# Actuarial Mathematics and Life-Table Statistics. Eric V. Slud Mathematics Department University of Maryland, College Park

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1 Actuarial Mathematics and Life-Table Statistics Eric V. Slud Mathematics Department University of Maryland, College Park c 2006

2

3 Chapter 6 Commutation Functions, Reserves & Select Mortality In this Chapter, we consider first the historically important topic of Commutation Functions for actuarial calculations, and indicate why they lose their computational usefulness as soon as the insurer entertains the possibility (as demographers often do) that life-table survival probabilities display some slow secular trend with respect to year of birth. We continue our treatment of premiums and insurance contract valuation by treating briefly the idea of insurance reserves and policy cash values as the life-contingent analogue of mortgage amortization and refinancing. The Chapter concludes with a brief section on Select Mortality, showing how models for select-population mortality can be used to calculate whether modified premium and deferral options are sufficient protections for insurers to insure such populations. 6.1 Idea of Commutation Functions The Commutation Functions are a computational device to ensure that net single premiums for life annuities, endowments, and insurances from the same life table and figured at the same interest rate, for lives of differing ages and for policies of differing durations, can all be obtained from a single table lookup. Historically, this idea has been very important in saving calculational labor when arriving at premium quotes. Even now, assuming that a govern- 149

4 150 CHAPTER 6. COMMUTATION & RESERVES ing life table and interest rate are chosen provisionally, company employees without quantitative training could calculate premiums in a spreadsheet format with the aid of a life table. To fix the idea, consider first the contract with the simplest net-singlepremium formula, namely the pure n-year endowment. The expected present value of \$1 one year in the future if the policyholder aged x is alive at that time is denoted in older books as n E x and is called the actuarial present value of a life-contingent n-year future payment of 1: A 1 = n E x = v n np x Even such a simple life-table and interest-related function would seem to require a table in the two integer parameters x, n, but the following expression immediately shows that it can be recovered simply from a single tabulated column: A 1 = vn+x l n+x v x l x = +n, D y v y l y (6.1) In other words, at least for integer ages and durations we would simply augment the insurance-company life-table by the column. The addition of just a few more columns allows the other main life-annuity and insurance quantities to be recovered with no more than simple arithmetic. Thus, if we begin by considering whole life insurances (with only one possible payment at the end of the year of death), then the net single premium is re-written A x = A 1 x: = v k+1 kp x q x+k = v x+k+1 (l x+k l x+k+1 ) v x l x = y=x v y+1 d y = M x, M x v y+1 d y y=x The insurance of finite duration also has a simple expression in terms of the same commutation columns M, D : A 1 = n 1 v k+1 d k+x = M x M x+n (6.2)

5 6.1. IDEA OF COMMUTATION FUNCTIONS 151 Next let us pass to to life annuities. Again we begin with the life annuitydue of infinite duration: ä x = ä x: = v k+xl k+x = N x, N x = v y l y (6.3) y=x The commutation column N x column: N x = turns is the reverse cumulative sum of the The expected present value for the finite-duration life-annuity due is obtained as a simple difference y=x D y ä = n 1 v k+x l x+k = N x N x+n There is no real need for a separate commutation column M x since, as we have seen, there is an identity relating net single premiums for whole life insurances and annuities: A x = 1 d ä x Writing this identity with A x and ä x replaced by their respective commutationfunction formulas, and then multupliying through by, immediately yields M x = dn x (6.4) Based on these tabulated commutation columns D, M, N, a quantitatively unskilled person could use simple formulas and tables to provide on-the-spot insurance premium quotes, a useful and desirable outcome even in these days of accessible high-powered computing. Using the case-(i) interpolation assumption, the m-payment-per-year net single premiums A (m)1 and ä (m) would be related to their single-payment counterparts (whose commutation-function formulas have just been provided) through the standard formulas A (m)1 = i ( ) i (m) A1, ä (m) = α(m) ä β(m) 1 A 1

6 152 CHAPTER 6. COMMUTATION & RESERVES Table 6.1: Commutation Columns for the simulated US Male Illustrative Life-Table, Table 1.1, with APR interest-rate 6%. Age x l x N x M x That is, A (m)1 = i i (m) M x M x+n ä (m) = α(m) N x N x+n β(m) (1 +n ) To illustrate the realistic sizes of commutation-column numbers, we reproduce as Table 6.1 the main commutation-columns for 6% APR interest, in 5-year intervals, for the illustrative simulated life table given on page Variable-benefit Commutation Formulas The only additional formulas which might be commonly needed in insurance sales are the variable-benefit term insurances with linearly increasing or de-

7 6.1. IDEA OF COMMUTATION FUNCTIONS 153 creasing benefits, and we content ourselves with showing how an additional commutation-column could serve here. First consider the infinite-duration policy with linearly increasing benefit IA x = (k + 1)v k+1 kp x q x+k This net single premium can be written in terms of the commutation functions already given together with R x = (x + k + 1)v x+k+1 d x+k Clearly, the summation defining IA x (x + k + 1)v k+1 kp x q x+k x can be written as v k+1 kp x q x+k = R x x M x Then, as we have discussed earlier, the finite-duration linearly-increasingbenefit insurance has the expression IA 1 = IA x k=n (k + 1)v k+x+1 d x+k = R x xm x R x+n xm x+n and the net single premium for the linearly-decreasing-benefit insurance, which pays benefit n k if death occurs between exact policy ages k and k + 1 for k = 0,..., n 1, can be obtained from the increasingbenefit insurance through the identity DA 1 = (n + 1)A 1 IA 1 Throughout all of our discussions of premium calculation not just the present consideration of formulas in terms of commutation functions we have assumed that for ages of prospective policyholders, the same interest rate and life table would apply. In a future Chapter, we shall consider the problem of premium calculation and reserving under variable and stochastic interest-rate assumptions, but for the present we continue to fix the interest rate i. Here we consider briefly what would happen to premium calculation and the commutation formalism if the key assumption that the same

8 154 CHAPTER 6. COMMUTATION & RESERVES life table applies to all insureds were to be replaced by an assumption involving interpolation between (the death rates defined by) two separate life tables applying to different birth cohorts. This is a particular case of a topic which we shall also take up in a future chapter, namely how extra ( covariate ) information about a prospective policyholder might change the survival probabilities which should be used to calculate premiums for that policyholder Secular Trends in Mortality Demographers recognize that there are secular shifts over time in life-table age-specific death-rates. The reasons for this are primarily related to public health (e.g., through the eradication or successful treatment of certain disease conditions), sanitation, diet, regulation of hours and conditions of work, etc. As we have discussed previously in introducing the concept of force of mortality, the modelling of shifts in mortality patterns with respect to likely causes of death at different ages suggests that it is most natural to express shifts in mortality in terms of force-of-mortality and death rates rather than in terms of probability density or population-wide relative numbers of deaths in various age-intervals. One of the simplest models of this type, used for projections over limited periods of time by demographers (cf. the text Introduction to Demography by M. Spiegelman), is to view age-specific death-rates q x as locally linear functions of calendar time t. Mathematically, it may be slightly more natural to make this assumption of linearity directly about the force of mortality. Suppose therefore that in calendar year t, the force of mortality µ x (t) at all ages x is assumed to have the form µ (t) x = µ (0) x + b x t (6.5) where µ (0) x is the force-of-mortality associated with some standard life table as of some arbitrary but fixed calendar-time origin t = 0. The age-dependent slope b x will generally be extremely small. Then, placing superscripts (t) over all life-table entries and ratios to designate calendar time, we calculate kp (t) x ( = exp k 0 ) µ (t) x+u du = k p (t) x exp( t k 1 j=0 b x+j )

9 6.2. RESERVE & CASH VALUE OF A SINGLE POLICY 155 If we denote B x = and assume that the life-table radix l 0 is not taken to vary with calendar time, then the commutation-function = (t) takes the form x y=0 b y D (t) x = v x l 0 x p (t) 0 = D (0) x e tbx (6.6) Thus the commutation-columns (0) (from the standard life-table) and B x are enough to reproduce the time-dependent commutation column D, but now the calculation is not quite so simple, and the time-dependent commutation columns M, N become M (t) x = N (t) x = y=x y=x v y (l (t) y l (t) y+1) = y=x ( D y (0) e tby 1 e t b y+1 q y (0) ) (6.7) D (0) y e tby (6.8) For simplicity, one might replace equation (6.7) by the approximation M (t) x = y=x ( ) D y (0) p (0) y + t b y+1 q y (0) e tby None of these formulas would be too difficult to calculate with, for example on a hand-calculator; moreover, since the calendar year t would be fixed for the printed tables which an insurance salesperson would carry around, the virtues of commutation functions in providing quick premium-quotes would not be lost if the life tables used were to vary systematically from one calendar year to the next. 6.2 Reserve & Cash Value of a Single Policy In long-term insurance policies paid for by level premiums, it is clear that since risks of death rise very rapidly with age beyond middle-age, the early premium payments must to some extent exceed the early insurance costs.

11 6.2. RESERVE & CASH VALUE OF A SINGLE POLICY 157 so that the terminal reserve coincides with the terminal (i.e., endowment) benefit to be paid at policy termination. Note that for simplicity, we do not consider here the insurances or endowment insurances with m > 1 possible payments per year. The reserves for such policies can be obtained as previously discussed from the one-payment-per-year case via interpolation formulas. The definition of net premium reserves is by nature prospective, referring to future payment streams and their expected present values. From the definition, the formulas for reserves under the term and endowment insurances are respectively given by: tv 1 = A 1 x+t:n t P 1 ä x+t:n t (6.9) tv = A x+t:n t P ä x+t:n t (6.10) One identity from our previous discussions of net single premiums immediately simplifies the description of reserves for endowment insurances. Recall the identity ä = 1 A d = A = 1 d ä Dividing the second form of this identity through by the net single premium for the life annuity-due, we obtain P = 1 ä d (6.11) after which we immediately derive from the definition the reserve-identity tv = ä x+t:n t ( P x+t:n t P ) = 1 äx+t:n t ä (6.12) Retrospective Formulas & Identities The notion of reserve discussed so far expresses a difference between expected present value of amounts to be paid out and to be paid in to the insurer dating from t years following policy initiation, conditionally given the survival of the insured life aged x for the additional t years to reach age x + t. It

12 158 CHAPTER 6. COMMUTATION & RESERVES stands to reason that, assuming this reserve amount to be positive, there must have been up to time t an excess of premiums collected over insurance acquired up to duration t. This latter amount is called the cash value of the policy when accumulated actuarially to time t, and represents a cash amount to which the policyholder is entitled (less administrative expenses) if he wishes to discontinue the insurance. (Of course, since the insured has lived to age x + t, in one sense the insurance has not been used at all because it did not generate a claim, but an insurance up to policy age t was in any case the only part of the purchased benefit which could have brought a payment from the insurer up to policy age t.) The insurance bought had the time-0 present value A 1, and the premiums paid prior to time t x:t had time-0 present value ä x:t P, where P denotes the level annualized premium for the duration-n contract actually purchased. To understand clearly why there is a close connection between retrospective and prospective differences between expected (present value of) amounts paid in and paid out under an insurance/endowment contract, we state a general and slightly abstract proposition. Suppose that a life-contingent payment stream (of duration n at least t) can be described in two stages, as conferring a benefit of expected time-0 present value U t on the policy-age-interval [0,t) (up to but not including policy age t ), and also conferring a benefit if the policyholder is alive as of policy age t with expected present value as of time t which is equal to F t. Then the total expected time-0 present value of the contractual payment stream is U t + v t tp x F t Before applying this idea to the balancing of prospective and retrospective reserves, we obtain without further effort three useful identities by recognizing that a term life-insurance, insurance endowment, and life annuity-due (all of duration n) can each be divided into a before- and after- t component along the lines of the displayed Proposition. (Assume for the following discussion that the intermediate duration t is also an integer.) For the insurance, the benefit U t up to t is A 1 x:t, and the contingent after-t benefit F t is A 1 x+t:n t. For the endowment insurance, U t = A 1 x:t and F t = A x+t:n t.

13 6.2. RESERVE & CASH VALUE OF A SINGLE POLICY 159 Finally, for the life annuity-due, U t = ä x:t and F t = ä x+t:n t. Assembling these facts using the displayed Proposition, we have our three identities: A 1 = A 1 x:t + v t tp x A 1 x+t:n t (6.13) A = A 1 x:t + v t tp x A x+t:n t (6.14) ä = ä x:t + v t tp x ä x+t:n t (6.15) The factor v t tp x, which discounts present values from time t to the present values at time 0 contingent on survival to t, has been called above the actuarial present value. It coincides with t E x = A 1, the expected x:t present value at 0 of a payment of 1 to be made at time t if the life aged x survives to age x+t. This is the factor which connects the excess of insurance over premiums on [0,t) with the reserves t V on the insurance/endowment contracts which refer prospectively to the period [t, n]. Indeed, substituting the identities (6.13), (6.14), and (6.15) into the identities A 1 = P 1 ä, A = P ä yields v t tp x t V 1 v t tp x t V [ ] = A 1x:t P 1 ä [ ] = A x:t P x:t ä (6.16) (6.17) The interpretation in words of these last two equations is that the actuarial present value of the net (level) premium reserve at time t (of either the term insurance or endowment insurance contract) is equal to the negative of the expected present value of the difference between the contract proceeds and the premiums paid on [0,t). Figure 6.1 provides next an illustrative set of calculated net level premium reserves, based upon 6% interest, for a 40-year term life insurance with faceamount 1, 000 for a life aged 40. The mortality laws used in this calculation, chosen from the same families of laws used to illustrate force-of-mortality curves in Figures 2.3 and 2.4 in Chapter 2, are the same plausible mortality laws whose survival functions are pictured in Figure 2.5. These mortality laws are realistic in the sense that they closely mirror the US male mortality in 1959 (cf. plotted points in Figure 2.5.) The cash-reserve curves t V 40:40 as functions of t = 0,..., 40 are pictured graphically in Figure 6.1. Note

14 160 CHAPTER 6. COMMUTATION & RESERVES that these reserves can be very substantial: at policy age 30, the reserves on the \$1000 term insurance are respectively \$459.17, \$379.79, \$439.06, and \$ Relating Insurance & Endowment Reserves The simplest formulas for net level premium reserves in a general contract arise in the case of endowment insurance t V, 1 as we have seen in formula (6.12). In fact, for endowment and/or insurance contracts in which the period of payment of level premiums coincides with the (finite or infinite) policy duration, the reserves for term-insurance and pure-endowment can also be expressed rather simply in terms of tv. Indeed, reasoning from first principles with the prospective formula, we find for the pure endowment tv 1 = v n t n tp x+t vn np x ä ä x+t:n t from which, by substitution of formula (6.12), we obtain V 1 = v n t n tp x+t v n np x ( 1 t V ) (6.18) Then, for net level reserves or cash value on a term insurance, we conclude V 1 = (1 v n np x ) t V + v n np x v n t n tp x+t (6.19) Reserves under Constant Force of Mortality We have indicated above that the phenomenon of positive reserves relates in some way to the aging of human populations, as reflected in the increasing force of mortality associated with life-table survival probabilities. A simple benchmark example helps us here: we show that when life-table survival is governed at all ages x and greater by a constant force of mortality µ, the reserves for term insurances are identically 0. In other words, the expected present value of premiums paid in within each policy year exactly compensates, under constant force of mortality, for the expected present value of the amount to be paid out for that same policy year.

15 6.2. RESERVE & CASH VALUE OF A SINGLE POLICY 161 Reserves for Term Insurance & Various Mortality Laws Cash Reserve Lognormal(3.491,.246^2) Weibull(3.653, 1.953e-6) Gamma(14.74, ) Gompertz(3.46e-3, ) Policy Age (years) Figure 6.1: Net level premium reserves as a function of policy age for a 40- year term insurance to a life aged 40 with level benefit 1000, at 6%, under the same mortality laws pictured in Figures 2.5, with median age 72 at death. For definitions of the mortality laws see the Examples of analytical force-ofmortality functions in Chapter 2.

16 162 CHAPTER 6. COMMUTATION & RESERVES The calculations here are very simple. Reasoning from first principles, under constant force of mortality A 1 = n 1 v k+1 e µ k (1 e µ ) = (e µ 1) (ve µ ) 1 (ve µ ) n 1 ve µ while ä = n 1 v k e µ k = 1 (ve µ ) n 1 ve µ It follows from these last two equations that P 1 = v (1 e µ ) which does not depend upon either age x (within the age-range where constant force of mortality holds) or n. The immediate result is that for 0 < t < n ) tv 1 = ä x+t:n t (P 1 x+t:n t P 1 = 0 By contrast, since the terminal reserve of an endowment insurance must be 1 even under constant force of mortality, the intermediate net level premium reserves for endowment insurance must be positive and growing. Indeed, we have the formula deriving from equation (6.12) tv = 1 äx+t:n t ä = 1 1 (ve µ ) n t 1 (ve µ ) n Reserves under Increasing Force of Mortality Intuitively, we should expect that a force-of-mortality function which is everywhere increasing for ages greater than or equal to x will result in intermediate reserves for term insurance which are positive at times between 0, n and in reserves for endowment insurance which increase from 0 at t = 0 to 1 at t = n. In this subsection, we prove those assertions, using the simple observation that when µ(x + y) is an increasing function of positive y, k kp x+y = exp( 0 µ ( x + y + z)dz) ց in y (6.20)

17 6.2. RESERVE & CASH VALUE OF A SINGLE POLICY 163 First suppose that 0 s < t n, and calculate using the displayed fact that k p x+y decreases in y, ä x+t:n t = n t 1 v k+1 kp x+t n t 1 v k+1 kp x+s < ä x+s:n s Therefore ä x+t:n t is a decreasing function of t, and by formula (6.12), tv is an increasing function of t, ranging from 0 at t = 0 to 1 at t = 1. It remains to show that for force of mortality which is increasing in age, the net level premium reserves tv 1 for term insurances are positive for t = 1,...,n 1. By equation (6.9) above, this assertion is equivalent to the claim that A 1 x+t:n t /ä x+t:n t > A 1 /ä To see why this is true, it is instructive to remark that each of the displayed ratios is in fact a multiple v times a weighted average of death-rates: for 0 s < n, A 1 { n s 1 x+s:n s v k kp x+s q } x+s+k = v ä n s 1 x+s:n s v k kp x+s Now fix the age x arbitrarily, together with t {1,...,n 1}, and define q = n 1 j=t v j jp x q x+j n 1 j=t v j jp x Since µ x+y is assumed increasing for all y, we have from formula (6.20) that q x+j = 1 p x+j is an increasing function of j, so that for k < t j, q x+k < q x+j and for all k {0,...,t 1}, q x+k < q (6.21) Moreover, dividing numerator and denominator of the ratio defining q by v t tp x gives the alternative expression v q = n t 1 a=0 v a ap x+t q x+t+a n t 1 a=0 v a ap x+t = P 1 x+t:n t

18 164 CHAPTER 6. COMMUTATION & RESERVES Finally, using equation (6.21) and the definition of q once more, we calculate A 1 ä ( t 1 = v vk kp x q x+k + n t 1 j=t v j jp x q x+j ) t 1 vk kp x + n t 1 j=t v j jp x ( t 1 = v vk kp x q x+k + q n t 1 j=t v j jp x ) t 1 vk kp x + n t 1 j=t v j jp x < v q = P 1 x+t:n t as was to be shown. The conclusion is that under the assumption of increasing force of mortality for all ages x and larger, tv 1 > 0 for t = 1,...,n Recursive Calculation of Reserves The calculation of reserves can be made very simple and mechanical in numerical examples with small illustrative life tables, due to the identities (6.13) and (6.14) together with the easily proved recursive identities (for integers t) A 1 x:t+1 = A 1 x:t + v t+1 tp x q x+t A x:t+1 = A x:t v t tp x + v t+1 tp x = A x:t dv t tp x ä x:t+1 = ä x:t + v t tp x Combining the first and third of these recursions with (6.13), we find = 1 v t tp x ( A 1 x:t+1 tv 1 = 1 v t tp x ( A 1 x:t P 1 ä x:t ) v t+1 tp x q x+t P 1 = v p x+t t+1 V 1 + v q x+t P 1 [ä x:t+1 v t tp x ]) The result for term-insurance reserves is the single recursion tv 1 = v p x+t t+1 V 1 + v q x+t P 1 (6.22) which we can interpret in words as follows. The reserves at integer policyage t are equal to the sum of the one-year-ahead actuarial present value of the reserves at time t + 1 and the excess of the present value of this

20 166 CHAPTER 6. COMMUTATION & RESERVES premium for a new insurance, the new face-amount F is determined as of the t policy-anniversary by the balance equation B tv x = F A x+t which implies that the equivalent face amount of paid-up insurance as of policy-age t is F = B ( ) / tv x = B 1 äx+t (1 d ä x+t) (6.24) A x+t ä x 6.3 Select Mortality Tables & Insurance Insurers are often asked to provide life insurance converage to groups and/or individuals who belong to special populations with mortality significantly worse than that of the general population. Yet such select populations may not be large enough, or have a sufficiently long data-history, within the insurer s portfolio for survival probabilities to be well-estimated in-house. In such cases, insurers may provide coverage under special premiums and terms. The most usual example is that such select risks may be issued insurance with restricted or no benefits for a specified period of time, e.g. 5 years. The stated rationale is that after such a period of deferral, the select group s mortality will be sufficiently like the mortality of the general population in order that the insurer will be adequately protected if the premium in increased by some standard multiple. In this Section, a slightly artifical calculation along these lines illustrates the principle and gives some numerical examples. Assume for simplicity that the general population has constant force of mortality µ, and that the select group has larger force of mortality µ. If the interest rate is i, and v = 1/(1+i), then the level yearly risk premium for a k-year deferred whole-life insurance (of unit amount, payable at the end of the year of death) to a randomly selected life (of any specified age x) from the general population is easily calculated to be Level Risk Premium = v k kp x A x+k /ä x = (1 e µ )v k+1 e µk (6.25) If this premium is multiplied by a factor κ > 1 and applied as the risk premium for a k-year deferred whole-life policy to a member of the select

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