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JIA 95 (1969) 79-113 79 A STOCHASTIC APPROACH TO ACTUARIAL FUNCTIONS BY A. H. POLLARD, M.Sc., M.SC.(ECON.), PH.D., F.I.A. AND J. H. POLLARD, B.Sc., PH.D., A.I.A. (Macquarie University and University of Cambridge*) * This paper was completed while the second author was a member of the Population Research and Training Centre at the University of Chicago. 1. INTRODUCTION THE classical approach to actuarial problems has been deterministic. That is, the assumption has been made that the number of policies in force is large, that chance fluctuations are unimportant, and that the use of mean values is adequate. The actual cost to a company of an annuity, for example, may in fact vary from nothing (if the annuitant dies immediately) to a large amount (if he lives to be a nonogenarian). On average, the cost will be ax, and for many purposes, this value is good enough. There are circumstances, however, when we could be interested in more than the average value of the annuity ax, and a knowledge of the likely variations about that average could be important. A study of the variance and high-order moments of actuarial functions and the possible applica- tion of these results has no doubt been avoided because of the numerical work involved. With the development of computers, however, this is no deterrent, and we can now learn a great deal more about a company’s business at little cost when computing facilities are available. Although the classical approach to actuarial problems has been deter- ministic, no life office is large enough to ignore completely chance fluctua- tions in mortality. Most actuarial functions are random variables, and should be regarded as such when the effects on a life office of chance fluctuations in mortality rates are being investigated. Actuarial random variables will be denoted in this paper by the usual (deterministic) symbols with tildes above them. Thus, ãx denotes the present value of an annuity of one per annum on a life aged x. This is a random variable with expected value ax,. Similarly, Ãx denotes the present value of an assurance of one on a life aged x. The expected value of Ãx is Ax. The moments of these actuarial random variables may be readily computed. Such calculations are discussed in Part 2, and certain proper- ties of the random variables involved are also described. Some numerical examples are given in Part 3, and these indicate certain uses to which the higher-order moments of life assurance functions may be put. Finally in Part 4, the problem of retention limits and reassurance arrangements is discussed. F 80 A Stochastic Approach to Actuarial Functions 2. THE MOMENTS OF CERTAIN ACTUARIAL FUNCTIONS An annuitant aged x will receive zero payments with probability (dx)/(lx), exactly one payment with probability (dx+1)/(lx), exactly two payments with probability (dx+2)(lx), etc., and these probabilities sum to unity. The present values of the payments received in each case are respectively. Mathematically, we may write (1) The rth moment of the annuity ãX about the origin is given by (2) and the first moment, or expected value, is (3) Similar results for assurances, endowment assurances, temporary and deferred annuities, etc. may easily be written down, and the extension to linear multiples of such functions when independent is straightforward. The equations for whole-life assurances, corresponding to equations (1), (2) and (3) are (4) and A life annuity and a whole of life assurance, both on a single life aged x, are not independent, and indeed it may be shown that one is a linear function of the other. Consider and (8) A Stochastic Approach to Actuarial Functions 81 Hence ãx is a linear function of Ãx. Similarly, it may be shown that is a linear function of We conclude therefore that the coefficients of correlation between ãx and Ãx and between and are both –1. The situation with temporary annuities and temporary assurances is different. Although are completely dependent, the relationship is not linear and hence the correlation coefficient between the two random variables has modulus less than unity. Another random variable which is of considerable interest is the policy value It satisfies the relation (9) (10) where Equations (8), (9) and (10) show that the three random variables and ãx+t are linearly dependent in pairs, and hence the correlation co- efficient between any two of the three random variables has modulus equal to unity. We now consider ãx, Ãx, and in greater detail. The expected values, variances, third central moments, fourth central moments, skewness coefficients and kurtosis coefficients ß2 were computed on TITAN, the computer of the Cambridge University Mathematical Laboratory at 2½% and 5% interest rates using A1949–52 ultimate mortality for all ages over 19 years. The covariance and correlation coefficient of ãx and Ãx were also computed. A summary of the results obtained is given in Table 1, and more extensive tables are included in the Appendix (Tables Al and A2). The entire computation at each rate of interest took 23 seconds. The following points should be noted: 1. The standard deviation of ãx is substantial, and for many ages is greater than half the mean value ax. Consider age 60 for example. At 2½% interest, the expected value of ã60 is 12·507 and its standard deviation is 5·681. For age 60 and 2½% interest, one annuity in twenty is likely to have a present value greater than about 22·879. 2. The expected value of ãx varies considerably with interest rate and so does the variance. 3. The correlation coefficient between ãX and ÃX is – 1. 4. The random variable Ãx is identical with the random variable ãx except for changes in sign, scale, and origin. Hence the coefficients of kurtosis are identical, and the coefficients of skewness are identical except for sign. 5. The expected value of ãX decreases with age, whereas the standard deviation increases monotonically up to a certain age and then decreases. Similar calculations were performed for endowment assurances maturing at age 45 and at age 60. Summaries of the results obtained are given in 82 A Stochastic Approach to Actuarial Functions Tables 2 and 3, and more extensive tables are included in the Appendix (Tables A3 to A6). Table 1. Summary table of moments and moment functions of ãx and Ãx at 2½% interest Moments and moment functions of ãx at 2½% interest Standard Age Expected Variance deviation a 20 28·28 21·29 4·61 –2·53 12·16 30 25·35 24·24 4·92 –1·74 7·18 40 21·61 30·20 5·50 –1·21 4·48 50 17·18 34·94 5·91 –0·76 3·06 60 12·51 32·28 5·68 –0·31 2·31 70 8·09 22·86 4·78 0·16 2·20 Moments and moment functions of Ãx at 2½% interest Standard Age Expected Variance deviation ß2 20 ·2858 ·0127 ·1125 2·53 12·16 30 ·03573 ·0144 ·1201 1·74 7·18 40 ·4484 ·0180 ·1340 1·20 4·48 50 ·5565 ·0208 ·1442 0·76 3·06 60 ·6706 ·0192 ·1386 0·31 2·31 70 ·1782 ·0136 ·1166 –0·16 2·20 Moments, moment functions and product moments of ãx,and Ãx Age E(ãx) E(Ãx) Var.(ãx) Var.(Ãx) Cov.(ãx, Ãx,) Correlation 20 28·28 ·2858 21·29 ·0127 –·5193 –1·00 30 25·35 ·3573 24·24 ·0144 –·5911 –1·00 40 21·61 ·4484 30·20 ·0180 –·7366 –1·00 50 17·18 ·05565 34·94 ·0208 –·8523 –1·00 60 12·51 ·6706 32·28 ·0192 –·7874 –1·00 70 8·09 ·7782 22·86 ·0136 –·5577 –1·00 The following points should be noted: 1. The coefficients of variation (= standard deviation/mean) of and are considerably smaller than the coefficients of variation of &· 2. The standard deviations of and decrease with age. 3. The expected values and standard deviations of and vary considerably with interest rate. 4. The correlation coefficient between and is –1, and so is the correlation coefficient between and 5. The random variable is identical with the random variable A Stochastic Approach to Actuarial Functions 83 except for changes in sign, scale and origin. Hence the co- efficients of kurtosis are identical, and the coefficients of skewness are identical except for sign. 6. The values of and ß2 are rather large. This phenomenon is caused by the unusual distribution of the distribution has most of its weight situated at the value We have not mentioned any calculations for Its moments may be computed as follows. Since Hence the variances of policy values may be obtained from the variances of annuities. From Tables 1 and 2 and the Appendix, we thus compute the information in Table 4. Table 2. Summary table of moments and moment functions of and at 2½% interest Moments and moment functions of at 2½% interest Standard Age Expected Variance deviation ß2 20 17·64 2·63 1·62 –7·87 68·78 25 14·81 1·59 1·26 –8·54 80·83 30 11·58 ·81 ·90 –9·38 97·36 35 7·90 ·31 ·56 –10·57 122·77 40 3·74 ·05 ·23 –13·25 189·74 Moments and moment functions of at interest Standard Age Expected Variance deviation ß2 20 ·5453 ·001564 ·0395 7·87 68·78 25 ·6145 ·400946 ·0308 8·54 80·83 30 ·6932 ·000486 ·0220 9·38 97·36 35 ·7827 ·000186 ·0137 10·57 122·77 40 ·08843 ·000031 ·0056 13·25 189·74 Moments, moment functions and product moments of and Var. Var. Age (annuity) (assurance) Covariance Correlation 20 17·64 ·5453 2·63 ·001564 –·06413 –1·00 25 14·81 ·6145 1·59 ·000946 –·03879 –1·00 30 11·58 ·6932 ·81 ·000486 –·01992 –1·00 35 7·90 ·7827 ·31 ·000186 –·00765 –1·00 40 3·74 ·8843 ·05 ·000031 –·00127 –1·00 84 A Stochastic Approach to Actuarial Functions Table 3. Summary table of moments and moment functions of and at 2½% interest Moments and moment functions of at 2½% interest Standard Age Expected Variance deviation ß2 20 24·00 8·02 2·83 –5·37 35·13 30 19·81 5·78 2·40 –4·94 30·35 40 14·42 4·10 2·02 –4·39 23·49 50 7·66 1·47 1·21 –4·43 22·83 Moments and moment functions of at 2½ interest Standard Age Expected Variance deviation ß2 20 ·3903 ·004771 ·0691 5·37 35·13 30 ·4925 ·003436 ·0586 4·94 30·35 40 ·6240 ·002437 ·0494 4·39 23·49 50 ·7887 ·000876 ·0296 4·43 22·83 Moments, moment functions and product moments of and Var. Var. Age E(annuity) E(assurance) (annuity) (assurance) Covariance Correlation 20 24·00 ·3903 8·02 ·004771 –·19563 –1·00 30 19·81 ·4925 5·78 ·003436 –·14088 –1·00 40 14·42 ·6240 4·10 ·002437 –·09990 –1·00 50 7·66 ·7887 1·47 ·000876 –·03593 –1·00 The following points may be noted: 1. The coefficients of variation of are considerably smaller than those of 2. The expected value of increases with duration whereas its standard deviation decreases with duration. 3. The expected value of increases with duration; up to a certain duration the standard deviation does also, but it then decreases. 4. The higher rate of interest produces lower expected values but higher coefficients of variation. 5. The higher rate of interest produces smaller standard deviations at the shorter durations but larger standard deviations at the longer durations. Similar computations may be performed for temporary assurances, pure endowments, joint-life annuities, and joint-life assurances. We do not discuss them here. The following theorem, however, should be noted. THEOREM. Let Ã(y) be an assurance and an annuity dependent upon the same status (y), and such that: A Stochastic Approach to Actuarial Functions 85 1. the sum assured is constant and independent of the mode of decrement, 2. the rate of interest is constant, 3. the premiums payable are at a level rate, and 4. the status is in force from time 0 until it comes to an end. Table 4. Coefficients of variation of policy values Coefficient of variation of at 2½% interest Coefficient of variation Standard = standard deviation/ Age = 20+t Expected Variance deviation expected 20 0 ·0266 ·163 30 ·104 ·0303 ·174 1·67 40 ·236 ·0378 ·l94 ·82 50 ·392 ·0437 ·209 ·53 60 ·558 ·0404 ·201 ·36 70 ·714 ·0286 ·169 ·24 at 5% interest 20 0 ·0138 ·117 30 ·057 ·0178 ·133 2·33 40 ·147 ·0285 ·169 1·15 50 ·278 ·0433 ·209 ·75 60 ·439 ·0505 ·225 ·51 70 ·614 ·0437 ·209 ·34 Coefficient of variation of at 2½% interest 20 0 ·0084 ·092 00 25 ·161 ·0051 ·071 ·44 30 ·344 ·0026 ·051 ·15 35 ·552 ·4010 ·032 ·06 40 ·788 ·00017 ·013 ·02 at 5% interest 20 0 ·0073 ·085 00 25 ·123 ·0050 ·071 ·58 30 ·280 ·0029 ·054 ·19 35 ·482 ·0013 ·036 ·07 40 ·741 ·00025 ·016 ·02 Then it may be shown that (i) the expectations of Ã(y) and obey the premium conversion relation 86 A Stochastic Approach to Actuarial Functions are linearly related in pairs, the linear relations being (11) and (12) (iii) the correlation coefficient between any two of the three random variables has modulus equal to unity. PROOF: Conditions (1) to (4) are the usual conditions for the ap- plicability of premium conversion relations. Therefore result (i) applies. (See, for example, Life and Other Contingencies, Vol. 2, pp. 75–6, by P. F. Hooker and L. H. Longley-Cook.) To prove (ii), we first note that we may construct a life table for the status. It is then possible to write down equations corresponding to (7), (8), (9) and (10). From these equations, we conclude that is a linear function of and that is a linear function of Result (iii) follows as a consequence of result (ii). Note 1. These results apply to joint-life annuities and assurances. Note 2. The time unit of a year is arbitrary. Similar results apply if premiums are payable m times per year, or if premiums are payable continuously. 3. APPLICATION OF SECOND-ORDER MOMENTS Example 1. A company issues life annuities to retired persons aged 60. The annuities, payable annually in arrear, are issued for multiples of £100 per annum, and the lives currently receiving annuities are listed in Table 5. Many of the annuities were issued some years ago when interest rates were high, but these have now fallen, and it is likely that they will remain at 2½% per annum for a long time. The annual instalments due have just been paid, and the fund is valued at £3,102,000. What is the probability that the fund will be insufficient to meet all annuity payments? Mortality is according to the A1949–52 (ultimate) Table. Let X be the random variable representing the present value of the liabilities of the fund, and let be the number of lives aged x currently receiving £(l00j) per annum. Then = 2,982,782, and = 2,638,352,204. Hence the standard deviation of X is 51,360. From the Central Limit Theorem, we known that X is approximately normally distributed. Hence the probability that the fund will be in- sufficient is A Stochastic Approach to Actuarial Functions 87 which is approximately ·01. Table 5. The age structure of the annuity fund in Example 1 of Part 3 Number of annuities of Age x £100 p.a. £200 p.a. £300 p.a. £400 p.a. £500 p.a. 60 15 25 27 33 3 61 10 15 40 20 5 62 8 15 30 18 2 63 10 20 31 20 0 64 8 10 12 13 10 65 33 47 77 20 1 66 26 24 39 33 23 67 21 29 31 17 5 68 15 17 19 14 12 69 9 13 10 11 1 70 7 12 9 12 0 71 5 11 2 4 0 72 4 9 1 3 1 73 4 3 2 1 0 74 0 2 0 0 0 75 0 0 0 0 0 76 0 1 0 0 0 77 0 1 0 0 0 A note concerning the variance of the sum of the present values of n annuities on n independent lives all aged x is relevant here. If all the annuities are for one per annum, the present value of the sum of the annuities has expected value näx and variance n σ2 where σ2 is the variance of By comparison, an annuity of n per annum on a life aged x has the same expected value, but a variance of n2 σ2.Thus the present value of an annuity of n per annum on one life (x) and the present value of n annuities of 1 per annum on each of n independent lives (x) have the same expectation näx but in the latter case the standard deviation is times that of the former. The present value of r annuities on r lives where the total annual payment is n, has a standard deviation between a and depending on how uneven the distribution is by size of the r annual payments. The following is a more detailed analysis. Consider n independent 88 A Stochastic Approach to Actuarial Functions lives all aged x. Let the total annual sum payable be S, the smallest annuity be a per annum and the largest b per annum. If we denote the variance of we require bounds for the variance of the sum of the present values. Consider first the lower bound. Let the ith annuity be of size xi per annum. At least one annuity must be for b per annum, and at least one must be for a per annum. Our problem is therefore the following: Minimize (13) subject to x1 = a, x2 = b, (14) and This is equivalent to minimizing (15) subject to (i = 4, 5, . . ., n). (16) The minimum value may be determined by differentiating (15) and equating to zero. The xi values so obtained satisfy constraint (16). The minimum value occurs when x1 = a, x2 = b, and xi = (S–b–a)/(n–2), and it is equal to [a2+b2+(S–a–b)2/(n–2)] σ2 To maximize the variance, we must maximize (17) subject to (18) and (i = 3, 4, . . ., n). (19) Let us start with the minimum value. It is possible to increase x3 and decrease xn equally in such a way that constraints (18) and (19) still apply. The sum of squares (17) is increased. This process must cease when either x3 = b or xn = a. However, it is possible to continue with different pairs of Xis until all except one of them are equal to either a or b. The A Stochastic Approach to Actuarial Functions 89 remaining xi will satisfy inequality (19). We then have the maximum value of the variance. When the maximum value occurs, k of the xis are equal to b, (n- k - 1) of the xis are equal to a, and one of them has the value c, where The following must be true: kb+(n–k–1)a+c=S. Therefore, c=S–(n–1)a–(b–a)k. But c satisfies the inequality (19), and we therefore obtain (20) This relation gives us k, since we know it must be an integer, and we can σ then determine c. The maximum variance is [kb2 + (n – k – 1)a2 + c2] 2. The difference between the maximum and minimum variances is equal to [kb2+(n–k–1)a2+c2–a2–b2–(S–a–b)2/(n–2)] σ2 [k(b2–a2)+(n–2)a2–(S–a–b)2/(n–2)] σ2 (since c b), [(S–na)(a+b)+(n–2)a2–(S–a–b)2/(n–2)] σ 2,(using(20)). This last result will only be of use when both a and b are small relative to s. Consider for example the case in which n2 = 120, S = 50,000, a = 200 and b = 500. The minimum variance = 20,887,373ó2, and the difference The standard deviation therefore lies between 4,570ó and 4,818ó. This range is only 5% of the minimum possible standard deviation. Example 2. It is desirable to have some idea of the likely range of variation in the actuarial liability of a life office or pension fund. The percentage variation from the mean could vary considerably from one office to another depending upon the distribution of type of policy written, age distribution, and distribution of policy size. The character of the business is influenced by such factors as method of payment of com- mission (on sum assured or on premium), retention limits, area of activity, etc. Some assessment of the range of variation is particularly desirable when working with a closed fund, and when large amounts are at risk for small premiums (e.g. accidental death benefits and term assurances). For an accurate assessment, two or more policies on the one life must be grouped, and moments may be calculated using the techniques of Part 2. A comparison of the random variable representing the total policy values with the company reserves can then be performed along the lines of Example 1. A numerical example is instructive. Consider a hypothetical life as- surance fund covering the 34 separate lives in Table 6. The policies are 90 A Stochastic Approach to Actuarial Functions Table 6. Hypothetical life assurance fund—policies in force (1) (2) (3) (4) (5) (6) (7) (8) 1 1,000 40 5 ·20732 ·012408 207·32 12,408 2 2,000 35 6 ·18982 ·011675 37964 46,700 3 2,500 33 3 ·08248 ·012791 206·20 79,944 4 2,000 42 1 ·04505 ·017201 90·10 68,804 5 2,000 33 5 ·14091 ·011919 281·82 47,676 6 1,500 41 9 ·41499 ·006720 622·49 15,120 7 1,000 35 6 ·18982 ·011675 189·82 11,675 8 1,500 27 4 ·08357 ·011378 125·36 25,601 9 1,500 42 2 ·09111 ·015979 136·67 35,953 10 2,500 39 7 ·27945 ·010409 698·63 65,056 11 3,000 33 5 ·14091 ·011919 422·73 107,271 12 1,000 41 6 ·26761 ·010847 267·61 10,847 13 2,500 42 2 ·09111 ·015979 227·78 99,869 14 2,000 39 7 ·27945 ·010409 558·90 41,636 15 1,000 27 3 ·06187 ·011744 61·87 11,744 16 500 25 1 ·01837 ·012400 9·19 3,100 17 1,500 42 1 ·04505 ·017201 67·58 38,702 18 2,000 35 6 ·18982 ·011675 379·64 46,700 19 2,500 39 6 ·23690 ·011500 592·25 71,875 20 2,500 25 1 ·01837 ·012400 45·93 77,500 21 1,000 39 7 ·27945 010409 279,45 10,409 22 1,500 42 1 ·04505 ·017201 67·58 38,702 23 2,000 27 3 ·06187 ·011744 123·74 46,976 24 2,000 41 6 ·26761 ·010847 535·22 43,388 25 1,000 39 6 ·23690 ·011500 236·90 11,500 26 1,000 25 1 ·01837 ·012400 18·37 12,400 27 2,500 27 4 ·08357 ·011378 208·92 71,113 28 3,000 42 2 ·09111 ·015979 273·33 143,811 29 2,500 35 6 ·18982 ·011675 474·55 72,969 30 1,000 33 3 ·08248 ·012791 82·48 12,791 31 3,000 40 5 ·20732 ·012408 621·96 111,672 32 2,500 35 6 ·18982 ·011675 474·55 72,969 33 2,000 40 5 ·20732 ·012408 414·64 49,632 34 1,000 41 9 ·41499 ·006720 414·99 6,720 TOTALS 9,798·21 1,623,233 A Stochastic Approach to Actuarial Functions 91 endowment assurances to age 60. Equation (9) gives a relation between and x+t A similar relation holds between and namely : (21) From this relation, we deduce: (22) and (23) Thus, for policy number 5, we have (using A1949–52 (ultimate) mortality and 2½% interest): and The mean policy value is therefore 2000 × 0·14091 = 281·82, while the variance of the policy value is (2000)2 × 0.01192 = 47,676. Summing the two appropriate columns in Table 6, we obtain the expected value and variance of the total policy values of the fund. The standard error of the total of the policy values is 1,273 and this is large compared with the total of expected policy values (9,798). An alternative approach is to carry out a stochastic analysis of the death strain. Most companies determine the expected death strain for the year and compare monthly the actual with the expected. Such an analysis would be of much greater value if the standard error were known, and if such an analysis with standard errors were dissected into branches, tables, etc. With modern digital computers, such computations should no longer be considered impossible. This problem is discussed in greater detail in Part 4. 4. RETENTION LIMITS AND REASSURANCE ARRANGEMENTS The prime purpose of setting retention limits or entering into reassurance arrangements is to reduce to an acceptable figure the likelihood of sto- chastic variations in mortality in any year affecting the bonus distribution through an unexpected fall in the surplus. A stochastic approach should therefore lead to a better assessment of retention limits and reassurance treaties than the deterministic approach provides. The process of selecting retention limits must be sequential. This is because the retention limit depends upon the policy structure of the company according to age, size of policy, type of policy, etc., and because when a new policy is written or a policyholder dies, the policy structure of the company is altered. In theory, it is possible to examine the policy structure of a company before writing each new policy. Clearly this is 92 A Stochastic Approach to Actuarial Functions impossible in practice. A practical procedure would be to examine the policies in force annually and decide then the company’s retention limits for the following year. Let us investigate this type of procedure. 4.1. REASSURANCES ON ORIGINAL TERMS Consider first a company whose policy has been to cede, on the original terms and for the full duration of the contract, that proportion of any contract which exceeds the retention limit in force when the contract was written. The problem is to determine the retention limits which should be set for new contracts during the coming year. Three points should be noted : 1. It would be incorrect to set limits based on the consequences of writing one large contract for, if a number of similar contracts were written, the actual death strain could easily exceed the permissible limit. 2. The number of small policies written in conjunction with the large policies is a material factor in determining the retention limit. 3. If we are concerned with yearly fluctuations in mortality, we should consider one year as a unit and determine retention limits to apply to a year’s new business considered as a single package. One could proceed in the following manner. Consider a new policy, written at age x for a sum assured S. The death risk during the first policy year may be taken as S; the expected value and variance of the death strain will be qxS and pxqxS2, respectively. Let V0denote the variance of the death strain of the in force due to mortality during the following policy year. Then the variance of the death strain of both in force and one year’s new business is V0 + assuming that no duplicate policies are included in the new business. The summation is over one year’s new business. We assume that the financial strength of the company will enable it to cope with an actual death strain which exceeds the expected by any amount up to X; that is to say, if the bonus is not to fall below a certain level because of mortality fluctuations, we require the chance p that the actual death strain exceeds the expected by X or more to be very small (say) ·001. If a large number* of policies is involved, the Central Limit Theorem establishes that the death strain will be normally distributed. * This requirement has not been expressed very precisely, to avoid cluttering up the text. The relative sizes of the policy death risks must be considered. If all the death risks are small, the Central Limit Theorem will certainly apply. If there is a reasonable number of policies all involving large sums at risk of approximately the same magnitude, the Central Limit Theorem will apply approximately. In almost all cases, however, even when there is a large deviation from normality, inequality (25) will be of use. We would then speak of an R-sigma limit rather than a p-limit. A Stochastic Approach to Actuarial Functions 93 The above condition may be expressed mathematically as follows: (25) where R is obtained from p using normal distribution tables. (If p =·001, R = 3.09.) From (25), (26) That is, X2/R2 is the upper limit to which the variance of the death strain of existing business and new business combined may be allowed to climb. This inequality may be satisfied by limiting the values of S in various ways. We could reassure enough policies in full-even a very large number of small policies-to bring the risk of a large death strain within acceptable limits. There are, however, a few logical criteria on which the selection of policies for reassurance might be based, perhaps the most obvious one being that the total sum assured should be as large as possible. Before dealing with these criteria and the results which flow from them, we shall study the variations in the variance of the death strain during the course of a life assurance contract. We need to do this firstly because it will throw some light on how v0 in the fundamental equation (26) is likely to vary with time and secondly because it may not be sufficient to base reassurance policy on the value of pqS2 at the time of entry into a contract. 4.2. THE VARIANCE OF THE DEATH STRAIN The variance of the death strain on a £100 non-participating policy during the course of the contract (based on A1949–52 (ultimate) tables at 2½% interest) follows the general pattern shown in Figure 1. We have selected three ages at entry namely 20, 40 and 60 and four types of contract namely endowment assurances maturing at 45, 65 and 80 and whole of life. For the whole of life, for example, age 20 at entry, the graph is actually that of pxqx(ä2x+1/ä220). As we wish to determine whether the sum assured retention limit S should be reduced below that based on the initial risk because of sub- sequent rises in the variance of the death strain, it is better to work with the standard deviation rather than the variance, since the latter varies with the square of S. Also, as we are comparing the position at entry with that later in the contract, it is more useful to present the standard deviation of the death strain at any time as a fraction of its initial value. This is shown in Figure 2. 94 A Stochastic Approach to Actuarial Functions FIG. 1. Variance of the death strain on a £100 non-participating policy during the course of the contract (Basis: A1949–52 (ultimate) 2½%) A Stochastic Approach to Actuarial Functions 95 FIG. 2. The standard deviation of the death strain on a £l00 non-participating policy during the course of the contract expressed as a fraction of its initial value. (Basis: A1949–52 (ultimate) 2½%) Several interesting facts emerge from these two figures: 1. The variance of the death strain passes through a maximum at age 70 for whole of life policies, at age 63 for endowment assurances to age 80 and at age 50 for endowment assurances to age 65. The age at which this maximum occurs is independent of the age at entry. This clearly follows from the formula since for any age at entry, the variance is proportional to pxqxyäx2+1 : where x is the age attained. 2. The shorter the term of the contract, the lower the variance. G 96 A Stochastic Approach to Actuarial Functions 3. The standard deviation of the death strain for whole of life policies rises to double its initial value in the case of middle and younger ages at entry. Substantial percentage increases also occur with endowment assurances maturing at age 80. For endowment assurances maturing at 65 or earlier there is no material increase over the initial value. 4.3 REASSURANCE ON ORIGINAL TERMS (Continued) At first sight it might appear that V0 in inequality (26) might, in certain circumstances, be greater than X²/R². This position would only arise in the case of a fund which consisted mainly of whole of life contracts (or equivalent very long-term endowment assurances) effected at ages under 45. This is a likely situation with a closed fund. Fortunately, in this case new business limitations do not arise, but the increasing variance of the death strain could make it desirable to reconsider the reassurance position of some existing contracts if this increase has not been allowed for when the contract was written. For an expanding office, X will be increasing with the size of the office and retention limits also should be increasing. It would be interesting to know at what rate X should be increasing in order that V0 will never be greater than X2/R2 in the future. Consider X growing with time at rate i per annum. In t years’ time it will take the value X(t) = (1 + i)tX. The upper bound for the variance in inequality (26) will be (1 + i)2t(X²/R²). Consider now the whole of life policy on a life aged 20 included in the year’s new business. A reference to compound interest tables of (1 + i)2n and to Figure 1 will show that the contribution of this policy to V0 in t years’ time is less than (1·015)2t times its first contribution to V0 for all t. Therefore, if X is increasing at 1½% or more per annum, the contribution of this policy to future V0 values will not cause any difficulty. At age 30, the rate of increase required is about 2½% per annum, at age 40 about 3½% per annum, falling to 1% at age 60. These are the figures for whole of life contracts; for other contracts the figures are considerably less. The magnitude of changes in the variance of the death strain during the course of the contract are such that they must at least be considered. Clearly however the problem is limited to whole of life contracts (or equivalent very long-term endowment assurances) where these are effected at ages between 30 and 55. With the usual distribution of contracts by age and by type and with an office growing at an average rate it is unlikely that special treatment for this factor will be required. Should some action be necessary several solutions are possible including: (i) When determining V0, policies where the variance now exceeds a certain figure could be detected and temporary reassurance effected, (ii) Reassurance limits generally could be reduced to allow for the possible increases in death strain variances with duration. Programmes suggested later in § 4.4 for determining limits can easily be modified to allow for this factor. A Stochastic Approach to Actuarial Functions 97 For simplicity only three ages at entry have been considered above and the associated necessary rates of growth of X have been quoted. In Appendix 2, a general treatment of this problem is given including the associated necessary rates of growth of X for any age and duration of contract. It is shown there (assuming A1949–52 (ultimate) mortality and 2½% interest) that the maximum rate of growth required is that necessary to cover whole of life policies entered into between ages 40 and 50 and is 3½% per annum—that is, a doubling of X every 20 years. This is an extreme case; for most contracts no increase or a small increase only is required. 4.4 RETENTION LIMITS If we were to base retention limits purely on the variance of the death strain at entry, then the retention limits would depend on age at entry only and not on type of contract and would be as follows (assuming an arbitrary figure of 25,000 for age 15): Age x √ (pxqx) Retention limit 15 ·0333 25,000 25 ·0334 24,900 35 ·0363 22,900 45 ·0573 14,500 55 ·1012 8,200 The overall limitation which must be met by the office is inequality (26). If portions of certain contracts are to be reassured in order to meet the limitations of inequality (26), the management of the office would usually have some overall policy which would affect the decision as to which policies are selected. The most common such criterion is that the policies selected in order to meet limitation (26) should be chosen so as to make the total remaining sum assured as large as possible. We then require V= V0+p1q1S12+p2q2S2²+. . . +prqrSr²+. .. to be reduced to less than X2/R2, subject to being kept as large as possible. If we select one particular policy, say the rth, and keep all the others constant, we have: Hence, if we reduce the sum assured on the rth policy, for every unit reduction in the sum assured S, the variance V will reduce by 2prqrS,. For the largest reduction in the variance for the smallest reduction in the sum assured, we therefore select those policies which have the largest values of prqrSr. On this criterion, retention limits should vary with age 98 A Stochastic Approach to Actuarial Functions at entry only (and not with type of contract) and vary as follows (assuming an arbitrary figure of 25,000 at age 15 for the particular office and A1949– 52 (ultimate) mortality) : Age x Pxqx Retention limit 15 ·001109 25,000 25 ·001119 24,780 35 ·001318 21,040 45 ·003289 8,430 55 ·010243 2,700 A more general criterion can be summed up in the mathematical statement that S = c1S1 +c2S2+ . . . +crSr+. .. should be a maximum. In this case, so we would select policies which have the largest values of If c1 = c2 = . . . = cr = . . ., we have the case of maximizing the new sum assured. If c1, c2, . . ., cr, . . . represent the rates of premium on the various policies, then we are maximizing the new premium income. Here, assuming premium rates do not vary appreciably with sum assured, ∂cr/ ∂Sr = 0, and we select policies which have the largest values of Prqrsr/cr. Using typical premium rates and A1949–52 (ultimate) mortality, we obtain under this criterion the following retention limits: Age Retention limits X Whole life Endowment assurance to 45 Endowment assurance to 65 15 25,000 58,100 32,100 25 31,900 91,900 42,800 35 36,400 162,000 51,700 45 20,600 33,200 55 9,900 22,100 The figures quoted above are all relative; to determine the absolute limits for a particular office, the steps are as follows: 1. Decide on the value of X which can be met without affecting the bonus; A Stochastic Approach to Actuarial Functions 99 2. Decide on a value for p; 3. Estimate from new business (before reassurances) in recent years ; 4. Calculate V0 for the business in force; 5. From these values calculate the retention limits. All the necessary data should be on the valuation cards and the cal- culations are possible if a computer is in use. 4.5 THE RISK PREMIUM METHOD The methods just discussed can clearly be applied to offices whose policy is to reassure on a year-to-year basis. In fact the application is straightforward since, with this policy, the problems of death strain variance increases with time do not arise. In this case we maximize Σ crSr subject to Σ prqrSr2 ≤ X2/R2 where S,, now stands for the death risk under a policy and the summation is over all policies both in force and new business. In the case c1 = c2 = . . .= cr = . . .) we are maximizing the death risk, which very roughly is maxi- mizing future premiums receivable. In other cases values of c, may be determined by equating crSr, to the values (for the particular reassurance criterion) of the quantity which it is desired to maximize, e.g. future premiums receivable, present value of future premiums, annual premium income, etc. Under this system an office retains a larger portion of its insurance portfolio. 4.6. STOP LOSS REINSURANCE The method of assessing retention limits for a company as outlined in the previous section required a knowledge of the total variance of the death strain of the individual policies. For companies whose actuarial departments are computer based such a calculation should be possible. However, when this information is available, a reinsurance arrangement alternative to the retention limit system is possible which should be of benefit to both the company and its reinsurers. We refer to stop loss reinsurance—that is, a reinsurance contract effected at the beginning of the year under which no individual policies are reinsured but under which the reinsurance company undertakes, if, and only if, the actual death strain for the ensuing year exceeds a certain figure Z, to pay to the insuring company the amount by which the actual death strain exceeds Z. In the absence of any isolated, really abnormally large policies, the death strain in any one year will have a distribution very close to normal. Let the expected death strain during the next year be £E, and the variance £V2. If we denote the death strain by W, then W Normal (E, V). 100 A Stochastic Approach to Actuarial Functions The company is prepared to meet an actual death strain of £Z (which depends upon its size, age, capital, extra reserves, etc.), but no more. The net premium payable to the reinsuring company to insure against any excess is (27) where f(x) = The net premium can therefore be readily computed using tables of the normal distribution, and should usually be small. The information necessary to make the calculation is available to the company and to its reinsurers. Such a treaty should enable a company, at small cost, to retain all its business and still limit the risk to a figure of its own choosing and which depends on its financial strength. This arrangement is superior to the retention limit system in that exceeding the upper limit is made impossible—not just very unlikely. The reinsurer should be able readily to assess its profit from the treaty. Four relevant matters should be discussed: 1. Several policies on one life. Where there are several policies on one life, the variance is underestimated. If there are four policies with equal amounts at risk on every insured life, then the standard deviation of the death strain will be double the figure obtained by treating policies sepa- rately. Some indication of the effect of multiple policies on the standard deviation of the death strain can be obtained by using the methods given in Part 3 after Example 1. However, with modern computers, multiple policies can and should be treated thoroughly. 2. What should be done about policies written during the year and after the annual payment to the reinsurers ? A temporary reassurance for the balance of the year of part of any large policies could of course be effected. The most satisfactory and most simple solution would be to cover the year’s new business under the same terms under the same contract and provide for a retrospective adjustment when the new calculation has been made at the end of the year. 3. Substandard lives. The treatment of substandard lives in the valuation will need to be considered. Where these are included in the normal portfolio, it is usual to include them at the rated-up age and, if this is so, appropriate allowance is made for the extra risk and higher variance. A Stochastic Approach to Actuarial Functions 101 4. The effect on the distribution of W of one (or more) isolated abnormally large policy. If there is one enormous policy involving a large possible death strain during the ensuing year which could affect the distribution of W, then the premium could be calculated as follows: Assume that the large policy is on a life now aged x and that the death strain is s. We first calculate the probability that the life (x) survives the year and that the death strain on the other policies is u. This probability is Similarly, the probability that the life (x) dies and the death strain on the other policies is u can be written down as The probability that the total death strain during the year is u therefore is The premium payable to the reassuring company, P, is therefore given by These integrals may be readily evaluated using tables of the Normal integral. This process may be extended to deal with several policies each involving a possible large death strain, and a computer can do the relevant calculations very rapidly. The death strain is the important criterion and not the actual policy size. When there are many large policies all with large expected death strains of comparable sizes, a Normal approximation should be accurate, and the above process should not be necessary. ACKNOWLEDGEMENTS We should like to thank the Director of the Cambridge University Mathematical Laboratory, for time on TITAN, and the Shell Company of Australia for the Research Scholarship it granted the second author. 102 A Stochastic Approach to Actuarial Functions APPENDIX 1 Appendix Table A1. Moments and moment functions of ãx and Ãx at 2½% interest Age (x) E(ãx) S.D.(ãx) (ãx) β 2(ãx) E(Âx) S.D.(Ãx) 20 28·28 4·61 –2·5 12·2 ·286 ·113 21 28·02 4·63 –2·4 11·6 ·292 ·113 22 27·76 4·65 – 2·4 11·0 ·299 ·113 23 27·48 4·68 –2·3 10·5 ·305 ·114 24 27·20 4·70 – 2·2 9·9 ·312 ·115 25 26·91 4·73 –2·1 9·4 ·319 ·115 26 26·61 4·76 –2·0 8·9 ·326 ·116 27 26·31 4·80 –2·0 8·5 ·334 ·117 28 26·00 4·84 –1·9 8·0 ·341 ·118 29 25·68 4·88 –1·8 7·6 ·349 119 30 25·35 4·92 –1·7 7·2 ·357 ·120 31 25·02 4·97 –1·7 6·8 ·365 ·121 32 24·61 5·02 –1·6 6·5 ·374 ·122 33 24·32 5·07 –1·5 6·1 ·382 ·123 34 23·96 5·13 –1·5 5·8 ·391 ·125 35 23·59 5·19 –1·4 5·6 ·400 ·127 36 23·21 5·25 –1·4 5·3 ·410 ·128 37 22·82 5·31 –1·3 5·1 ·419 ·130 38 22·43 5·37 –1·3 4·9 ·429 ·131 39 22·02 5·43 – 1·2 4·7 ·438 ·133 40 21·61 5·50 –1·2 4·5 ·448 ·134 41 21·20 5·56 –1·2 4·3 ·459 ·135 42 20·77 5·61 –1·1 4·1 ·468 ·136 43 20·34 5·67 –1·1 4·0 ·480 ·138 44 19·90 5·72 –1·0 3·8 ·490 ·139 45 19·46 5·77 –1·0 3·7 ·501 ·141 46 19·01 5·81 –· 9 3·5 ·512 ·142 47 18·56 5·84 –·9 3·4 ·523 ·143 48 18·11 5·87 –·9 3·3 ·534 ·143 49 17·65 5·90 –·8 3·2 ·545 ·143 50 17·18 5·91 –· 8 3·1 ·556 ·144 51 16·72 5·92 – ·7 3·0 ·568 ·144 52 16·25 5·92 – ·7 2·9 ·579 ·144 53 15·79 5·92 –·6 2·8 ·591 ·144 54 15·32 5·91 –· 6 2·7 ·602 ·144 A Stochastic Approach to Actuarial Functions 103 Age (x) E(ãx) S.D.(ãx) (ãx) β2(ãx) E(Ãx) S.D.(Ãx) 55 14·85 5·89 –·5 2·6 ·613 ·144 56 14·38 5·86 –·5 2·5 ·625 ·143 57 13·91 5·83 –·5 2·5 ·636 ·142 58 13·44 5·78 –·4 2·4 ·648 ·141 59 12·97 5·74 –·4 2·4 ·659 ·140 60 12·51 5·68 –·3 2·3 ·671 ·139 61 12·04 5·62 –·3 23 ·682 ·137 62 11·58 5·55 –·2 2·2 ·693 ·135 63 11·13 5·18 –·2 2·2 ·704 ·134 64 10·68 5·39 –· 1 2·2 ·715 ·132 65 10·23 5·31 –· 1 2·2 ·726 ·129 66 9·79 5·21 ·0 2·2 ·737 ·127 67 9·35 5·11 ·0 2·2 ·747 ·125 68 8·93 5·01 ·1 2·2 ·758 ·122 69 8·51 4·90 ·1 2·2 ·768 ·119 70 8·09 4·78 ·2 2·2 ·778 ·117 71 7·69 4·66 ·2 2·2 ·788 ·114 72 7·30 4·54 ·3 2·3 ·798 ·110 73 6·92 4·41 ·3 2·3 ·807 ·106 74 6·54 4·28 ·4 2·3 ·816 ·104 75 6·18 4·15 ·4 2·4 ·825 ·101 76 5·83 4·01 ·4 2·4 ·833 ·098 77 5·49 3·88 ·5 2·5 ·842 ·095 78 5·17 3·74 ·5 2·6 ·850 ·091 79 4·86 3·60 ·6 2·7 ·857 ·088 80 4·56 3·46 ·6 2·7 ·864 ·084 81 4·27 3·32 ·7 2·8 ·872 ·081 82 3·99 3·19 ·7 2·9 ·878 ·078 83 3·73 3·05 ·8 3·0 ·885 ·074 84 3·48 2·92 ·8 3·1 ·891 ·071 85 3·25 2·79 ·9 3·2 ·896 ·068 86 3·03 2·66 ·9 3·3 ·902 ·065 87 2·82 2·53 ·9 3·3 ·907 ·062 88 2·62 2·41 1·0 3·4 ·912 ·059 89 2·43 2·28 1·0 3·4 ·916 ·056 90 2·25 2·16 1·0 3·4 ·921 ·053 91 2·09 2·04 1·0 3·3 ·925 ·050 92 1·93 1·91 1·0 3·2 ·929 0·47 93 1·78 1·79 ·9 3·0 ·932 ·044 94 1·63 1·65 ·9 2·8 ·936 ·040 104 A Stochastic Approach to Actuarial Functions Appendix Table A2. Moments and moment functions of ãx and Ãx at 5% interest Age (x) E(ãx) S.D.(ãx) (ãx) β2(ãx) E(Ãx) S.D.(Ãx) 20 17·97 2·11 –4·5 29·5 ·097 ·101 21 17·89 2·13 –4·4 28·2 ·101 ·101 22 17·80 2·14 –4·3 26·8 ·105 ·102 23 17·72 2·17 –4·1 25·4 ·109 ·103 24 17·62 2·19 –4·0 24·0 ·113 ·104 25 17·52 2·22 –3·8 22·6 ·118 ·106 26 17·42 2·25 –3·7 21·2 ·123 ·107 27 17·31 2·28 –3·5 19·8 ·128 ·109 28 17·20 2·32 –3·4 18·5 ·I33 ·110 29 ·17·08 2·36 –3·2 17·2 ·139 ·112 30 16·95 2·40 –3·1 16·0 ·145 ·114 31 16·82 2·45 –2·9 14·8 –151 ·116 32 16·68 2·49 –2·8 13·8 ·158 ·119 33 16·54 2·55 –2·7 12·8 ·165 ·122 34 16·39 2·61 –2·6 11·8 ·172 ·124 35 16·23 267 –2·5 11·0 ·180 ·127 36 16·06 2·74 –2·4 10·2 ·187 ·I30 37 15·89 2·81 –2·3 9·6 ·196 ·134 38 15·71 2·88 –2·2 8·9 ·204 ·137 39 15·52 2·96 –2·1 8·3 ·213 ·141 40 15·32 3·03 –2·0 7·8 ·223 ·144 41 15·12 3·11 –1·9 7·3 ·232 ·148 42 14·91 3·19 –1·8 6·9 ·242 ·152 43 14·69 3·26 – 1·8 6·5 ·253 ·155 44 14·47 3·34 – 1·7 6·1 ·263 ·159 45 14·23 3·41 –1·6 5·7 ·275 ·163 46 14·00 3·48 –1·6 5·4 ·286 ·166 47 13·75 3·55 –1·5 5·1 ·298 ·169 48 13·50 3·62 –1·4 4·8 ·310 ·I73 49 13·24 3·68 –1·4 4·6 ·322 –175 50 12·98 3·74 –1·3 4·3 ·334 ·178 51 12·71 3·79 –1·2 4·1 ·347 ·181 52 12·43 3·84 –1·2 3·9 ·360 ·183 53 12·16 3·88 –1·1 3·7 ·374 ·185 54 11·87 3·92 –1·0 3·5 ·387 ·187 55 11·58 3·95 –· 9 3·3 ·401 ·188 56 11·29 3·98 –·9 3·1 ·415 ·190 57 1099 4·01 – ·8 3·0 ·429 ·191 58 10·69 4·02 –·8 2·9 443 ·192 59 1038 4·03 –·7 2·7 ·458 ·192 A Stochastic Approach to Actuarial Functions 105 Age (x) E(ãx) S.D.(ãx) (ãx) β2(ãx) E(Ãx) S.D.(Ãx) 60 10·07 4·04 –·6 2·6 ·473 ·192 61 9·76 4·04 –·6 2·5 ·487 ·192 62 9·44 4·03 –·5 2·4 ·502 ·192 63 9·13 4·02 –·5 2·4 ·517 ·191 64 8·82 4·00 –· 4 23 ·532 ·190 65 8·50 3·97 – ·4 2·2 ·548 ·189 66 8·18 3·94 –· 3 2·2 563 ·188 67 7·87 3·90 –· 3 2·1 ·578 ·186 68 7·55 3·86 – ·2 2·1 ·592 ·184 69 I·24 3·81 – ·2 21 ·608 ·181 70 6·93 3·16 –·1 2·1 ·622 ·179 71 6·62 3·70 ·0 2·1 ·637 ·176 72 6·32 3·63 ·0 2·1 ·652 ·173 73 6·02 3·56 ·1 2·1 ·666 ·169 74 5·73 3·48 ·1 2·1 ·680 ·166 75 5·44 3·40 ·2 2·1 ·693 ·162 76 5·16 3·32 ·2 2·2 ·707 ·158 77 4·88 3·23 ·3 2·2 ·720 ·154 78 4·61 3·14 ·3 2·3 ·733 ·150 79 4·36 3·05 ·4 2·3 ·745 ·145 80 4·11 2·95 ·4 2·4 ·757 ·141 81 3·86 2·85 ·5 2·4 ·768 ·136 82 3·63 2,76 ·6 2,5 ·779 ·131 83 3·41 2·66 ·6 2·6 ·790 ·126 84 3·19 2·56 ·7 2·7 ·800 ·122 85 299 2·46 ·7 2·8 ·810 ·117 86 2,79 2·36 ·7 2·9 ·819 ·112 87 2·61 2·26 ·8 2·9 ·828 ·108 88 2·43 2·16 ·8 3·0 ·836 ·103 89 2·27 2·06 ·9 3·0 ·844 ·098 90 2·11 1·96 ·9 3·1 ·852 ·093 91 1·96 1·86 ·9 3·0 ·859 ·089 92 1·82 1·76 ·9 3·0 ·866 ·084 93 1·68 1·65 ·9 2·9 ·873 ·078 94 1·54 1·54 ·8 2·7 ·879 ·073 106 A Stochastic Approach to Actuarial Functions Appendix Table A3. Moments and moment functions of and Age S.D. S.D. (x) E (annuity) (annuity) β2(annuity) (assurance) 20 17·64 1·62 –7·9 68·8 ·545 ·040 21 17·19 1·55 –8·0 70·9 ·558 ·038 22 16·55 1·48 –8·1 73·2 ·572 ·036 23 15·99 1·41 –8·3 75·5 ·586 ·034 24 15·40 1·33 –8·4 78·1 ·600 ·033 25 14·81 1·26 –8·5 80·8 ·614 ·031 26 14·19 1·19 –8·7 83·7 ·629 ·029 27 13·56 1·12 –8·9 86·8 645 ·027 28 12·92 1·05 –9·0 90·1 ·661 ,026 29 12·26 ·97 –9·2 93·6 ·677 ,024 30 11·58 ·90 –9·4 97·4 ·693 ·022 31 10·88 ·83 –9·6 101·4 ·710 ·020 32 10·16 ·76 –9·8 105·9 ·728 ·019 33 9·43 ·70 – 10·0 110·8 ·746 ·017 34 8·68 ·63 – 10·3 116·4 ·764 ·015 35 7·91 ·56 –10·6 122·8 ·783 ·014 36 7·11 ·49 – 10·9 130·3 ·802 ·012 37 6·31 ·43 –11·3 139·6 ·822 ·010 38 5·47 ·36 –11·8 151·4 ·842 ·009 39 4·62 ·29 – 12·4 167·3 ·863 ·007 40 3·14 ·23 –13·3 189·7 ·884 ·006 41 284 ·16 – 14·5 224·5 ·906 ·004 42 1·92 ·10 – 16·4 283·2 ,929 ·003 43 ·97 ·05 – 19·6 384·1 ·952 ·001 A Stochastic Approach to Actuarial Functions 107 Appendix Table A4 Moments and moment functions of and at 5% interest Age S.D. S.D. (x) (annuity) (annuity) β2(annuity) (assurance) 20 13·63 1·16 –8·4 78·4 ·303 ·055 21 13·33 1·12 –8·5 80·4 ·318 ·054 22 13·01 1·08 –8·6 82·5 ·333 ·052 23 12·68 1·04 – 8·7 84·8 ·349 ·050 24 12·33 1·00 –8·8 87·2 ·365 ·408 25 11·96 ·96 –9·0 89·7 ·383 ·406 26 11·57 ·92 –9·1 92·4 ·401 ·044 27 11·16 ·81 –9·3 95·3 ·421 ·42 28 10·73 ·83 –9·4 98·4 ·441 ·039 29 10·28 ·78 –9·6 101·7 ·463 ·037 30 9·81 ·73 –9·7 105·2 ·485 ·035 31 9·31 ·69 –9·9 109·0 ·509 ,033 32 8·79 ·64 – 10·1 113·1 ·534 ,030 33 8·24 ·59 – 10·3 117·7 ·560 ·028 34 7·66 ·54 – 10·5 122·9 ·588 ·026 35 7·06 ·49 – 10·8 128·8 ·616 ·023 36 6·42 ·43 –11·1 136·0 ·647 ·021 37 5·75 ·38 –11·5 144·8 ·679 ·018 38 5·04 ·33 – 12·0 156·2 ·712 ·016 39 4·30 ·27 –12·6 171·5 ·747 ·013 40 3·53 ·21 – 13·4 193·4 ·784 ·010 41 2·71 ·16 – 14·6 227·5 3823 ·007 42 1·85 ·10 –16·4 285·3 ·864 ·005 43 ·95 ·05 – 19·6 384·1 ·907 ·002 108 A Stochastic Approach to Actuarial Functions Appendix Table A5. Moments and moment functions of and Age S.D. S.D. (x) (annuity) (annuity) β2(annuity) (assurance) 20 23·99 2·83 – 5·4 35·1 ·390 ·069 21 23·63 2·79 –5·3 34·9 ·399 ·068 22 23·24 2·74 –5·3 34·6 409 ·067 23 22·85 2·70 –5·3 34·3 ·418 ·066 24 22·45 2·65 –5·2 33·9 ·428 ·065 25 22·04 2·61 – 5·2 33·4 –438 ·064 26 21·61 2·57 –5·2 32·9 448 ·063 27 21·18 2·52 –5·1 32·3 ·459 ·062 28 20·73 2·48 –5·1 31·7 ·410 ·061 29 20·27 2·44 – 5·0 31·1 ·481 ·060 30 19·81 2·40 –4·9 30·4 ·493 ·059 31 19·32 2·31 –4·9 29·6 ·504 ·058 32 18·83 2·33 –4·8 28·9 516 ·057 33 18·32 2·29 –4·8 28·1 ·529 ·056 34 17·81 2·26 –4·7 21·4 ·541 ·055 35 17·27 2·22 –4·6 26·7 ·554 ·054 36 16·73 2·19 –4·6 26·0 ·568 ·053 37 1617 2·15 –4·5 25·3 ·581 ·052 38 1560 2·11 –4·5 24·6 ·595 ·051 39 15·02 2·07 –4·4 24·0 609 ·050 40 14·42 2·02 –4·4 23·5 ·624 ·049 41 13·81 1·97 –4·4 23·0 ·639 ·048 42 13·18 1·92 –4·3 22·6 ·654 ·047 43 12·54 1·86 –4·3 22·2 ·670 ·045 44 11·89 1·79 –4·3 21·9 ·686 ·044 45 11·22 1·72 –4·3 21·7 ·702 ·042 46 10·54 1·63 –4·3 21·7 ·719 ·040 41 9·84 1·54 –4·3 21·7 ·736 ·038 48 9·13 1·44 –4·3 21·9 ·753 ·035 49 8·41 1·33 –4·4 22·2 ·771 ·032 50 7·66 1·21 –4·4 22·8 ·789 ·030 51 6·90 1·09 –4·5 23·7 ·807 ·027 52 6·12 ·95 –4·7 25·0 ·826 ·023 53 5·32 ·82 –4·8 26·8 ·846 ·020 54 4·50 ·67 –5·1 29·4 ·866 ·016 55 3·66 ·53 –5·5 33·4 ·886 ·013 56 2·79 ·38 –6·0 39·6 ·908 ·009 57 1·89 ·24 –6·8 50·4 ·930 ·006 58 ·96 ·11 –8·2 69·0 ·952 ·003 A Stochastic Approach to Actuarial Functions 109 Appendix Table A6. Moments and moment functions of and Age S.D. S.D. (x) (annuity) (annuity) β2(annuity) (assurance) 20 16·63 1·68 –6·4 48·6 ·160 ·080 21 16·48 1·67 –6·3 48·1 ·168 ·080 22 16·32 1·66 –6·3 47·6 ·175 ·079 23 16·16 1·65 –6·2 47·0 ·183 ·078 24 15·99 1·63 –6·2 46·2 ·191 ·078 25 15·80 1·62 –6·1 45·4 ·200 ·011 26 15·61 1·61 –6·0 44·4 ·209 ·017 27 15·41 1·60 – 5·9 43·4 ·218 ·076 28 15·20 1·59 –5·8 42·3 ·228 ·076 29 14·98 1·58 –5·8 41·0 ·239 ·015 30 14·75 1·57 –5·7 39·8 ·250 ·015 31 14·50 1·57 –5·6 38·4 ·262 ·075 32 14·25 1·56 –5·5 37·1 ·274 ·074 33 13·98 1·56 –5·4 35·1 ·287 ·074 34 13·69 1·55 –5·3 34·3 ·300 ·074 35 13·40 1·55 – 5·2 33·0 ·314 ·074 36 13·08 1·54 –5·1 31·7 ·329 ·073 37 12·76 1·54 – 5·0 30·5 ·345 ·073 38 12·42 1·53 –4·9 29·4 ·361 ·073 39 12·06 1·52 –4·8 28·3 ·378 ·073 40 11·68 1·51 –4·7 27·3 ·396 ·072 41 11·29 1·50 –4·7 26·4 ·415 ·071 42 10·88 1·48 –4·6 25·7 ·434 ·070 43 10·45 1·45 –4·5 25·0 ·454 ·069 44 10·00 1·42 –4·5 24·4 ·476 ·068 45 9·53 1·38 –4·5 24·0 ·499 ·066 46 9·04 1·34 –4·5 23·6 ·522 ·064 47 8·53 1·28 –4·4 23·5 ·546 ·061 48 7·99 1·21 –4·5 23·5 ·572 ·058 49 7·43 1·14 –4·5 23·7 ·599 ·054 50 6·84 1·05 –4·5 24·1 ·627 ·050 51 6·23 ·96 –4·6 24·8 ·656 ·046 52 5·58 ·85 –4·8 26·0 ·686 ·041 53 4·91 ·74 –4·9 27·7 719 ·035 54 4·20 ·62 –5·2 30·2 ·753 ·029 55 3·45 ·49 –5·5 34·0 ·788 ·023 56 2·66 ·36 –6·0 40·2 ·826 ·017 57 1·82 ·23 –6·9 50·7 ·866 ·011 58 094 ·11 –8·3 69·0 ·908 ·005 110 A Stochastic Approach to Actuarial Functions APPENDIX 2 We aim, for a £100 whole of life policy, for any age at entry and any duration, 1. to determine the actual value of the standard deviation of the death strain; 2. to determine the standard deviation of the death strain as a fraction of its initial value; and 3. to express this change in standard deviation over a period as an equivalent continuous rate of growth. In Figure 3, we have the graphs of 100 and of äx according to A1949–52 (ultimate) mortality with 2½% interest from ages 15 to 95. Let us pro- ceed as follows: 1. Select any age at entry x0 (say) and draw new horizontal and vertical axes through the point on the äx curve and use the same logarithmic scale with this point as origin. That portion of the other curve to the right of x0 now represents the standard deviation of the death strain for a whole of life policy aged x0 at entry. This is due to the fact that the actual height of this curve from the new origin (i.e. on a linear scale) is and hence on the logarithmic scale used the height is which is the standard deviation of the death strain at age x of a £100 policy. 2. If we carry out a similar change of axes and use as our new origin the point on the upper curve in Figure 3 with abscissa x0, then this curve represents the standard deviation of the death strain as a fraction of its initial value. This is due to the fact that the actual height of this curve from the new origin (i.e. on a linear scale) is and hence on the logarithmic scale used the height is which is the standard deviation of the death strain at age x as a fraction of its initial value. 3. If a quantity of size a increases to size b over time t, then the instantaneous rate of growth δ is given by That is A Stochastic Approach to Actuarial Functions 111 FIG. 3. Graphs of 100 and äx from which may be obtained (as explained in Appendix 2) the standard deviation of the death strain both in absolute terms and as a fraction of its initial value and also the equivalent continuous rate of growth. If we join the point on the upper curve which has abscissa x0 to any subsequent point on the curve-say that with abscissa x-then the actual slope of this line is 112 A Stochastic Approach to Actuarial Functions A Stochastic Approach to Actuarial Functions 113 which therefore equals the equivalent continuous rate of growth corresponding to the change in standard deviation during this period. This slope must be measured on a linear scale which is given on the right-hand side of the graph for this purpose. The following points should be noted: (i) The procedure for endowment assurances is similar, and the relevant graphs are given in Figure 4. (ii) From these graphs we can readily determine, for any age at entry, the standard deviation of the death strain, either in actual value or as a fraction of its initial value and the equivalent continuous rate of growth. In practice, it is convenient to use a loose sheet of transparent paper or plastic with a logarithmic scale and axes on it to facilitate the change of axes. For the theoretical purposes of this paper a glance at the curves is sufficient. (iii) The maximum rate of growth is about 3½% per annum and this occurs with whole of life policies with ages at entry 40–50 and where the age attained is less than 50–55.

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