EXPLORATION OF ACTUARIAL MATHEMATICS WITH RECOGNITION OF NUCLEAR HOLOCAUST HAZARD. Cecil J. Nesbitt

Transcription

1 EXPLORATION OF ACTUARIAL MATHEMATICS WITH RECOGNITION OF NUCLEAR HOLOCAUST HAZARD by Cecil J. Nesbitt ~ Jntroduction The immense development of nuclear means of destroying life challenges the world to develop constructive means of preserving life. Each science has its contribution to make in searching out and communicating to the public its truths in this environment. Actuarial science is not excepted, and its core, actuarial mathematics, should lead its efforts. The goal of this paper is to begin the study of actuarial mathematics with explicit recognition of nuclear holocaust hazard. It is simplest to consider the case of total nuclear holocaust wherein insurers and insureds are ~ompletely destroyed. While such a case may be over-dramatic, it is not beyond the realm of possibility in our present world. Many individuals, both distinguished persons and those of less renown, have concerned themselves about nuclear holocaust and are striving for means to obviate this tremendous risk of our lives. I shall refer to this risk as nuclear holocaust hazard. A notation section lays the ground work for study of a discrete model extending over a lifetime and based on interest, nuclear holocaust hazard, and mortality. The analysis in this paper will make much use of discount rates d, and their complements, 1 - d - v. The sections and concepts thereafter are related to the presentation in Lebensversicherungsmathematik, [1]. -67-

2 As a first application, we discuss financial transactions based on only two discount factors, namely, for interest and for nuclear holocaust hazard. While for calculation purposes, one can consider nuclear holocaust hazard as leading to an additional component rate of discount, its operation is not identical with that of discounting for interest. There is a contingency element involved. If holocaust occurs, all transactions are terminated without further payments. A next section discusses survival theory with recognition of nuclear holocaust hazard. Now such hazard discounts survival itself, and expectations of life convert into life annuity. values with interest rates equivalent to the rates of nuclear holocaust. Alternatively, one can consider that mortality rates have been increased by recognition of nuclear holocaust hazard. For another view of the impact of nuclear holocaust hazard on survival, the effect on median survival ages is illustrated. In Sections 5-9, all three factors, interest, nuclear holocaust hazard, and mortality are considered, and the implications for net premiums and reserves are indicated explicitly. A brief summary section states principal conclusions from this exploration and indicates areas for further study if the ideas take root. The time-period for the model shall be the lifespan of (x), and, if x - 0, approximates 110 years. For each year, a triad of rates (for symmetry labelled as discount rates) and their complements will be defined, corresponding to the three basic factors of interest, nuclear holocaust hazard, and mortality. Because these three rates operate differently, it seems best to define them separately, albeit somewhat redundantly

3 - 69- dl(k) - the effective rate of discount for the year (k,k+l) in regard to interest. In some cases, we may use the equivalent effective rate of interest, il(k+l) - dl(k)/[l-dl(k)]. (2.1) k-l k-l vl(h,k) - IT vl(g,g+l) - IT [l-dl(g)]. g-h g-h (2.2) These formulas allow for full variability of the rates of discount year-by-year in the life span. If dl(k) is a constant, d l ' then k-h vl(k,k+l) d l - v l ' and vl(h,k) - v l d 2 (k) - The probability that a nuclear holocaust will occur in the year (k,k+l), conditional on non-occurrence before the beginning of the year. (2.3) k-l k-l v (h,k) - IT v (g,g+1) - IT [l-d (g)] g-h g-h (2.4) Note that v 2 (h,k) discounts a unit (due at time k) down to time h for the contingency that a nuclear holocaust occurs in the interval (h,k), not having occurred before time h. It is also the conditional probability that a nuclear holocaust does not happen in the interval (h,k). Also, we define (2.5) and

4 (2.6) k-l v l,2(h,k) - IT v l 2(g,g+1) - v l (h,k)v 2 (h,k). g-h ' (2.7) Formula (2.6) seems most easily interpreted in the form d (k) + dl(k)[l-d (k)], where 2 2 dl(k) is multiplied by the probability of non-occurrence of nuclear holocaust in the year (k,k+l). Observe that in formulas (2.5) and (2.6), dl(k) and d (k) 2 are operating as absolute (or independent) rates. d 3 (k) - The probability according to a given life table that (x) will die in the year (k,k+l), given survival (according to that table) of (x) to age x+k. Thus, k-l v 3 (h,k) - IT v 3 (g,g+1) - k-hpx+h' g-h (2.10) v l 3(k,k+l) - v l (k,k+l)v 3 (k,k+l) - [l-dl(k)]px+k' (2.11), k-l k-l v l 3(h,k)..;, IT v l 3(g,g+1) -, g-h' IT [l-dl(g)]k_hpx+h' g-h (2.12) -70 -

5 Combining nuclear holocaust and ordinary mortality factors, we have v 2 3(k,k+l) ~ v 2 (k,k+l)v 3 (k,k+l),, (2.13) and (2.14) The probability that (x) will die in year (k,k+l), that is, Pr(K-k) where K is the curtate future lifetime of (x), is then (2.15) and w-x L [v 2,3(O,k) - v 2,3(O,k+l)] - v 2,3(O,O) - 1. It is useful to rearrange (2.15) in the form (2.16) where the first term represents the probability that nuclear holocaust occurs in year (k,k+l) and the second, the probability that nuclear holocaust does not occur but ordinary death does

6 Finally, we define (2.17). and k-l vi 2 3(h,k) - n v l 2 3(g,g+1). J g-h' f (2.18) In case dl(k), d 2 (k) are constants d l and d 2, respectively, then (2.19) One may question why discount notation is used for all three factors, interest, nuclear holocaust hazard, and mortality, since for the latter two, probability comes in naturally. This is a matter of taste. The mathematical effect of nuclear holocaust hazard is somewhat like that of a discount rate, since it does not trigger any benefit pay-out but instead terminates all such. For this reason and for symmetry, discount notation is used for all rates. Another question is why not use forces of discount and continuous functions. In some ways that would be easier for mathematically informed readers, but if proved sound, these ideas should go beyond such readers, and the discrete model may be more readily grasped. Also, it is directly computable, and very general

7 ~ 1- MAthematics 2t Inyestment From the preceding section, we have v l,2(h,k) as the discounted value at time h (under interest and nuclear holocaust hazard) of 1 payable at time k. Correspondingly, l/v l 2(h,k), at time k of 1 invested at time h and subject to interest is the appreciated value accumulation and nuclear holocaust hazard in the interval (h,k). For year (k,k+l), the effective interest (in arrears) rate i l,2(k+l), equivalent to d l,2(k) can be expressed as - 1 d l (k)+d 2 (k)-d l (k)d 2 (k) [l-dl(k)] [1-d 2 (k)] - il(k+l) [1+i 2 (k+l)]+i 2 (k+l) [l+i l (k+l)]-i l (k+l)i 2 (k+l) (3.1) where il(k+l) and i 2 (k+l) are equivalent to dl(k) and d 2 (k), respectively. Further (3.2) In this environment, we have or more generally a l,2(o,n) - n-l L n-l L v l,2(h,k); k-h (3.3) -73 -

8 and or n - l 5 l,2(o,n) - L [l/v l,2(k,n)], (3.4) If d l 2(k) - d l 2 ' a constant, then,, calculated at d - d l 2, where The analogue to the relation da;:, + v n - 1 n with d - d l 2.., is n-l L d 1,2(k)v 1,2(O,k) n-1 - L [1-v 1,2(k,k+1)]v 1,2(O,k) n-1 - L [v 1,2(O,k)-v 1,2(O,k+l)] + v 1 2(O,n) - 1., -74 -

9 A loan A(O), to be amortized by payments r h, h - 1,2,...,n has starting relation A(O) - n L r h v 1,2(O,h). h-1 (3.5) The outstanding principal A(h) after h payments is then n A(h) - L r k v 1,2(h,k), k-h+1 and it may be shown that rh+1 - A(h) - A(h+1) + i 1,2(h+1) A(h). (3.6) A final observation is in regard to the recursion formula (3.7) If this is rearranged as then multiplied by v 1 (O,k) and summed, there results a 1,2(O,n) - n-1 a 1 (O,n) - L v 1 (O,k+1) d 2 (k) a 1,2(k+1,n). (3.8) Formula (3.8) expresses the impact of nuclear holocaust hazard on the annuity value by the sum term representing the annuity values that would be lost by holocaust. -75-

10 A note of caution which will be echoed several times throughout this paper concerns the interpretation of d 2 (k), k - 0,1,... As long as we are dealing with first moments, formulas are the same whether we consider d (k) as a discount rate or as a probability. For the former, there is no 2 consideration of higher moments but for the latter there is, in particular, of variance. For example, the present value random variable of 1 due at time n has expected value and variance (3.9) when v 2 (D,n) is considered as a probability. Until we know more about the estimation of nuclear holocaust hazard, it seems unnecessary to elaborate on variance formulas but they do exist. ~ Survival Theory Let us consider v 2,3(h,k) - v 2 (h,k) v 3 (h,k). It represents the probability at time h that (x) having reached age x + h will survive nuclear holocaust hazard and ordinary mortality to attain age x + k at time k. It also represents the value at time h of 1 due at time k and discounted in regard to nuclear holocaust hazard and ordinary mortality. If d (k) is a constant d then 2 2 (4.1) We note also that suggests -76 -

11 1 l+i 2 (k+l) (1-d 2 (k»)p x + k P x + k (4.2) as the effective accumulation rate in year (k,k+l) in regard to nuclear holocaust hazard and ordinary mortality. We have from (2.15) that - v 2 3(O,k) (d 2 (k)+d 3 (k)-d 2 (k)d 3 (k»)., The curtate expectation of life is then given by ",-x- l e 2,3(O,,,,-x) - E(K) - L k(v 2,3(O,k)-V 2,3(O,k+l») I "'-x ",-x -l - -k v 2, 3(O,k) 0 + L v 2, 3(O,k+l) ",-x-l - L v 2,3(O,k+l). (4.3) e 2,3(O,,,,-x) - ",-x-l L (4.4) - ax' based on the given life table It follows that e 2, 3(O,,,,-x) < ex based on the given life table, the difference depending on the level of discount for nuclear holocaust hazard

12 To illustrate survivorship with recognition of nuclear holocaust hazard, Table 1 shows curtate expectations of life based on the U.S. Life Tables : : Total Population [2], both alone and also with recognition of nuclear holocaust hazard with i , or equivalently, with d / column of [3]. The latter expectations are given by the a x Table 1 CURTATE EXPECTATION OF LIFE (a) (b) The same as (a) except for U.S. Life Tables : recognition of nuclear Age : Total holocaust hazard with x Population i2-0.01, or d Another measure of survival is the curtate median survival age x + m(x) such that m(x) is the greatest integer with or 1 2 S v 2,3[0,m(x)]. (4.5) For d 2 (k) - d 2 ' this becomes! t S (l-d )m(x)t 2 x 2 x+m(x) (4.6) -78 -

13 -ro- The curtate median survival age is such that (x) has approximately 50% chance of surviving thereto. It is easy to compute and may be more meaningful than the much quoted expectation of life. In preparation to calculate curtate median survival ages. Table 2 illustrates values of (l-d )llo-x 2. ranging from to Tables: : Total Population are shown. for x - O and 75 and d 2 Also. values of tx from the U.S. Life Here. (l-d )llo-x _ v 110.x 2 2 is the probability that a nuclear holocaust will not happen over the future lifespan of (x) if w and the geometric average of v 2 (k.k+l). k x. is v 2. For d v such probability is close to 1 for all ages. but for d it is less than 0.11 for a newborn. Table 3 illustrates curtate median survival ages based on the values and t in Table 2. x The underlined figures indicate cases where the curtate median survival ages are unchanged by recognition of nuclear holocaust hazard. This occurs especially at advanced ages x. But if d recognition of nuclear holocaust hazard reduces the curtate median survival age for a newborn by 22 years.

14 Table 2 ASSUMED VALUES OF d 2 AND ' tx AND RESULTING VALUES OF PROBABILITY, (1_d )ll0-x, THAT NO NUCLEAR HOLOCAUST WILL HAPPEN OVER 2 FUTURE LIFESPAN OF (x). ~2 x (l-d ) 1l0-x 2 t x , , , ,799 * From the U.S. Life Tables: , Total Population * Table 3 CURTATE MEDIAN SURVIVAL AGES (Based on assumed values of d 2 and tx in Table 2.) ~2 x x + m(x) II II ,50 l!l. l!l A more refined illustration would use a generation rather than a current life table, and would consider the. possibility of w > 110. However, the median survival ages in Table 3 are unaffected by the setting of w, which illustrates an advantage of such ages

15 By now. we have discussed two of.the major consequences of recognizing nuclear holocaust hazard. namely. payments due in the future should have an additional discount because of that hazard, and survival itself should be discounted. To test the model further, we go on to discuss the recognition of nuclear holocaust hazard in the calculation of net single premiums for life insurances and annuities, and in the evaluation of net level pre~iums and net premium reserves. ~ ~ Insurance As in [1] and [4]. for a whole life insurance of 1 payable at the end of the year of death of (x). we define a present value random variable, Z, but now we recognize nuclear holocaust hazard by considering two cases when K - k: ~ l. Nuclear holocaust occurs in the year (k,k+l). Since K - k, (x) was surviving at the beginning of the year. Then Z - 0, and the probability of this case is v 2,3 (O,k) d 2 (k). ~ 2. Nuclear holocaust does not occur in the year (k,k+l) and the insured dies from ordinary mortality. Then Z - vl(o,k+l), and the probability of this case is v 2,3(0,k) [1-d 2 (k)] d 3 (k). We recall from.(2.l6) that

16 It follows that the net single premium, A I,2,3(O,w-x), for this insurance is given by A I,2,3(O,w-x) - E(Z) - L vl(o,k+l) v 2,3(O,k) ~1-d2(~)1 d 3 (k) - L v l,2(o,k+l) v 3 (O,k) d 3 (k). (5.1) This formula indicates it could have been established more directly by defining Z - v I,2(O,K+I), where K is the curtate future lifetime according to the given life table, and nuclear holocaust is recognized in the discount process. However, we shall see that higher moments of Z and Z do not agree and a distinction appears between d 2 as a discount rate and as a probability. For constant d l,2(k) - d l,2 (5.1) becomes ~,2,3(O,w-x) -A x with interest at (5.2) This illustrates again the additional discounting of future payments if nuclear holocaust is recognized. Algebraically, or by general reasoning, one can establish the recursion formula where is the value at time k of the remaining whole life insurance originally issued to (x) at time 0 and who is now aged x+k. _. Multiplication by v l,2(o,k) and summation over k yields -82-

17 A l,2,3(o,w-x) - L v l,2(o,k+l) (5.4) which expresses Al 2 3(O-w-x) in terms of future net costs of insurance.,, This is analogous to (6.6) of Chapter 3 of [1]. We note also that E(Z2) - L [V (O,k+l)]2 v (O,k+l) v (O,k) d (k). l (5.5) For constant this equals Ax calculated with or i - d/(l-d). Note that E(Z2) - L [V,2(O,k+l)]2 v (O,k) d (k) l 3 3 differs from E(Z2). The formulas of this section illustrate that it is easy to adapt standard formulas for net single premiums for life insurances to recognize nuclear holocaust hazard but care must be taken in regard to higher moments. ~ Life Annuities Analogous to initial concepts in [1], Chapter 4, we define K Y - L vl(o,k) - L vl(o,k) I(~k), where I(~k) is the indicator for the event that K ~ k. Then - 83-

18 8 1,2,3(O,w-X) - E(Y) _. L vl(o,k) v 2,3(O,k) - L v l,2(o,k) v 3 (O,k) (6.1) - L v l,2,3(o,k). The reader may be interested in establishing the middle formula (6. 1) by K defining Y - L v l,2(o,k). For d l 2(k) - a constant d l 2 ' we have,, a l,2,3(o,w-x) - L (V l,2)k kpx - ax (6.2) calculated with d - d l,2 or i - d/(l-d). The analogue of the relation 1 - da x + A x is L d l, 2(k) v l, 2, 3(O,k)+ L v l, 2(O,k+l) - L (l-v l 2(k,k+l»)v 1 2 3(O,k)+ L v l 2 (O,k+l)v 3 (O,k)(l-v 3 (k,k+l») t,, t, - L [v l,2,3(o,k) - v 1,2,3(O,k+l)] (6.3)

19 One might wonder if a relation exists with dl(k) as the annuity rent. Such a relation would require repayment of 1 at the end of the year of death of (x) by ordinary or nuclear means. But payout in the latter case would not occur, 80 the relation fails. By &l,2,3(k,w-x) we denote the value at time k of the remaining payments of the annuity originally provided to (x) at age 0, and who' is surviving at age x+k. The recursion formula (6.4) on substitution of v l,2(k,k+l) (1-d 3 (k)} for v l,2,3, (k,k+l) becomes Multiplication by v l,2(o,k) and summation over k gives &l,2,3(o,w-x) - &l,2(o,w-x)- L v l,2(o,k+l)d 3 (k)&l,2,3(k+l,w-x), (6.5) which shows that the present value of the whole life annuity is less than the present value of an (w-x)-year annuity, with payments discounted for interest and nuclear holocaust, by the present value of annuities released by ordinary death. One might also proceed as above to express v l,2,3(k,k+l) in (6.4) as vl(k,k+l) (1-d 2,3(k)}, and obtain &l,2,3(o,w-x) - &l(o,w-x)- L vl(o,k+l) d 2,3(k) &l,2,3(k+l,w-x). (6.6) Cf. (3.8)

20 Formula (6.6) compares the life annuity value. based on all three factors. with the value of an annuity - certain for (w-x) years with payments discounted by the rates dl(k). k - O.l.... The difference arises in regard to annuity payments terminated by ordinary death or by nuclear holocaust Net Annual Premiums The reader may have observed that in defining the present value raridom variable. Y. for a whole life annuity. it was unnecessary to distinguish two cases when K - k. as was done for life insurance. Now. it is again necessary to consider the two cases when we define a loss variable. L. in proceeding to define the net level annual premium, w. Case 1. Nuclear holocaust occurs in year (k,k+l). Then with probability v 2 3(O,k) d 2 (k).. k L w L vl(o.j), (7.1) j-q If this occurs. there is no payout for the insured. but the insurer gained by having the use 'of the net annual premiums w during the k years. including the premium at the beginning of the final year. Case Z. Nuclear holocaust does not occur,in the year (k.k+l) and the insured dies from ordinary mortality in that year. Then. k L - vl(o.k+l) - w L vl(o.j) j-o with probability v 2 3(O.k) [l-d 2 (k)] d 3 (k). (7.2) - 86-

21 By comparing (7.1) and (7.2) with the definition of Z at the k beginning of Section 5, and noting from Section 6 that Y - L vl(o,j) when j~ L - Z - ~Y. (7.3) Then E(L) - E(Z) - ~ E(Y) (7.4) and setting E(L) - 0, we obtain, not unexpectedly, the formula ~ - Al 2 3(O,w-x)/i l 2 3(O,w-x).,,, t (7.5) This illustrates for the simple form of life insurance the process by which net annual premiums may be defined for the various forms of life insurance and annuities. ~ Net Premium Reserves It should be clear by now how to define a loss variable U for the years from k to w-x for a whole life insurance issued to (x), so that E(U) - A l,2,3(k,w-x) - ~ i l,2,3(k,w-x), (8.1) is the reserve at time k with recognition of nuclear holocaust hazard. -87-

22 The mathematics of net premium reserves would be a particularly rich area to elaborate on with recognition of nuclear holocaust hazard, but by now we have gone far enough to see that such mathematics exists and could be developed systematically. One aspect needs emphasis. The discrete model we have explored is on a generation basis for the, years allows for full variability of the year-by-year discount rates. (0,w-x), and The bulky computations that would ensue seem well within the capacity of modern computers. It would be possible, however, to merge generations after a given period of years if that should prove desirable.!q... Summary Initially, the main thesis of this paper was to assert that nuclear holocaust hazard is an ever present force for the extinction of life, and thereby has significant impact on survival probabilities and on measures such as expectation of life and median survival age. To test the concept further, a discrete mathematical model, involving three variable discount factors, has been developed which permits explicit recognition of nuclear holocaust hazard in common forms of long-term transactions. It is clear from this model that nuclear holocaust hazard brings in an additional discount beyond the usual ones for the time value of money and for survival contingencies. There remain many areas for study for example, joint-life theory, risk theory, pension funding and Social Security. But first there should be an evaluation of the concept of nuclear holocaust hazard, and some rational means for its estimation. The research and education opportunities are a challenge for actuarial science to contribute to the living of our time

23 REFERENCES Gerber, H.U., Lebensversicherungsmatbematik, Springer-Verlag Berlin Heidelberg, Vereinigung Schweizerischer Versicherungsmathematiker. u.s. Decennial Life Tables, , Volume I, National Center for Health Statistics, Wade, A., Actuarial Tables Based on the U.S. Life Tables: Actuarial Study No. 96, Social Security Administration, Office of The Actuary, Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., Actuarial Mathematics, Society of Actuaries, Itasca, IL, I -89-