provided per unit on deposit and per time unit at time T . Computation goes by the scheme described above, with the ζtjk = f˜tk − f˜tj obtained from (15.12) subject to (15.13) with hj = (r∗ − rj )+ . C. Second example: The interest guarantee in insurance. A more practically relevant example is an interest rate guarantee on a life insurance policy. Premiums and reserves are calculated on the basis of a prudent so-called first order assumption, stating that the interest rate will be at some fixed (low) level r ∗ throughout the term of the insurance contract. Denote the corresponding first order reserve at time t by Vt∗ . The (portfolio-wide) mean surplus created by the first order assumption in the time interval [t, t+dt) is (r ∗ −rt )+ t p∗x Vt∗ dt. This surplus is currently credited to the account of the insured as dividend, and the total amount of dividends is paid out to the insured at the term of the contracts at time T . Negative dividends are not permitted, however, so at time T the insurer must cover Z T R T H = e s r (r∗ − rs )+ s p∗x Vs∗ ds . 0

The intrinsic value of this claim is Z T Rs ˜t = E ˜ e− 0 r (r∗ − rs )+ s p∗x Vs∗ ds Ft H =

0 R − 0s r

(r∗ − rs )+ s p∗x Vs∗ ds + e−

0 r

Itj ftj ,

where the ftj are the state-wise expected values of future guarantees, discounted at time t, Z T R j ∗ ∗ − ts r ∗ + ˜ ft = E e (r − rs ) s px Vs ds Yt = j . t

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Working along the lines of Section 15.5, we determine the ftj by solving X k d j ˜ jk , f = −(r∗ − rj )+ t p∗x Vt∗ + rj ftj − (ft − ftj )λ dt t j k∈Y

subject to fTj = 0 .

(15.8)

The intrinsic value has dynamics (15.4) with ζtjk = f˜tk − f˜tj . From here we proceed as described in Paragraph A. D. Computing the risk. Constructive differential equations may be put up for the risk. As a simple example, for an interest rate derivative the state-wise risk is Z TX X gk gk 2 ˜ tj = dτ . R pjg λ ψτ τ −t t

k;k6=g

Differentiating this equation, we find X jk jk 2 Z T X d jg X gk 2 d ˜j dτ , ψτ λ ψt + Rt = − p dt dt τ −t k;k6=g t g k;k6=j and, using the backward version of (15.2), X jh hg d jg λ ps−t + λj· pjg p =− s−t , dt s−t h;h6=j we arrive at

X jk k X jk jk 2 d ˜j ˜ t + λj· R ˜ tj . λ R λ ψt − Rt = − dt k;k6=j

k;k6=j

15.9

The Vandermonde matrix in finance

A. The Vandermonde matrix. Let An denote the generic n × n matrix of the form j=1,...,n , A n = e αi β j i=1,...,n

(15.1)

where α1 , . . . , αn and β1 , . . . , βn are reals. This is a classic in matrix theory, known as β the generalized Vandermonde matrix (usually its elements are written in the form xi j with xi > 0). It is well known that it is non-singular iff all αi are different and all βj are different, see Gantmacher [22] p. 87.

B. Purpose of the study. The matrix An in (15.1) and its close relative j=1,...,n An − 1n 10n = eαi βj − 1 , i=1,...,n

(15.2)

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arise naturally in zero coupon bond prices based on spot interest rates driven by certain hom*ogeneous Markov processes. It turns out that, in such bond markets, the issue of completeness is closely related to the rank of the two archetype matrices. Roughly speaking, non-singularity of matrices of types (15.1) or (15.2) ensures that any simple T -claim can be duplicated by a portfolio consisting of the risk-free bank account and a sufficiently large number of zero coupon bonds. The non-singularity results are proved in Section 15.10, and applications to bond markets are presented in Section 15.11.

15.10

Two properties of the Vandermonde matrix

A. The main result. We take the opportunity here to provide a short proof of the quoted result on nonsingularity of the Vandermonde matrix in (15.1), and will supply a similar result about its relative defined in (15.2). Theorem (i) If the αi are all different and the βj are all different, then An is non-singular. (ii) If, furthermore, the αi and the βj are all different from 0, then An − 1n 10n is non-singular.

Proof: The proof goes by induction. Let Hn be the hypothesis stated in the two items of the lemma. Trivially, H1 is true. Assuming that Hn−1 is true, we need to prove Hn . Addressing first item (i) of the the hypothesis, it suffices to prove that det(An ) 6= 0. Recast this determinant as   · · · e(α1 −αn )βn e(α1 −αn )β1   · · · · · !   n Y   · · · · · αn β j   e det  det(An ) =  · · · · ·   j=1  e(αn−1 −αn )β1 · · · e(αn−1 −αn )βn  =

n Y

j=1

e αn β j

n−1 Y i=1

e(αi −αn )βn

1 !

det

An−1 10n−1

1n−1 1

(15.1)

where j=1,...,n−1 . An−1 = e(αi −αn )(βj −βn ) i=1,...,n−1

(15.2)

The determinant appearing in (15.1) remains unchanged upon subtracting the n-th row of the matrix from all other rows, which gives An−1 − 1n−1 10n−1 0n−1 An−1 1n−1 = det det 0 0 1n−1 1 1n−1 1 0 (15.3) = det An−1 − 1n−1 1n−1 .

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Now, since the αi are all different and also the βj are all different, the matrix An−1 in (15.2) is of the form required in item (ii) of the lemma and so, by the assumed hypothesis Hn−1 , det(An−1 − 1n−1 10n−1 ) 6= 0. It follows from (15.1) and (15.3) that det(An ) 6= 0, hence item (i) of Hn holds true. Next, we turn to item (ii) of Hn . Preparing for an ad absurdum argument, assume that An is as specified in item (ii) of the lemma and that An − 1n 10n is singular. Then there exists a vector c = (c1 , . . . , cn )0 6= 0n such that An c = 1n 10n c . Introducing the function f (α) =

n X

(15.4)

cj eαβj ,

j=1

and putting α0 = 0, we can spell out (15.4) as f (α0 ) = f (α1 ) = · · · = f (αn ) ,

(15.5)

that is, f assumes the same value at n + 1 distinct values of α. Since f is continuously differentiable, Rolle’s theorem implies that the derivative f 0 of f is 0 at n distinct values α∗1 , . . . , α∗n (say) of α. Now, f 0 (α) =

n X

cj βj eαβj ,

j=1

and since some cj are different from 0 and all βj are different from 0, it follows that ∗ j=1,...,n the matrix A∗n = eαi βj should be singular. This contradicts the previously i=1,...,n

established item (i) under Hn , showing that the assumed singularity of An − 1n 10n is absurd. We conclude that also item (ii) of Hn holds true.

B. Remarks. In fact, if α1 < · · · < αn and β1 < · · · < βn , then det(An ) > 0 (see [22]). If we take this fact for granted, (15.1) and (15.3) show that also det(An − 1n 10n ) > 0, implying that the latter is non-singular under the hypothesis of item (ii) in the theorem. The sign of a general Vandermonde determinant is, of course, the product of the signs of the row and column permutations needed to order the αi and the βj by their size.

15.11

Applications to finance

A. Zero coupon bond prices. A zero coupon bond with maturity T , or just T -bond in short, is the simple contingent claim of 1 at time T . Taking an arbitrage-free financial market for granted, the price process {p(t, T )}t∈[0,T ] of the T -bond is i h RT ˜ e− t ru du Ft , (15.1) p(t, T ) = E ˜ denotes expectation under some martingale measure, and Ft is the information where E available at time t.

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We will provide some examples where the results in Section 15.10 are instrumental for establishing linear independence of price processes of bonds with different maturities. The issue is non-trivial only in cases where the bond prices are governed by more than one source of randomness, of course, so we have to look into cases where the spot rate of interest is driven by more than one martingale. B. Markov chain interest rate. Referring to Chapter 7, let us model the spot rate of interest {rt }t≥0 as a continuous time, hom*ogeneous, recurrent Markov chain with finite state space {r 1 , . . . , rn }. We are working under some martingale measure given by an infinitesimal matrix ˜ = (λ ˜ jk ) of the Markov chain, that is, the transition intensities are λ ˜ jk , j 6= k, and Λ P jj jk ˜ ˜ λ = − k;k6=j λ . The price at time t ≤ T of a zero coupon bond with maturity T is n X Itj pj (t, T ) , p(t, T ) = j=1

where

Itj

= 1[rt = r ] and h R i ˜ e− tT ru du rt = rj . pj (t, T ) = E

The vector of state-wise prices,

p(t, T ) = (pj (t, T ))j=1,...,n , is given by (15.15), ˜ − R)(T − t)}1 = Φ Diag(eρj (T −t) ) Ψ1 , p(t, T ) = exp{(Λ where R = Diag(r j ) is the n × n diagonal matrix with the entries r j down the principal diagonal, 1 is the n-vector with all entries equal to 1, ρ˜j , j = 1, . . . , n, are the ˜ − R, and Φ and Ψ are the n × n matrices formed by the right and eigenvalues of Λ left eigenvectors, respectively. The price processes of m zero coupon bonds with maturities T1 < · · · < Tm are linearly independent only if the matrix j

(eρ˜

Ti j=1,...,n )i=1,...,m

has rank m. From item (i) in the theorem in Paragraph 15.10A we conclude that this is the case if there are at least m distinct eigenvalues ρ˜j . It also follows that the Rt market consisting of the bank account with price process exp 0 rs ds and the m zero coupon bonds is complete for the class of all FTr1 -claims only if both the number of distinct eigenvalues and the number of bonds are no less than the maximum number of states that can be directly accessed from any single state of the Markov chain. C. Mixed Vaciˇ cek interest rate. The Vasiˇcek model takes the spot rate of interest to be an Ornstein-Uhlenbeck process given by ˜t. drt = α (ρ − rt ) dt + σ dW

(15.2)

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Here ρ is the stationary mean of the process, α is a positive mean reversion parameter, ˜ is a standard Brownian motion under a σ is a positive volatility parameter, and W martingale measure. The dynamics of the discounted T -bond price, p˜(t, T ) = e−

0 ru du

p(t, T ) ,

(15.3)

σ −α (T −t) ˜t, e − 1 dW α

(15.4)

is d˜ p(t, T ) = p˜(t, T ) ˜

see e.g. [9]. Obviously, any FTW claim can be duplicated by a self-financing portfolio in the T -bond and the bank account, and so the completeness issue is trivial in this model. To create an example where one bond is not sufficient to complete the market, let us concoct a mixed Vasiˇcek model by putting rt =

n X

rtj ,

j=1

where the rj are independent Ornstein-Uhlenbeck processes, ˜ tj , drtj = αj (ρj − rtj ) dt + σ j dW ˜ j are independent standard Brownian motions. We assume j = 1, . . . , n, and the W j that the α are all distinct (otherwise we could gather all processes r j with coinciding mean reversion parameter into one Ornstein-Uhlenbeck process). The mixed Vasiˇcek process is not mean-reverting in the same simple sense as the traditional Vasiˇcek process. It is stationary, however, and is apt to describe interest that is subject to several random phenomena, each of mean-reverting type. By the assumed independence, the price of the T -bond is just p(t, T ) =

n Y

pj (t, T ) ,

j=1

˜ e− where pj (t, T ) = E

RT t

i ru du j rt , and the discounted price is p˜(t, T ) =

n Y

p˜j (t, T ) ,

j=1

where p˜j (t, T ) is the “j-analogue” to (15.3). By virtue of (15.4), we conclude that the discounted T -bond price has dynamics d˜ p(t, T ) = p˜(t, T )

n X σ j −αj (T −t) ˜ tj . e − 1 dW αj j=1

(15.5)

Now, consider the market consisting of the bank account and m zero coupon bonds with maturities T1 < · · · < Tm . From (15.5) it is seen that this market is complete for ˜ ˜ the class of FTW11 ,...,Wn -claims if and only if the matrix

e−α

(Ti −t)

−1

i=1,...,m j=1,...,n

(15.6)

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CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

has rank n. By virtue of item (ii) in the theorem in Paragraph 15.10A, we conclude that this is the case if m ≥ n. D. Mixed Poisson-driven Ornstein-Uhlenbeck interest rate. Referring to [40], let us replace the Brownian motions in Paragraph C above with independent compensated Poisson processes, that is, ˜ tj = dNtj − λj dt , dW where each N j is a Poisson process with intensity λj . Instead of (15.5) we obtain d˜ p(t, T ) = p˜(t−, T )

n X j=1

exp

σ j −αj (T −t) ˜ tj . e − 1 − 1 dW αj

(15.7)

It is seen from (15.7) that the market consisting of the bank account and m zero ˜ ˜ coupon bonds with maturities T1 < · · · < Tm is complete for the class of FTN11 ,...,Nn claims if and only if the matrix

exp

i=1,...,m σ j −αj (T −t) e − 1 − 1 αj j=1,...,n

has rank n. By item (ii) in the theorem in Paragraph 15.10A, we know that the matrix (15.6) has full rank. Thus, completeness of a market consisting of the bank account and at least n bonds would be established – and we would be done – if we could prove that the n × m matrix (eγji − 1) has full rank whenever (γji ) has full rank. With this conjecture our study of these problems will have to halt for the time being.

15.12

Martingale methods

A. Preliminaries. Referring to Sections 5.4 and 7.3 we shall see examples of how martingale methods can be used to prove results that throughout the text have been obtained by the direct backward argument. The policy was described as a time-continuous Markov chain Z with finite state space and transition intensities µjk , j 6= k. We introduced Ij (t) = 1[Z(t) = j] , the indicator of the event that Z is in state j at time t, and Njk (t) = ]{s; 0 < s ≤ t, Z(s−) = j , Z(s) = k} , the number of direct transitions of Z from state j to state k (6= j) in the time interval (0, t]. Taking Z to be right-continuous, the same goes for the indicator processes Ij and the counting processes Njk . Let FtZ = σ{Zs ; 0 ≤ s ≤ t}, t ≥ 0, be the filtration generated by Z. The compensated counting processes Mjk , j 6= k, defined by dMjk (t) = dNjk (t) − Ij (t) µjk dt

(15.8)

and Mjk (0) = 0, are zero mean, square integrable, mutually orthogonal martingales w.r.t. the filtration.

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241

We need a couple of general results: 1. Let X be a real-valued random variable such that E|X| < ∞. Then the process M defined by M (t) = E[X|Ft ] is a martingale. This follows by the rule of iterated expectation and the filtration property, Fs ⊂ Ft for s < t : E[M (t) | Fs ] = E{E[X|Ft ] | Fs } = E[X | Fs ] = M (s) . 2. A martingale M with paths that are (almost surely) continuous and of finite variation in every finite interval is constant as a function of time; M (t) = M (0) for all t. This is seen as follows. Since M has finite variation, it obeys the rules of ordinary calculus and, in particular, Z t M (s) dM (s) . M 2 (t) = M 2 (0) + 2 0

Since M is continuous, it is also predictable so that the integral martingale. It follows that E M 2 (t) = M 2 (0) .

Rt 0

2M (s) dM (s) is a

Since E[M (t)] = M (0), we conclude that

Var[M (t)] = 0 , hence M is constant. Now to the martingale technique:

B. First example: Thiele’s differential equation. Consider the standard multi-state insurance policy defined in Paragraph 5.4.A and recall the form of the payment function, X X bjk (t) dNjk (t) , (15.9) Ij (t) dBj (t) + dB(t) = j

j6=k

where the Bj and the bjk are deterministic functions. Motivated by Section 5.6, we allow the interest rate to depend on the current state of the Markov process: X r(t) = rZ(t) = Ij (t) rj . (15.10) j

(This does not complicate matters.) Define the martingale Z n Rτ e− 0 r dB(τ ) Ft M (t) = E 0−

e−

0− t

e 0−

Rτ

0 r

R − 0τ r

dB(τ ) + e−

dB(τ ) + e

0 r

R − 0t r

X j

n t

e−

Rτ

t r

dB(τ ) Ft

Ij (t)Vj (t) .

(15.11)

242

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

Rτ Rτ (Recall the short-hand 0 r = 0 r(s) ds.) The last step follows from the Markov property of Z and the fact that the payments at any time are (functionally) independent of the past: Z n R Z n R τ τ E e− t r dB(τ ) Ft = E e− t r dB(τ ) Zt = VZ(t) (t) t t X Ij (t)Vj (t) . = j

Now apply Itˇ o’s formula to (15.11): dM (t)

e−

0 r

+ e−

dB(t) + e−

0 r

(−r(t) dt)

Ij (t) Vj (t)

Ij (t) dVj (t) + e−

0 r

X j6=k

dNjk (t) (Vk (t) − Vj (t−)) .

The last term on the right takes care of the jumps of the Markov process: upon a jump from state j to state k the last term in (15.11) changes immediately from the discounted value of the reserve in state j just before the jump to the value of the reserve in state k at the time of the jump. Since the state-wise reserves are deterministic functions with finite variation, they have at most a countable number of discontinuities at fixed times. The probability that the Markov process jumps at any such time is 0. Therefore, we need not worry about possible common points of discontinuity of the Vj (t) and the Ij (t). For the same reason we can also disregard the left limit in Vj (t−) in the last term. We proceed by inserting the expressions (15.9) for dB(t), (15.10) for r(t), and the expression dNjk (t) = dMjk (t) − Ij (t) µjk (t) dt obtained from (15.8), and gather   Rt X X Ij (t) dBj (t) − rj Vj (t) dt + dVj (t) + µjk dt Rjk (t) dM (t) = e− 0 r j

R − 0t r

k; k6=j

Rjk (t) dMjk (t) ,

(15.12)

j6=k

where Rjk (t) = bjk (t) + Vk (t) − Vj (t)

is recognized as the sum at risk in respect of transition from j to k at time t. Since the last term on the right of (15.12) is the increment of a martingale, the first term of the right is the difference between the increments of two martingales and is thus itself the increment of a martingale. This martingale has finite variation and, as will be explained below, is also continuous, and must therefore be constant. For this to be true for all realizations of the indicator functions Ij , we must have X dBj (t) − rj Vj (t) dt + dVj (t) + µjk dt Rjk (t) = 0 . (15.13) k; k6=j

This is nothing but Thiele’s differential equation. We also obtain that Rt X dM (t) = e− 0 r Rjk (t) dMjk (t) , j6=k

which displays the dynamics of the martingale M .

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

243

Finally, we explain why the (15.13) is the increment at t of a continuous function. The dt terms are continuous increments, of course. Outside jump times of the Bj both the Bj themselves and the Vj are continuous. At any time t where there is a jump in some Bj the reserve Vj jumps by the same amount in the opposite direction since Vj (t−) = ∆Bj (t) + Vj (t). Thus, Bj + Vj is indeed continuous.

C. Second example: Revisiting Exercise 56, the geometric Poisson price process. Martingale methods are certainly not needed in this case since the Poisson process is sufficiently well structured to allow of direct computation of all functions asked for in the exercise. But just for the sake of the example: Introduce V (t) = E[S(t)] . Let Ft , t ≥ 0, be the filtration generated by the Poisson process N . Fix a time T > 0 and introduce the martingale M (t)

= = =

E [ S(T ) | Ft ] i h eαt+βN (t) E eα(T −t)+β(N (T )−N (t) Ft S(t) V (T − t) .

Here we have made use of the fact that the Poisson process has stationary and independent increments. d Write V 0 = dt V and apply Itˇ o: dM (t) = S(t) α dt V (T − t) + S(t) V 0 (T − t)(−dt) + (S(t) − S(t−)) V (T − t)(15.14) , the last term accounting of jumps. Rewrite S(t) − S(t−)

= = = =

eαt+βN (t) − eαt+βN (t−)

eαt+β(N (t−)+∆N (t)) − eαt+βN (t−) S(t−) eβ∆N (t) − 1 S(t−) eβ − 1 dN (t) ,

where the last two step is due to the zero-or-one nature of the increments of counting process N . Upon inserting this into (15.14) and introducing the martingale dMN (t) = dN (t) − λ dt , we get dM (t)

h i S(t) α V (T − t) − V 0 (T − t) + eβ − 1 λ V (T − t) dt + S(t−) eβ − 1 dMN (t) ,

Here we have used the fact Rthat S(t−) dt = S(t) dt which is to be understood in R integral form: f (t−) dt = f (t) dt since the integral is not affected by a change of the integrand at an at most countable set of points. Arguing as in the previous example, the drift term in the dynamics of M must vanish, and we arrive at the differential equation h i V 0 (t) = α + eβ − 1 λ V (t) ,

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

244

The solution, subject to the obvious condition V (0) = 1 , is V (t)

exp

α +

i eβ − 1 λ t .

To rehearse the technique, you should use it to solve Item (c) in exercise 56. Start from the martingale Z T Z t S −1 (τ ) dτ + e−αt−βN (t) V (T − t) , M (t) = E S −1 (τ ) dτ Ft = 0

where

V (t) = E

S −1 (τ ) dτ

D. Third example: Revisiting pricing of the unit-linked term insurance with guarantee, pages 166-167. We need to determine π = E

n 0

1 ∨ e−

Rτ

0 r

g f (τ ) dτ .

Let Ft = σ{Ys ; 0 ≤ s ≤ t}, t ≥ 0, be the filtration generated by the ’economy process’ Y . Start from the martingale i h Rτ M (t) = E 1 ∨ e− 0 r g f (τ ) dτ Ft Z n R Z t Rτ Rτ Rt t e 0 r ∨ e− t r g f (τ ) dτ Ft 1 ∨ e− 0 r g f (τ ) dτ + e− 0 r E = t

Rτ Rt X Rτ 1 ∨ e− 0 r g f (τ ) dτ + e− 0 r Ie (t) We (t, e 0 r ) , e

where We (t, u) = E

n t

R − tτ r u∨e g f (τ ) dτ Y (t) = e .

∂ ∂ Assuming that the partial derivatives ∂t We (t, u) and ∂u We (t, u) exist, apply Itˇ o: Rt Rt R X τ dM (t) = 1 ∨ e− 0 r g f (t) dt + e− 0 r (−r(t) dt) Ie (t) We (t, e 0 r ) e

+ e− +e

Rt 0

X r

R − 0t r

e6=f

Ie (t)

Rτ Rτ Rτ ∂ ∂ We (t, e 0 r ) dt + We (t, e 0 r ) e 0 r r(t) dt ∂t ∂u

dNef (t) (Wf (t, e

Rτ

0 r

) − We (t, e

Substitute dNef (t) = dMef (t) + λef dt ,

Rτ

0 r

)) .

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

245

Rτ

where the Mef are martingales, and put U (t) = e 0 r . Arguing along the lines of Paragraph B, we conclude that 1 ∨ U (t)−1 g f (t) − U (t)−1 re We (t, U (t)) ∂ ∂ We (t, U (t)) + We (t, U (t)) U (t) re (t) + U (t)−1 ∂t ∂u X −1 λef (Wf (t, U (t)) − We (t, U (t))) = 0 . + U (t) f ; f 6=e

for all realizations of Y . We end up with the partial differential equation ∂ ∂ (u ∨ g) f (t) − re We (t, u) + We (t, u) + We (t, u) u re (t) ∂t ∂u X λef (Wf (t, u) − We (t, u)) = 0 , + f ; f 6=e

which are to be solved subject to the obvious conditions We (n, u) = 0 .

E. Remark on the technique. In all three examples we needed to determine the expected value of some random variable W that depends on the development of a stochastic process. Here is an outline of the method: Start from the martingale M (t) = E [ W | Ft ]. Inspect W and try to write it in the form W = W (T (t), U (t)) , where T (t) depends only on the future development of the process and U (t) depends only on the past history Ft . The random variable U (t) is the called the ’state variable(s)’ (it may be multi-dimensional). How to proceed from here depends on the properties of the driving stochastic process. In our situations (let us keep the example in Paragraph D in mind) we use the Markov property of Y to conclude that the conditional expected value M (t) must be a function only of the current time, state, and value of the state variable: X Ie (t) Fe (t, U (t)) . M (t) = FY (t) (t, U (t)) = e

Assuming continuous differentibility of the functions Fe with respect to t and u, use Itˇ o to form the dynamics of M : X ∂ ∂ dM (t) = Ie (t) Fe (t, U (t)) dt + Fe (t, U (t)) dU c (t) ∂t ∂u e X + [Ie (t) Fe (t, U (t)) − Ie (t−) Fe (t−, U (t−))] . e

Here U denotes the continuous part of U . The jump part may have contributions from possible jumps of U outside jump times for Y , but these will always cancel out and vanish in the end, however, see the example in Paragraph B above. In any case the jump part will consist of the following terms due to jumps of Y : X dNef (t) [Ff (t, U (t)) − Fe (t, U (t−))] . f 6=e

CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE

Now, insert dNef (t) = dMef (t) + λef dt and identify the martingale part and the drift part with the factor dt in the expression for dM (t). The drift part must vanish, and we arrive at a set of constructive non-stochastic differential equations from which the functions Fe (t, u) can be solved (at least numerically).

Bibliography [1] Aase, K.K. and Persson, S.-A. (1994). Pricing of unit-linked life insurance policies. Scand. Actuarial J., 1994, 26-52. [2] Aczel J. Lectures on Functional Equations and their Applications. Academic Press, 1966. [3] Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer-Verlag, New York, Berlin, Heidelberg. [4] Anderson, J.L. and Dow, J.B. (1948). Actuarial Statistics, Vol. II: Constructions of Mortality and other Tables. Cambridge University Press. [5] Barlow, R.E. and Proschan, F (1981): Statistical Theory of Reliability and Life Testing, Holt, Reinhart and Winston Inc. [6] Berger, A. (1939): Mathematik der Lebensversicherung. Verlag von Julius Springer, Vienna. [7] Bibby J.M., Mardia, K.V., and Kent J.T. Multivariate Analysis. Academic Press, 1979. [8] Bj¨ ork, T., Kabanov, Y., Runggaldier, W. (1997): Bond market structures in the presence of marked point processes. Mathematical Finance, 7, 211-239. [9] Bj¨ ork, T. (1998): Arbitrage Theory in Continuous Time, Oxford University Press. [10] Black, F., Scholes, M. (1973): The pricing of options and corporate liabilities. J. Polit. Economy, 81, 637-654. [11] Bowers, N.L. Jr., Gerber, H.U., Hickman, J.C., and Nesbitt, C.J. (1986). Actuarial Mathematics. The Society of Actuaries. Itasca, Illinois. [12] Cox, J., Ross, S., Rubinstein, M. (1979): Option pricing: A simplified approach. J. of Financial Economics, 7, 229-263. [13] Delbaen, F., Schachermayer, W. (1994): A general version of the fundamental theorem on asset pricing. Mathematische Annalen, 300, 463-520. [14] De Pril, N. (1989). The distributions of actuarial functions. Mitteil. Ver. Schweiz. Vers.math., 89, 173-183. [15] De Vylder, F. and Jaumain, C. (1976). Expos´e moderne de la th´eorie math´ematique des op´erations viag`ere. Office des Assureurs de Belgique, Bruxelles. [16] Dhaene, J. (1990). Distributions in life insurance. ASTIN Bull. 20, 81-92. [17] Donald D.W.A. Compound Interest and Annuities Certain. Heinemann, London, 1970.

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[18] Eberlein, E., Raible, S. (1999): Term structure models driven by general L´evy processes. Mathematical Finance, 9, 31-53. [19] Elliott, R.J., Kopp, P.E. (1998): Mathematics of financial markets, SpringerVerlag. [20] Esary, J.D. and Proschan, F (1972): Relationships among some notions of bivariate dependence, Annals of Mathematical Statistics, 43, 651-655. [21] F¨ ollmer, H., Sondermann, D. (1986): Hedging of non-redundant claims. In Contributions to Mathematical Economics in Honor of Gerard Debreu, 205-223, eds. Hildebrand, W., Mas-Collel, A., North-Holland. [22] Gantmacher, F.R. (1959): Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin. [23] Gerber, H.U. (1995). Life Insurance Mathematics, 2nd edn. Springer-Verlag. [24] Harrison, J.M., Kreps, D.M. (1979): Martingales and arbitrage in multiperiod securities markets. J. Economic Theory, 20, 1979, 381-408. [25] Harrison, J.M., Pliska, S. (1981): Martingales and stochastic integrals in the theory of continuous trading. J. Stoch. Proc. and Appl., 11, 215-260. [26] Hoem, J.M. (1969): Markov chain models in life insurance. Bl¨ atter Deutsch. Gesellschaft Vers.math., 9, 91–107. [27] Hoem, J.M. (1969): Purged and partial Markov chains. Skandinavisk Aktuarietidskrift, 52, 147–155. [28] Hoem, J.M. and Aalen, O.O. (1978). Actuarial values of payment streams. Scand. Actuarial J. 1978, 38-47. [29] Jordan, C.W. (1967). Life Contingencies. The Society of Actuaries, Chicago. [30] Karlin, S., Taylor, H. (1975): A first Course in Stochastic Processes, 2nd. ed., Academic Press. [31] Kellison S. (1970). The theory of interest. Richard D. Irwin, Inc. Homewood, Illinois. [32] Merton, R.C. (1973): The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141-183. [33] Merton, R.C. (1976): Option pricing when underlying stock returns are discontinuous. J. Financial Economics, 3, 125-144. [34] Møller, T. (1998): Risk minimizing hedging strategies for unit-linked life insurance. ASTIN Bull., 28, 17-47. [35] Norberg, R. (1985). Lidstone in the continuous case. Scand. Actuarial J. 1985, 27-32. [36] Norberg, R. (1991). Reserves in life and pension insurance. Scand. Actuarial J. 1991, 1-22. [37] Norberg R. (1993). Identities for present values of life insurance benefits. Scand. Actuarial J., 1993, p. 100–106. [38] Norberg, R. (1995): Differential equations for moments of present values in life insurance. Insurance: Math. & Econ., 17, 171-180. [39] Norberg, R. (1995): A time-continuous Markov chain interest model with applications to insurance. J. Appl. Stoch. Models and Data Anal., 245-256.

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[40] Norberg, R. (1998): Vasiˇcek beyond the normal. Working paper No. 152, Laboratory of Actuarial Math., Univ. Copenhagen. [41] Norberg, R. (1999): A theory of bonus in life insurance. Finance and Stochastics., 3, 373-390 [42] Norberg, R. (2001): On bonus and bonus prognoses in life insurance. Scand. Actuarial J. [43] Pliska, S.R. (1997): Introduction to Mathematical Finance, Blackwell Publishers. [44] Ramlau-Hansen, H. (1991): Distribution of surplus in life insurance. ASTIN Bull., 21, 57-71. [45] Sverdrup, E. (1969): Noen forsikringsmatematiske emner. Stat. Memo. No. 1, Inst. of Math., Univ. of Oslo. (In Norwegian.) [46] Thomas, J.W. (1995): Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag. [47] Width, E. (1986): A note on bonus theory. Scand. Actuarial J. 1986, 121-126.

Appendix A

Calculus A. Piecewise differentiable functions. Being concerned with operations in time, commencing at some initial date, we will consider functions defined on the positive real line [0, ∞). Thus, let us consider a generic function X = {Xt }t≥0 and think of Xt as the state or value of some process at time t. For the time being we take X to be real-valued. In the present text we will work exclusively in the space of so-called piecewise differentiable functions. From a mathematical point of view this space is tiny since only elementary calculus is needed to move about in it. From a practical point of view it is huge since it comfortably accommodates any idea, however sophisticated, that an actuary may wish to express and analyse. It is convenient to enter this space from the outside, starting from a wider class of functions. We first take X to be of finite variation (FV), which means that it is the difference between two non-decreasing, finite-valued functions. Then the left-limit Xt− = lims↑t Xs and the right-limit Xt+ = lims↓t Xs exist for all t, and they differ on at most a countable set D(X) of discontinuity points of X. We are particularly interested in FV functions X that are right-continuous (RC), that is, Xt = lims↓t Xs for all t. Any probability distribution function is of this type, and any stream of payments accounted as incomes or outgoes, can reasonably be taken to be FV and, as a convention, RC. If X is RC, then ∆Xt = Xt − Xt− , when different from 0, is the jump made by X at time t. For our purposes it suffices to let X be of the form Z t X Xt = X 0 + xτ dτ + (Xτ − Xτ − ) . (A.1) 0

0 q). A single premium π is paid at time 0. Determine the premium π by the principle of equivalence, assuming that the interest rate r is constant. Find the reserve at time t < n − q for an insured who is disabled and is currently receiving the disability benefit. (d) Consider the Markov chain model in Figure 7.3. Fill in appropriate statements in the enclosed program ’prores1’ to make it compute the transition probabilities pai (t, 10) and pii (t, 10) for t = 0, 1, . . . , 10 in the case where all the intensities are constant: µ = ν = σ = 0.01 and ρ = 0.

Exercise 52 (a) We adopt the usual notation and assumptions of the theory of multi-life insurance policies and consider two independent lives (x) and (y) with remaining life lengths Tx and Ty , respectively. Assume that the benefit is an assurance of 1 payable at time Ty if Tx + n/2 < Ty < n and that premium is payable at constant rate π until time min(Tx , Ty , n/2), where n is the term of the contract (fixed). Determine the equivalence premium π.

Exercise 53 The employees of a firm are automatically members of a pension scheme with salary dependent premiums and benefits defined as follows. Consider an employee (x), who enters the scheme x years old at time 0 (say), retires at pensionable age 65 at time m = 65 − x, and earns salary at rate S(t) per time unit at time t ∈ [0, m]. Contingent on survival, premium is payable continuously at rate πS(t) at time t ∈ [0, m], and pension is received as an endowment of 5S(m) upon retirement at time m. Assume that the economy is governed by a continuous time Markov chain Y (t), t ≥ 0, with state space J = {1, . . . , J}, constant intensities of transition λjk , j 6= k, and initial state Y (0) = i, say. At any time t ≥ 0 the accumulation factor U (t) of the investment portfolio is given by Z t X U (t) = exp r(s) ds , r(s) = Ij (s)rj , 0

and the salary rate is given by Z t S(t) = exp a(s) ds , 0

a(s) =

Ij (s)aj .

Here, Ij (t) = 1[Y (t) = j], and the rj and the aj are known, fixed numbers; rj is the interest rate and aj is the rate at which salary increases per time unit and per unit of salary when the economy is in state j.

APPENDIX F. EXERCISES

(a) Demonstrate how to determine the premium rate π that makes expected discounted benefits less premiums equal to 0 at time 0; Construct differential equations for benefits and for premiums and specify appropriate side conditions. (b) Explain that, if aj = rj for all states j so that salary is perfectly linked to investments, then equivalence can be attained in a large (in principle infinitely large) portfolio of identical policies.

Exercise 54 (a) Describe briefly the main characteristics of with-profit insurance and unit-linked insurance. (b) Consider a pure endowment benefit at time n (the term of the contract) against a single premium at time 0. Assume that the investment portfolio of the insurance company bears interest at rate r(t) at time t, where r is driven by a Markov chain as described in Question 6. Show how to determine the single premium for a unit-linked contract with sum insured max(U (n), g), where U (n) is the index of the investment portfolio given by U (t) as defined in Question 6, and g is a guaranteed minimum sum insured.

Exercise 55 We adopt the usual notation and assumptions of the theory of multi-life insurance policies and consider two independent lives (x) and (y) with remaining life lengths Tx and Ty , respectively. (a) Assume that the benefit is an assurance of 1 payable at time Ty if 2Tx < Ty < n and that premium is payable at constant rate π until time min(Tx , Ty , n/2), where n is the term of the contract (fixed). Determine the equivalence premium π. (b) Propose a method for computing the premium numerically. (Hint: One possibility is to treat t/2 px as a survival function t p˜x with intensity µ ˜ x+t , which you would need to express in terms of µ, and then solve a Thiele differential equation numerically.) (c) Determine the reserve at any time t, assuming that the insurer currently knows the complete past history of the two lives. You need to distinguish between various cases, whether (y) is alive or dead, whether t is before or after time n/2, and whether x is alive or dead and, if dead, when. Is the reserve always non-negative? (d) What is the variance of the present value of the benefit? SURPLUS, BONUS, WITH PROFIT, GUARANTEES, UNIT-LINKED A. Preliminaries. We are going to restate the theory of surplus and bonus and related problems in the framework of the simple, still fairly general, single life contract treated in Section 4.4 and add material on interest guarantees and unit-linked insurance.

APPENDIX F. EXERCISES

The terms of the contract are set out in the expression for the prospective reserve, Z n R Rn τ e− t (ru +µx+u ) du (µx+τ bτ − πτ ) dτ + e− t (ru +µx+u ) du bn , Vt = t

and the equivalence relation V0 = π 0 ,

(F.10)

where π0 is the lump sum premium payment collected upon the inception of the policy (it may be 0, of course). The equivalence relation (F.10) can be cast as Z t R Rt τ π0 + e− 0 (ru +µx+u ) du (πτ − µx+τ bτ ) dτ − e− 0 (ru +µx+u ) du Vt = 0 . (F.11) 0

The first two terms in (F.11) are the expected present value at time 0 of premiums less benefits up to and including time t. The second term is minus the expected present value at time 0 of benefits less premiums after time t. Thus, the equivalence principle ensures that, at any time, net incomes in the past provide precisely the amount needed to meet net liabilities in the future. Thiele’s differential equation is d Vt = πt − µx+t bt + (rt + µx+t ) Vt , dt

(F.12)

with the boundary condition Vt− = bn .

(F.13)

B. With profit contracts (participating policies); Surplus and Bonus. Insurance policies are long term contracts, with time horizons wide enough to capture significant variations in interest and mortality. Therefore, at time 0 when the contract is written with benefits and premiums binding to both parties, the future development of (rt , µx+t ), t > 0, is uncertain, and it is impossible to foresee which premium level will satisfy (F.11) and establish equivalence in the end. If it should turn out that, due to adverse development of interest and mortality, premiums are insufficient to cover benefits, then there is no way the insurance company can avoid a loss; it cannot reduce the benefits and it cannot increase the premiums since these were irrevocably set out in the contract at time 0. The only way the insurance company can prevent such a loss, is to charge a premium ’on the safe side’, high enough to be adequate under all likely scenarios. Then, if everything goes well, a surplus will accumulate. This surplus belongs to the insured and is to be repaid as so-called bonus, e.g. as increased benefits or reduced premiums. The usual way of setting premiums to the safe side is to base the calculation of the premium level and the reserves on a provisional first order basis, (rt∗ , µ∗x+t ), t > 0, which represents a worst case scenario and leads to higher premium and reserves than are likely to be needed. We follow common practice and take the first order interest rate to be constant, r ∗ . (From a mathematical point of view this is just a matter of notation.) The reserve based on the prudent first order assumptions is called the first order reserve, and we denote it by Vt∗ . It satisfies Thiele’s differential equation d ∗ Vt = πt + r∗ Vt∗ − µ∗x+t (bt − Vt∗ ) , dt

(F.14)

∗ subject to the natural side condition Vn− = bn . The premiums are determined so as to satisfy the first order equivalence relation V0∗ = π0 .

APPENDIX F. EXERCISES

Taking our stand at a given time t after the inception of the policy, the development of interest and mortality in the past, (rτ , µx+τ ), τ ≤ t, is now known and can be invoked in an updated calculation of the net incomes up to time t. The future development of interest and mortality, (rτ , µx+τ ), τ > t, remains uncertain, however, so for the assessment of future liabilities one must stick to the conservative first order basis, (r∗ , µ∗x+τ ), τ > t. Thus, instead of the balance equation (F.11), which cannot be set up since it involves unknown future rates of interest and mortality, we have the following expression for the mean surplus per policy at time t: Z t R Rt τ e− 0 (ru +µx+u ) du (πτ − µx+τ bτ ) dτ − e− 0 (ru +µx+u ) du Vt∗ . (F.15) St = π 0 + 0

If the factual interest and mortality in the past were more favourable than the pessimistic first order basis, then this surplus is positive.

C. Emergence of surplus. To see how the surplus is emerges, we need to study the dynamics of St . Differentiating (F.15), using (F.14), and rearranging terms, we obtain Rt d St = e− 0 (ru +µx+u ) du ct , dt

where ct = (rt − r∗ ) Vt∗ + (µ∗x+t − µx+t )(bt − Vt∗ ) .

(F.16)

Obviously, ct is the rate at which surplus emerges per survivor and per time unit at time t. Interpret the two terms on the right hand side of (F.16) as surplus emerging from safety margins in the interest rate and in the mortality rate, respectively. We can reasonably say that a first order element is set on the safe side if the corresponding contribution to the surplus is positive. By inspection of the first term on the right of (F.16), we see that r ∗ is on the safe side as long as it is less than the true rt (provided that the first order reserve is positive, as it should be for any meaningful contract). By inspection of the second term on the right of (F.16), we see that the sign of the sum at risk by death, bt − Vt∗ , determines how to set first order mortality to the safe side: If the sum at risk is positive (e.g. term assurance or endowment assurance), then µ∗x+t is on the safe side if it is bigger than µx+t . If the sum at risk is negative (as is the case for e.g. a pure endowment, a deferred annuity, or some other savings insurance with bt = 0), then µ∗x+t is on the safe side if it is less than µx+t . D. Redistribution of surplus as bonus. The word bonus is Latin and means ’good’. In insurance terminology it denotes various forms of repayments to the policyholders of that part of the company’s surplus that stems from good performance of the insurance portfolio, a sub-portfolio, or the individual policy. In the present context of life insurance it denotes the repayments of surplus stemming from favourable development of interest and mortality. Let us denote such repayments by ˜b in general. For the sake of concreteness, suppose bonuses are paid back continuously at rate ˜bt per survivor for 0 < t < n and possibly with a lump sum ˜bn per survivor at time t = n. By statute, surplus is to be repaid in its entirety, which means that equivalence is to be re-established on basis of the true interest and mortality conditions when these are ultimately known at the term of the contract: Z n R Z n R τ τ e− 0 (ru +µx+u ) du ˜bτ dτ e− 0 (ru +µx+u ) du cτ dτ = 0

+ e−

0 (ru +µx+u ) du

˜bn .

(F.17)

APPENDIX F. EXERCISES

In the following Paragraphs E - G we will study some commonly used bonus schemes. E. Cash Bonus. This means that surplus is being repaid continually as it emerges, i.e. ˜bt = ct , 0 < t < n, and ˜bn = 0. It may for instance take the form of a premium deductible payable at rate ct as long as the insured is alive during the contract period. In the case of a term assurance contract it could reasonably take the form of an additional payment ˆbt upon death at time t ∈ (0, n), and a natural choice is ˆbt = ct /µx+t . Exercise 1-12 Verify that the two cash bonus schemes described above comply with the ultimate equivalence requirement (G.36). Construct a scheme that is a combination of the two proposed here. F. Terminal Bonus. This means that surplus is repaid as a lump sum ˜bn to survivors at the end of the term, and ˜bt = 0, 0 < t < n. Exercise 1-13 Determine ˜bn by (G.36). G. Purchase of Additional Insurance. Under this bonus scheme the surplus is spent on purchase of additional insurance. Additional insurance is written on the first order basis and will therefore also generate surplus, which in its turn will be used for further purchase of additional insurance, and so on. The scheme is non-trivial and requires a bit of theoretical reasoning: For a policy in force at time t let Vt∗+ denote the expected present value, on the first order basis, of future benefits only; Z n R Rn ∗ τ ∗ ∗ ∗ e− t (r +µx+u ) du µx+τ bτ dτ + e− t (r +µx+u ) du bn . Vt∗+ = t

It satisfies the Thiele’s differential equation d ∗+ V = r∗ Vt∗+ − µ∗x+t (bt − Vt∗+ ) , dt t

(F.18)

∗+ with natural side condition Vn− = bn . ∗+ The quantity Vt is the single premium payable at time t if the insured then were to purchase an additional insurance for the balance of the term, with the same benefits as in the original contract. Spending the surplus ct dt generated in [t, t + dt) as a single premium for additional benefits of the form specified in the original contract, will buy the insured a fraction qt dt of future benefits given by

ct = qt Vt∗+ .

(F.19)

At any time t ∈ (0, n) the death benefits from the original contract and from the additional benefits purchased during (0, t] total (1 + Qt ) bt ,

(F.20)

where Qt =

qτ dτ . 0

(F.21)

APPENDIX F. EXERCISES Likewise, the total endowment benefit at the term of the contract is (1 + Qn ) bn .

(F.22)

At time t the total surpluses from the original contract and the additional benefits purchased during (0, t] emerge at rate ct = (rt − r∗ ) (Vt∗ + Qt Vt∗+ ) + (µ∗x+t − µx+t ) (1 + Qt )bt − Vt∗ − Qt Vt∗+ . (F.23)

Now all elements needed are in place, and a dynamic computation will deliver the solution. First, at the time of the inception of the contract, the functions Vt∗ and Vt∗+ and the equivalence premium πt are determined by use of the program ’prores1.pas’ (or ’prores2.pas’). The computation goes backwards starting from the side conditions ∗+ ∗ Vt− = bn and Vt− = bn . Then, as time goes by and surpluses are being observed and disposed of, one computes simultaneously the functions Vt∗ and Vt∗+ (again) and the random function Qt as solutions to the differential equations (F.14), (F.18), and (rewrite (F.19)) d 1 Qt = ∗+ ct , dt Vt

(F.24)

with ct given by (F.23). The computation goes forwards, starting from time t = 0 with the initial conditions V0∗ = 0 , V0∗+ = V0∗+ (picked from the first computation), and Q0 = 0 . Having determined Q, the benefits under this bonus scheme are now given by (F.20) and (F.22). H. Prognostication of bonus. At regular times (typically annually) the customer receives a statement of his policy account, informing about bonus earned from surplus in the past and also predicting future bonuses based on a qualified guess as to the future development of interest and mortality. Exercise 1-14 Outline such a statement with these pieces of information for the standard contract considered so far, including the relevant formulas, and basing the prognostication of future surplus on the assumption that rτ = r∗ + ∆r and µx+τ = µ∗x+τ − ∆µ for some given positive ∆r and ∆µ. Exercise 1-15 Apply the present theory to a pension insurance policy for which benefits are an m year deferred life annuity payable at level rate 1 per year in n years, and premiums are payable at level rate during the deferred period. Write out all relations and formulas that differ from the corresponding ones above. Will surplus emerge also in the benefit period [m, m + n)?

APPENDIX F. EXERCISES

Exercise 1-16 Extend the theory so as to include expenses. Consider an endowment insurance with constant sum insured b and constant gross premium rate π 0 (no down payment π00 at time 0). Assume that true expenses incur with a lump sum α0 + α00 b at time 0 and 0 thereafter continuously at rate βt0 + βt00 π 0 + γt0 + γt00 b + γt000 Vt∗ at time t ∈ (0, n) as ∗0 long as the policy is in force. Here Vt denotes the gross premium reserve on first order basis. First order assumptions specify that expenses incur with a lump sum α∗ b at time 0 and thereafter continuously at constant rate β ∗ π 0 + γ ∗ b. Discuss how the first order elements can be set on the safe side. Observe that there is a lump sum contribution to surplus at time 0. I. Stochastic interest. The uncertain development of the second order elements can be built into the model by describing the interest rate and the (parameters of the) mortality rate as stochastic processes. To keep things simple, we will focus on interest, which is the more important of the two, and assume that the mortality is perfectly predicted by the first order basis: µ∗x+t = µx+t . This means that contributions to surplus stem only from interest gains, so that ct = (rt − r∗ )Vt∗ .

(F.25)

As a simple, but flexible, model for stochastic interest, we will assume that {rt }t≥0 is generated by the Markov chain model in Section 7.8 of BL, see also Exercise 2. To save space, we will write Yt and rt instead of Y (t) and r(t) and, since subscripts are now used for the time variable t, denote the state-wise interest rate by r e . The statement of account, which is regularly sent to the insured, usually comes with a prognosis of future bonuses on the insurance. Such a prognosis must be based on a qualified guess about the future development of the factual valuation basis – in our simplified situation about r. This guess may be exogenous to the model, e.g. based on combined opinions of experts in the finance department of the company. Having adopted a stochastic model for r, the insurer can make an endogenous, model-based forecast of future bonus payments. Thus, consider a policy which is still in force at time t, and suppose the insurer wants to inform the insured about the conditional expected value of future bonuses, given that the current interest rate is rt = re (which means Yt = e, assuming that all r e are different). We will consider a few examples. J. Cash bonus: The rate at which bonus will be paid at some fixed future time u, provided the insured is then alive, is W = (ru − r∗ )Vu∗ . At time t < u, given rt = re , W is predicted by its conditional expected value We (t) = E[W | rt = re ] . Exercise 1-17 Show that the functions We (t) are the solution to the differential equations X d We (t) = λef (We (t) − Wf (t)) , dt `;f 6=e

APPENDIX F. EXERCISES subject to the conditions

We (u) = (r e − r∗ )Vu∗ ,

e = 1, . . . , J .

K. Terminal bonus: Bonus payable as a lump sum at the term of the contract n, provided the insured is then alive, is Z n R n e τ rs ds (rτ − r∗ )Vτ∗ dτ W = 0 Z t R t = Wt0 e τ rs ds (rτ − r∗ )Vτ∗ dτ + Wt00 , 0

where Wt0

Wt00

e t rs ds , Z n R n e τ rs ds (rτ − r∗ )Vτ∗ dτ . t

The random variables Wt0 and Wt00 , which are unknown at time t, are predicted by We0 (t)

We00 (t)

E[Wt0 | rt = re ] ,

E[Wt00 | rt = re ] .

Exercise 1-18 Writing Wt0

0 ert dt Wt+dt ,

Wt00

00 , Wt0 (rt − r∗ ) Vt∗ dt + Wt+dt

show that the functions We0 (t) and We00 (t) are the solution to the differential equations d 0 W (t) dt e

d 00 We (t) dt

−re We0 (t) +

f ;f 6=e

λef (We0 (t) − Wf0 (t)) ,

−We0 (t)(re − r∗ )Vt∗ +

f ;f 6=e

λef (We00 (t) − Wf00 (t)) ,

subject to the conditions We0 (n−) = 1 , We00 (n−) = 0 , e = 1, . . . , J Y . L. Additional benefits: At a fixed future time u bonus is paid as a multiple Qu of the contractual benefits provided the insured is then alive. At time t we decompose Qu into Qt , which is known, and Qu − Qt , which is unknown, and we need to predict the latter. Recalling (F.19) and (F.25), start from the differential equation d Vt∗ Qt = (rt − r∗ ) ( ∗+ + Qt ) dt Vt

APPENDIX F. EXERCISES and use the technique with integrating factor to obtain Qu = Wt0 Qt + Wt00 , where Wt0

Wt00

∗

e t (rs −r ) ds , Z u R u ∗ Vτ∗ e τ (rs −r ) ds (rτ − r∗ ) ∗+ dτ . Vτ t

Exercise 1-19 Derive differential equations for the state-wise predictions We0 (t) and We00 (t) of Wt0 and Wt00 . Exercise 1-20 (a) Predict discounted future cash bonuses given survival to n, Z n R τ e− t rs ds (rτ − r∗ )Vτ∗ dτ . t

(b) Predict discounted future cash bonuses, Z n R τ e− t (rs +µx+s ) ds (rτ − r∗ )Vτ∗ dτ . t

(c) Find differential equations for the conditional variance, given rt = re , of the future cash bonuses. You may try your hand also on the conditional variance of future terminal bonus.

As we have said before, the Markov model proposed here can hardly be 100 per cent realistic. Now, the usefulness of a model depends on its purpose. The sole purpose of the interest rate model is to provide the insured with a reasonable guess as to his future prospects of bonus, and for that purpose a rough model can certainly be adequate. Anyway, at the end of the day the bonus payments will be determined entirely by the factual interest rate and will not depend on the assumptions in our model. M. Guaranteed interest. Recall the basic rules of the ’with profit’ insurance contract: On the one hand, any surplus is to be redistributed to the insured. On the other hand, benefits and premiums set out in the contract cannot be altered to the insured’s disadvantage. This means that negative surplus, should it occur, cannot result in negative bonus. Thus, the with profit policy comes with an interest rate guarantee to the effect that bonus is to be paid as if factual interest were no less than first order interest, roughly speaking. For instance, cash bonus is to be paid at rate (rt − r∗ )+ Vt∗ per survivor at time t, hence the insurer has to cover (r∗ − rt )+ Vt∗ .

(F.26)

APPENDIX F. EXERCISES

Similarly, terminal bonus (typical for e.g. a pure endowment benefit) is to be paid as a lump sum Z n R n e τ rs ds (rτ − r∗ )Vτ∗ dτ 0

per survivor at time n, hence the insurer has to cover Z n R n e τ rs ds (r∗ − rτ )Vτ∗ dτ 0

(F.27)

(We write a+ = max(a, 0) = a ∨ 0.) An interest guarantee of this kind represents a liability on the part of insurer. It cannot be offered for free, of course, but has to be compensated by a premium. This can certainly be done without violating the rules of game for the participating policy, which lay down that premiums and benefits be set out in the contract at time 0. Thus, for simplicity, suppose a single premium is to be collected at time 0 for the guarantee. The question is, how much should it be? Being brought up with the principle of equivalence, we might think that the expected discounted value of the liability is an agreeable candidate for the premium. However, the rationale of the principle of equivalence, which was to make premiums and benefits balance on the average in an infinitely large portfolio, does not apply to financial risk. Interest rate variations cannot be eliminated by increasing the size of the portfolio; all policy-holders are faring together in one and the same boat on their once-in-a-lifetime voyage through the troubled waters of their chapter of economic history. This risk cannot be averaged out in the same way as the risk associated with the lengths of the individual lives. None the less, in lack of anything better, let us find the expected discounted value of the interest guarantee, and just anticipate here that this actually would be the correct premium in an extended model specifying a so-called complete financial market. Those who are familiar with basic arbitrage theory know what this means. Those who are not should just imagine that, in addition to the bank account with the interest rate rt , there are some other investment opportunities, and that any future financial claim can be duplicated perfectly by investing a certain amount at time 0 and thereafter just selling and buying available assets without any further infusion of capital. The initial amount required to perform this duplicating investment strategy is, quite naturally, the price of the claim. It turns out that this price is precisely the expected discounted value of the claim, only under a different probability measure than the one we have specified in our physical model. With these reassuring phrases, let us proceed to find the expected discounted value of the interest guarantee. (a) Cash bonus with gurantee given by (F.26): Given that r0 = re (say), the price of the total claims under the guarantee, averaged over an infinitely large portfolio, is Z n R τ E e− 0 r (r∗ − rτ )+ Vτ∗ τ px dτ r0 = re . (F.28) 0

A natural starting point for creating some useful differential equations by the backward construction is the ’price of future claims under the guarantee’ in state e at time t, Z n R τ (F.29) e− t r (r∗ − rτ )+ Vτ∗ τ px dτ rt = re , We (t) = E t

e = 1, . . . , J Y , 0 ≤ t ≤ n. The price in (F.28) is precisely We (0).

APPENDIX F. EXERCISES

Conditioning on what happens in the time interval (t, t + dt] and neglecting terms of order o(dt) that will disappear in the end anyway, we find X e λef dt Wf (t) . We (t) = (1 − λe· dt) (r∗ − re )+ Vt∗ t px dt + e−r dt We (t + dt) + f ; f 6=e

From here we easily arrive at the differential equations X d We (t) = −(r ∗ − re )+ Vt∗ t px + re We (t) − λef dt (Wf (t) − We (t)) , dt

(F.30)

f ; f 6=e

which are to be solved subject to the conditions We (n−) = 0 .

(F.31)

(b) Terminal bonus at time n given by (F.27): Given r0 = re , the price of the claim under the guarantee, averaged over an infinitely large portfolio, is # " Z n R R n e − 0n rs ds rs ds ∗ ∗ τ E e e (r − rτ )Vτ dτ n p x r0 = r 0 + " Z # n Rτ e =E (F.32) e− 0 r (r∗ − rτ )Vτ∗ dτ r0 = r n p x . 0 +

Let us try and copy the method of Item (a) and look at the ’price of the claim at time t’, which should be the conditional expected discounted value of the claim, given what we know at the time: # " Z n R R n r ds ∗ ∗ − tn rs ds r ; 0 ≤ τ ≤ t e τ s (r − rτ )Vτ dτ p E e τ n x 0 + # " Z n R − tτ r ∗ ∗ e (r − rτ )Vτ dτ Ut + =E rτ ; 0 ≤ τ ≤ t n p x . t +

(F.33)

where Ut =

τ r

(r∗ − rτ )Vτ∗ dτ .

The quantity in (F.33) is more involved than the one in (F.29) since it depends effectively on the past history of interest rate through Ut . We can, therefore, not hope to end up with the same simple type of problem as in Item (a) above and in all other situations encountered so far, where we essentially had to determine the conditional expected value of some function depending only on the future course of the interest rate. Which was easy since, by the Markov property, we could look at state-wise conditional expected values We (t), e = 1, . . . , J Y , say. These are deterministic functions of the time t only and can be determined by solving ordinary differential equations. Let us proceed and see what happens. Due to the Markov property (conditional independence between past and future, given the present) the expression in (F.33) is a function of t, rt and Ut . Dropping the uninteresting factor n px , consider its value for given Ut = u and rt = re , # " Z n R e − tτ r ∗ ∗ e (r − rτ )Vτ dτ u+ We (t, u) = E rt = r . t +

APPENDIX F. EXERCISES Use the backward construction: We (t, u) = (1 − λe· dt)E +

∗

u + (r − r

)Vt∗

dt + e

−r e dt

t+dt

Rτ − t+dt r

∗

(r −

λef dt Wf (t, u) =

f ; f 6=e

(1 − λe· dt)e−r Insert here e±r

e dt

We (t + dt, er

u + (r∗ − re )Vt∗ dt) +

rτ )Vτ∗

# e dτ rt+dt = r +

λef dt Wf (t, u) .

f ; f 6=e

= 1 ± re dt + o(dt), e

We (t + dt, er dt u + (r∗ − re )Vt∗ dt) = ∂ ∂ We (t, u)(u r e + (r∗ − re )Vt∗ ) dt + o(dt) , We (t, u) + We (t, u) dt + ∂t ∂u and proceed in the usual manner to arrive at the partial differential equations X ∂ ∂ λef (Wf (t, u)−We (t, u)) = 0 . We (t, u)+(ur e +(r∗ −re )Vt∗ ) We (t, u)−r e We (t, u)+ ∂t ∂u f ; f 6=j

These are to be solved subject to the conditions We (n−, u) = u+ , e = 1, . . . , J Y . Since the functions we are interested in involved both t and Ut , we are lead to state-wise functions in two arguments and, therefore, quite naturally end up with partial differential equations for those. N. Unit linked insurance. We have been discussing the participating (or with profit) policy, characteristic of which is that benefits and premiums are set out in nominal amounts in the contract at time 0. Thus, For the fairly general contract described in the introduction to this note, the functions bt and πt would be deterministic, not dependent on the development of the interest rate over the term of the contract. Introduce Rt Ut = e 0 ru du , which is the value at time t of a unit deposited in the investment portfolio at time 0. We may call it the price index of the investment portfolio. Recast the equivalence relation (F.11) as Z n (F.34) −π0 + Uτ−1 τ px (µx+τ bτ − πτ ) dτ + Un−1 n px bn = 0 . 0

With bt and πt fixed at time 0 there is no way one can make them fulfill (F.34) for all possible future courses of the interest rate process. Depending on the economic development there will be inequality in the one or the other direction. The financial risk thus introduced is hedged (hopefully perfectly) by setting premiums on a prudent first order basis, i.e. replacing the unknown rt in (F.34) by some r ∗ set to the ’safe side’.

APPENDIX F. EXERCISES

An alternative scheme for management of financial risk in life insurance is known as unit linked insurance (also called variable life insurance). The idea of this concept is to link benefits and premiums to the performance of the investment portfolio, that is, let contractual payments be inflated by the index U instead of being fixed nominal amounts. Under a perfect unit linked contract we would have bt = Ut b◦t and πt = Ut πt◦ for some ’baseline’ benefits b◦t and premiums πt◦ , t ∈ [0, n], determined at time 0. Inserting this into (F.34) gives Z n ◦ ◦ ◦ −π0 + (F.35) τ px (µx+τ bτ − πτ ) dτ + n px bn = 0 . 0

We see that, for a given baseline benefit function b◦t , the equivalence relation can be fulfilled by a suitable choice of baseline premium rate πt◦ . The future course of the interest rate process has disappeared from the relation upon discounting the indexed payments and, thus, the problem with financial risk has been resolved by the perfect unit linked device. However, in practice unit-linked contracts are usually not perfect in the sense described above. Typically, only the benefits are linked to the investment index, whereas premiums are not. Furthermore, the unit linked contract is typically equipped with a guarantee specifying that the benefit cannot fall below a certain pre-specified nominal minimum. Such modifications to the perfect linking re-introduce financial risk, of course. Before we return to mathematics, we dare to suggest that guarantees, whether they apply to benefits under unit linked contracts or to interest under with profit contracts, are remains of the social security concern that traditionally was paramount in life insurance. They introduce a discrimination between various forms of saving; unlike those who invest in stocks, bonds, or real estate, those who invest in life or pension insurance are granted the privilege of gaining from booms without loosing from recessions. However, parity can be restored by letting the insured pay for the guarantee. Thus we proceed to determine its right price. For an example, let us try and determine the single premium payable at time 0 for a term insurance with sum bt = (Ut ∨ g) at time t ∈ (0, n), where g is the guaranteed minimum sum insured specified at time 0. The premium is Z n R R Z n Rτ τ τ π=E 1 ∨ e− 0 r g fτ dτ , e− 0 r e 0 r ∨ g τ px µx+τ dτ = E 0

where we have abbreviated

ft = t px µx+t . Following the recipe in Item (b) of Paragraph M, consider the ’price of future claims at time t’, Z n R R τ τ E e− t r e 0 r ∨ g fτ dτ rτ ; 0 ≤ τ ≤ t t Z n R − tτ r (F.36) =E Ut ∨ e g fτ dτ rτ ; 0 ≤ τ ≤ t . t

Arguing as before, the expression in (F.36) is a function of t, rt and Ut . Consider its value at time t for given Ut = u, and rt = re , Z n R e − tτ r We (t, u) = E u∨e g fτ dτ rt = r . t

APPENDIX F. EXERCISES When r0 = re , the premium we seek is We (t, 1) Now use the backward construction, this time leaving details aside: We (t, u) = Z e (1 − λe· dt)E (u ∨ g) ft dt + e−r dt +

λef dt Wf (t, u)

n t+dt

u ∨ e−

f ; f 6=e

Rτ

t+dt r

g fτ dτ rt+dt = re

X e e λef dt Wf (t, u) . = (1 − λe· dt) (u ∨ g) ft dt + e−r dt We (t + dt, er dt u) + f ; f 6=e

Insert e±r

= 1 ± re dt + o(dt) and

We (t + dt, er

= =

We (t + dt, u + ur e dt) + o(dt) ∂ ∂ We (t, u) + We (t, u) dt + We (t, u) u r e dt + o(dt) , ∂t ∂u

and fill in some details to arrive at the partial differential equations X ∂ ∂ (u∨g)ft −re We (t, u)+ We (t, u)+ We (t, u) u re + λef (Wf (t, u)−We (t, u)) = 0 . ∂t ∂u f ; f 6=e

These are to be solved subject to the conditions We (n−, u) = 0 , e = 1, . . . , J Y .

O. Salary dependent premiums and benefits. The employees of a firm are enrolled in a pension scheme with salary dependent premiums and benefits. Consider an employee (x), who enters the scheme x years old at time 0 (say), retires at pensionable age 65 at time m = 65 − x, earns salary at rate S(t) per time unit at any time t < m, and will receive pension continuously at level rate Q (yet to be determined) for n years after retirement. Let us first work under the assumption that the interest rate r is constant and known for the entire term of the contract up to time m + n. (a) We will first consider a ’defined contributions’ scheme under which a fixed proportion of the salary is used as premium for additional pension benefits. It will turn out that equivalence is automatically attained regardless of the development of the salary. In any small time interval [t, t + dt), t < m, the insured earns S(t) dt. A fixed proportion πS(t) dt, 0 < π < 1, of this salary is used as a single premium for a pension of q(t) dt per time unit in the time interval [m, m + n]. By the principle of equivalence, qt is given by π S(t) dt = q(t) dt m−t|n a ¯x+t , that is, S(t) . a ¯x+m n The total rate of pension per time unit purchased by a survivor at time m is Z m Z m S(τ ) q(τ ) dτ = π Q= dτ . E a ¯x+m n m−τ x+τ 0 0 q(t) = π

S(t)

¯x+t m−t|n a

= π

m−t Ex+t

APPENDIX F. EXERCISES

It should be fairly obvious that the equivalence requirement is fulfilled by this scheme since, no matter how much or little salary the insured will earn and no matter if it can be predicted or not at the outset, the benefits are entirely determined by the salary-dependent contributions. Let us, however, just check: For any given salary function, S, the expected discounted premiums are Z m Z m R τ e− 0 (r+µx+s ) ds S(τ ) dτ = π π τ Ex S(τ ) dτ , 0

and the expected discounted benefits are Z m Z m S(τ ) Q m|n a ¯x = π dτ m Ex a ¯x+mn = π τ Ex S(τ ) dτ , ¯x+m n m−τ Ex+τ a 0 0 where we have used the well-known identity

m Ex

= τ Ex m−τ Ex+τ .

(b) Suppose now that, instead of letting the contributions determine the benefits, the benefits are linked to the salary whereas the premiums are not. More specifically, suppose pension is payable continuously at rate 0.75S(m) (i.e. 75% of the salary rate at the time of retirement) for n years after retirement, and that premium is payable at a prefixed level rate π while active (both contingent on survival, of course). To determine the premium level π at time 0, we now need to make assumptions about the future development of the salary. Let us also abandon the unrealistic assumption that the future development of the interest rate is known: Assume that the economy is governed by a continuous time Markov chain Y (t), t ≥ 0, with state space J = {1, . . . , J}, constant intensities of transition λjk , j 6= k, and initial state Y (0) = i, say. At any time t ≥ 0 the accumulation factor U (t) of the investment portfolio is given by Z t X U (t) = exp r(s) ds , r(s) = Ij (s)rj , (F.37) 0

and the salary rate is given by Z t a(s) ds , S(t) = exp 0

a(s) =

Ij (s)aj .

Exercise 1-21 Determine the premium rate π that makes expected discounted benefits less premiums equal to 0 at time 0; Construct differential equations for benefits and for premiums and specify appropriate side conditions.

Exercise 1-22

APPENDIX F. EXERCISES

Consider the Markov chain interest model outlined in Figure F.5. Fill in appropriate statements in the h program R ’prores2’ to make iit compute the state-wise expected dis 5 count factors E exp − 0 r(s) ds Y (0) = j , j = 1, 2, 3. Exercise 1-22 Let the benefit be an n years unit-linked pure life endowment with guaranteed sum insured max(U (n), g), and let premium be payable continuously at level rate π during the insurance period. Show how to determine π. Exercise 56 Let {N (t)}t≥0 be Poisson process with intensity λ. This is a counting process of even simpler type than the counting processes associated with a Markov chain; N is not only Markov, but also has independent increments. Thus, in any small time interval [t, t + dt) the process N makes a jump of 1 with probability λ dt regardless of the past history of the process in [0, t). Let the price S(t) of a share of stock at time t be modelled as a so-called geometric Poisson process with drift, S(t) = exp (αt + βN (t)) , t ≥ 0. If β = 0, then S(t) is just the accumulation factor for a bank account with fixed interest rate. The Poisson term in the exponent adds jumps at random times, and a jump at time t makes the stock price jump from S(t−) to S(t) = S(t−) eβ . Thus, γ = eβ − 1 is the relative change (S(t) − S(t−))/S(t−) in the stock price at the jump time. Between the jumps the stock price increases at fixed “rate of interest” α. (a) Find the expected value E[S(t)] at time 0 of the stock price at time t, and do this in two ways: First, work directly with the Poisson distribution of N (t) and, second, solve a differential equation obtained by the direct backward construction (condition on “what happens in the small time interval [0, dt)”). Explain that, having determined the expected value, higher order moments are easily obtained. (b) Find the dynamics dS(t) of the stock price by applying the change of variable rule, see Appendix A. (c) Using the direct backward construction, show that the expected present value of a perpetuity (an everlasting annuity), is Z ∞ E S −1 (τ ) dτ = (α + λ(1 − e−β ))−1 . 0

(d) Let {N1 (t)}t≥0 and {N2 (t)}t≥0 be independent Poisson processes with intensities λ1 and λ2 , respectively. Let the price S(t) of a share of stock at time t be S(t) = exp (αt + β1 N1 (t) + β2 N2 (t)) , t ≥ 0. Letting β1 and β2 have opposite signs, we have created a stock price which may increase or decrease with instantaneous jumps. Write out the straightforward analogs

APPENDIX F. EXERCISES of the formulas in Items (a) - (c) for this more general model.

Exercise 57 (a) Let A and A0 be two payment functions with retrospective reserves denoted by U and U 0 , respectively. Assuming that the interest rate is always positive, verify the following rather obvious assertion: If At ≤ A0t for all t, then Ut ≤ Ut0 for all t. (In particular, any advancement of deposits will produce an increase of the retrospective reserve if the interest rate is positive. This is the general circ*mstance underlying ¯n ≤ a ¨n .) results like the following about ordering of expected present values: an ≤ a (b) Let the payment stream A represent deposits less withdrawals on an n years savings account that bears interest with strictly positive interest rate. It is required that Ut ≥ 0 for all t, with strict inequality for some t, and that Un = 0. Prove that An < 0, and explain this result. Exercise 58 (a) Show that, for t ≤ s ≤ u, pjj (t, u) = pjj (t, s) pjj (s, u), which is obvious. (b) Given start in state a at time 0, write up the probability that the process remains in a during the time interval [0, t1 ), then jumps to state i in [t1 , t1 + dt1 ), then remains in i during the time interval [t1 + dt1 , t2 ), then jumps to state a in [t2 , t2 + dt2 ), then remains in a during the time interval [t2 + dt2 , t3 ), and finally jumps to state d in the time interval [t3 , t3 + dt3 ). This is the probability of one particular full specification of the history of the process.

Exercise 59 Read Sections 9.1 and 9.2 in ’Basic Life Insurance Mathematics’. Suppose n independent lives, which follow the same Gompertz-Makeham mortality law with intensity µ(t) = α + β exp(γt), are observed from birth until death. Find the equations for the Maximum likelihood estimators for the parameters α, β, and γ, and find the asymptotic properties of the estimators.

Exercise 60 In the situation of Paragraphs 9.1.B-E, consider the problem of estimating µ from the Di alone, the interpretation being that it is only observed whether survival to z takes place or not. Show that the likelihood based on Di , i = 1, . . . , n, is q N (1 − q)n−N , with q = 1−e−µz , the probability of death before z. (Trivial: it is a binomial situation.) Note that N is now sufficient, and that the class of distributions is a regular exponential class. The MLE of q is q∗ =

N n

APPENDIX F. EXERCISES with the first two moments Eq ∗ = q, Varq ∗ =

q(1 − q) . n

It is UMVUE in the class of estimators based on the Di . The MLE of µ = − ln(1 − q)/z is µ∗ = − ln(1 − q ∗ )/z. Apply standard asymptotic results about MLE to show that q µ∗ ∼as N µ, 2 . nz (1 − q) The asymptotic efficiency of µ ˆ relative to µ∗ is !2 2 µz µz e 2 − e− 2 sinh(µz/2) asVarµ∗ = = asVarˆ µ µz µz/2 (sinh is the hyperbolic sine function defined by sinh(x) = (ex − e−x )/2). This function measures the loss of information suffered by observing only death/survival by age z as compared to inference based on complete observation throughout the time interval (0, z). It is ≥ 1 and increases from 1 to ∞ as µz increases from 0 to ∞. Thus, for small µz, the number of deaths is all that matters, whereas for large µz, the life lengths are all that matters. Reflect over these findings. Exercise 61 We refer to the disability model. (a) Consider an x years old insured who enters an insurance scheme at time 0. The Rt probability paa (0, t) = exp(− 0 (µx+s + σx+s ) ds) can be viewed as the probability (0) paa (0, t) of being active at time t after having been disabled 0 times. Derive forward (1) differential equations for the probability pai (0, t) of being disabled for the first time (1) at time t and for the probability paa (0, t) of being active at time t after having been disabled once. (b) Find the probability of being disabled for the first time at time t and that the disability has lasted for at least q years. (c) At time 0 an active person aged x buys a disability pension insurance with the following terms: The benefit is a pension payable at level rate 1 during the first disability, but only after it has lasted for at least q years (the qualifying period). Premium is payable at level rate π as long as the insured is active and has not yet been disabled, but not after time n − q, where n is the contract period (n > q). Determine the premium π by the principle of equivalence, assuming that the interest rate r is constant. Find the reserve at time t < n − q for an insured who is disabled for the first time and is currently receiving the disability benefit.

Exercise 62 The two-state Markov chain Z sketched in Fig. F.1 can be given many interpretations; it could describe transitions of a person into and out of the work-force (’active’ means

APPENDIX F. EXERCISES

employed, ’invalid’ means unemployed, and mortality is disregarded), or the transitions of a person between marital states (’active’ means single, ’invalid’ means married), or the transitions of a machine or mechanical device between states of functioning (’active’ means intact, ’invalid’ means out of order), and so on. Let us take the model as a description of a lamp, which is always switched on (it is installed in a submarine), and which is ’active’ when the bulb is intact and ’invalid’ or ’inactive’ when the bulb is burnt-out. One can assume that the life-time of a bulb is exponentially distributed, and it also seems reasonable to assume that the lapse of time from a bulb burns out until the failure is discovered and the bulb is replaced, is exponentially distributed. Then the intensities σ and ρ are constant (σ is the ’mortality’ intensity of the life length of a bulb or the expected number of ’deaths’ per time unit for an burning bulb, and ρ is the expected number of maintenance inspections per time unit.). We follow the lamp from time 0 when it is active. Let Nai (t) and Nia (t) denote the number of failures and replacements of bulb, respectively, in the time interval [0, t], and let Ia (t) and Ii (t) be the indicators of the events ’active at time t’ and ’inactive at time t’, respectively. Of course, Ii (t)+Ia (t) = 1, and Nia (t)−Nai (t) is either 0 or 1. (a) Find E[Ii (t)] = pai (0, t) by solving Kolmogorov’s forward equations, using paa (0, t) = 1 − pai (0, t). (b) Find the expected time as inactive in [0, t], E[ the function pai (0, τ ) in (a). Explain why.

Rt 0

Ii (τ ) dτ ]. You may just integrate

(c) Find E[Nai (t)]. You may also find the answer by just integrating the function paa (0, τ ) in (a) and multiplying with σ. Explain why. (d) Divide the expected values in (b) and (c) by t and find the limit as t → ∞. Discuss the expressions as functions of σ and ρ. (e) Find the failure intensity σ ˜ (t), 0 < t < n, for the Markov chain Z, conditional on Z(n) = i. (f) If σ = ρ, then Nai (t) + Nia (t) is a Poisson process with intensity σ. Explain why. What is then Nai (t) in terms of the Poisson process? Exercise 63 The situation is as described in Exercise 62. Suppose data are available for m independent lamps observed over the same time interval [0, n], all active at time 0. (a) Assume complete histories have been recorded for each lamp. Find the MLE for σ and find its asymptotic distribution.

Exercise 64 (b) The parameter ρ is subject to control since it is the frequency with which the lamps are being checked by maintenance personnel. Show that the asymptotic variance of the MLE in (a) is a decreasing function of ρ.

APPENDIX F. EXERCISES

(c) Suppose that, at time n, data are available only for lamps that are inactive at the time. Then it is the conditional process in 7.8e which is the relevant stochastic model for the individual histories. Find the MLE in this situation and find also its asymptotic variance.

Exercise 65 Go through Paragraphs 9.1 A-E and G in ’BL’ and add all details in proofs.

Exercise 66 At your disposal are data from m independent lives insured under one and the same scheme. You know for each individual the time of entry into the scheme, the age upon entry, the length of the observation period, whether he/she died during the observation period, and - if died - the age at death. Find the ML estimators under the assumption of piece-wise constant intensities and explain how they can be fitted by analytic graduation by a Gompertz Makeham function. Exercise 67 Consider the continuous time version of an n-year endowment insurance of 1 (say) against level premium π during the insurance period. Assume that the interest rate r is constant. Prove that the premium rate π is a decreasing function of n and that the premium reserve Vt for fixed t is a decreasing function of n for n > t. Note that the results are valid for all specifications of interest rate and mortality rate.

Exercise 68 We refer to Basic Life Insurance Mathematics (BL), Section 7.5, Paragraph J, Widow’s pension. The situation is depicted in Figure F.4. At time t = 0 (say) the husband (x) is x years old and the wife (y) is y years old. Their remaining life lengths after time 0 are denoted by S and T , respectively. The joint distribution of S and T is determined by the mortality rates named in the figure: for instance, at any time t ≥ 0, (x) has mortality rate µ(t) if (y) is still alive and µ0 (t) if (y) is dead. (a) Find integral expressions for the marginal survival probabilities P[S > s], P[T > t], and for the joint survival probability P[S > s , T > t]. (b) Prove that, if the mortality rates are independent of marital status, µ = µ0 and ν = ν 0 , then S and T are stochastically independent: P[S > s , T > t] = P[S > s] P[T > t] . Adopt the independence assumption in (b) and assume that x = y = 30 and that (x) and (y) are subject to the same law of mortality, G82M, irrespective of marital status. Thus, at any time t ≥ 0, µ(t) = ν(t) = µ0 (t) = ν 0 (t) = 0.0005 + 0.000075858 × 1.09144 30+t .

(F.38)

APPENDIX F. EXERCISES

The couple buys a combined life insurance and widow’s pension policy specifying that a pension is to be paid with intensity b = 0.5 as long as the wife is alive and husband is dead, a life assurance with sum s = 1 is due immediately upon the death of the husband if the wife is already dead (a benefit to their dependents), and the equivalence premium is to be paid with level intensity c as long as both husband and wife are alive. The policy terminates at time n = 30. Interest is earned on the reserve at constant rate r = ln(1.045). (c) Using the direct backward argument, derive the differential equation for the noncentral moments of first and second order of the discounted future benefits in state 0. (The first order moment is just the reserve.) (d) Using the program ’prores1.pas’, compute the three first moments of discounted benefits less premiums at times 0,5,...,30. Do the same for a modified contract, by which 50% of the current reserve (in state 0) is to be paid back immediately to the husband in case the wife dies before him during contract period. (e) Let us now drop the independence assumption in (b) and instead assume that, for each party, the mortality rate increases upon the death of the spouse (a ’grief effect’): µ0 > µ ,

ν0 > ν .

(F.39)

In a forthcoming note ’Dependent lives’ it will be proved that, under the assumption (F.39), S and T are positively dependent in the sense that P[S > s , T > t] > P[S > s] P[T > t] . Give some numerical evidence in support of this result by e.g. computing P[S > 30 , T > 30], P[S > 30], and P[T > 30] in the case where µ and ν are as in (F.38) and µ0 (t) = µ(t) + 0.0005 and ν 0 (t) = ν(t) + 0.0005. (f) Under the assumptions of Item (e) find the covariance between the present values at time 0 of a term insurance of 1 in 30 years on (x) and a similar insurance on (y). (The relationship Cov(X + Y ) = 21 [Var(X + Y ) − Var(X) − Var(Y )] may be useful.) Exercise 69 The uncertain development of interest can be accounted for by letting the interest rate r(t) at any time t depend on the “state of the economy” Y (t) modeled as a stochastic process. We will assume that {Y (t)}t≥0 is a continuous time Markov chain with finite state space J Y = {1, . . . , J Y }. The probability of transition from state e to state f in the time interval from t to u is denoted by pef (t, u) = P[Y (u) = f | Y (t) = e] .

(F.40)

We assume that the process is hom*ogeneous, which means that the transition probabilities pef (t, u) depend on t and u only through u − t, the length of the time interval. This implies that the process has constant intensities of transition, λef = lim

dt↓0

pef (t, t + dt) . dt

(F.41)

Just like the mortality intensity, λef is a ”probability of transition per time unit”. The total intensity of transition out of state e at time t is denoted by X λe· = λef . f ; f 6=e

APPENDIX F. EXERCISES

0 Both alive

µ -

1 Husband dead

ν0

2 Wife dead

3 µ0 Both dead

Figure F.4: Sketch of a Markov model for two lives.

r1 = 0.02

λ12 = 0.5 λ21 = 0.25

r2 = 0.05

λ23 = 0.25 λ32 = 0.5

r3 = 0.08

Figure F.5: Sketch of a simple Markov chain interest model. (a) Use the ”direct backward argument” to derive the so-called backward Kolmogorov differential equations for the transition probabilities: X ∂ pef (t, u) = λeg (pef (t, u) − pgf (t, u)) . ∂t g;g6=e The side conditions are pef (u, u) = δef , i.e. 1 if e = f and 0 otherwise (the Kroenecker delta). These differential equations can easily be solved numerically with a suitable variation of the program ’prores1.pas’ or ’prores2.pas’. Stochastic interest is now modeled by letting the interest rate assume J Y possible values r1 , . . . , rJ Y and, at any time t, the interest rate is r(t) = rY (t) (dependent on the state of the economy). Figure F.5 shows a flow-chart of a simple Markov chain interest rate model with three states, 0.02, 0.05, 0.08. Direct transition can only be made to a neighbouring state, and the total intensity of transition out of any state is 0.5, that is, the interest rate changes once in two years on the average. By symmetry, the long run average interest rate is 0.05.

APPENDIX F. EXERCISES

(b) Use the direct backward argument to derive differential equations for the state-wise expected discount factors We (t) = E[e−

t r(s) ds

| Y (t) = e] ,

t ∈ [0, n), What are the side conditions? Remark: It has been said that ”all models are wrong, but some are useful”. The Markov model proposed here is, of course, unable to mimic perfectly the development of the true interest rate. We can, however, by judicious choice of the state space (sufficiently big) and the transition intensities, make it catch the basic features of the real world interest fairly well. (c) Consider the model in Fig. F.5. Fill in appropriate statements in the h program R i 5 ’prores2.pas’ to make it compute the state-wise expected discount factors E exp − 0 r(s) ds Y (0) = e , e = 1, 2, 3. Exercise 70 Poisson processes, which are totally memoryless, can of course be generated from continuous time Markov chains, which are more general. For instance, let {Y (t)}t≥0 be a Markov chain on the state space {1, 2} with intensities of transition µ12 (t) = µ21 (t) = λ. Then {N (t)}t≥0 defined by N (t) = N12 (t) + N21 (t) (the total number of transitions in (0, t]) is a Poisson process with intensity λ; transitions counted by N occur with intensity λ at any time regardless of the past history of the process. Two independent Poisson processes, {N1 (t)}t≥0 with intensity λ1 and {N2 (t)}t≥0 with intensity λ2 , can be generated by letting {Y (t)}t≥0 be a Markov chain on the state space {1, 2, 3, 4} with intensities µ12 (t) = µ21 (t) = µ34 (t) = µ43 (t) = λ1 , µ13 (t) = µ31 (t) = µ24 (t) = µ42 (t) = λ2 , µ14 (t) = µ41 (t) = µ23 (t) = µ32 (t) = 0, and defining N1 (t) = N12 (t)+N21 (t)+N34 (t)+N43 (t) and N2 (t) = N13 (t)+N31 (t)+N24 (t)+N42 (t). Three independent Poisson processes can be generated from a Markov chain with 8 states (work out the details), and, in general, k independent Poisson processes can be generated from a Markov chain with 2k states. (a) Let {Y (t)}t≥0 be a Markov chain on the state space {1, 2}, and take µ12 (t) = µ21 (t) = 1. Denote the corresponding indicator processes and counting processes be Ie (t) and Nef (t). The total time spent by Y in state 1 during the time interval (t, n] is Z n T1 (t, n] = I1 (τ ) dτ , t

and the total number of transitions made from state 1 to state 2 in that interval is N12 (t, n] = N12 (n) − N12 (t) .

(These quantities can be viewed as present values of benefits of annuity type and assurance type, respectively, for a two-state policy with no interest.) (a) Assume Y (0) = 1. What are the interpretations of the random variables T1 (t, u] and N12 (t, u] in terms of the Poisson process N (t) = N12 (t) + N21 (t)?

APPENDIX F. EXERCISES

(b) Find, by solving the relevant differential equations analytically, explicit expressions for the first two state-wise conditional moments Ve(q) (t) = E[T1q (t, n] | Y (t) = e] , q We(q) (t) = E[N12 (t, n] | Y (t) = e] ,

e = 1, 2, q = 1, 2, and find also the corresponding variances. (You should obtain e.g. E[T1 (0, 1] | Y (0) = 1] =

1 1 + 2µ − e−2µ , 4µ

1 1 + 4µ − (2 − e−2µ )2 . 16µ2 (c) Using the program prores1.pas (or prores2.pas), solve the differential equations also numerically and compare the results with the exact solutions obtained in (b). Var[T1 (0, 1] | Y (0) = 1] =

Exercise 71 Let {N (t)}t≥0 be Poisson process with intensity λ. This is a counting process of even simpler type than the counting processes associated with a Markov chain; N is not only Markov, but also has independent increments. Thus, in any small time interval [t, t + dt) the process N makes a jump of 1 with probability λ dt regardless of the past history of the process in [0, t). Let the price S(t) of a share of stock at time t be modelled as a so-called geometric Poisson process with drift, S(t) = exp (αt + βN (t)) , t ≥ 0. If β = 0, then S(t) is just the accumulation factor for a bank account with fixed interest rate. The Poisson term in the exponent adds jumps at random times, and a jump at time t makes the stock price jump from S(t−) to S(t) = S(t−) eβ . Thus, γ = eβ − 1 is the relative change (S(t) − S(t−))/S(t−) in the stock price at the jump time. Between the jumps the stock price increases at fixed “rate of interest” α. (a) Find the expected value E[S(t)] at time 0 of the stock price at time t, and do this in two ways: First, work directly with the Poisson distribution of N (t) and, second, solve a differential equation obtained by the direct backward construction (condition on “what happens in the small time interval [0, dt)”). Explain that, having determined the expected value, higher order moments are easily obtained. (b) Find the dynamics dS(t) of the stock price by applying the change of variable rule, see Appendix A. (c) Using the direct backward construction, show that the expected present value of a perpetuity (an everlasting annuity), is Z ∞ E S −1 (τ ) dτ = (a + (1 − e−β ))−1 . 0

APPENDIX F. EXERCISES

t ≥ 0. Letting β1 and β2 have opposite signs, we have created a stock price which may increase or decrease with instantaneous jumps. Write out the straightforward analogs of the formulas in Items (a) - (c) for this more general model. Exercise 72 (a) Using your Turbo Pascal program with the Danish basis, compute the net premium rate π and the net premium reserve Vt (at selected times t) for an endowment insurance with age at entry x = 30, term n = 30, sum insured bt = bn = b = 1, and premium payable continuously at constant rate throughout the contract period. (b) Compute the gross premium rate π 0 and the gross premium reserve Vt0 assuming that administration expenses consist of a lump sum cost of 0.003 + 0.001 b at time 0, costs incurring continuously at rate 0.0001 + 0.01 π 0 + 0.005 Vt0 at any time t in the insurance period, a cost of 0.002 due immediately upon possible payment of the death benefit, and a cost of 0.0001 due at time n upon possible payment of the endowment benefit. Compare with the quantities net of expenses found in Item (a).

Exercise 73 In connection with a pension insurance there is an additional benefit which is a sum insured to possible dependent children less than 18 years old at the time of death of the insured. I the technical basis we therefore need to make assumptions about births. We have to distinguish by sex, and in the following we are going to consider insured women only. The Figure below shows a flowchart describing a life history with births (at most J). To keep things simple, we will assume that the process is Markov, that all participants enter the insurance scheme in state 0 at age 0, and that the individual life histories are independent replicates of the process. Assume furthermore that the mortality intensity µj (t) and the birth intensity φj (t) for a t year old woman, who has given birth to j children, are given by µj (t)

αj + βct , j = 0, . . . , J,

(F.42)

φj (t)

ηj f (t) , j = 0, . . . , J − 1,

(F.43)

where c and the function f are known and the parameters αj , β, and ηj are unknown. (a) Assume for the time being that, at the time of consideration, we have complete information about the past history for all individuals that have previously been or currently are insured under the scheme (i.e. we know the exact times of possible births and death). Put up the equations for determining the ML (maximum likelihood) estimators for the unknown parameters and, to the extent possible, find explicit expressions for the estimators. Explain also how to determine the asymptotic covariance matrix. (b) In our model there is non-differential mortality if α0 = · · · = α J .

(F.44)

Derive the likelihood ratio test for the null hypothesis (F.44) and determine the rejection limit such that the asymptotic level is 10%.

APPENDIX F. EXERCISES

Assume now that the past history of births and death is being observed only upon death of the insured, when the additional benefit to the possible dependents is due. Suppose that the statistical data comprise only those who are dead at the time of consideration and that for each of those there is a complete record of the times of possible births and of death. In these data the observed life history of a woman, who entered the scheme u years ago, is governed by a Markov process as described above, but with intensities µ∗j (t)

φ∗j (t)

1 , pjd (t, u) pj+1,d (t, u) φj (t) . pjd (t, u) µj (t)

(F.45) (F.46)

It is to be proved that, if the mortality increases with the number of births, that is,

then Introduce

µj (t) ≤ µj+1 (t) , t > 0, j = 0, . . . , J − 1,

(F.47)

φ∗j (t) ≥ φj (t) , t > 0, j = 0, . . . , J − 1 .

(F.48)

pj (t, u) = 1 − pjd (t, u) ,

(F.49)

the probability that a t year old with j births will survive to age u. Build the proof as follows: (c) Prove that, for t ≤ τ ≤ u, X X pjk (t, τ )pk (τ, u) . pjk (t, u) = pj (t, u) =

(F.50)

k≥j

(d) Prove that pj (t, u) = e−

t µj (t,ϑ)dϑ

(F.51)

where µj (t, u) =

k≥j

pjk (t, u)µk (u)

k≥j

pjk (t, u)

(F.52)

the mortality intensity at age u for a woman who is in state j at time t < u). (e) Show (preferably by direct reasoning), that Z u R Ru τ pj (t, u) = e− t (φj +µj ) + e− t (φj +µj ) φj (τ )pj+1 (τ, u) dτ .

(F.53)

(d) Use the results (F.50) – (F.53) to show that (F.47) implies pj+1 (t, u) ≤ pj (t, u) , j = 0, . . . , J − 1 ,

(F.54)

which by (F.46) and (F.49) implies (F.48). The relationship (F.54) is easy to prove for j = J − 1, and the result then follows by induction.

APPENDIX F. EXERCISES

φ0 -

Alive 0 births µ0 ?

···

φj−1 -

φj -

Alive j births

···

µj ?

φJ−1 -

J Alive J births µJ ?

d. Dead

Commment: The inequality (F.48) means that the fertility rates will be overestimated if one uses the estimators for the φ∗j based on diseased participants in the scheme. If the inequalities (F.47) are reversed, then also the inequality (F.48) will be reversed, and the estimators the φ∗j will underestimate the fertility. In particular it follows that, under the hypothesis of non-differential mortality, the fertility rates will be unbiasedly estimated from the selected material of diseased participants. Exercise 74 We adopt the usual notation and assumptions of the theory of multi-life insurance policies and consider two independent lives (x) and (y) with remaining life lengths Tx and Ty , respectively. (a) Assume that the benefit is an assurance of 1 payable at time Ty if 2Tx < Ty < n and that premium is payable at constant rate π until time min(Tx , Ty , n/2), where n is the term of the contract (fixed). Determine the equivalence premium π. (b) Propose a method for computing the premium numerically. (Hint: One possibility is to treat t/2 px as a survival function t p˜x with intensity µ ˜ x+t , which you would need to express in terms of µ, and then solve a Thiele differential equation numerically.) (c) Determine the reserve at any time t, assuming that the insurer currently knows the complete past history of the two lives. You need to distinguish between various cases, whether (y) is alive or dead, whether t is before or after time n/2, and whether x is alive or dead and, if dead, when. Is the reserve always non-negative? (d) What is the variance of the present value of the benefit?

Exercise 75 Find an expression for the mortality intensity of the life length of the last-survivor status x1 . . . xr . Do this by direct reasoning and also the hard way by calculating µx1 ...xr (t) = −

d lnt px1 ...xr . dt

APPENDIX F. EXERCISES

Exercise 76 An example of an insurance policy with benefits that are ’path-dependent’, that is, dependent on the past history of the policy: (a) Consider two independent lives (x) and (y). Find the expected present value at time 0 of a life annuity payable continuously at rate 1 from time Txy until time max(Txy + 20 , Txy + 10). In words, payments start at the time of the first death and continues thereafter for a term of 20 years or until 10 years after the death of the survivor, whichever is the longer period. (b) Suppose premium is payable continuously at level rate π from time 0 until time Txy . What is the reserve for this policy? This question is difficult and will be addressed later, but you can start thinking about it. Exercise 77 Another example of ’path-dependent’ payments: Consider three independent lives (x), (y), and (z). Find the expected present value of a sum insured of 1 payable at time Tx if Tx < min(Tz , Ty + 20). State in words what this contractual benefit is.

Exercise 78 Use the program ’prores1.pas’ to compute π=

A¯20,20,n , a ¯20,20,n

assuming that both lives follow the mortality law G82M and that the interest rate is r = ln(1.05). What are the terms of the policy for which π is the level equivalence premium rate? Exercise 80 We refer here to Section 7.9 ’Dependent lives’ in BL. (a) Prove the rather obvious statements PQD(T, T ), AS(T, T ), and RTI(T |T ). (b) Prove that the Definitions PQD – RTI are equivalent to the modified definitions obtained upon replacing the strict inequalities S > s and T > t with the non-strict inequalities S ≥ s and T ≥ t. For instance, for PQD prove that (7.92) is equivalent to P[S ≥ s , T ≥ t] ≥ P[S ≥ s] P[T ≥ t] for all s and t. (c) Negative dependence in the PQD sense: Prove that PQD(−S, T ) is equivalent to P[S > s , T > t] ≤ P[S > s] P[T > t] for all s and t. (d) Negative dependence in the AS sense: Prove that AS(−S, T ) is equivalent to Cov(g(S, T ), h(S, T )) ≤ 0 for all real-valued functions g and h that are decreasing in S and increasing in T (and for which the covariance exists). (e) Negative dependence in the RTI sense: Prove that RTI(−S|T ) is equivalent to RTD(S|T ).

APPENDIX F. EXERCISES

(f) Consider the model in Figure 7.6, and let µ = µ0 and ν = ν 0 . We have shown in Exercise 1 that S and T are then independent. Now, add a cause of simultaneous death (due to ’catastrophe’) with intensity µ03 = κ. Does it follow that RTI(S|T )? Assume instead, maybe more reasonably, that catastrophe risk is present independently of the state of the marriage: µ01 (t) = µ(t), µ02 (t) = ν(t), µ03 (t) = κ(t), µ13 (t) = ν(t) + κ(t), µ23 (t) = µ(t) + κ(t). Prove that RTI(S|T ), hence PQD(S, T ).

Exercise 81 We refer to Chapter 8 of BL and Exercise paper No. 6. Note the following correction to Exercise paper No. 6, page 8, line 2 from top: ”Recalling (10) and (14),...”. A 30 years old buys a pure life endowment of 1 in n = 30 years against premium payable continuously at level rate as long as the policy is in force. The first order basis specifies G82M mortality and interest at instantaneous rate 0.02. The second order basis specifies the same mortality G82M and stochastic interest as given in Fig. 1 of Exercise 2. (a) See Exercise 20a. Suppose surpluses are to be repaid immediately as cash bonus. Compute the state-wise expected present values (under second order interest) at time 0 of future bonuses, given that the insured survives 30 years. Use the program ’prores2.pas’. It may be a good idea to define two ’alive’ states which do not communicate, one for computation of the first order reserve, the other for computation of the expected present values of bonuses. (b) See Exercise 19. Suppose instead that that surpluses are currently spent on purchase of additional benefits. Compute the state-wise expected values at time 0 of the additional benefit Q30 . OK TO HERE (b) Give an alternative proof of (3.16) along the following lines: Write Z b (T ∧ b) − (T ∧ a) = It dt , a

where It = I{T >t} . By Itˆ o’s formula, Z

It dt a

b a

Is ds

k−1

It dt .

Use Is It = It for s < t and Itk = It , and take expectation.

Exercise 2 Find the q-th non-central moment of P V a;m|n in (4.21). Start from ! q 1 X q p(Tx ∧m) (q−p)(T ∧(m+n)) a;m|n q v v . (V ) = q r p=0 p Exercise 4 Dependence of expected present values on age, duration, and technical basis.

APPENDIX F. EXERCISES

Commence with the pure endowment treated in Paragraph A. The expected present value in (4.2) can be written as t Ex

= e−rt−

0 µx+s ds

(F.55)

The dependence of t Ex on the deferred period t, the age x, and the technical basis r and µ is to be discussed. Some qualitative aspects are obvious by inspection of (F.55) and also on intuitive grounds: t Ex decreases with increasing t; t Ex decreases with increasing x if µ is an increasing function; t Ex decreases with increasing r (or interest rate i); t Ex decreases by a general increase of µ. A closer study, also of the quantitative aspects, can be based on the derivatives ∂ = − t Ex (r + µx+t ), (F.56) t Ex ∂t ∂ = − t Ex (µx+t − µx ) (F.57) t Ex ∂x Rt R x+t Rx (the derivative of 0 µx+s ds = 0 µy dy − 0 µy dy with respect to x is µx+t − µx ), ∂ t Ex = −t Ex t, ∂r

(F.58)

and, if µx = µ(x, α) is a differentiable function of some finite-dimensional parameter α = (α1 , . . . , αr ), Z t ∂ ∂ µ(x + s, α) ds. (F.59) t Ex = − t Ex ∂αj ∂αj 0 In particular, if the force of mortality is of G-M type, µx = α + βcx , then Z

µx+s ds

αt + βcx (ct − 1)/ ln c

αt + (µx+t − µx )/ ln c.

One easily finds (check the details) Z t ∂ µx+s ds ∂α 0 Z t ∂ µx+s ds ∂β 0

= = =

∂ ∂c

µx+s ds

t, cx (ct − 1) ln c µx+t − µx , β ln c

(x + t)ct − x ct − 1 βcx−1 − ln c ln2 c 1 1 x− (µx+t − µx ) + t(µx+t − α) . c ln c ln c

APPENDIX F. EXERCISES

Thus, in the G-M case (F.59) specializes to ∂ t Ex ∂α ∂ t Ex ∂β ∂ t Ex ∂c

= − t Ex t, 1 (µx+t − µx ), β ln c 1 1 = − t Ex x− (µx+t − µx ) + t(µx+t − α) . c ln c ln c

= − t Ex

(F.60) (F.61) (F.62)

The expressions in (F.59) and (F.58) are the same, of course, since t Ex depends on r and α only through their sum r + α. In general, a constant change in the force of mortality is equivalent to a change in the force of interest. A comparison of (F.57) and (F.61) reveals that a change of β is essentially the same as a change of x. In fact, replacing β by β 0 = βch (say) is equivalent to a shift from x to x + h. The expression in (F.62) is not so transparent, but one may say that a change of c to c0 = ch (say) amounts to an expansion or contraction in age by a factor h. All right hand side expressions in (F.56)–(F.62) include − t Ex as a factor. Division by − t Ex gives the derivative of − ln t Ex , which is the relative decrease of t Ex (in units of t Ex ) by a unit increase in the argument. The factors multiplying − t Ex form a basis for comparing the arguments with respect to the import of changes. Roughly speaking, for common values of the arguments, they can be ordered as follows according to their impacts on the value of t Ex , starting with the less important: x, t and α, β, and c. In fact, a change in t is generally of greater importance than a similar change in x. This result is immediately meaningful since t and x are measured in the same units. Other comparisons that can be made are not so lucid since the arguments are not ”commensurable”. For instance, even though changes in β are more important than ∂ ∂ changes in x in the sense that ∂β t Ex > ∂x t Ex (for common values of the arguments, that is), it must be kept in mind that the relevant changes in these two arguments are of quite different order of magnitude. Usually β is confined to values below 10−3 , whereas x typically ranges between 20 and 70. To account for this aspect, one should ∂ ∂ rather compare the two differentials ∂β t Ex db and ∂x t Ex dx for representative changes db and dx, and (presumably) conclude that x produces the greater changes in t Ex . Here is another comment along the same line: Despite the fact that r and α play equivalent roles in t Ex from a purely mathematical point of view, the interest rate is widely held to the most important element in the technical basis. The reason is, of course, that the interest rate nowadays varies by percentages whereas the mortality is fairly stable and varies by less than per milles. Now, consider an n-year temporary level life annuity with expected present value given by (4.14). Taking derivatives under the integral sign, one obtains results about a ¯x n by just inserting the expressions from (F.56)–(F.62). The dependence on n is obtained immediately from (4.14). The results are ∂ a ¯x n ∂n ∂ a ¯x n ∂x ∂ a ¯x n ∂r

n Ex ,

¯1 −A

− {1 − (r + µx ) a ¯ x n − n Ex } , ∂ a ¯x n = −(I¯ a ¯ )x n , ∂α

(F.63) + µx a ¯x n (F.64) (F.65)

APPENDIX F. EXERCISES ∂ a ¯x n ∂i ∂ a ¯x n ∂β ∂ a ¯x n ∂c

−(I¯ a ¯)x n v,

−

= =

(F.66)

1 {1 − (r + µx ) a ¯ x n − n Ex } , (F.67) β ln c 1 1 ¯ x − α I¯ a ¯ 1 − µx a − x− A ¯x n + I¯ A ¯ xn xn xn c ln c ln c 1 1 [ x− − {1 − (r + µx )¯ a x n − n Ex } c ln c ln c ¯a)x n − n n Ex ]. +a ¯x n − (r + α)(I¯ (F.68)

¯ 1 (and A¯x n ) on the arguments is obtained from the Finally, the dependence of A xn ¯ results above upon putting A 1 = 1 − r¯ ax n − n Ex . Work out the details. xn Clearly, since the expressions in (F.56)–(F.62) are all negative when µ is increasing, so are also the expressions in (F.63)–(F.68). As a by-product one obtains that 1 > (r + µx )¯ a x n + n Ex , which can easily be established by direct calculation. Exercise 2 In the situation of the present paragraph consider the problem of estimating µ from the Di alone, the interpretation being that it is only observed whether survival to z takes place or not. Show that the likelihood based on Di , i = 1, . . . , n, is q N (1 − q)n−N , with q = 1−e−µz , the probability of death before z. (Trivial: it is a binomial situation.) Note that N is now sufficient, and that the class of distributions is a regular exponential class. The MLE of q is q∗ =

N n

with the first two moments Eq ∗ = q, Varq ∗ =

q(1 − q) . n

It is UMVUE in the class of estimators based on the Di . The MLE of µ = − ln(1 − q)/z is µ∗ = − ln(1 − q ∗ )/z. Apply (D.6) in Appendix D to show that q µ∗ ∼as N µ, 2 . nz (1 − q) The asymptotic efficiency of µ ˆ relative to µ∗ is !2 2 µz µz asVarµ∗ e 2 − e− 2 sinh(µz/2) = = asVarˆ µ µz µz/2 (sinh is the hyperbolic sine function defined by sinh(x) = (ex − e−x )/2). This function measures the loss of information suffered by observing only death/survival by age z as compared to inference based on complete observation throughout the time interval

APPENDIX F. EXERCISES

(0, z). It is ≥ 1 and increases from 1 to ∞ as µz increases from 0 to ∞. Thus, for small µz, the number of deaths is all that matters, whereas for large µz, the life lengths are all that matters. Reflect over these findings. Exercise 3 Use the general theory of the present section to prove the special results in Section 11.1.

Exercise 4 Work out the details leading to (11.42) – (11.42). Exercise 5 An insurance company is to carry out a mortality study based on complete records for n life insurance policies with unlimited term period. Policy No. i was issued zi years ago to a person who was then aged xi . The actuary sets out to maximize the likelihood Z xi +Ti n Y µ(s, θ)ds , µ(xi + Ti , θ)Di exp i=1

where the notation is selfexplaining. One employee in the department objects that the method represents a neglect of information; it is known that the insured have survived, not only the period they were insured, but also the period from birth until entry into the scheme. Thus, he claims, the appropriate likelihood is rather Z xi +Ti n Y µ(xi + Ti , θ)Di exp µ(s, θ)ds . i=1

Settle this apparent paradox. (Hint: A suitable framework for discussing the problem is an enriched model with three states, ”uninsured”, ”insured”, and ”dead”.)

Exercise 6 (a) Modify the formulas to the situation where person No. i entered the study zi years ago at age xi . (b) Find explicit expressions for the entries of the asymptotic covariance matrix of the MLE.

Exercise 1 Prove (C.6) in the theorem by induction: Verify that it is true for r = 1 and, assuming it is true for a given r, prove that it is true also for r + 1.

Exercise 2 Derive the binomial distribution by applying the theorem to the situation where A1 , . . . , Ar are independent and equally probable, P[Aj ] = p, j = 1, . . . , r.

APPENDIX F. EXERCISES Exercise 3 Use the theorem to find E[Q] and V[Q] expressed in terms of the Zp . Exercise 4 Find the probability that at least 3 out of 4 events occur. Let the price at time t of a stock be S(t) = eαt+βN (t) ,

where N (t) is a Poisson process with intensity λ. The money market account bears interest at spot rate r. At time 0 our hero (x) purchases an n-year unit linked pure life endowment with sum insured S(n) ∨ g against a single premium π. Here g is the guaranteed minimum sum insured introduced to protect the insured against poor performance of the stock; if β is negative (in which case α should certainly be positive), then a Poisson event at time t represents a sudden drop in the stock price from S(t−) to S(t) = S(t−)eβ (a crash in the stock market if the absolute value of β is big). Combining basic principles in finance (no arbitrage) and insurance (equivalence), π should be the expected discounted value of the claim under a suitable probability measure (equivalent martingale measure for the market and physical measure for the life length): (F.69) π = E e−rn (S(n) ∨ g) 1[Tx > n] = E [S(n) ∨ g] e−rn n px . We need to find the expected value appearing in the last expression. Start as usual from the conditional expected value of the random variable S(n) ∨ g, given everything that is known by time t: E[S(n) ∨ g | N (τ ); 0 ≤ τ ≤ t] i h = E eαn+βN (n) ∨ g N (τ ); 0 ≤ τ ≤ t i h = E eαt+βN (t) eα(n−t)+β(N (n)−N (t)) ∨ g N (τ ); 0 ≤ τ ≤ t .

Here we have separated out what pertains to the past (known under the conditioning) and what pertains to the future (remains random under the conditioning), and it is seen that we can work with the function h i W (t, u) = E u eα(n−t)+β(N (n)−N (t)) ∨ g . Preparing for a backward construction, write h i W (t, u) = = E u eαdt+β(N (t+dt)−N (t)) eα(n−t−dt)+β(N (n)−N (t+dt)) ∨ g ,

and proceed as usual, conditioning on what happens in (t, t + dt): W (t, u) = (1 − λ dt) W t + dt, ueαdt + λ dt W t + dt, ueαdt+β + o(dt) = W t + dt, ueαdt − λ dt W (t, u) + λ dt W t, ueβ + o(dt) = W (t + dt, u + u α dt) − λ dt W (t, u) + λ dt W t, ueβ + o(dt) =

∂ ∂ W (t, u) dt + W (t, u) u α dt ∂t ∂u − λ dt W (t, u) + λ dt W t, ueβ + o(dt) .

W (t, u) +

APPENDIX F. EXERCISES We arrive at ∂ ∂ W (t, u) + W (t, u) u α − λ W (t, u) + λ W t, ueβ = 0 , ∂t ∂u

which is to be solved subject to the condition

W (n, u) = u ∨ g . Remark: We could have written (F.69) as h i π = E eβN (n) ∨ ge−α n e(α−r) n n px ,

and, redefining W (t, u) accordingly, essentially get rid of e−α t . We have chosen the present approach since it gives us an opportunity to see the different roles of the (non-stochastic) smooth function e−α t and the (stochastic) jump process N (t).

Appendix G

Solutions to exercises Exercise 1 R1 (a) Exact value 0 t2 dt = 1/3 = 0.3333.... Can be computed by solving numerically the ordinary differential equation v 0 (t) = a(t) + b(t) v(t) by ’Ode-1.pas’. Take v(t) =

s2 ds ,

(G.1)

v 0 (t) = t2 ,

hence a(t) = t2 and b(t) = 0. Compute forwards (’F’) starting from v(0) = 0. Insert the following statements in the program: (*SPECIFY DIMENSION OF v !*) dim := 1; (*SPECIFY DIRECTION OF COMPUTATION - FORWARD (’F’) OR BACKWARD (’B’):!*) BF := ’F’; (*SPECIFY THE TIME INTERVAL [0,T] BY INSERTING T IN ”term” AND - IF NEEDED - THE AGE OF THE LIFE AT TIME 0 ! *) term := 1; (*SPECIFY BOUNDARY CONDITION(S) AT TIME t = 0 IF FORWARD (BF = ’F’) AND AT t = T if BACKWARD (BF = ’B’): !*) v[1] := 0; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a’ and B’ THAT ARE NOT 0 !*) a[1] := t∗t; a1[1] := 2∗t;

APPENDIX G. SOLUTIONS TO EXERCISES Similar for

R1 0

t−1/2 dt = 0.5. Take v(t) =

s−1/2 ds ,

v 0 (t) = t−1/2 ,

√

hence a(t) = 1/ t and b(t) = 0. Statements as above, except (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a’ and B’ THAT ARE NOT 0 !*) a[1] := 1/sqrt{t}; a1[1] := -0.5∗a[1]/t; (b) Since Φ(−x) = 1 − Φ(x), we need only compute Φ(x) for positive x. Moreover, Φ(0) = 0.5, hence 2 Z x t 1 exp − Φ(x) = 0.5 + √ dt . 2 2π 0 Put 2 1 t 0 v(t) = Φ(t) , v (t) = √ exp − , 2 2π 2

and use ’Ode-1.pas’ for (G.1) with a(t) = √12π exp(− t2 ) and b(t) = 0, working forwards (’F’) starting from v(0) = 0.5. Statements as above, except (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a’ and B’ THAT ARE NOT 0 !*) a[1] := (1/sqrt{2∗pi})∗exp(-t∗t/2); a1[1] := a[1]∗(-t); (c) Ut : Use ’Ode-1.pas’ forwards for the differential equation Ut0 = rUt + 1. Vt : Use ’Ode-1.pas’ backwards for the differential equation Vt0 = rVt − 1. Exercise 3 Fig 3.1: The functions we are interested in are Z t eγ t − 1 ¯ v1 (t) = F (t) = exp − µ(s) ds = exp −α t − β , γ 0 v2 (t) = f (t) = F¯ (t) µ(t) = v1 (t) v3 (t) , v3 (t)

µ(t) = α + β eγ t .

They can be computed directly since they are given by explicit formulas. Thus, the program ’Ode-1.pas’ is not really needed, but it is still useful since it can produce a nicely arranged output. One could drop everything that has to do with the difference scheme and just put in the following statements beginning from line 27: dim := 3; (*auxiliary quantities from Danish life table :*) alpha := 0.0005; (*Gompertz-Makeham parameters*) beta := 0.00007585775;

APPENDIX G. SOLUTIONS TO EXERCISES

gamma := 0.038*ln(10); for l := 0 to 100 do (*Do not confuse the letter l with the number 1*) begin writeln; writeln(odeout); (*Line shift*) t := l; v[3] := alpha + beta∗exp(gamma∗t); v[1] := exp ( - alpha∗t - beta∗( exp(gamma∗t) - 1)/gamma ); v[2] := v[1]∗v[3]; write(t:4,’ ’); write(odeout,t:4,’ ’); for j := 1 to dim do begin write(’ ’,v[j]); write(odeout,’ ’,v[j]); end; end; close(odeout); end. Alternatively, one could use the program as it is to compute F¯ and add statements to compute f and µ. We have v 0 1 (t) = b11 (t) v1 (t) , where b11 (t) = −α − β exp(γ t)) .

Having computed b11 (t) and v1 (t), we compute v3 (t)

v2 (t)

−b11 (t) ,

v1 (t) v3 (t) .

Statements: (*SPECIFY DIMENSION OF v !*) dim := 1; (* SPECIFY DIRECTION OF COMPUTATION - FORWARD (’F’) OR BACKWARD (’B’): !*) BF := ’F’; (*auxiliary quantities from Danish life table :*) alpha := 0.0005; (*Gompertz-Makeham parameters*) beta := 0.00007585775; gamma := 0.038*ln(10); (*SPECIFY THE TIME INTERVAL [0,T] BY INSERTING T IN ”term” AND - IF NEEDED - THE AGE OF THE LIFE AT TIME 0 ! *) x:= 0; term:= 100; (*SPECIFY STEPS IN DIFFERENCE SCHEME AND OUTPUT !: *) steps := 10000; (*number of steps in difference method*)

APPENDIX G. SOLUTIONS TO EXERCISES

outp := 100; (*number of values in output*) h := term/steps; (*steplength*) count := steps/outp - 0.0005; (* SPECIFY BOUNDARY CONDITION(S) AT TIME t = 0 IF FORWARD (BF = ’F’) AND AT t = term if BACKWARD (BF = ’B’): !*) v[1] := 1; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a’ and B’ THAT ARE NOT 0 !*) B[1,1] := - alpha - beta∗exp(gamma∗(x+t)); B1[1,1] := - beta∗exp(gamma*(x+t))∗gamma; (Then comes the difference scheme, and we do not need to do anything until we come to the output at times t = 0,1,... :) if countout > count then begin v[3] := - B[1,1]; v[2] := v[1]∗v[3]; (*this comes extra*) countout := 0; writeln; writeln(odeout); write(t:4,’ ’); write(odeout,t:4,’ ’); for j := 1 to 3 do (*NB! 3 instead of dim*) begin write(’ ’,v[j]); write(odeout,’ ’,v[j]); end; end; end; (*difference scheme*) close(odeout); end. Exercise 4 (a) A special case of (b) - just put α = 1. (b) Write out the defining expression in each interval where there is one analytic expression:   0 α , t ≤ 0 , 1 − 1 − ωt F (t) = , 0 < t < ω,  1 , t ≥ ω.  , t ≤ 0,  1 ω−t α , 0 < t < ω, F¯ (t) = ω  0 , t ≥ ω. α−1 α ω−t α−1 − ω1 = ω , 0 < t < ω, −α ω−t ω ω f (t) = 0 else. The mortality intensity is µ(t) = f (t)/(1 − F (t)), defined for all t such that the denominator is positive: α ω−t −1 α = ω−t , 0 < t < ω, ω ω µ(t) = undefined , t ≥ ω.

APPENDIX G. SOLUTIONS TO EXERCISES

Note that µ(t) % +∞ as t % ω (as is always the case if ω < ∞ and µ is nondecreasing). For x and x + t both in (0, ω), α ω−x−t F¯ (x + t) , = F¯ (t|x) = ¯ ω−x F (x) the same type of distribution as F¯ , only with ω replaced by ω − x. Thus we find f (t|x) and µ(t|x) by just replacing ω with ω − x in the formulas above. (c) Do as above. Exercise 5 (a) t p0

m f f = P[T > t] = P[M ] P[T > t|M ] + P[F ] P[T > t|F ] = sm 0 t p0 + s 0 t p0 .

sm t = P[M |T > t] =

P[M ∩ (T > t)] P[M ] P[T > t|M ] sm t pm = = m m0 0 f f . P[T > t] P[T > t] s0 t p0 + s 0 t p0

(b) µt

d f d f m d f f f m m p0 sm sm t p0 + s0 dt t p0 0 0 t p 0 µt + s 0 t p 0 µt − dt t = − mdt m = f f m t p0 s0 t p0 + sf0 t pf0 sm 0 t p0 + s 0 t p0

m f f sm t µt + s t µt ,

a weighted average of the mortalities at age t for males and females, the weights being the conditional probabilities of being male and female, respectively, given survival to age t. (c) sm t =

sm 0

sm sm 0 hR 0 i, = f f m t f f + s 0 t p 0 /t p 0 sm (µm s − µs ) ds 0 + s0 exp 0

f a decreasing of t if µm t − µt > 0 for all t > 0. R ∞ mfunction f If 0 (µs − µs ) ds = ∞, then sm t → 0 as t → ∞.

(d)

t px

x+t p0 x p0

f f m sm f f m 0 x+t p0 + s0 x+t p0 = sm x t px + s x t px . f f m sm 0 x p0 + s 0 x p0

(From this expression we could have derived the mortality intensity µx upon forming limt&0 (1 − t px )/t, which would give the result in (b).) Similarly f f m m m|n qx = sx m|n qx + sx m|n qx , n Ex

f f m = sm x n Ex + s x n Ex .

APPENDIX G. SOLUTIONS TO EXERCISES Exercise 6 F¯ (t) = a0 + a1 t + a2 t2 + a3 t3 , (a)

t ∈ [0, ω] ,

f (t) = −F¯ 0 (t) = −a1 − 2a2 t − 3a3 t2 , 2

f (t) a1 + 2a2 t + 3a3 t µ(t) = ¯ , = − a0 + a 1 t + a 2 t2 + a 3 t3 F (t)

a3 6= 0, . t ∈ [0, ω] . t ∈ [0, ω] .

Proof of f (t) lim µ(t) = lim ¯ = ∞ : t%ω F (t)

(G.2)

t%ω

We know that limt%ω F¯ (t) = 0. If limt%ω f (t) 6= 0, then (G.2) holds. If limt%ω f (t) = 0, then we have a 0/0 expression in the limit and use l’Hospital’s rule: lim µ(t) = − lim

t%ω

f 0 (t) . f (t)

If limt%ω f 0 (t) 6= 0, then (G.2) holds. If limt%ω f 0 (t) = 0, use l’Hospital again: lim µ(t) = − lim

t%ω

f 00 (t) . f 0 (t)

But f 00 (t) = −6a3 6= 0, and so (G.2) holds. Observe that F¯ (t|x), considered as function of t, is also trinomial. (b)

F¯ (0) = 1 : F¯ (ω) = 0 :

a0 = 1 ,

f (ω) = 0 :

a1 + 2a2 ω + 3a3 ω 2 = 0 , a1 2 a2 3 a3 4 a0 ω + ω + ω + ω = e0 . 2 3 4

a 0 + a1 ω + a2 ω 2 + a3 ω 3 = 0 ,

F¯ (t) dt = e0 :

Exercise 7 The probability distribution of the random variable Tx is given by P[Tx ≤ t] = t px , t ≥ 0. We are interested in the probability distribution of a present value P V (Tx ) which is just a real-valued function of the random variable Tx : P[P V (Tx ) ≤ u], u ∈ (−∞, ∞). Thus for each value of u we need to determine the set of values of Tx that make P V (Tx ) ≤ u and determine its probability. It may be helpful to draw a graph of the function, with Tx on the horizontal axis and P V (Tx ) on the vertical axis, and for each given u on the vertical axis determine the set on the horizontal axis where the graph is under u. We summarize the results: (a) Pure endowment benefit of 1: P V (Tx ) = e

−rn

1[Tx > n] =

0 e−rn

, ,

Tx ≤ n , Tx > n .

APPENDIX G. SOLUTIONS TO EXERCISES A very simple non-decreasing function with just   0 1 − n px P[P V (Tx ) ≤ u] =  1

two values. , , ,

u < 0, 0 ≤ u < e−rn , e−rn ≤ u .

(b) Term insurance with sum 1:

P V (Tx ) = e−r Tx 1[Tx ≤ n] =

e−r Tx 0

Tx ≤ n , Tx > n .

, ,

A simple non-increasing function starting from its maximum value 1 at Tx = 0, decreasing exponentially for Tx ∈ (0, n], dropping to 0 at Tx = n and remaining 0 thereafter.  0 , u < 0,    , 0 ≤ u < e−rn , n px P[P V (Tx ) ≤ u] = px , e−rn ≤ u < 1 ,    − ln u / r 1 , 1 ≤ u.

The third line on the right is the only one that takes a small bit of calculation: For e−rn ≤ u < 1 solve e−rt = u to find t = − ln u / r, and conclude that P V (Tx ) ≤ u is equivalent to Tx ≥ t. (c) Endowment insurance with sum 1: Same problem as (b), only simpler:  , u < e−rn ,  0 p , e−rn ≤ u < 1 , P[P V (Tx ) ≤ u] = x  − ln u / r 1 , 1 ≤ u.

(d) Life annuity of 1 per year:

  0 1 − P[P V (Tx ) ≤ u] =  1

− ln(1−ru) / r px

, , ,

u < 0, 0≤u t] and the rule of iterated expectation E[X] = EE[X | Y ]. (Seems like shooting sparrows with cannons in these simple examples, but serves to illustrate a technique that may be useful): Life endowment: m+n Im+n = v m v n E [Im+n ] , m+n Ex = E v

and (just to illustrate) E [Im+n ]

= =

E [E[Im+n | Im ]] = m px E[Im+n | Im = 1] + m qx E[Im+n | Im = 0] m px n px+m

Deferred annuity: ¯x m|n a

= = =

m+n

v t It dt

Z m+n vm E E v (t−m) It dt Im m Z m+n m v m px E E v (t−m) It dt Im = 1 m Z m+n v (t−m) It dt Im = 0 +v m m qx E E m

v m m px a ¯x+m n .

Similar for deferred endowment insurance. (b) Here we use the second method, which is general. Let P V(t,u] denote the random present value at time t of benefits less premiums in (t, u], and abbreviate P Vt = P V(t,∞]. We have P Vt = P V(t,u] + v u−t P Vu . Then

(G.3)

Vt = E[P Vt | It = 1] = E[P V(t,u] | It = 1] + v u−t E[P Vu | It = 1] .

Here E[P V(t,u] | It = 1] = V(t,u] , and

E[P Vu | It = 1] = E [ E[P Vu | It = 1, Iu ] | It = 1] = u−t px+t Vu + u−t qx+t · 0 . From this you gather the stated result. (c) Second method is the superior one. We start from (G.3) for payments deferred in m years: P V0 = v m P Vm . Denote variance by V. We have V[P V0 ] = v 2m V[P Vm ] and V[P Vm ]

= = =

V [ E[P Vm | Im ]] + E [ V[P Vm | Im ]]

V [ Im E[P Vm | Im = 1]] + E [ Im V[P Vm | Im = 1]]

V[Im ] (E [P Vm | Im = 1])2 + E[Im ] V [P Vm | Im = 1] ,

APPENDIX G. SOLUTIONS TO EXERCISES hence

V[P V0 ] = v 2m m px m qx (E [P Vm | Im = 1])2 + m px V [P Vm | Im = 1] = m Ex(2r) − m Ex2 (E [P Vm | Im = 1])2 + m Ex(2r) V [P Vm | Im = 1] . (G.4)

For instance, for an m year deferred n year annuity (G.4) gives the variance (see (4.16) in BL) (2r) (2r) 2 a ¯x+m n − a ¯2x+m n . ¯x+m n − m Ex2 a m Ex r Apply the result to the pure life endowment (a deferred benefit by its very definition) and to a deferred life insurance. (You could put up the expressions immediately using the trick shown in Chapter 4.) Exercise 9 Nt2 = N02 +

Nτ dNτ +

Nτ − dNτ .

For t < T : All terms on both sides are 0, so the equation holds. R t For t ≥ T : On the left Nt2 = 1. On the right N02 = 0, 0 Nτ dNτ = 1, and Rt Nτ − dNτ = 0, so the equation holds. Ignoring the left limit gives 2 on the right 0 when t > T (the jump of N at T ) has been “counted twice”). The relationship Z Z t

Nτ dNτ =

(Nτ − + 1) dNτ

is true for any counting process, not only the simple one considered here. By definition Z t X N (τ )(N (τ ) − N (τ −)) Nτ dNτ = 0

τ ≤t

The sum on the right ranges effectively over the (finite number of) time points τ where N jumps, and at such a time τ it jumps (by 1) from the value it had just before the jump, N (τ −), to the value at the jump time, N (τ ) = N (τ −) + 1. In particular, if N has only one jump, then N (τ −) = 0 at the time jump time τ , and the integral is therefore 0.

Exercise 10 (c) Set v1 (t) = t p0 , v2 (t) = e¯0:t . They satisfy the differential equations v 0 1 (t) = −v1 (t) µt , v 0 2 (t) = −v2 (t) v1 (t) ,

with side conditions v1 (0) = 1, v2 (0) = 0. Statements: (*SPECIFY DIMENSION OF v !*) dim := 2;

(* SPECIFY DIRECTION OF COMPUTATION - FORWARD (’F’) OR BACKWARD (’B’): !*)

APPENDIX G. SOLUTIONS TO EXERCISES

BF := ’F’; (*auxiliary quantities from Danish life table :*) alpha := 0.0005; (*Gompertz-Makeham parameters*) beta := 0.00007585775; gamma := 0.038*ln(10); (*SPECIFY THE TIME INTERVAL [0,T] BY INSERTING T IN ”term” AND - IF NEEDED - THE AGE OF THE LIFE AT TIME 0 ! *) x:= 0; term:= 100; (*SPECIFY STEPS IN DIFFERENCE SCHEME AND OUTPUT !: *) steps := 10000; (*number of steps in difference method*) outp := 100; (*number of values in output*) h := term/steps; (*steplength*) count := steps/outp - 0.0005; (* SPECIFY BOUNDARY CONDITION(S) AT TIME t = 0 IF FORWARD (BF = ’F’) AND AT t = term if BACKWARD (BF = ’B’): !*) v[1] := 1; v[2] := 0; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a’ and B’ THAT ARE NOT 0 !*) B[1,1] := - alpha - beta∗exp(gamma∗(x+t)); B1[1,1] := - beta∗exp(gamma*(x+t))∗gamma; B[2,1] := 1;

Exercise 13 Premium payable at rate π in the deferred period t < m and pension payable at rate 1 in the pension period m < t < m+n, say. Use Thiele’s differential equation: Inspect it directly, or integrate it from 0 to t and from t to n to obtain the retrospective formula for t in the deferred period and the prospective formula in the benefit payment period: R ( R t t π 0 exp τ (r + µx+u ) du dτ , 0 ≤ t < m, Vt = R m+n Rτ exp − (r + µ ) du dτ , m ≤ t < m + n. x+u t t

It is seen that Vt is an increasing function for t < m, no matter if µ is increasing or not; increasing t gives a bigger integrand integrated over a longer interval. We know from the theory in Chapter 4 of BL that, for t ≥ m, Vt = a ¯x+t m+n−t is decreasing if µ is increasing.

Exercise 14 (a) Vt

= = =

E P V(t,n] Tx > t i h E P V(t,t+dt] + e−r dt P V(t+dt,n] Tx > t h i (1 − µx+t dt) E P V(t,t+dt] + e−r dt P V(t+dt,n] Tx > t + dt

APPENDIX G. SOLUTIONS TO EXERCISES

h i + µx+t dt E P V(t,t+dt] + e−r dt P V(t+dt,n] t < Tx < t + dt h i h i (1 − µx+t dt) −πt dt + e−r dt Vt+dt + µx+t dt bt + e−r dt · 0 + o(dt)

−πt dt + (1 − µx+t dt)(1 − r dt)(Vt +

− πt dt + Vt − (µx+t + r) dt Vt +

d Vt dt) + µx+t dt bt + o(dt) dt

d Vt dt + µx+t dt bt + o(dt) dt

Cancel Vt , divide by dt, and let dt go to 0, to arrive at Thiele’s diff. eq. (b) (2)

2 Tx > t E P V(t,n] h i E (P V(t,t+dt] )2 + 2 P V(t,t+dt] e−r dt P V(t+dt,n] + e−2r dt (P V(t+dt,n] )2 Tx > t h i (1 − µx+t dt) E (P V(t,t+dt] )2 + 2 P V(t,t+dt] e−r dt P V(t+dt,n] + e−2r dt (P V(t+dt,n] )2 Tx > t + dt h i + µx+t dt E (P V(t,t+dt] )2 + 2 P V(t,t+dt] e−r dt P V(t+dt,n] + e−2r dt (P V(t+dt,n] )2 t < Tx < t + dt i h (2) (1 − µx+t dt) (−πt dt)2 + 2(−πt dt)e−r dt Vt+dt + e−2r dt Vt+dt i h +µx+t dt b2t + 2bt e−r dt · 0 + e−2r dt · 02 + o(dt) d d (2) (2) (1 − µx+t dt) 2 (−πt dt) (1 − r dt) Vt + Vt dt + (1 − µx+t dt)(1 − 2r dt) Vt + Vt dt dt dt

= = =

+µx+t dt b2t + o(dt)

(2)

−2πt dt Vt + (1 − (µx+t + 2r) dt)Vt

(2)

Cancel Vt

d (2) V dt + µx+t dt b2t + o(dt) dt t

, divide by dt, let dt go to 0, and rearrange a bit to arrive at d (2) (2) V = 2πt Vt + (µx+t + 2r)Vt − µx+t b2t . dt t

APPENDIX G. SOLUTIONS TO EXERCISES

Exercise 15 (a) Let τ1 , τ2 , .P . . denote the times of transition of Z listed in chronological order. The process Nt = i 1[τi ≤ t], t ≥ 0, which counts the total number of transitions, is a hom*ogeneous Poisson process with intensity µ. (Write µ instead of λ to make the formulas given in item (b) meaningful.) The times τ1 , τ3 , . . . are the times where N takes odd values, and τ2 , τ4 , . . . are the times where N takes even values. Thus N12 (t, u] is the number odd numbered of Poisson events between time t and time u. h occurrences i h i N (t) In particular, N12 (t) = N (t)+1 . Likewise, N (t) = . T (t, u] is the total time 21 1 2 2 in (t, u] with odd value of N . (q)

(q)

(b) We could derive the backward differential for the functions Vj (t) and Wj (t), but they are really special cases of Thiele’s differential equation for the reserve and its generalization to higher order moments: T1 (t, n] is the present value at time t of an n-year annuity of 1 per time unit running in state 1 in the present simple Markov model, and with no interest. Similarly, N12 (t, n] is the present value at time t of an n-year insurance 1 payable upon every transition from state 1 to state 2. Thiele for the annuity: d (1) V (t) dt 1 d (1) V (t) dt 2

(1)

−1 − µ (V2 (t) − V1 (t)) ,

− µ (V1 (t) − V2 (t)) ,

(1)

(G.5) (G.6)

subject to (1)

(1)

V1 (n) = V2 (n) = 0 .

(G.7)

There are many ways of solving these equations and we mention a few: (1) Differentiate (G.5) and substitute expressions for the first order derivatives from (1) (G.5) and (G.6) to obtain a second order differential equation in V1 (t) subject to (1) (1) d conditions V1 (n) = 0 and dt V1 (n) = −1, the latter obtained from (G.5). There are standard methods for this problem, see solution to Exercise No 15 in Exercise paper No 5. (2) Add (G.5) and (G.6) to obtain d (1) (1) (V + V2 )(t) = −1 , dt 1 (1)

subject to (V1

(1)

+ V2 )(n) = 0. Solution: (1)

(1)

+ V2 (t) = n − t .

(G.8)

Subtract (G.6) from (G.5) to obtain d (1) (1) (1) (1) (V − V2 )(t) = −1 − 2 µ (V1 − V2 )(t) , dt 1 (1)

(1)

subject to (V1 − V2 )(n) = 0. Solution (we recognize the differential equation and side condition for the reserve on a deterministic annuity of 1 at interest rate 2 µ): (1)

(1)

V1 (t) − V2 (t) =

e−2 µ(τ −t) dτ dτ =

1 − e−2 µ (n−t) . 2µ

APPENDIX G. SOLUTIONS TO EXERCISES We find (1)

V1 (t) =

n−t 1 − e−2 µ (n−t) + , 2 4µ

(G.9)

1 − e−2 µ (n−t) n−t − . 2 4µ (3) Already before we arrived at relation (G.8) we ought to have realized the following: Given Z(t) = 1, T1 (t, n] is the time that remains to spend in the current state. Given Z(t) = 2, T1 (t, n] is the time that remains to spend in the other state. Given Z(t) = 1, T2 (t, n] = (n − t) − T1 (t, n] is the time that remains to spend in the other state. Due to symmetry, the two last mentioned cases are probabilistically identical, so we can conclude that (1) (1) V2 (t) = (n − t) − V1 (t) , (1)

V2 (t) =

which is (G.8). Substituting this into (G.5), we get

d (1) (1) V (t) = −1 + µ ((n − t) − 2 V1 (t)) , dt 1 (1)

which is to be solved subject to V1 (n) = 0. We easily arrive at the solution above. Now to non-central second order moments, and we proceed with method (3) using the general differential equation, which specializes to d (2) (2) (1) (2) V (t) = µ V1 (t) − 2 V1 (t) − µ V2 (t) , dt 1 subject to

(2)

V1 (n) = 0 . Using the symmetry again, we realize that (2)

(1)

(2)

V2 (t) = (n − t)2 − 2 (n − t) V1 (t) + V1 (t) . Substituting this and reorganizing a bit, the differential equation above becomes d (2) (1) V (t) = 2 (µ (n − t) − 1)V1 (t) − µ (n − t)2 . dt 1 Inserting the expression (G.9), we get Z n 1 − e−2 µ (n−τ ) n−τ (2) + − µ (n − τ )2 dτ . V1 (t) = − 2 (µ (n − τ ) − 1) 2 4µ t Substituting n − τ , we are left with the simple task of integrating some standard functions. One should finally arrive at  2  −2µ (n−t) 1 − 2 − e 1   (2) (1) Var[T1 (t, n] | Z(t) = 1] = V1 (t)−(V1 (t))2 = n − t + . 4µ 4µ Exercise 16 (a) Obviously, for t < s < u, paa (t, u) = paa (t, s)paa (s, u) .

APPENDIX G. SOLUTIONS TO EXERCISES Details: P[Z(τ ) = a, τ ∈ (t, u] | Z(t) = a]

= P[Z(τ ) = a , τ ∈ (t, s] | Z(t) = a]P[Z(τ ) = a , τ ∈ (s, u] | Z(t) = a , Z(τ ) = a , τ ∈ (t, s]] ;

The first factor here is paa (t, s), and the second factor is (due to the Markov property) paa (s, u). For instance forward argument (put u and u + du in the roles of s and u): paa (t, u + du) = paa (t, u)paa (u, u + du) = paa (t, u)(1 − (µ(u) + σ(u))du) . From this we get ∂ paa (t, u) = −paa (t, u)(µ(u) + σ(u)) . ∂u Integrating, using the side condition paa (t, t) = 1, we arrive at the answer. (b) Using the Markov property: paa (0, t1 ) σ(t1 ) dt1 pii (t1 + dt1 , t2 ) ρ(t2 ) dt2 paa (t2 + dt2 , t3 ) µ(t3 ) dt3 (plus o(dt1 ) + o(dt2 ) + o(dt3 ), strictly speaking). Due to the factors dti we can replace the arguments ti + dti appearing here by ti . (d) The Kolmogorov forward equations with constant intensities, letting differentiation w.r.t. t be denoted by primes: p0aa (t) = −paa(t)(µ + σ) + pai (t)ρ , p0ai (t)

= paa (t)σ − pai (t)(ν + ρ) .

(G.10) (G.11)

Side conditions: paa (0) = 1 ,

pai (0) = 0 .

(G.12)

Differentiate (G.10): p00aa (t) = −p0aa (t)(µ + σ) + p0ai (t)ρ .

(G.13)

Substitute here p0ai (t) from (G.11): p00aa (t) = −p0aa (t)(µ + σ) + (paa (t)σ − pai (t)(ν + ρ))ρ .

(G.14)

Now solve pai (t) from (G.10) and substitute into (G.14) and rearrange a bit to arrive at p00aa (t) + p0aa (t)((µ + σ) + (ν + ρ)) + paa (t)((µ + σ)(ν + ρ) − σρ) = 0 .

(G.15)

This is a simple hom*ogeneous second order ordinary differential equation, which is to be solved subject to the conditions paa(0) = 1 ,

p0aa (0) = −(µ + σ) ,

(G.16)

the latter obtained by setting t = s in (G.10). The general solution to (G.15) - (G.16) is paa (t) = c1 er1 t + c2 er2 t ,

APPENDIX G. SOLUTIONS TO EXERCISES

when r1 and r2 are distinct solutions to r2 + r((µ + σ) + (ν + ρ)) + ((µ + σ)(ν + ρ) − σρ) = 0 . Here c1 and c2 are constants that are to be determined so as to match the side conditions (G.16): c1 + c 2

c 1 r1 + c 2 r2

−(µ + σ) .

If the roots r1 and r2 coincide, r1 = r2 = r (say), then the general solution is of the form (c1 + c2 t)ert. You find p (µ + σ) + (ν + ρ) ± ((µ + σ) − (ν + ρ))2 + 4σρ r1 =− r2 2 Fill in the details yourself.

Exercise 17 Recall the forward equations: ∂ paa (0, t) = −paa(0, t)(µx+t + σx+t ) + pai (0, t)ρx+t , ∂t

(G.17)

∂ pai (0, t) = paa (0, t)σx+t − pai (0, t)(νx+t + ρx+t ) . ∂t

(G.18)

Side conditions paa (0, 0) = 1 ,

pai (0, 0) = 0 .

(G.19)

Note the following: The forward differential equations for the transition probabilities are usually easier to work with than the backward differential equations. Under the forward construction we work with the functions pjk (t, · ), k = 1, 2, ..., J, for fixed j and t. These functions sum to one and therefore we need only solve J − 1 equations. Under the backward construction we work with the functions pjk ( · , u), j = 1, 2, ..., J, for fixed k and u, which do not sum to 1 or anything else that could be helpful. If one should never the less want to use the backward equations in the present situation, one would face a quasi-difficulty which is due to the notation: e.g. in t paa x , which means paa (x, x + t), x appears in both time variables. To apply the backward construction one must parametrize time as in the general theory, letting starting time and ending time be functionally unrelated. (a) When ρx+t = 0, (G.17) reduces to ∂ paa (0, t) = −paa (0, t)(µx+t + σx+t ) , ∂t which subject to the first condition in (G.19) integrates to Z t (µx+s + σx+s ) ds , paa (0, t) = exp − 0

APPENDIX G. SOLUTIONS TO EXERCISES

the same as the occupancy probability paa (0, t), of course. The equation (G.18) reduces to ∂ pai (0, t) = paa (0, t)σx+t − pai (0, t) νx+t , ∂t or ∂ pai (0, t) + pai (0, t) νx+t = paa (0, t)σx+t , ∂t R t where the function on the right is now known. Multiply by integrating factor exp 0 νx+s ds , integrate from 0 to t, use the second condition in (G.19), and arrive at Z t Z τ Z t νx+s ds dτ , (µx+s + σx+s ) ds σx+τ exp − exp − pai (0, t) = τ

pai (0, t) =

aa τ px

σx+τ t−τ pii x+τ dτ .

This expression can be read aloud; Under the integral sign is the probability that the policy stays in a until an intermediate time τ , then makes the transfer to state i in the time interval (τ, τ + dτ ], and thereafter stays in state i until time t, and the integral sums up the probabilities of these mutually exclusive events. See Section 7.F of BL. The reason why we get explicit expressions here is that there is no return to a state once the policy has left it. It should be made clear, however, that computation of the probabilities goes by numerical solution to the differential equations, which is just as easy with recovery as without. The explicit expressions are useful mainly because they can be directly understood and also because they give us a possibility of discussing how the transition probabilities depend on the intensities. In the same manner as for paa (0, t) we find in the case without recovery that Z t νx+s ds . pii (0, t) = exp − 0

(b) The conditional probability of being active at time t, given alive at time t (and start as active at time 0), is ˜aa tp x

P[Z(t) = a | Z(t) ∈ {a, i}] =

paa(0, t) . paa (0, t) + pai (0, t)

P[Z(t) = a ∩ Z(t) ∈ {a, i}] P[Z(t) ∈ {a, i}]

Likewise, pai (0, t) . paa (0, t) + pai (0, t) = paa (0, t) + pai (0, t) and using (G.17) and (G.18), one finds ˜ai tp x

Differentiating t p[x] µ[x]+t

∂ t p[x] − ∂t t p[x]

− ∂t

∂

= = =

∂ paa (0, t) + ∂t pai (0, t) paa (0, t) + pai (0, t) −paa(0, t)(µx+t + σx+t ) + pai (0, t)ρx+t + paa (0, t)σx+t − pai (0, t)(νx+t + ρx+t ) − paa (0, t) + pai (0, t) paa (0, t)µx+t + pai (0, t)νx+t paa (0, t) + pai (0, t)

˜aa tp x

µx+t + t p˜ai x νx+t ,

APPENDIX G. SOLUTIONS TO EXERCISES

a weighted average of mortality for active and mortality for invalid, the weights being the conditional probabilities of being active and invalid, respectively, given survival. The select mechanism is in this case due to the fact that the insured is known to be, not just alive and x years old at time 0, but also in active state. (c) If µx+t = νx+t for all t ≥ 0, then µ[x]+t = µx+t , of course, and Z t µx+s ds . t p[x] = exp − 0

We have ˜aj tp x

aj = t paj x /t p[x] = t px exp

µx+s ds

j = a, i. Differentiating w.r.t. t, and rearranging a bit, you will see that the t p˜aj x , j = a, i, satisfy the differential equations (G.17) - (G.18) with ρ = 0, which are the Kolmogorov forward equations for in the partial model. They also satisfy the side conditions (G.19). Thus, in the case of non-differential mortality, the transition probabilities can be obtained by first determining the transition probabilities in the simpler partial model with only two states, and then multiplying them with the survival probability. (d) Statements: (* SPECIFY NON-NULL PAYMENTS AT TIME t ! *) bi[1,3] := 1; bi[2,3] := 1; ca[1] := 1; (* SPECIFY MAXIMUM ORDER OF MOMENTS AND NUMBER OF STATES ! *) q := 1; (*moments*) JZ := 3; (*number of states of the policy*) (* SPECIFY TRANSITION INTENSITIES FOR POLICY Z ! Here Danish basis extended with recovery; States 1 = active, 2 = disabled, 3 = dead:*) alpha[1,3] := 0.0005; beta[1,3] := 0.00007585775; gamma[1,3] := 0.038*ln(10); alpha[2,3] := 0.0005; beta[2,3] := 0.00007585775; gamma[2,3] := 0.038*ln(10); alpha[1,2] := 0.0004; beta[1,2] := 0.000003467368; gamma[1,2] := 0.06*ln(10); alpha[2,1] := 0.005; (*SPECIFY AGE x, TERM t, INTEREST RATES AND NON-NULL LIFE ENDOWMENTS ! *) x:= 30; (*age*) t := 30; (*term*) r := ln(1+0.045); (*interest rate*)

APPENDIX G. SOLUTIONS TO EXERCISES be[1] := 0; be[2] := 0; (*endowments at term of contract*)

(*SPECIFY LUMP SUM PREMIUM AT TIME 0: PUT c0 := 1 IF ALL OTHER PREMIUMS ARE 0 AND ONLY MOMENTS OF BENEFITS ARE WANTED ! *) c0 := 0; b0 := 0; (*SPECIFY NUMBER OF STEPS IN RUNGE-KUTTA (OPTIONAL) ! *) steps := 3000; (*number of steps*) h := t/steps; (*steplength*) count := 30; (*number of output times*) count := steps/count - 0.5; Result: state:

time:

30.00

NC 1:

0.000000

time: NC 1:

29.00 0.009740

0.013915

time: NC 1:

28.00 0.017861

0.025918

time: NC 1:

27.00 0.024547

0.036236

.....

time: NC 1:

15.00 0.038582

0.081542

.....

time: NC 1:

1.00 0.002763

0.066309

time: NC 1:

0.00 0.000000

0.064598

pi = 0.004336

Exercise 18 The process must start somewhere, so let us say Z(0) = 1. (a) P

"r+1 \ i=0

Z(ti ) = ji

r \

i=0

Z(ti ) = ji

P Z(tr+1 ) = jr+1

# r \ Z(ti ) = ji i=0

APPENDIX G. SOLUTIONS TO EXERCISES

r \

Z(ti ) = ji

i=0

pjr ,jr+1 (tr , tr+1 ) .

We have here used the Markov property of the process. Repeating this, we obtain "r+1 # r+1 \ Y P Z(ti ) = ji = p1j0 (0, t0 ) pji−1 ,ji (ti−1 , ti ) . (G.20) i=0

i=1

Next, using (G.20), # " r \ Z(ti ) = ji Z(t0 ) = j0 , Z(tr+1 ) = jr+1 P

i=1

= =

T P r+1 i=0 Z(ti ) = ji P [Z(t0 ) = j0 , Z(tr+1 ) = jr+1 ] Q p1j0 (0, t0 ) r+1 i=1 pji−1 ,ji (ti−1 , ti ) p1j0 (0, t0 ) pj0 ,jr+1 (t0 , tr+1 ) Qr+1 i=1 pji−1 ,ji (ti−1 , ti ) . (G.21) pj0 ,jr+1 (t0 , tr+1 )

(b) For s = t0 < t1 < · · · < tr < tr+1 = t: r−1 # " \ Z(ti ) = ji , Z(s) = i, Z(t) = j P Z(tr ) = jr i=1 " # r−1 \ = P Z(tr ) = jr Z(ti ) = ji , Z(t0 ) = j0 , Z(tr+1 ) = jr+1 i=1 Tr+1 P i=0 Z(ti ) = ji i hT = P i=0,...,r−1,r+1 Z(ti ) = ji Q p1j0 (0, t0 ) r+1 i=1 pji−1 ,ji (ti−1 , ti ) Q = p1j0 (0, t0 ) i=1,...,r−1 pji−1 ,ji (ti−1 , ti ) pjr−1 ,jr+1 (tr−1 , tr+1 ) =

pjr−1 ,jr (tr−1 , tr ) pjr ,j (tr , t) . pjr−1 ,j (tr−1 , t)

(G.22)

For given tr = t, jr+1 = j (and t0 = s, j0 = i) this is just a function p˜jr−1 ,jr (tr−1 , tr ) (say) of jr−1 , jr , tr−1 , and tr , showing that the conditional Markov chain is itself Markov. The intensities of the conditional Markov chain are µ ˜gh (τ )

= = =

p˜gh (τ, τ + dτ ) dτ pgh (τ, τ + dτ ) phj (τ + dτ, t) lim dτ &0 dτ pgj (τ + dτ, t) phj (τ, t) . µgh (τ ) pgj (τ, t) lim

dτ &0

 µ (t)  ,  µgh (t) µhj gj (t) lim µ ˜gh (τ ) = ∞, τ %t   0,

g 6= j, h 6= j , g 6= j, h = j , g = j, h 6= j .

(G.23)

APPENDIX G. SOLUTIONS TO EXERCISES

The expression in the case g 6= j, h 6= j is obtained upon writing phj (τ, t) phj (τ, t)/(t − τ ) = pgj (τ, t) pgj (τ, t)/(t − τ ) before taking the limit. Think about these results - they are quite natural. (e) σ ˜ (τ ) = σ(τ ) µ ˜(τ ) = µ(τ )

pii (τ, t) , pai(τ, t)

pdi (τ, t) = 0. pai(τ, t)

Constant intensities and no recovery: σ ˜ (τ )

σRt τ

(

1 e−ν (t−τ ) = Rt −(µ+σ)(s−τ ) −ν (t−s) (ν−µ−σ)(s−τ ) ds e σe ds e τ

ν−µ−σ e(ν−µ−σ)(t−τ ) −1 1 , t−τ

ν−µ−σ = 6 0, ν − µ − σ = 0.

Exercise 19 (a) An n-year endowment insurance with sum insured 1 against level premium c per time unit, continuous time, age of insured upon issue of contract is x. (b) Integration is book-work, see lecture notes. One obtains Z n R Rn τ e− t (r+µx+s ) ds (µx+τ − c) dτ + e− t (r+µx+s ) ds Vt = t

¯x+t n−t − c a A ¯x+t n−t .

Determine c by equivalence requirement V0 = 0 (no down payment at time0): c =

A¯x n . a ¯x n

(b) One must first prove ¯x+t n−t = 1 − r a A ¯ x+t n−t , which is book-work (there are several ways). See lecture notes. Then write Vt = 1 − r a ¯ x+t n−t − c a ¯x+t n−t = 1 − (r + c) a ¯ x+t n−t . Thus, the problem reduces to proving that if µx+t is an increasing function of t, then a ¯x+t n−t is a decreasing function of t. This is book-work, see lecture notes.

APPENDIX G. SOLUTIONS TO EXERCISES

To construct an example where Vt is not an increasing function, i.e a ¯ x+t n−t is not a decreasing function, we must obviously take a µx+t that is not increasing. Fix t < n and write Z n R τ e− 0 (r+µx+s ) ds dτ a ¯x n = 0 Z t R Z n R Rt τ τ = e− 0 (r+µx+s ) ds dτ + e− 0 (r+µx+s ) ds e− t (r+µx+s ) ds dτ 0

¯x+t n−t . a ¯ x t + t Ex a

Keep µx+s fixed for t ≤ s ≤ n. By increasing µx+s for 0 ≤ s ≤ t, we can obviously make a ¯x t and t Ex arbitrarily small. In particular we can arrange that a ¯ x t < 0.5 a ¯x+t n−t and t Ex < 0.5, hence a ¯x n < a ¯x+t n−t . Exercise 20 (a) Book-work. See lecture notes. (b) t p0 is a survival function since it is decreasing and 0 p0 = 1, ∞ p0 = 0. The mortality intensity is d γw(t) t p0 = ··· = . (G.24) µt = − dt δ + W (t) t p0 γ γ δ + W (x) δ + W (x) x+t p0 = = . t px = δ + W (x + t) (δ + W (x)) + (W (x + t) − W (x) x p0 This is of the same form as t p0 , only with W (t) and δ replaced by W (x + t) − W (x) and δ + W (x), respectively.

P[W (T ) + δ > x] = P[T > W

−1

(x − δ)] =

δ W (W −1 (x − δ)) + δ

γ δ = , x

x > δ, a Pareto distribution with shape parameter γ and truncation parameter δ. (c) Use (F.5) with G(t) = t2 , hence dG(t) = 2 t dt Z ∞ 1 2t dt . E[T 2 ] = (1 + t2 /δ)γ 0 Substitute u = 1 + t2 /δ. We have du = 2 t/δ dt and u is an increasing function of t, u = 1 for t = 0, and u = ∞ for t = ∞. Thus Z ∞ δ u−γ du = E[T 2 ] = δ . γ − 1 1 One could also observe directly that Z ∞ Z ∞ −γ+1 −γ δ d 1 + t2 /δ 2 t 1 + t2 /δ dt = −γ + 1 0 0 h −γ+1 −γ+1 i δ 1 + ∞2 /δ − 1 + 02 /δ = −γ + 1 δ = . 1−γ

APPENDIX G. SOLUTIONS TO EXERCISES (d) Recall (G.24). We need ln µt = ln γ + ln w(t) − ln(δ + W (t)) and

T ∧z

µt dt =

Log likelihood is ln Λ

T ∧z

w(t) dt = γ (ln(δ + W (T ∧ z)) − ln δ) . δ + W (t)

m; Tm < zm

−γ

X m

[ln γ + ln w(Tm ) − ln(δ + W (Tm ))]

[ln(δ + W (Tm ∧ z)) − ln δ] .

First order derivatives: X X ∂ [ln(δ + W (Tm ∧ z)) − ln δ] , γ −1 − ln Λ = ∂γ m m; T < z m

X ∂ ln Λ = − ∂δ m; T < z m

X 1 − γ δ + W (Tm ) m

1 1 − δ + W (Tm ∧ z) δ

Put N (z) = ]{m; Tm < z}, the number of deaths within age z. The ML equations are γˆ = P X

m; Tm < zm

N (z) , ˆ m ln(1 + W (Tm ∧ z)/δ)

X 1 + γˆ ˆ δ + W (Tm ) m

(G.25)

1 1 − δˆ + W (Tm ∧ z) δˆ

= 0.

(G.26)

These equations have to be solved numerically. Insert the expression (G.25) for γ ˆ into ˆ Substitute the solution into (G.25) to find γ (G.26) and solve the latter w.r.t. δ. ˆ. ∂2 To find the asymptotic variance matrix, form the second derivatives ∂γ 2 ln Λ, ∂2 ∂δ 2

∂ ln Λ, find their expected values, change sign, and invert the informaln Λ, ∂γ∂δ tion matrix.

(e) If δ is known, then (G.25) with δˆ replaced by δ is an explicit expression for γ ˆ. Divide by n in denominator and numerator, let n go to ∞, and use the law of large numbers to conclude that γˆ →

P[T1 ≤ z] . E[ln(1 + W (T1 ∧ z)/δ)]

(G.27)

From (F.6) we have P[T1 ≤ z] = 1 − t p0 = 1 −

δ W (z) + δ

Use (F.5) with t p0 given by (F.6) and G(t) = ln(1 + W (t ∧ z)/δ), for which G(0) = 0 and ( w(t) dt , 0 < t < z , δ+W (t) dG(t) = 0 t > z.

APPENDIX G. SOLUTIONS TO EXERCISES

We find E[ln(1 + W (T1 ∧ z)/δ)]

γ δ w(t) dt W (t) + δ δ + W (t) 0 Z z w(t) δγ dt (W (t) + δ)γ+1 0 1 1 − δγ γ (W (0) + δ)γ (W (z) + δ)γ γ δ γ 1− W (z) + δ Z

= = = =

Inserting these expressions in (G.27), we find that the limit is γ. (f) OE rate in year j is µ ˆ=

Nj , Wj

P where Nj = ]{m; j − 1 < Tm ≤ j} and Wj = m ((Tm ∧ j) − (j − 1)) ∨ 0. Choose representative age τj ∈ (j − 1, j] and weights wj and minimize Q =

n X j=1

wj (µ(τj ; γ, δ) − µ ˆ j )2 =

2 n X γw(τj ) −µ ˆj . δ + W (τj ) j=1

Minimize by forming the partial derivatives, n X ∂ w(τj ) γw(τj ) Q = − µ ˆj , wj 2 ∂γ δ + W (τ δ + W (τj ) j) j=1 n X γw(τj ) −γ w(τj ) ∂ wj 2 Q = − −ˆ µj , ∂δ δ + W (τ (δ + W (τj ))2 j) j=1

setting them equal to zero and solving for γ ∗ and δ ∗

Exercise 21 (a) Direct backward argument, conditioning on what happens in the small time interval (t, t + dt) and splitting payments in (t, z] into payments in (t, t + dt] and payments in (t + dt, z]: Va (t)

(1 − µx+t dt − σx+t dt)(−c dt + e−r dt Va (t + dt))

+ σx+t dt (O(dt) + e−r dt Vi (t + dt)) + µx+t dt O(dt) + o(dt) ,

where O(dt) is of order dt and o(dt) is of order less than dt (i.e o(dt)/dt → 0 as dt → 0). Insert (Taylor expansion to first order) e−r dt = 1 − r dt + o(dt) , Vj (t + dt) = Vj (t) +

d Vj (t) dt + o(dt) , j = a, i, dt

APPENDIX G. SOLUTIONS TO EXERCISES and collect all terms of order dt2 or less in o(dt): Va (t) = − c dt + (1−µx+t dt−σx+t dt − r dt) Va (t) +

d Va (t) dt + σx+t dt Vi (t) + o(dt) . dt

Subtract Va (t) on both sides, divide by dt and let dt go to zero, to arrive at d Va (t) = rVa (t) + c − σx+t (Vi (t) − Va (t)) + µx+t Va (t) . dt Similarly d Vi (t) = r Vi (t) − b + νx+t Vi (t) . dt Side conditions Va (z) = Vi (z) = 0. (b) The differential equations now become d Va (t) = rVa (t) + c − σx+t (Vi (t) − Va (t)) . dt d Vi (t) = r Vi (t) − b + νx+t Vi (t)) . dt Side conditions remain the same. (c) Same product as in (a) plus an endowment insurance with sum 1. (d) paa (s, t)

pii (s, t) pai (s, t) pad (s, t)

e−

s (µx+u +σx+u ) du

e− s νx+u du

e−

Rτ

s (µx+u +σx+u ) du

σx+τ e−

τ νx+u du

dτ ,

1 − paa (s, t) − pai (s, t) .

(e) As usual, let Ij (t) and Njk (d) denote, respectively, the number of policies in state j at time t and the total number of transitions j → k up to and including time t. The log likelihood function is Z z Z z Z z 0 ln Λ = ln σ dNai (t) + ln(α + βeγ(x+t) ) dNad (t) + ln(α + βeγ (x+t) ) dNid (t) 0 0 0 Z z Z z γ(x+t) γ 0 (x+t) − (α + βe + σ) Ia (t) dt − (α + βe ) Ii (t) dt . 0

1st derivatives: ∂ ln Λ ∂α

∂ ln Λ ∂β

1 dNad (t) + α + βeγ(x+t)

− Z

1 dNid (t) α + βeγ 0 (x+t)

eγ (x+t) dNid (t) α + βeγ 0 (x+t)

(Ia (t) + Ii (t)) dt ,

eγ(x+t) dNad (t) + α + βeγ(x+t)

−

γ(x+t)

Ia (t) + e

γ 0 (x+t)

Ii (t)) dt ,

APPENDIX G. SOLUTIONS TO EXERCISES ∂ ln Λ ∂γ ∂ ln Λ ∂γ 0 ∂ ln Λ ∂σ

= = =

Z Z Z

z 0 z 0 z 0

βeγ(x+t) (x + t) dNad (t) − α + βeγ(x+t) 0

βeγ (x+t) (x + t) dNid (t) − α + βeγ 0 (x+t) Z z 1 dNai (t) − Ia (t) dt . σ 0

βeγ(x+t) (x + t)Ia (t) dt ,

βeγ

(x+t)

(x + t)Ii (t) dt ,

Set equal to 0 and solve for the ML estimators.

Exercise 23 Assume that X is non-decreasing. Note that ∞ D(X) = ∪∞ m=1 ∪n=1 Dm,n ,

where Dm,n = {t ; t ≤ n , Xt+ − Xt− ≥ ∞ > X n ≥ X0 +

t∈Dm,n

1 }. m

Since

(Xt+ − Xt− ) > X0 +

1 ]Dm,n , m

we conclude that ]Dm,n is finite. Thus, being a countable union of finite sets, D(X) is (at most) countable. Exercise 25 q dXtq = qXtq−1 xct dt + (Xt− + xdt )q − Xt− dNt . Exercise 26 (See also Exercise 56. Here notation is changed: a and b are called α and β) dS(t) = eαt+βN (t) αdt + dN (t) eαt+β(N (t−)+1) − eαt+βN (t−) = eαt+βN (t−) αdt + dN (t)eαt+βN (t−) eβ − 1 = S(t−) α dt + eβ − 1 dN (t) = S(t−) α + λ eβ − 1 dt + S(t−) eβ − 1 dM (t) ,

where M (t) = N (t) − λt, a so-called martingale (a process with conditionally zero mean and uncorrelated increments, here actually independent increments). The rest is straightforward .

Exercise 34 Balance equation 14 X j=0

(1 + i)24−j c `55+j −

24 X

j=15

(1 + i)24−j b `55+j = 0

APPENDIX G. SOLUTIONS TO EXERCISES gives

P24

j=15 (1

c = P14

j=0 (1

+ i)−j `55+j + i)−j `55+j

= 0.381 .

Exercise 50 (a) Kolmogorov forward, using the obvious paa (t, u) = 1 − pai (t, u): pai (t, u + du) = pai (t, u)(1 − ρ du) + (1 − pai (t, u))σ du, ∂ pai (t, u) + pai (t, u)(ρ + σ) = σ ∂u ∂ pai (t, u)e(ρ+σ)u = σe(ρ+σ)u ∂u Integrate from t to u using pai (u, u) = 0 to find σ (ρ+σ)u pai (t, u)e(ρ+σ)u = e − e(ρ+σ)t , ρ+σ

hence the claimed formula. It depends only on u − t due to hom*ogeneity.

(b) Either direct integration Z n pai (0, τ ) dτ = πn = 0

σ ρ+σ

1 − e−(ρ+σ)n n− , ρ+σ

or use Thiele’s differential equations. Single premium per unit of time insured is πn σ 1 − e−(ρ+σ)n = 1− . n ρ+σ (ρ + σ)n The function g(x) =

1 − e−x x

(G.28)

has derivative

1 + x − ex e−x 1 − e−x − = e−x , 2 x x x2 which is < 0. It follows that, as n % +∞, g 0 (x) =

πn σ % . n ρ+σ

Reasonable: Increasing function of σ, decreasing function of ρ. (c) Likelihood

Λ = σ Nai ρNia e−σWa −ρWi , (m) , `=1 Tn

the total time spent in active state (and Wi = nm − Wa the where Wa = total time spent in inactive state). Nai ∂ ln Λ = − Wa , ∂σ σ

APPENDIX G. SOLUTIONS TO EXERCISES ∂2 Nai ln Λ = − 2 , ∂σ 2 σ plus similar expressions for derivatives w.r.t. ρ, and ∂2 ln Λ = 0 . ∂σ∂ρ Thus, the ML estimators are the occurrence-exposure rates σ ˆ=

Nai , Wa

ρˆ =

Nia , Wi

which are asymptotically independent, normally distributed and unbiased, and as.Varˆ σ=

σ2 σ , = ENai mn(1 − πn /n)

where we have used Z n Z ENai = m paa (0, τ )σ dτ = mσ 0

(1 − pai (0, τ )) dτ = mnσ(1 −

πn ), n

see item (b). For fixed w = mn (i.e. m = w/n) as.Varˆ σ=

σ w(1 −

πn ) n

which is an increasing function of the time span n by the results in item (b). We also find ρ2 ρ as.Varρˆ = = πn , ENia w n where we have used ENia = m

pai (0, τ )ρ dτ = mnρ 0

πn . n

We see that as.Varˆ ρ is a decreasing function of the time span n for fixed w. Comment: The asymptotic variance of an intensity estimator is better the longer the total expected time spent in the state from which the the relevant transition is made. All policies start from state a at time 0. For fixed total exposure the estimation of σ will be good for many policies observed over a short time (when they are likely to remain active), and estimation of ρ will be good for few policies observed over a long time when they can make it to inactive state. (d) We find p˜jk (t, u)

= = =

P[Z(s1 ) = j1 , . . . , Z(sr ) = jr , Z(t) = j, Z(u) = k, Z(n) = i] P[Z(s1 ) = j1 , . . . , Z(sr ) = jr , Z(t) = j, Z(n) = i] paj1 (0, s1 ) · · · pjr−1 jr (sr−1 , sr )pjr j (sr , t)pjk (t, u)pki (u, n) paj1 (0, s1 ) · · · pjr−1 jr (sr−1 , sr )pjr j (sr , t)pji (t, n)

pjk (t, u)pki (u, n) . pji (t, n)

APPENDIX G. SOLUTIONS TO EXERCISES Corresponding intensities: σ ˜ (t) = lim

u&t

pii (t, n) pai (t, u) pii (u, n) =σ u − t pai (t, n) pai (t, n)

and, similarly, ρ˜(t) = ρ

pai (t, n) , pii (t, n)

with pai (t, n) and pii (t, n) given by expressions in item (a). Full expression for σ ˜ (t) is useful later: ρ 1 − e−(σ+ρ)(n−t) 1 − σ+ρ σ + ρe−(σ+ρ)(n−t) = . σ ˜ (t) = σ σ (1 − e−(σ+ρ)(n−t) ) 1 − e−(σ+ρ)(n−t) σ+ρ (e) When ρ = 0,

σ . 1 − e−σ(n−t) Let the occurrence-exposure rate in year j = 1, . . . , n be σ ˜ (t) =

σ ˆj =

Nai;j , Wa;j

where Nai;j and Wa;j are the number of claims and the total time at risk in year j (time interval [j −1, j)). Graduation means estimating σ by minimizing the weighted squares differences between the observed rates σ ˆ j and the theoretical intensities σ ˜ (j −0.5), with 2 ˆ σj = Nai;j /Wa;j weights equal to the inverse estimated variances Varˆ : 2 n 2 X Wa;j σ σ ˆj − . Nai;j 1 − e−σ(n−j+0.5) j=1 Exercise 51 (a) (1)

leads to

(1)

pai (0, t + dt) = pai (0, t)(1 − (ν(t) + ρ(t)) dt) + paa (0, t) σ(t) dt

d (1) (1) p (0, t) = −pai (0, t)(ν(t) + ρ(t)) + paa (0, t) σ(t) , dt ai with side condition (1) pai (0, 0) = 0 . (Integrating gives the following integral expression, which could be put up by direct reasoning: Z t Z s Z t (1) pai (0, t) = exp(− (µ + σ))σ(s) ds exp(− (ν + ρ)) .) 0

Next, leads to

(1)

(1) p(1) aa (0, t + dt) = paa (0, t)(1 − (µ(t) + σ(t)) dt) + pai (0, t) ρ(t) dt

d (1) (1) paa (0, t) = −p(1) aa (0, t)(µ(t) + σ(t)) + pai (0, t) ρ(t) , dt

APPENDIX G. SOLUTIONS TO EXERCISES with side condition

p(1) aa (0, 0) = 0, .

(Integral expression, which could be put up by direct reasoning: Z t Z t (1) (µ + σ)) .) pai (0, s) ρ(s) ds exp(− p(1) aa (0, t) = s

Repeating the argument for k = 2, 3, ...: d (k) (k) p (0, t) = −pai (0, t)(ν(t) + ρ(t)) + p(k−1) (0, t) σ(t) , aa dt ai (k)

pai (0, 0) = 0 , and

d (k) (k) p (0, t) = −p(k) aa (0, t)(µ(t) + σ(t)) + pai (0, t) ρ(t) , dt aa p(k) aa (0, 0) = 0 . (0)

Introducing paa (0, t) = paa (0, t) would save work. Obviously,

∞ X

(k)

pai (0, t) = pai (0, t) .

k=1

(b)

P [Z(τ ) = i; τ ∈ [t − q, t] | Z(0) = a] = pai (0, t − q)pii (t − q, t) .

(c)

π= Reserve:

R t+q t

n q

e−rτ pai (0, τ − q)pii (τ − q, τ ) dτ .

e−r(τ −t)pii (t, τ ) dτ +

t+q

e−r(τ −t)pii (t, τ − q)pii (τ − q, τ ) dτ .

Exercise 52 Expected PV at time 0 of benefits is Z n e−rτ (1 − τ −n/2 px )τ py µy+τ dτ . n/2

Expected PV at time 0 of premiums is π times Z n/2 e−rτ τ pxτ py dτ . 0

Equivalence premium π is the ratio between these expressions.

Exercise 53 (a) Expected present value at time 0 of benefits is 5 times

APPENDIX G. SOLUTIONS TO EXERCISES

i h Rm W b = E e 0 (−r(s)+a(s)−µ(x+s)) ds Y (0) = i .

(G.29)

Consider the functions

h Rm i Wjb (t) = E e t (−r(s)+a(s)−µ(x+s)) ds Y (t) = j ,

t ∈ [0, m], j = 1, . . . , J. We need to determine W b = Wib (0). Backward construction: X λjk dtWkb (t) + o(dt) Wjb (t) = (1 − λj· dt)e(−rj +aj −µ(x+t)) dt Wjb (t + dt) + k;k6=j

leads to X d b λjk Wkb (t) . Wj (t) = (λj· + rj − aj + µ(x + t))Wjb (t) − dt k; k6=j

Solve backwards subject to conditions Wjb (m) = 1 , j = 1, . . . , J. Expected present value at time 0 of premiums is π times Z m R τ Wc = E e 0 (−r(s)+a(s)−µ(x+s)) ds dτ Y (0) = i .

(G.30)

Consider the functions Wjc (t)

e t

Rτ

t (−r(s)+a(s)−µ(x+s)) ds

dτ Y (t) = j .

t ∈ [0, m], j = 1, . . . , J. We need to determine W c = Wic (0). Backward construction: X λjk dtWkc (t) + o(dt) Wjc (t) = (1 − λj· dt) dt + e(−rj +aj −µ(x+t)) dt Wjc (t + dt) + k;k6=j

leads to X d c λjk Wkc (t) . Wj (t) = −1 + (λj· + rj − aj + µ(x + t))Wjc(t) − dt k; k6=j Solve backwards subject to conditions Wjc (m) = 0 , j = 1, . . . , J. Solve both systems numerically by e.g. ’prores2’ and determine π = W b /W c . (b) If aj = rj for all j, then a(t) = r(t) for all t and they cancel out of the expressions for the present values when we drop the expectation. Thus, equivalence in the sense defined can be attained.

APPENDIX G. SOLUTIONS TO EXERCISES Exercise 54

(a) With profit contract: Stipulating benefits and premiums in nominal values, binding to both parties. Charge a premium ’on the safe side’, typically by using conservative technical (first order) valuation basis. If everything goes well, surplus will accumulate. This surplus belongs to the insured and is to be repaid as so-called bonus, e.g. as increased benefits or reduced premiums. Unit linked: Unit linked contract: Stipulating benefits and (possibly) premiums in units of a share of the investment portfolio, that is, let contractual payments be inflated by the ’price index’ of the investment portfolio rather than being fixed nominal amounts. With all payments perfectly linked, the financial risk will be eliminated in a large portfolio. (b) Single premium, given Y (0) = i, is h Rn i π = E e− 0 r (U (n) ∨ g) Y (0) = i n px .

Disregarding the uninteresting term n px henceforth, we need to determine h i Rn W = E (1 ∨ e− 0 r g) Y (0) = i .

The starting point is the ’price’ of the claim at time t, given the current information about the past, i h i h Rn Rn E e− t r (U (n) ∨ g) Y (τ ); 0 ≤ τ ≤ t = E U (t) ∨ e− t r g Y (τ ); 0 ≤ τ ≤ t .

Due to the Markov property this expression is a function of t, Y (t) and U (t). Consider its value at time t for given U (t) = u, and Y (t) = j, i h Rn Wj (t, u) = E u ∨ e− t r g Y (t) = j .

The premium we seek is Wi (0, 1) Now use the backward construction, disregarding terms of order o(dt): i h Rτ X λjk dt Wk (t, u) Wj (t, u) = (1 − λj· dt)E u ∨ e−rj dt e− t+dt r g Y (t + dt) = j + k; k6=j

(1 − λj· dt) e

−rj dt

(1 − λj· dt) e

−rj dt

rj dt

∨e

Wj (t + dt, ue

Rτ − t+dt r

rj dt

i X λjk dt Wk (t, u) g Y (t + dt) = j + k; k6=j

λjk dt Wk (t, u)

k; k6=j

Insert e±rj dt = 1 ± rj dt + o(dt) and Wj (t + dt, erj dt u)

= =

Wj (t + dt, u + urj dt) + o(dt) ∂ ∂ Wj (t, u) + Wj (t, u) dt + Wj (t, u) u rj dt + o(dt) , ∂t ∂u

and fill in some details to arrive at the partial differential equations −rj Wj (t, u) +

X ∂ ∂ Wj (t, u) + Wj (t, u) u rj + λjk (Wk (t, u) − Wj (t, u)) = 0 . ∂t ∂u k; k6=j

APPENDIX G. SOLUTIONS TO EXERCISES

These are to be solved subject to the conditions Wj (n, u) = (u ∨ g) , j = 1, . . . , J.

Exercise 55 (a) Expected PV at time 0 of benefits: Z n e−rτ (1 − τ /2 px )τ py µy+τ dτ . 0

Expected PV at time 0 of premiums is π times Z n/2 e−rτ τ pxτ py dτ . 0

Equivalence premium π is the ratio between these expressions. (b) (Do not spend too much time on this.) A straightforward method for computation is to define v1 (t) = 1 − t/2 px , v2 (t) = t py , Z t v3 (t) = e−rτ (1 − τ /2 px )τ py µy+τ dτ . 0

and solve numerically the system of differential equations v10 (t) = µx+t/2 (1/2) v1 (t) , v20 (t) = −µy+t v2 (t) ,

v30 (t) = e−rt v1 (t) v2 (t) µy+t , by a forward difference scheme starting from the conditions v1 (0) = 0, v2 (0) = 1, v3 (0) = 0. A more sophisticated method hinted at in the problem: Observe that ! Z t Z t/2 1 1 µ(x + s) ds , µ(x + s) ds = exp − t/2 px = exp − 2 0 0 2 formally a survival function with intensity µ ˜(t) = µ(x + t/2)/2. Then the single premium is the difference between the single premiums of two well-known simple products, which may be computed by solving their Thiele differential equations numerically. Or compute by e.g. the program ’prores1’ the expected discounted value of an assurance of 1 payable upon transition from state 1 to state 3 in a four states Markov model on {0, 1, 2, 3}, starting from state 0, with transition intensities µ01 (t) = µ23 (t) = µ(x + t/2)/2, µ02 (t) = µ13 (t) = µ(y + t), and all other intensities 0. (c) Reserve Vt at time t ∈ [0, n] depends on what is currently known about (x) and (y): (y) dead: Vt = 0. (y) alive, t ≥ n/2, Tx > n/2: Vt = 0.

APPENDIX G. SOLUTIONS TO EXERCISES

Rn (y) alive, t ≥ n/2, Tx ≤ n/2: Vt = 2Tx ∨t e−r(τ −t) τ −t py+tµy+τ dτ . Rn (y) alive, t < n/2, Tx ≤ t: Vt = 2Tx e−r(τ −t) τ −t py+t µy+τ dτ . R n −r(τ −t) R n/2 (y) alive, t < n/2, Tx > t: Vt = 2t e (1−τ /2−t px+t ) τ −t py+t µy+τ dτ −π t e−r(τ −t)τ −t px+t τ −t py+t dτ .

(d) In general, for a unit due at some random time, the non-central 2nd moment of present value is the same as the expected value, only with 2r instead of r. Exercise 21 (b) The expected present value at time 0 of the benefits is 0.75 times Wb

= =

E E

S(m) e

Rm 0

m+n m

a(s) ds

U (τ )−1 τ px dτ Y (0) = i

m+n

e−

Rτ

0 (r(s)+µ(x+s)) ds

dτ Y (0) = i .

Proceed along the lines of Problems 4.7 - 4.11. The conditional expected value of the discounted benefits, given what is known at time t ∈ [0, m + n], is 0.75 times R Z m+n R τ m e− 0 (r(s)+µ(x+s)) ds dτ Y (τ ); 0 ≤ τ ≤ t . W b (t) = E e 0 a(s) ds m

The gist of the matter in the backward construction is to separate out what relates only to the past and what relates to the future here, and to work with the conditional expected values of the latter, given the relevant pieces of current information. Firstly, for m ≤ t ≤ m + n, Z t R Rm τ W b (t) = e 0 a(s) ds e− 0 (r(s)+µ(x+s)) ds dτ m h i Rm Rt ˜ b (t) | Y (τ ); 0 ≤ τ ≤ t , + e 0 a(s) ds− 0 (r(s)+µ(x+s)) ds E W where ˜ b (t) = W

m+n t

e−

Rτ

t (r(s)+µ(x+s)) ds

dτ .

In the expression for W b (t) we need to determine the conditional expected value ˜ b (t) in the last term. Due to the Markov property (conditional independence of W between past and future, given the present), this conditional expected value depends only on the time t and the current state of the economy, Y (t). Therefore, we can restrict attention to the state-wise conditional expected values, i h ˜ jb (t) = E W ˜ b (t) Y (t) = j , W m ≤ t ≤ m + n, j = 1, . . . , J. These are the solution to the differential equations X d ˜b ˜ jb (t) − 1 − ˜ kb (t) − W ˜ jb (t)) = 0 , Wj (t) = (rj + µ(x + t)) W λjk (W (G.31) dt k; k6=j

subject to the conditions ˜ jb (m + n) = 0 , W

(G.32)

APPENDIX G. SOLUTIONS TO EXERCISES

j = 1, . . . , J. Outline of details (which you shouldn’t spell out unless you are explicitly asked to do so): Z t+dt R τ ˜ b (t) = W e− t (r(s)+µ(x+s)) ds dτ t

+ e− =

R t+dt t

(r(s)+µ(x+s)) ds

m+n

e−

t+dt

Rτ

t+dt (r(s)+µ(x+s)) ds

dτ

˜ b (t + dt) + o(dt) , eO(dt) dt + e−(r(t)+µ(x+t)) dt W

where O(dt) is of order dt and o(dt) is of order (dt)2 , hence ˜ b (t) = dt + (1 − (r(t) + µ(x + t)) dt) W ˜ b (t + dt) + o(dt) . W (We have lumped negligible terms into o(dt): For instance, by Taylor expansion, ˜ kb (t + dt) = dt (W ˜ kb (t) + O(dt)) = dt exp(O(dt)) = dt (1 + O(dt)) = dt + o(dt) and dt W b ˜ dt Wk (t)+o(dt).) Now, condition on what happens in the small time interval (t, t+dt): i h ˜ jb (t) = (1 − λj · dt) E dt + (1 − (r(t) + µ(x + t)) dt) W ˜ b (t + dt) + o(dt) Y (τ ) = j, τ ∈ [t, t + dt] W h i X ˜ b (t + dt) + o(dt) Y (t) = j, Y (t + dt) = k λjk dt E dt + (1 − (r(t) + µ(x + t)) dt) W + k; k6=j

˜ jb (t + dt)] + (1 − λj · dt) [ dt + (1 − (rj + µ(x + t)) dt) W

˜ jb (t + dt) − (rj + µ(x + t) + λj · ) dt W ˜ jb (t) + dt + W

˜ kb (t) + o(dt) λjk dt W

k; k6=j

˜ kb (t) + o(dt) . λjk dt W

k; k6=j

(The first step in the above derivation is made here just to make sure you understand the argument - it need not be given in an answer to the Problem.) Subtracting ˜ jb (t + dt) on both sides, dividing by dt, and letting dt go to 0, we arrive at (G.31). W Another way to do it is to set ˜ jb (t + dt) = W ˜ jb (t) + d W ˜ b (t) dt + o(dt) , W dt j ˜ jb (t) on both sides, and finally divide by dt. cancel W Secondly, for 0 ≤ t ≤ m, h i Rt ˜ b (t)| Y (τ ); 0 ≤ τ ≤ t , W b (t) = e 0 (a(s)−r(s)−µ(x+s)) ds E W

where now

˜ b (t) = e W

Rm t

(a(s)−r(s)−µ(x+s)) ds

m+n m

e−

Rτ

m (r(s)+µ(x+s)) ds

dτ .

The state-wise conditional expected values of this function are i h ˜ b (t) Y (t) = j , ˜ jb (t) = E W W

0 ≤ t ≤ m, j = 1, . . . , J. These are the solution to the differential equations X d ˜b ˜ jb (t)) = 0 . ˜ kb (t) − W ˜ jb (t) − λjk (W Wj (t) = (rj + µ(x + t) − aj ) W dt k; k6=j

(G.33)

APPENDIX G. SOLUTIONS TO EXERCISES

˜ b (t) in [0, m] and [m, m + n] coincide, At time t = m the defining expressions for W ˜ jb (m) obtained after having so we just proceed backwards in [0, m] from the values W completed the backward scheme in [m, m + n]. Brief outline of details of the derivation of the differential equations: ˜ b (t) W

= =

˜ jb (t) W

R t+dt

˜ b (t + dt) W ˜ b (t + dt) ; (1 + (a(t) − r(t) − µ(x + t)) dt) W

(a(s)−r(s)−µ(x+s)) ds

˜ jb (t + dt) (1 − λj · dt) (1 + (aj − rj − µ(x + t)) dt)W X ˜ kb (t + dt) + o(dt) , λjk dt W +

k; k6=j

and the rest is trivial. The expected present value of future benefits at time 0, when the economy starts in state i, is ˜ ib (0) . Wb = W The expected present value at time 0 of the premiums is π times

= =

E U (τ ) τ px dτ Y (0) = i 0 Z m R τ e− 0 (r(s)+µ(x+s)) ds dτ Y (0) = i . E Z

−1

The situation is now simpler than in the case of the benefits. For t ∈ [0, m] it suffices to look at the state-wise conditional expected values Z m R τ e− t (r(s)+µ(x+s)) ds dτ Y (t) = j , Wjc (t) = E t

0 ≤ t ≤ m, j = 1, . . . , J. These we determine as the solution to the differential equations (copy arguments for the case t ∈ [m, m + n] above) X d c λjk (Wkc (t) − Wjc (t)) = 0 , Wj (t) = (rj + µ(x + t)) Wjc (t) − 1 − dt k; k6=j

(G.34)

subject to the conditions Wjc (m) = 0 ,

(G.35)

j = 1, . . . , J. The expected present value at time 0, when the economy starts in state i, is W c = Wic (0) . Finally, determine π from the equivalence relation π Wic (0) = 0.75 Wib (0): π = 0.75 Wib (0)/Wic (0) . This solution was filled with details that would not make any good in an exam; you should just write the essential steps. To check your understanding of these things,

APPENDIX G. SOLUTIONS TO EXERCISES

try your hand on the modified contract where the benefit is a lump sum of 4 S(m) payable upon retirement at time m. (A simpler problem, of course.) Problem 4.15 This is a small perturbation of Problem 4.11, and actually it is simpler. We treat only the benefits (for premiums we do as in Problem 4.13). Suppose Y (0) = i. The expected discounted value of the benefits is n px times h Rn i h i Rn W = E e− 0 r (U (n) ∨ g) Y (0) = i = E (1 ∨ e− 0 r g) Y (0) = i .

A suitable starting point is the ’price’ of the claim at time t, given the current information about the past, h Rn i i h Rn E e− t r (U (n) ∨ g) Y (τ ); 0 ≤ τ ≤ t = E U (t) ∨ e− t r g Y (τ ); 0 ≤ τ ≤ t . Arguing as in Problem 4.10 of the Exercises, we realize that this expression is a function of t, Y (t) and U (t). Consider its value at time t for given U (t) = u, and Y (t) = j, h i Rn Wj (t, u) = E u ∨ e− t r g Y (t) = j .

The value W we seek is Wi (0, 1). Now use the backward construction, disregarding terms of order o(dt): h i Rτ X Wj (t, u) = (1 − λj· dt)E u ∨ e−rj dt e− t+dt r g Y (t + dt) = j + λjk dt Wk (t, u) k; k6=j

(1 − λj· dt) e

−rj dt

(1 − λj· dt) e

−rj dt

rj dt

∨e

Wj (t + dt, ue

Rτ − t+dt r

rj dt

i X λjk dt Wk (t, u) g Y (t + dt) = j + k; k6=j

λjk dt Wk (t, u) .

k; k6=j

Now do as in 4.11 and fill in some details to arrive at the partial differential equations −rj Wj (t, u) +

X ∂ ∂ Wj (t, u) + Wj (t, u) u rj + λjk (Wk (t, u) − Wj (t, u)) = 0 . ∂t ∂u k; k6=j

These are to be solved subject to the conditions Wj (n, u) = (u ∨ g) , j = 1, . . . , J. The expected discounted premiums are found as in Problem 4.13. Exercise 57 (a) Direct from the Poisson distribution: E[S(t)]

= =

h i eαt E eβN (t) eαt

∞ X

n=0

eβn

(λt)n −λt e n!

APPENDIX G. SOLUTIONS TO EXERCISES

= = =

∞ X (eβ λt)n n! n=0 eαt e−λt exp eβ λt exp αt + (eβ − 1)λt .

eαt e−λt

Backward construction, particularly simple here since αt + bN (t) has stationary and independent increments: h i V (t) = E eαt+βN (t) =

= =

(1 − λdt)eαdt V (t − dt) + λ dt eαdt+β V (t − dt) + o(dt)

eαdt V (t − dt) − λ dt eαdt V (t − dt) + λ dt eαdt+β V (t − dt) + o(dt) d (1 + α dt)(V (t) − V (t) dt) dt −λ dt eαdt V (t − dt) + λ dt eα dt+β V (t − dt) + o(dt) .

Cancel V (t), divide by dt and let dt go to 0 to obtain a simple differential equation, to be solved subject to V (0) = 1. For non-central q-th moments, just replace α and β with qα and qβ. (b) dS(t)

= = =

eαt+βN (t) αdt + dN (t) eαt+β(N (t−)+1) − eαt+βN (t−) eαt+βN (t−) αdt + dN (t)eαt+βN (t−) eβ − 1 S(t−) αt + λ eβ − 1 dt + S(t−) eβ − 1 dM (t) ,

where M (t) = N (t) − λt, a so-called martingale (a process with conditionally zero mean and uncorrelated increments, here actually independent increments). (c) Replace α and β in (a) with −α and −β, and integrate the expression for the expected value, to obtain the claimed answer (misprint: a λ is missing). (d) This is now trivial. BEGIN From 305exsol Exercise 55 (a) Expected PV at time 0 of benefits: Z n e−rτ (1 − τ /2 px )τ py µy+τ dτ . 0

Expected PV at time 0 of premiums is π times Z n/2 e−rτ τ pxτ py dτ . 0

APPENDIX G. SOLUTIONS TO EXERCISES

e−rτ (1 − τ /2 px )τ py µy+τ dτ .

and solve numerically the system of differential equations v10 (t) = µx+t/2 (1/2) v1 (t) , v20 (t) = −µy+t v2 (t) ,

v30 (t) = e−rt v1 (t) v2 (t) µy+t , by a forward difference scheme starting from the conditions v1 (0) = 0, v2 (0) = 1, v3 (0) = 0. A more sophisticated method hinted at in the problem: Observe that ! Z t Z t/2 1 1 µ(x + s) ds = exp − µ(x + s) ds , t/2 px = exp − 2 0 2 0 formally a survival function with intensity µ ˜(t) = µ(x + t/2)/2. Then the single premium is the difference between the single premiums of two well-known simple products, which may be computed by solving their Thiele differential equations numerically. Or compute by e.g. the program ’prores1’ the expected discounted value of an assurance of 1 payable upon transition from state 1 to state 3 in a four states Markov model on {0, 1, 2, 3}, starting from state 0, with transition intensities µ01 (t) = µ23 (t) = µ(x + t/2)/2, µ02 (t) = µ13 (t) = µ(y + t), and all other intensities 0. (c) Reserve Vt at time t ∈ [0, n] depends on what is currently known about (x) and (y): (y) dead: Vt = 0. (y) alive, t ≥ n/2, Tx > n/2: Vt = 0. Rn (y) alive, t ≥ n/2, Tx ≤ n/2: Vt = 2Tx ∨t e−r(τ −t) τ −t py+tµy+τ dτ . R n −r(τ −t) (y) alive, t < n/2, Tx ≤ t: Vt = 2Tx e τ −t py+t µy+τ dτ . R n −r(τ −t) R n/2 (y) alive, t < n/2, Tx > t: Vt = 2t e (1−τ /2−t px+t ) τ −t py+t µy+τ dτ −π t e−r(τ −t)τ −t px+t τ −t py+t dτ .

(d) In general, for a unit due at some random time, the non-central 2nd moment of present value is the same as the expected value, only with 2r instead of r.

Exercise 75 We work under the independence hypothesis (should have been stated in the exercise text.) First the brute force method: Y (1 − t pxj ) . t px1 ...xr = 1 − j

APPENDIX G. SOLUTIONS TO EXERCISES

Using the rule for differentiating a product (special case of Itˇ o), Y X X d Y d (1 − t pxj ) = − − t p xk (1 − t pxj ) . t px ...x = − t pxk µxk +t dt 1 r dt j; j6=k k k j; j6=k It follows that µx1 ...xr (t) =

Q µxk +t j; j6=k (1 − t pxj ) Q . 1 − j (1 − t pxj )

k t p xk

Second, direct reasoning: The conditional probability that the last survivor dies in (t, t + dt), given that there are survivors at time t, is dt times the expression above.

Exercise 76 (a) Apology: Problems of this kind are one of the favorite sports of classical actuaries, not because they are so common in practice (market share in the per mille range), but rather because they can entertain and stimulate the brains of actuaries. The proposed product is not totally inconceivable, however: it might be useful for a couple that needs to secure economically the last survivor and also their children after the possible early death of the last survivor. It is also of some theoretical interest beyond that of mere parlor games as it is an example of a product where payments are dependent on the past history of the driving process. This is seen clearly if the problem is formulated in the set-up of the Markov chain models for two lives. Now to work: As is almost always the case, the best method is to find the expected value of the discounted payment in each small time interval (τ, τ +dτ ) and then sum over all times. For τ ≤ 20 the benefit is running if (and only if) Txy < τ . For τ > 20 the benefit is running if τ − 20 < Txy < τ or if Txy < τ − 20 and τ − 10 < Txy < τ . We gather the following expected value of future discounted payments at time 0: Z 20 e−rτ (1 − τ px τ py ) dτ 0 Z ∞ + e−rτ [τ −20 px τ −20 py − τ px τ py ] dτ 20 Z ∞ + e−rτ [(1 − τ −20 px ) (τ −10 py − τ py ) + (1 − τ −20 py ) (τ −10 px − τ px )] dτ . 20

(b) The premium rate π is the ratio between the expected present value in item (a) and the expected present value Z ∞ e−rτ τ px τ py dτ . 0

The reserve is a long and tedious story. One must, at each time of consideration t, distinguish between all possible past histories of the two lives along. For instance, if Txy > t, then the reserve is simply the first expression above minus π times the second expression above, with x and y replaced by x + t and y + t. Exercise 77 Z

20 0

e−rτ τ px µx+τ τ pz dτ +

∞ 20

e−rτ τ px µx+τ τ −20 py τ pz dτ .

APPENDIX G. SOLUTIONS TO EXERCISES

A benefit of 1 payable immediately upon the death of (x) if (z) is then still alive and (y) was alive 20 years ago. Exercise 78 Let us say n = 30. Use the set-up of the four states Markov chain for two lives. Set age x = 0 (we could have taken x = 20 since, accidentally, the two start at same age, but the x in the program, which refers to a single life, doesn’t really apply to the general case). Account of starting age of the two by writing α + βeγ (20+t) = α + β ∗ eγ t with β ∗ = β eγ 20 : (* SPECIFY NON-NULL PAYMENTS AT TIME t ! *) bi[2,4] := 1; bi[3,4] := 1; ca[1] := 1; (* SPECIFY MAXIMUM ORDER OF MOMENTS AND NUMBER OF STATES ! *) q := 1; (*moments*) JZ := 4; (*number of states of the policy*) (* SPECIFY TRANSITION INTENSITIES FOR POLICY Z ! *) alpha[1,2] := 0.0005; gamma[1,2] := 0.038*ln(10); beta[1,2] := 0.00007585775*exp(gamma[1,2]*20); alpha[1,3] := 0.0005; gamma[1,3] := 0.038*ln(10); beta[1,3] := 0.00007585775*exp(gamma[1,3]*20); alpha[2,4] := 0.0005; gamma[2,4] := 0.038*ln(10); beta[2,4] := 0.00007585775*exp(gamma[2,4]*20); alpha[3,4] := 0.0005; gamma[3,4] := 0.038*ln(10); beta[3,4] := 0.00007585775*exp(gamma[3,4]*20); (*SPECIFY AGE x, TERM t, INTEREST RATES AND NON-NULL LIFE ENDOWMENTS ! *) x:= 0; (*age*) t := 30; (*term*) r := ln(1+0.045); (*interest rate*) be[1] := 0; be[2] := 0; (*endowments at term of contract*) (*SPECIFY LUMP SUM PREMIUM AT TIME 0: PUT c0 := 1 IF ALL OTHER PREMIUMS ARE 0 AND ONLY MOMENTS OF BENEFITS ARE WANTED ! *) c0 := 0; b0 := 0;

Exercise 1-12 Pure verification - just insert the appropriate expressions on the right hand side. Any combination of cash bonus at rate ˜bt = αt ct and additional death benefit of ˆbt = (1 − αt ) ct /µx+t , 0 ≤ α ≤ 1, produces a right hand side equal to the left hand

APPENDIX G. SOLUTIONS TO EXERCISES side.

Exercise 1-13 ˜bn must solve Z

e−

Rτ

0 (ru +µx+u ) du

cτ dτ = e−

0 (ru +µx+u ) du

˜bn ,

(G.36)

hence ˜bn =

Exercise 1-14 Relation (7) becomes

e 0

τ (ru +µx+u ) du

cτ dτ .

(G.37)

ct = ∆r Vt∗ + ∆µ(bt − Vt∗ ) .

Here are some examples of time t prognosis of future bonuses, assuming that the insured will survive the term of the contract: 1. Rate of cash bonus payments at time u ∈ (t, n) is just cu defined above. 2. Present value of future cash bonuses: Z n ∗ e−(r +∆r)(τ −t) cτ dτ , t

3. Value of terminal bonus (not discounted): Z n R n ∗ ∗ e τ (µx+s + ∆µ + r + ∆r) ds cτ dτ . 0

Exercise 1-15 All that is needed is to put m + n in the role of n and work with the general formulas. Here Vt∗ is given by d ∗ V dt t d ∗ V dt t

(µ∗x+t + r∗ ) Vt∗ + π ,

0 < t < m,

(µ∗x+t + r∗ ) Vt∗ − 1 ,

m < t < m + n.

This is the only place where the particulars of the contract matter: Thiele’s differential equation is needed for the computation of Vt∗ alongside that of ct . Since Vt∗ > 0 for all t ∈ (m, m + n), also ct > 0 throughout this time interval. Exercise 1-16 Expenses can be treated as benefits in addition to those specified in the contract (see Chapter 5). We need the differential equation for the first order gross reserve, 0 0 d ∗0 Vt = Vt∗ r∗ + π − β ∗ π − γ ∗ b − µ∗x+t (b − Vt∗ ) dt 0

∗ (with side condition Vn− = b), and the equivalence relationship, 0

V0∗ = −α∗ b ,

(G.38)

APPENDIX G. SOLUTIONS TO EXERCISES which determines π. The discounted mean surplus per policy at time t is now St

−(α0 + α00 b) Z t R τ 0 e− 0 (rs +µx+s ) ds (π − µx+τ b − βτ0 − βτ00 π 0 − γτ0 − γτ00 b − γτ000 Vτ∗ ) dτ +

− e−

0 (rs +µx+s ) ds

It is seen that

Vt∗ .

S0 = −(α0 + α00 b) − V0∗ = α∗ b − (α0 + α00 b) ,

which is the surplus arising immediately upon issue of the contract due to prudent first order assumptions about the initial cost. It is positive (and indeed prudent) if α∗ b > (α0 + α00 b) , which means that the first order initial cost is set on the safe side. (This cannot be achieved for all b > 0 if α0 > 0; one then has to assume that b is greater than a certain minimum, which is certainly the case in practice.) The dynamics of the surplus is dSt

e−

0 (rs +µx+s ) ds

+ e− 0

Inserting dVt∗ =

0 (ru +µx+u ) du

0 d V∗ dt t

(π − µx+t b − βt0 − βt00 π 0 − γt0 − γt00 b − γt000 Vt∗ ) dt 0

(rt + µx+t ) Vt∗ − e−

0 (ru +µx+u ) du

dVt∗ .

dt from (G.38), we gather dSt = e−

0 (rs +µx+s ) ds

ct dt ,

where ct

(rt − r∗ )Vt∗ + (β ∗ π 0 − βt0 − βt00 π 0 ) 0

+ (γ ∗ Vt∗ − γt0 − γt00 b − γt000 Vt∗ ) + (µ∗x+t − µx+t )(b − Vt∗ )

is the mean contribution to surplus per survivor at time t. This contribution decomposes into gains stemming from safety loadings in the various first order elements – interest, expenses of β type, expenses of γ type, and mortality – and how these elements can be set on the safe side. Just as for the initial cost, there is a problem with the safety loading on expenses of β and γ type: if e.g. γt0 > 0, then there will inevitably be a loss on the γ expenses for small t since the gross reserve starts from a negative value. This loss has to be compensated by setting other first order elements sufficiently to the safe side to make ct (or at least St ) non-negative for all t ∈ (0, n). Exercise 1-17 This is a trivial one, and the same goes for the conditional expected value of any random variable that depends only on the state of Y at some fixed future time. Starting from X We (t) = (1 − λe · dt)We (t + dt) + λef dt Wf (t + dt) + o(dt) ,

we get

`;f 6=e

We (t) − We (t + dt) = −λe · We (t + dt) +

`;f 6=e

λef dt Wf (t + dt) + o(dt) ,

APPENDIX G. SOLUTIONS TO EXERCISES

and, dividing by dt and letting dt & 0, we arrive at the answer. The side conditions are obvious (as always).

Exercise 1-18 We start with We0 and supply details (to be precise, add a term o(dt) on the right of the two expressions given for Wt0 and Wt00 in the exercise): i h 0 We0 (t) = (1 − λe · dt)E ert dt Wt+dt Yτ = e , t ≤ τ ≤ τ + dτ h i X 0 + λef dt E ert dt Wt+dt Yt = e , Yt+dt = f + o(dt) f ;f 6=e

(1 − λe · dt)er

We0 (t + dt) +

λef dt eO(dt) Wf0 (t + dt) + o(dt) ,

f ;f 6=e

where O(dt) signifies a term of order dt (i.e. such that O(dt)/dt is bounded as dt & 0). Inserting the Taylor expansions er

We0 (t + dt)

eO(dt) Wf0 (t

+ dt)

1 + re dt + o(dt) , d We0 (t) + We0 (t) dt + o(dt) , dt 1 + O(dt) ,

Wf0 (t) + O(dt) ,

multiplying out, gathering all o(dt) terms and rearranging a bit, one obtains the differential equations for the functions We0 . Next, the We00 , a bit more sketchy and gathering o(dt) terms currently as they arise without further mentioning: We00 (t) = (1 − λe · dt) We0 (t) (re − r∗ ) Vt∗ dt + We00 (t + dt) X + λef dt We00 (t) + o(dt) . f ;f 6=e

Proceeding as above, we obtain the differential equations for the functions We00 . The side conditions are obvious.

Exercise 1-19 Goes along the lines of Exercise 18.

Exercise 1-20 (a) Same thing again. Introduce Z Wt = and write

n t

e−

Rτ

t rs ds

(rτ − r∗ )Vτ∗ dτ ,

Wt = (rt − r∗ )Vt∗ dt + e−rt dt Wt+dt + o(dt) .

APPENDIX G. SOLUTIONS TO EXERCISES Apply the direct backward construction to We (t) = E[Wt | Yt = e]. Start from e We (t) = (1 − λe · dt) (re − r∗ )Vt∗ dt + e−r dt We (t + dt) X λef dt Wf (t) + o(dt) , + f ;f 6=e

and do a small piece of paper-work to arrive at X d λef (Wf (t) − We (t)) . We (t) = We (t) re − (re − r∗ ) Vt∗ − dt f ;f 6=e

Side conditions are: We (n−) = 0, e = 1, . . . , J . (b) Basically the same exercise as (a). (c) For discounted cash bonuses use the general formula for higher order moments of present values of payment streams with state-dependent payment intensity and interest intensity. Forget about the variance of terminal bonus (too cumbersome).

Exercise 80 Items (a) - (e) are rather theoretical and not typical exam questions in ST305. Anyway, we offer something for students who accept only statements that have been firmly proved. (a) RTI(T |T ) is easy to prove: P[T > s | T > t] =

P[T > max(s, t)] = P[T > t]

(

P[T >s] P[T >t]

, ,

t < s, t ≥ s.

This is obviously an increasing (non-decreasing) function of t for fixed s. PQD(T, T ) and AS(T, T ) then follow. We could also prove PQD(T, T ) directly: P[T > s , T > t] = P[T > max(s, t)] ≥ P[T > s] P[T > t] . A direct proof of AS(T, T ) goes as follows: Let g(s, t) and h(s, t) be increasing ˜ functions in both arguments. Then g˜(t) = g(t, t) and h(t) = h(t, t) are increasing functions in t. Thus, marginal association is enough, see notes ’depend-l.pdf’. (b) Proof of the continuity property of probabilities: Let An , n = 1, 2, . . . be an increasing sequence of events, that is, An ⊆ An+1 for all n. Write ∪∞ j=1 Aj = A1 ∪

∞ [

j=2

(Aj ∩ Acj−1 ) ,

which is a union of mutually exclusive sets. (Here Ac denotes the complement of the event A.) Then P ∪∞ j=1 Aj

P[A1 ] +

∞ X j=2

(P[Aj ] − P[Aj−1 ])

APPENDIX G. SOLUTIONS TO EXERCISES

= hence

lim

n→∞

P[A1 ] +

n X j=2

(P(Aj ) − P[Aj−1 ])

P ∪∞ j=1 Aj = lim P[An ] . n→∞

Let An , n = 1, 2, . . ., be a decreasing sequence of events, that is, An ⊇ An+1 for all n. Then P ∩∞ j=1 Aj = lim P[An ] . n→∞

The two statements are equivalent. For instance, the latter follows by applying the former to the increasing sequence Acn : c c c P ∩∞ = 1 − P ∩∞ = 1 − P ∪∞ j=1 Aj j=1 Aj j=1 Aj = 1 − lim P [An ] n→∞

1 − lim (1 − P[An ]) = lim P[An ] . n→∞

n→∞

Now, P[S > s , T > t] = P hence

and

[ n

1 1 S ≥ s+ , T ≥ t+ n n

1 1 = lim P S ≥ s + , T ≥ t + , n→∞ n n

1 1 P T ≥t+ , P[S > s]P[T > t] = lim P S ≥ s + n→∞ n n

P[S ≥ s , T ≥ t] = P

\ n

S ≥s−

1 1 , T ≥ t− n n

1 1 , = lim P S ≥ s − , T ≥ t − n→∞ n n

hence

1 1 P[S ≥ s]P[T ≥ t] = lim P S > s − P T >t− . n→∞ n n It follows that the strict inequalities > in the definition of PQD and RTI can be replaced by ≥. (The definition of AS is no issue here.) (c) PQD(−S|T ) means P[−S > −s , T > t] ≥ P[−S > −s] P[T > t] for all s (or all −s, which is the same, of course) and all t. This is the same as P[S < s , T > t] ≥ P[S < s] P[T > t] , which is the same as P[T > t] − P[S ≥ s , T > t] ≥ (1 − P[S ≥ s]) P[T > t] , which is the same as P[S ≥ s , T > t] ≤ P[S ≥ s] P[T > t] . Due to the result in (b), this is the same as the asserted result.

APPENDIX G. SOLUTIONS TO EXERCISES (d) Misprint in the exercise text: ≤ should be ≥. Now, AS(−S, T ) means that C(g(−S, T ), h(−S, T )) ≥ 0

for all g and h that are increasing in both arguments. But this is equivalent to the asserted result. (e) RTI(−S|T ) means that P[−S > −s | T > t] is increasing in t for fixed s. Rewriting P[−S > −s | T > t] = 1 − P[S ≥ s | T > t] and recalling the result in (b), we arrive at the asserted result. (f) The Markov model is sketched in the figure below. Only small amendments are needed in the calculations made in the theory (’depend-l.pdf’), but the results are a bit surprising. We will discuss the matter under the more general assumption that µ0t ≥ µt and νt0 ≥ νt . First the case s ≤ t: Rt Rt 0 R t Rτ e− 0 µ+ν+κ + s e− 0 µ+ν+κ µτ e− τ ν dτ Rt Rt P[S > s | T > t] = R t Rτ 0 e− 0 µ+ν+κ + 0 e− 0 µ+ν+κ µτ e− τ ν dτ R s − R τ µ+ν+κ−ν 0 e 0 µτ dτ 0 Rt = 1− . R t Rτ 0 0 e− 0 µ+ν+κ−ν + 0 e− 0 µ+ν+κ−ν µτ dτ

1 µ - Husband dead

Both alive

ν ?

@ κ @ R @

ν0 ?

3 µ0 Both dead

Wife dead

We need to discuss this expression as a function of t, which appears only in the denominator. The derivative of the denominator is e−

0 µ+ν+κ−ν

(νt0 − νt − κt ) .

It follows that, in the presence of a positive κt , P[S > s | T > t] is not in general an increasing function function of t if νt0 ≥ νt . For νt0 = νt it is actually decreasing. We have thus already answered the question and need not look into the case s < t. The second part of the question is now easily sorted out in the case s < t by setting µ0t = µt + κt and νt0 = νt + κt in the result above. We find that P[S > s | T > t] is

APPENDIX G. SOLUTIONS TO EXERCISES

independent of t for s < t. Therefore, the RTI issue is so far unsettled and we need to investigate the case s > t: Rs Rs R s Rτ e− 0 µ+ν+κ + t e− 0 µ+ν+κ ντ e− τ µ+κ dτ Rt Rt P[S > s | T > t] = R t Rτ e− 0 µ+ν+κ + 0 e− 0 µ+ν+κ µτ e− τ ν+κ dτ =

e−

t κ

P∗ [S > s | T > t] ,

where P∗ denotes probability under the independence hypothesis µ0t = µt and νt0 = νt (see expression in ’depend-l.pdf’). This is an increasing function of t, and we have proved RTI(S|T ).

Exercise 81 See ’EXERC22A.pas’ on public folder. END From 305exsol Exercise 57 (a) Inspect (2.16) in BL. It should be quite obvious that the cash balance at any time will get bigger if at any time a bigger amount has been deposited on the account (when interest is positive). In particular this is true if a given amount of deposits is being advanced, i.e. payed earlier. (b) If Ut > 0 for some t ∈ (0, n), then, by right-continuity of U , Uτ > 0 for τ ∈ [t, t+), R t+dt some non-degenerate interval, hence t Uτ rτ dτ R n > 0 if r is strictly positive. If, moreover, Ut ≥ 0 for all t ∈ (0, n), it follows that 0 Uτ rτ dτ > 0. Then, if Un = 0, it follows from (2.15) that Z n

0 = An +

Uτ rτ dτ ,

and so An < 0. Think of a savings account: Deposits are made first, interest is earned on these, and at the end one can withdraw more than the Rtotal deposited. For instance, a unit n deposited at time 0 grows with interest to exp( 0 r) in n years, which can then be withdrawn to make R n the balance nil at time n. In this case At = 1 for 0 ≤ t < n and An = 1 − exp( 0 r), which is negative if interest is positive.

Exercise 58 (b) paa (0, t1 ) σ(t1 ) dt1 pii (t1 , t2 ) ρ(t2 ) dt2 paa (t2 , t3 ) µ(t3 ) dt3 + o(dt) ,

where dt = max(dt1 , dt2 , dt3 ). We have used the differentiability of the transition probabilities to write e.g. pii (t1 + dt1 , t2 ) = pii (t1 , t2 ) + o(dt1 ). Exercise 61 (a) (1)

leads to

(1)

pai (0, t + dt) = pai (0, t)(1 − (ν(t) + ρ(t)) dt) + paa (0, t) σ(t) dt d (1) (1) p (0, t) = −pai (0, t)(ν(t) + ρ(t)) + paa (0, t) σ(t) , dt ai

APPENDIX G. SOLUTIONS TO EXERCISES with side condition

(1)

pai (0, 0) = 0 . Integrating gives the following integral expression, which could be put up by direct reasoning: Z t Z s Z t (1) pai (0, t) = exp(− (µ + σ))σ(s) ds exp(− (ν + ρ)) . 0

Next,

(1)

leads to

(1) p(1) aa (0, t + dt) = paa (0, t)(1 − (µ(t) + σ(t)) dt) + pai (0, t) ρ(t) dt

d (1) (1) p (0, t) = −p(1) aa (0, t)(µ(t) + σ(t)) + pai (0, t) ρ(t) , dt aa with side condition p(1) aa (0, 0) = 0, . Integral expression, which could be put up by direct reasoning: Z t Z t (1) (µ + σ)) . pai (0, s) ρ(s) ds exp(− p(1) aa (0, t) = s

As an exercise, repeat the argument for k = 2, 3, ... to find differential equations (k) for the probability pai (0, t) of being disabled for the k-th time and the probability (k) paa (0, t) of being active after having been disabled k times. One could attack this problem by redefining the state-space of the process as indicated in Figure G.1, where the notation speaks for itself:

a(0)

σx+t -

µx+t ?

ρx+t -

i(1)

a(1)

νx+t ?

µx+t ?

σx+t -

νx+t µx+t

i(> 1) 6 ρx+t σx+t ? a(> 1)

Figure G.1: Problem 7.7 (b) (1)

pai (0, t − q)pii (t − q, t) =

t−q

paa (0, τ ) σx+τ pii (τ, t) dτ .

You should be able to interpret the integral expression as a sum of probabilities of mutually exclusive favourable events.

APPENDIX G. SOLUTIONS TO EXERCISES

Rn q

(1)

e−rτ pai (0, τ − q)pii (τ − q, τ ) dτ Rn . e−rτ paa (0, τ ) dτ 0

Comment: The premium plan is unacceptable in practice since it will produce a negative reserve if the insured is in premium paying state after time n − q (then, for sure, no benefits will be received, but premium will still be paid). Therefore, the premiums should beRpaid only over a shorter period R n and certainly not after time n − q. t+q Reserve: t e−r(τ −t)pii (t, τ ) dτ + t+q e−r(τ −t)pii (t, τ − q)pii (τ − q, τ ) dτ . Exercise 62 (a) Use Kolmogorov forward, taking advantage of the obvious relationship paa (t, u) = 1 − pai (t, u): pai (t, u + du) = pai (t, u)(1 − ρ du) + (1 − pai (t, u))σ du . You obtain a (very simple) differential equation for pai (t, · ), which you integrate from t to u using pai (u, u) = 0, and find pai (t, u) =

σ σ − e−(ρ+σ)(u−t) . ρ+σ ρ+σ

(G.39)

Then calculate paa (t, u) = 1 − pai (t, u): paa (t, u) =

ρ σ + e−(ρ+σ)(u−t) . ρ+σ ρ+σ

(G.40)

The transition probabilities depend only on u − t due to hom*ogeneity. Discuss the probabilities as functions of u − t and look at the limits as u − t tends to +∞. (b) Thiele is actually not a good idea (it is doable, of course, but being a backward equation it does not make use of the fact that paa (t, u) + pai (t, u) = 1). Since we have an explicit expression for pai (t, u), the easiest way is to calculate Z t Z t Z t σ σ −(ρ+σ)t pai (0, τ ) dτ = E[Ii (τ )] dτ = Ii (τ ) dτ = t− E 1 − e (G.41) . ρ+σ (ρ + σ)2 0 0 0 (c) E [Nai (t)] = E

t 0

Z dNai (t) =

paa (0, τ ) σ dτ = 0

σ2 ρσ t + 1 − e−(ρ+σ)t (G.42) . 2 ρ+σ (ρ + σ)

(d) From (G.41) the expected proportion of inactive time in t years is σ 1 − e−(ρ+σ)t σ − . 2 ρ+σ (ρ + σ) t

(G.43)

The function g(x) =

1 − e−x x

(G.44)

APPENDIX G. SOLUTIONS TO EXERCISES is 1 for x = 0 (l’Hospital). It has derivative g 0 (x) =

1 − e−x ex − 1 − x e−x − = −e−x , 2 x x x2

which is < 0 for x > 0 (Taylor expansion), hence g is decreasing. Thus, as t → +∞, the proportion in (G.43) is increasing from 0 to σ . ρ+σ Reasonable: Increasing function of σ, decreasing function of ρ. From (G.42) the expected number of onsets of invalidity per time unit in t years is σ2 1 − e−(ρ+σ)t ρσ + . ρ+σ (ρ + σ)2 t As t increases from 0 to +∞ the value of this expression decreases from σ (of course!) to (ρ σ)/(ρ + σ). It is interesting to note that the limiting expression here is symmetric in ρ and σ. You should try and figure why. (e) If ρ were 0, we could pick the result from Item 7.2 (e) above, setting µ = ν = 0: σ ˜ (t) =

σ , 1 − e−σ (n−t)

0 < t < n.

(G.45)

In general we have σ ˜ (t) = σ

pii (t, n) σ + ρe−(ρ+σ)(n−t) = , pai (t, n) 1 − e−(ρ+σ)(n−t)

0 < t < n.

(G.46)

Here we have used (with a view to (G.40), just switch roles of σ and ρ) pii (t, u) =

ρ σ + e−(ρ+σ)(u−t) . ρ+σ ρ+σ

(f) Constant intensity of transition (whether a → i or i → a) means that the transitions are generated by a Poisson process. Nai and Nia count every second transition: Nai counts transition No. 1,3,... and so on, i.e. Nai (t) is distributed as [N (t) + 1]/2, where N (t) is a Poisson variate with parameter σ t. Nia counts transition No. 2,4,... and so on, i.e. Nia (t) is distributed as [N (t)]/2, where N (t) is a Poisson variate with parameter σ t.

Exercise 64 Problem 9.3 (a) To simplify notation, put Nai = Nai (n) and Nia = Nia (n). The likelihood is Λ = σ Nai ρNia e−σWa −ρWi , P (m) , the total time spent in active state (and Wi = nm − Wa the where Wa = m `=1 Tn total time spent in inactive state). ∂ Nai ln Λ = − Wa , ∂σ σ

Nai , Wa

ρˆ =

Nia , Wi

which are asymptotically independent, unbiased, and normally distributed with asymptotic variances as.Varˆ σ=

σ2 , ENai

as.Varˆ ρ=

ρ2 . ENia

(G.47)

By (G.40), ENai

paa (0, τ ) σ dτ n σ ρ + e−(ρ+σ)τ dτ mσ ρ+σ ρ+σ 0 ρ σ −(ρ+σ)n 1 − e , mσ n + ρ+σ (ρ + σ)2

(G.48)

= =

(G.49)

hence as.Varˆ σ=

σ (ρ + σ)

m ρn + Similarly as.Varˆ ρ=

σ ρ+σ

(1 − e−(ρ+σ)n )

ρ (ρ + σ) mσ n −

1 ρ+σ

(1 − e−(ρ+σ)n )

(G.50)

(G.51)

(b) One can discuss the explicit expression (G.51), but there is an easier way: By (G.47) and (G.48) as.Varˆ σ=

Rn 0

σ , paa (0, τ ) dτ

(G.52)

so it suffices to show that paa (0, τ ) is an increasing function of ρ or, what is the same, an increasing function of ρ + σ. By (G.40) paa (0, τ ) =

ρ σ 1 − e−(ρ+σ) τ + e−(ρ+σ) τ = 1 − σ τ , ρ+σ ρ+σ (ρ + σ) τ

so you need only recall that the function g in (G.44) is decreasing. (c) General comment: The theory of conditional Markov chains worked out in Problem 7.2 is often needed in statistical analysis of insurance data where one does not have

APPENDIX G. SOLUTIONS TO EXERCISES

access to the complete ’life histories’ of the policies. It is often the case that one must work with data that are selected somehow. For instance, suppose data on disabilities can be seen only from the claims records of those that are currently disabled. Then the relevant probabilities and intensities are the conditional one, given that the process is now in disabled state. In the present situation you must therefore work with the intensities in 7.8 (e).

Exercise 66 To conform with the notation in ’Basic Life Insurance Mathematics’, let us call the number of lives n instead of m. We will throughout refer to formulas in the general theory in ’Basic Life Insurance Mathematics’. Assume piece-wise constant mortality intensity, see (9.58): q − 1 ≤ t < q , j = 1, 2, . . .

µ(t) = µq , The log likelihood (9.53) is ln Λ =

X q

(ln µq Nq − µq Wq ) ,

where Nq and Wq are, respectively, the number of deaths and the total time spent alive in the age interval [q − 1, q). Each µq is a parameter which is functionally unrelated to all the others, so there are many parameters in this model! For instance, if we are interested in mortality up to age 100 and have data in the age range from 0 to 100, there are 100 parameters, which is quite a lot. Remember, however, that this model is just a first step in a twostage procedure where the second step is to graduate (smoothen) the ML estimators resulting from the present naive model with piece-wise constant intensity. The ML estimators are the occurrence-exposure rates Nq , Wq

µ ˆq =

µq , E[Wq ]

(G.53)

where the expected exposure is E[Wq ] =

n Z X

m=1

p(m) (τ ) dτ ,

q−1

APPENDIX G. SOLUTIONS TO EXERCISES

For instance, suppose we have observed each individual life from birth until death or until attained age 100, whichever occurs first (i.e. censoring at age 100). Then, for τ ∈ [q − 1, q) with q = 1, . . . , 100, we have Z τ (m) µ(s) ds p (τ ) = exp − 0

exp −

q−1 X p=1

µp − (τ − (q − 1)) µq

(G.54)

hence E[Wq ]

q q−1

exp −

n exp −

q−1 X

q−1 X p=1

µp

p=1

dτ

1 − exp (−µq ) , µq

and σq2 =

µ2q 1 P , n exp − q−1 µp (1 − exp (−µq )) p=1

(G.55)

You should look at other censoring schemes and discuss the impact of censoring on the variance. Take e.g. the case where person No m enters at age z (m) and is observed until death or age 100, whichever occurs first (all z (m) less than 100). Estimators σ ˆ q2 of the variances are obtained upon replacing the µj in (G.55) by the estimators µ ˆ j . Simpler estimators are obtained by just replacing µq and E[Wq ] in (G.53) with their straightforward empirical counterparts: σ ˆ q2 = µ ˆq /Wq = Nq /Wq2 . Now to graduation. The occurrence-exposure rates will usually have a ragged appearance. Assuming that the real underlying mortality intensity is a smooth function, we therefore will fit a suitable function to the occurrence-exposure rates. Suppose we assume that the true mortality rate is a Gompertz Makeham function, µ(t) = α+βe γt . Then, take some representative age ξq (typically ξq = q − 0.5) in each age interval and fit the parameters α, β, γ by minimizing a weighted sum of squared errors X 2 Q= aq µ ˆq − α − βeγt . q

This is a matter of non-linear regression. Optimal weights aq are the inverse of the variances, but since these are unknown, we plug in the estimators and use aq = 1/ˆ σq2 . Belongs to what?: For fixed w = mn we have n = w/m and as.Varˆ σ=

σ πw/m , w(1 − w/m )

which is a decreasing function of m by the results in item (b). We also find ρ2 ρ as.Varρˆ = = πw/m , ENia w w/m

APPENDIX G. SOLUTIONS TO EXERCISES where we have used ENia = m

pai (0, τ )ρ dτ = mnρ 0

πn . n

We see that as.Varˆ ρ is an increasing function of m for fixed w. Comment: The asymptotic variance of an intensity estimator is better the longer the total expected time spent in the state from which the the relevant transition is made. All policies start from state a at time 0. For fixed total exposure the estimation of σ will be good for many policies observed in a short time (when they are likely to remain active), and estimation of ρ will be good for few policies observed over a long time when they can make it to inactive state.

Exercise 67 If r is constant, then π=

1 −r, a ¯x n

which is a decreasing function of n. For t < n the premium reserve is Vt = 1 −

a ¯x+t n−t ¯x 1 x+t | n−t a =1− . a ¯x n ¯x t + x+t | n−t a ¯x t Ex a

For fixed t this is a decreasing function of n. In particular, a whole life insurance has smaller premium than an n-year temporary endowment insurance, and it has also smaller reserve for t < n. Exercise 3 Follows from () and the fact that U is right-continuous, hence strictly positive in some interval. The result says that total withdrawals exceed total deposits, which is due to earned interest. Exercise 3 (a) Prove (3.11) along the following lines: By definition, Z ∞ E[G(T )] = G(0)F (0) + G(t) dF (t) .

(G.56)

Observe that Z

∞ 0

G(t) dF (t) = −

∞

G(t)dF¯ (τ ) .

Integrate by parts to obtain Z n Z ¯ ¯ ¯ G(t) dF (t) = G(n)F (n) − G(0)F (0) − 0

F¯ (t−) dG(t) .

Assuming first that G is non-negative and non-decreasing, we have Z ∞ Z ∞ G(t) dF (t) → 0 , dF (t) ≤ G(n)F¯ (n) = G(n) n

(G.57)

(G.58)

APPENDIX G. SOLUTIONS TO EXERCISES

hence G(n)F¯ (n) → 0, as n → ∞. The same holds true for an integrable G of finite variation, which is the difference between two integrable non-decreasing functions. Thus, letting n → ∞ in (G.58), we obtain Z ∞ Z ∞ G(t) dF¯ (t) = −G(0)F¯ (0) − F¯ (t−) dG(t) . (G.59) 0

Now combine (G.56) – (G.59) to arrive at (3.11). Exercise 1 Assume that X is non-decreasing. Note that ∞ D(X) = ∪∞ m=1 ∪n=1 Dm,n ,

where Dm,n = {t ; t ≤ n , Xt+ − Xt− ≥ ∞ > X n ≥ X0 +

t∈Dm,n

1 }. m

Since

(Xt+ − Xt− ) > X0 +

1 ]Dm,n , m

we conclude that ]Dm,n is finite. Thus, being a countable union of finite sets, D(X) is (at most) countable. Exercise 2 q dXtq = qXtq−1 xct dt + (Xt− + xdt )q − Xt− dNt .

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