CHAPTER IX JOINT CONTINGENT FUNCTIONS PART I. THEORETICAL DEVELOPMENT 63. Joint Contingent Functions.—By a joint contingent func- tion we shall understand a contingent function whose present value depends upon two or more lives. Thus we have joint endowments, joint annuities, and joint insurances. In general, we shall limit the discussion to contingent functions depending upon two lives with hints as to extension when more than two lives are involved. In the first part of the present chapter the theoretical formulas for the present values of certain joint contingent functions will be developed; and in the second part will be considered the method, by means of the Gompertz-Makeham law, for finding the numer- ical present values of joint contingent functions. 64. Joint Endowments.—When two lives are involved, we may distinguish three forms of joint endowment: 1. A pure joint endowment to (x) and (y) is a sum of money payable at the end of n years if both (a;) and (y) are then living. The present value of a pure joint endowment of $1 to (a;) and (y), payable in n years, is represented by the symbol nExs. 2. A reversionary joint endowment to (x) and (y) is a sum of money payable at the end of ra years if (x) is then dead and (y) is then living. The present value of a reversionary joint endowment of $1 is represented by the symbol nEx\v The lives may be reversed, and then the symbol for the present value is written nEy\,. S. A survivorship joint endowment to (x) and (y) is a sum of money payable at the end of n years if (x) and (y) are not then both dead. The present value of a survivorship joint endowment of $1 is represented by the symbol n^xy- 88 §64] JOINT CONTINGENT FUNCTIONS 89 The formulas for the present values of joint endowments are developed on the basis of the principal of mathematical expecta- tion (Sec. 27), and it is, therefore, necessary to know how to express the probability that the events in question happen. Thus, n-E',1, = V^nPx nPv = V\p^, (1) since the probability that both (x) and (y) are living at the end of n years is the compound probability consisting of the product of the separate probabilities that each lives n years. This probability is denoted by ,,?,„. Similarly, when more than two lives are involved. For a reversionary joint endowment we have, nE^y = V(l — nP,)nPy = V\py — V"np, np,, = V^nPv - V^nPxv = nE, — nE^y, (2) since the probability that (x) is dead and (y) is living at the end of the nth year is the compound probability denoted by (1 - nPx)nPy. For a survivorship j oint endowment, nE^ = »»[! - (1 - „;,,)(! - ,?„)] = y(np, + „?„ - »?,„) = nE, + nEy - „£',„, (3) since the probability that (x) and (y) are not both dead at the end of the »th year is denoted by 1 — (1 — np,)(l — npx). In Eq. (2), if both (x) and (y) are living at the end of the nth year, both nE, and nE^y yield $1 and their difference is 0; if (a;) is living and (y) is dead, both yield 0 and their difference is 0; but if (x)is dead and (y) is living, ^Ey yields $1 and nE^y yields 0 and their difference is $1, as it should be. Similarly, in Eq. (3), if both (a;) and (y) are living at the end of the nth year, nE-^ yields $1 + $1 - $1 = $1; if both are dead, nE^y yields 0; if (a;) is dead and (y) is living,,^ yields 0+$1 - 0 = $1; and if (x) is living and (y) is dead, nE-xy yields $1 + 0 - 0 = $1. 90 MATHEMATICS OP LIFE INSURANCE [§65 With three lives (x), (y), and (z) several forms of joint endowment may be imagined, of which the following are examples: nE,\y, = y-(l — npx)npvnP. = nEy. - nE^.. (4) nEx\y; = y»(l - »p,)[l - (1 - »p»,)(l - »p,)] = »»(! - nPx)(nP» + »?, - nPv.) = V"(»P» + nP. — nP»t — nPx. — nPxs + nPz»«) = En + JE, — nE,, — nE,t — nE^y + Em. (5) Note that if (x) is living at the end of the nth year, nEx\yz yields 0. Similarly, the yield is 0 if both (y) and (2) are dead, whether (a;) is living or dead. Again, nE^.= V[l - (1 - „?,)(! - np»)(l - »p,)] = nEf + nE, + nE, — nEy, — nE,, — nE^y + nExy,. (6) Note that nEw yields $1, provided all three of the lives have not ceased during the term of n years. Obviously, the discussion may include as many lives as may be desired and in as many combinations of living and dying as may be desired. In general, we can say: Any joint endowment consists of a linear combination of a series of pure endowments. Exercises 1. Interpret in words the symbols n.E'*|»«, vEx^z, nExyi- 2. Interpret in words the symbol nEiy\, and express its value in terms of pure endowments. 65. Each Person's Share of the Present Value of a Survivor- ship Endowment.—When a survivorship joint endowment is in the form of a bequest, it becomes important to determine each bene- ficiary's share of the present value of the endowment. Especially is this true when inheritance taxes are levied on the present values of bequests. Consider, for example, the endowment nExy, and suppose (a;) and (y) share alike if both survive the term of n years. It is clear, then, that (x) should be responsible for as much of the present value as would amount to $1 if he survives the term and §66] JOINT CONTINGENT FUNCTIONS 91 (y) does not, and would amount to $0.50 if both survive the term. Hence, it follows that (a;)'s share = -^» + J^ = «Ex - inS,, 1, , , (y)'s share = in-E,, + nE^y == n^» - ^nE,» \ ' ' In case of the endowment nEx\Tn on ^e supposition of equal shares if (y) and (2) survive the term, we have (y)'s share = ^rJ3i\y, + n^'a:»l» = n-E» — ^nE,. — nE^y + ^nE,y, 1 ^ ..„< (2)'s share = ^nE^y. + ^B^y,, = nE. - ^nEy. - ,£',. + ^nE,,. I Again, in case of the endowment nE^., on the supposition that survivors share alike,
Note that these shares yield the proper amounts at the end of the nth year, whatever eventually happens to the lives involved at, or previous, to that time; and that the sum of the shares is the total present value of the endowment. 66. Joint Annuities.—A joint annuity is usually payable annu- ally and its present value is the sum of the present values of cor- responding joint endowments extending for 1, 2, 3, . . . years until all the lives involved have vanished. Formulas for present values of joint annuities can be derived from the corresponding formulas for the present values of joint endowments by sub- stituting the symbol a for the symbol nE. Thus,
= ^JE^W
xv — ^/rxv
is the present value of a pure joint annuity paying $1 at the end of each year as long as both (x) and (y) are living. a,\, = ^Ai, = ^(nEs - „£'„,) = a» - o,» (2) 92 MATHEMATICS OF LIFE INSURANCE [§67 is the present value of a reversionary joint annuity paying $1 at the end of each year after the death of (,x) and as long as (y) lives thereafter. <txy = ^nE^y = ^(,nE. + „.£?, - £',,) = a, + a, - a,» (3) is the present value of a survivorship joint annuity paying $1 at the end of each year as long as (a;) or (y) or both (,x) and (y) are living. Note that the present value indicated in Eq. (3) is suffi- cient to sustain an annuity of $1 as long as at least one of the lives involved is in force. Similar formulas hold when more than two lives are involved. Thus, "xyz = al + "r + a' ~ a^ - a,. - a,» + a.». (4) is the present value of a survivorship joint annuity paying $1 as long as at least one of the three lives involved is in force. Bequests are often made in the form of survivorship annuities. Each person's share of the present value of a joint survivorship annuity follows from the corresponding formula in the preceding section. Thus, in case of the annuity a-, (x)'s share = a, — ^a.y, (y)'s share = a, - ia,y, (5) assuming equal shares during the joint lives of (x) and (y). 67. Joint Annuities-due.—A joint annuity-due has a present value indicated by the full-faced letter a. Thus, a-xy = 1 + a*,,. (1) a,i»= a, - a,» = (1 + dy) - (1 + a,,) = a» - <i,». (2) The present value of the annuity-due in this case does not differ from the present value of the ordinary annuity a^y, since a dollar is not due at the beginning of the first year, (x) being then alive. a^ = a, + a,, - &^ = (1 + a,) + (1 + a») - (1 + a,,) (3) = 1 + a, + By —— Ctxy = 1 + tt^-. Corresponding to the statement made at the close of Sec. 64, we have:—Any joint annuity can be expressed as a linear combina- tion of a series of pure annuities. §68] JOINT CONTINGENT FUNCTIONS 93 Exercises 1. Interpret in words the symbol ai\y, and express its value in terms of pure annuities. 2. Write the formula for the shares of (y) and (z) in case of the annuity o,iy,, assuming (y) and (?) share the annuity equally throughout their joint lives after the death of (x). 68. Joint Insurances.—By the symbol Axy we shall understand the present value of $1 payable at the end of the year in which the first death (either (a) or (y)) occurs. The dollar will fall due at the end of the nth year, provided both (a;) and (y) live the preceding n — 1 years and that both do not live the nth year. The prob- ability that these events occur is the compound probability denoted by n-lP, n-lP»(l — Pt+n-l P»+n-l) = n—lPxv — nPxv Hence, the mathematical expectation is f"(n-lPt» - nPzv). This expression must be summed for all integral values of n from 1 to the end of the mortality table. But ^v»-ip*» = y^v-^-ip.i, = tw,», and ^y'np.1, = a,,. Therefore, A,» = va..» - a,y (cf. Sec. 33, Eq. (1)). (1) By the symbol Axy we shall understand the present value of $1 payable at the end of the year in which the last death (either (x) or (y)) occurs, or in which both (a) and (y) die. The probability that the dollar will be paid at the end of the nth year consists of the total probability that (y) dies during the nth year, (a;) having pre- viously died; or (a:) dies during the nth year, (y) having previously died; or both (x) and (y) die during the nth year. This probability is denoted by (I - »-lP,)(n-lP» - nP») + (1 - n-lP»)(»-lP, - »?,) + (n-lPt - nPx) (n-lPs - nPv), 94 MATHEMATICS OF LIFE INSURANCE [§69 which reduces (»-lP-c - „?,) + (,-lP,, - np,) - {n-lPxy - nP,»). Multiplying by v" and summing for all integral values of n from 1 to the end of the mortality table, we get A-xy = A, + Ay — A,». (2) So long as (x) and (y) are both living, the insurances A,, Ay, and A,» do not become payable. If (a;) dies first, A, and A,y become payable at the end of the year and balance each other in the formula, leaving an insurance of $1 on the after lifetime of (y). If (,y) dies first, the insurances Ay and A,,, become payable and balance each other, leaving an insurance of Sl on the after life- time of (x). If both (a;) and (y) die the same year, the insurances Ax, Ay, and Asey become payable and their algebraic sum is $1. A,y and A^y are net single premiums chargeable for an insurance of $1. The net annual premiums are found by dividing the net single premiums by the corresponding annuities-due. Thus,
and
P —— n-CV /HI -1*1' ~ „ W <*»v
p—— c"n\ txv=—— (4)
Exercises 1. Interpret in words the symbols A^ and A^—/a—. 2. Prove that A^=Ax+Av+A,- A^ - A,, - A^+A^ 69. Instalment Insurances.—As has been remarked, the pri- mary purpose of insurance is the protection of a beneficiary against financial consequences arising from the death of the insured. A possible financial consequence is the loss of the entire insurance through poor or ill-advised investment in case the insurance is paid in one sum. To guard against this possible consequence, insurance companies issue instalment insurance policies which provide an income (payable annually or oftener) certainly payable for a period of years, or payable throughout the after lifetime of §69] JOINT CONTINGENT FUNCTIONS 95 the beneficiary, or certainly payable for a period of years and then as long as the beneficiary shall live thereafter. 1. If (a;) desires to establish a fund that will furnish m annual instalments of $1000 certainly payable, first payment at the end of the year in which he dies, the present value of the fund is represented by lOOOA.(a^), since the m instalments constitute an annuity-due certain whose present value is 1000(aro[), and the net single premium chargeable for the entire assurance is the expression written above. The net annual premium payable throughout the life of (x) is lOOOA^^^^^^ ^
For example,
N.
^2o(a2oi)Mm _„ —»—— = iv^zoD- &20 A'20
At 3% per cent, a^oj = 14.709837. We have already found Mw/Nw = 0.013477. The net annual premium is, therefore, 0.013477 X 14.709837 = 0.19825. A net annual premium of $198.25 will secure an instalment insurance of $1000 certainly payable for 20 years, first payment at the end of the year in which the insured of age twenty dies. The annuity-certain may be accumulated, of course, at a differ- ent rate of interest from that upon which net premiums are based. 2. If (a) desires to establish a fund that will furnish an annual income of $1000 to a beneficiary (y) as long as (y) may live after the death of (a;), first payment at the end of-the year in which (x) dies, the present value of the fund is 1000a,i,, = 1000 (a, - a,,), and the net annual premium, ^ ^ 10000,1,.^ &xV The annuity-due &x» is here used, so that the premiums automati- cally cease at the death of (a;) or (y) or both, for, if (x) dies before (y), the payment of premiums cannot be conveniently shifted to (y); and, if (y) dies before (x), it is not desirable to expect (x) to con- tinue paying premiums for a benefit that has ceased. 96 MATHEMATICS OP LIFE INSURANCE [§69 3. If (x) desires to establish a fund that will furnish an annual income of $1000 to a beneficiary (y) certainly payable for m years and as long thereafter as (y) may live, the present value of the fund
IS
1000[A,an.| + ^Ey a,|y+,»].
For the first term of this expression is sufficient to furnish $1000 certainly payable for m years. To continue the payments, a fund of 1000 ax\y+m will then be necessary, since (y) will then be TO years older than he would have been had no term of m years been interposed. This fund must be discounted over the period of m years, under the proviso of the continuance of the life of (y), to obtain its present value. Hence the factor JS,. The net annual premium throughout the life of (x) is „ , 1000[A^+^.^,^J ^ a-x If vi represents the annual premium for the m instalments cer- tainly payable, and a-a represents the annual premium chargeable for the instalments to (y) should he survive (x) and the period of m years, Eq. (3) may be written 71-.= 71-1 +7r2--"'- (4) ax If (y) dies before (x), the annual premium automatically drops to what it would have been had no continuance of instalments been proposed beyond those certainly payable, since the annuity a,, ceases payments on the death of (y). Exercises 1. Compute the numerical value of lo^oolao. 2. How much money must A invest now at rate 0.035 to furnish his daughter a sum of $1000 on her twenty-first birthday, provided he should die prior to that time? A is now forty and his daughter is eleven. Clearly, annual instalments may be commuted to semi-annual quarterly, or monthly instalments; or they may be changed to a single sum, as the insured may desire. Annual premiums may be §70] JOINT CONTINGENT FUNCTIONS 97 limited to a period of years instead of extending throughout life, or they may be paid oftener than once a year. Joint contingent functions may be intercepted so as to cover only a portion of the time from the present until all lives involved have ceased. Thus, \n(lxv = \nax + \"aV ~ \<taISl (5) \^XV = |nA. + |,A, - \nA^, (6) \r>ax\y = |nffl» — \nCt,y, (7) are equations easily proved and whose meanings are clear from previous discussions. PART II. NUMERICAL COMPUTATION 70. Joint Endowments.—There is no difficulty in computing the numerical present values of joint endowments. Thus, at 3^ per cent, r> in v^ss-lso /,,, 10^25,20 = V1" 10P25,20 = —i———7—— (i) (25 120 _ 0.7089188 X 81,822 X 85,441 89,032 X 92,637 = 0.5872. A fund of $587.20 at 3% per cent compound interest will yield $1000 at the end of 10 years if two persons now of ages twenty-five and twenty are then living. 10^?25|20 = 10^20 — 10" 25,20 (2) = 0.65384 - 0.58720 = 0.06664. A fund of $66.64 will yield $1000, provided a person now of age twenty-five dies before the expiration of a term of 10 years and a person now of age twenty is living at the expiration of the term. 10-^25^0 = IO-E'25 + wEtO ~ 10^25,20 (3) = 0.65151 + 0.65384 - 0.58720 = 0.71815. A fund of $718.15 will yield $1000, provided at least one of two persons now of ages twenty-five and twenty is living at the expira- tion of a term of 10 years. On the basis of equal shares, if both 98 MATHEMATICS OF LIFE INSURANCE [§71 are living at the end of 10 years, the older person's share of the $718.15 is $357.91 and the younger person's share is $360.24. Similarly, if more than two ages are involved, it is possible to compute the numerical present value of any ordinary endowment with relatively few arithmetical operations. If many lives are involved, the operations necessary to compute present values become tedious; and, in case of joint annuities, quite out of the question. 71. Commutation Symbols for Two or More Lives.—To obviate tedious arithmetical computations, actuaries sought to define commutation symbols for two or more lives analogous to those used in case of a single life. Thus, if (a;) is the older of two lives (a;) and (y), we have the Davies commutation symbol D,y = V'Uf, (1) so called because introduced by the actuary Griffith Davies. But this definition leads to certain difficulties, because the exponent of v depends only upon the older age. To obviate these difficulties, the mathematician De Morgan suggested the definition D,, = v^U,. (2) With the Davies definition, p _ Vix+nly+n _ V'^lx^ly+n_D^+n,y+n ,„> " ~ 7 7 ~ ——««77——— - ——n———' W vx^v V ixly i-fxy and with the De Morgan commutation symbol. x+n+y+n El _ v_______ tx+nlm-n Vx+n,v+n ,,\ ^ - ———^———— = -p^-. (4) v 2 U, With either definition, if we have a table giving the value of Bis for each possible pair of ages throughout the mortality table, the computation of the numerical value of nE^y is reduced to a simple division of one number by another, just as in the case of A. Either the Davies or the De Morgan definition is extensible immediately to cases involving more than two lives. But the main difficulty with these definitions is the number of necessary entries in a table of such commutation symbols. There are 3741 §72] JOINT CONTINGENT FUNCTIONS 99 pairs of ages (including equal ages) from ages ten to ninety-five. And, if we are to include commutation symbols for more than two ages and for several rates of interest, almost a small library would be needed for the numerical computation of present values of joint contingent functions. If x = y, the Davies and De Morgan definitions coincide. Thus, Davies definition: Dxx = v'Ux = vl^. x+x De Morgan definition: Dxx = v 2 1x1, == vl^. If, then, we can find a pair of equal ages w, w that will replace a pair of unequal ages x, y so that nPww == nPxy for all values of n, the number of entries in a table of commutation symbols for two ages from ages ten to ninety-five will be reduced to eighty-six, the same as for a single life. The method for finding the equal age w rests upon the Gompertz-Makeham law of mortality. 72. Force of Mortality.—The/orce of mm-taUty is the instantane- ous rate of mortality at a given age. At age thirty, the annual rate of mortality is 330 = 720/85,441 = 0.00843. If a table of mortality were constructed exhibiting the number of deaths each month, it would be easy to compute a monthly rate of mortality. Theoretically, we can imagine rates of mortality at a given age for smaller and smaller intervals of time and thus arrive at the notion of an instantaneous rate of mortality. The Greek letter y. is used to indicate force of mortality. Thus, fi.x means the force of mor- tality at age x. In the absence of a table exhibiting mortality rates at small intervals of time, some assumption must be made that will agree substantially with mortality rates at yearly intervals and that will furnish a basis for computing mortality rates at small inter- vals of time. In 1825 Benjamin Gompertz assumed that fix increases with the age in a geometric ratio; that is, that the forces which cause death accumulate like compound interest. This amounts to assuming that ^ = Be', (1) 100 MATHEMATICS OF LIFE INSURANCE [§73 where B and c are constants to be determined from yearly mor- tality rates as exhibited in a mortality table. This is the Gom- pertz law of mortality. But it was found to be inadequate, principally because the forces which cause death are not entirely due to what may be called the regular "wear-out" processes of life. Fatal diseases and accidents are not due to wear-out processes, although they may be invited by such processes and thus hasten death. In 1860 Mr. Makeham suggested the addition of a constant A to the right-hand side of the Gompertz law. Thus the Gompertz- Makeham law of mortality is expressed by the equation' ^ = A + Bc1. (2) It was found that the constants A, B, and c can be determined so that the Gompertz-Makeham law accords very well with mortality rates as exhibited in a mortality table. A remarkable circum- stance is that log c is very nearly equal to 0.04000 for all tables of mortality. But the values of the constants A, B, and c need not detain us here, since Arthur Hunter has computed a table of values for juz for the American Experience Table (Table IV). 73. Equal Age.—The age w such that two or more equal ages w, w, w, . . . will replace two or more unequal ages x,y,z, . . . is an age such that the force of mortality ju,, is the average of the forces fi,, fty, p.., . . For example, jttso = 0.00835 ju4o = 0.00977
20.01812
Average = /i» = 0.00906. This value of ju«, lies between i^se = 0.00902 and /lar = 0.00916. By interpolation, we find w = 36.25. ' The force of mortality may be expressed thus dh,. ltx = - dx^' the negative sign because I, is a monotone decreasing function of x and, therefore, dl,/dx is continuously negative. §74] JOINT CONTINGENT FUNCTIONS 101 Again, juso = 0.00786 fiw = 0.00977 Mso = 0.01384 3|0.03147 Average = JL»» = 0.01049. By interpolation in the table, we find w = 42.7. In general, we have V., = A + 5C", JUy = A+ jgg",____________ ^ + ft. = 2A + B(v +c"), Average = /!„ = A + 5(ct^+-cl') But also /x» = A + .Be", and, therefore, we have the important relation, c* + c»
or
2c" = c* + c". (I) Similarly, for three ages x, y,and z,we have 3c" = c* + c" + c'; (2) and so on for any number of ages. 74. The Law of Uniform Seniority.—If we multiply both sides of Eq. (1) of the preceding section by c", we have 2(;t»+n = gi+n -(- (;»+n_ /]\ Similarly from Eq. (2), 3c«'+» =(;*+"+ (;"+' + c'+"; (2) and so on for any number of ages. Hence, 102 MATHEMATICS OF LIFE INSURANCE [§7i If w is the equal age for any number of given ages x, y, z, . . then w + n is the equal age for the ages x + n, y+ n, z + n, . . This is known as the Law of Uniform Seniority. Exercises 1. Compute the equal age for ages ten and forty. For ages ten thirty, and fifty. 2. From Exercise 1, how would one determine the equal age to; ages fifteen and forty-five? For ages twenty-five, forty-five, anc sixty-five? 3. Compute 2)40,30 by the Davies definition, by the De Morgar definition, and by the equal-age method and compare results, usin^ interest at rate 0.035. 75. A Makehamized Table of Mortality.—From the Makehan law of mortality (Sec. 72), it can be shown that1 ;, = ks'g'',(1] where k, s,and gare three new constants whose values can be determined, together with the value of c, so that Eq. (1) wil substantially repeat any mortality table, at least between the ages twenty and eighty. For the American Experience Table, Mr. Hunter found the following values for the four constants k, s, g,and c: log c = 0.04579609. log s = -0.003296862. log g= -0.00013205. log k =5.03370116. ' For -log (, = f(A + Bc^dx +log k,(log kbeing a constant of integration). Hence, / 7? \ log (. = -Ax -(iog-Jc* + log k, or l^k^e-^-(e-^10^)'", = ks'y', where * = e-* and g= e-s/lw. Therefore, A= -log s/log eand B = —log?- log c/log e. §76] JOINT CONTINGENT FUNCTIONS 103 To Makehamize a mortality table it isnecessary to find the values of the four constants and then reconstruct the table by means of Eq. (1). When a table has been Makehamized, it is said to be graduatedby the Makeham formula. 76. Probability of Living.—We must now show that the equal age w, found by means of the Gompertz-Makeham law, satisfies the necessary condition npw . . . =np,w . . . , (1) for any value of n. To begin with two unequal ages xand y,we have —'F-^-'''-". and '-'T-^-'-"-' Therefore, .P.» = s^'^^'-^ =s2"^"<'"-i> (Sec. 73) = sngrw^"-!) . gngw(c"-l) = nP» nP». Obviously, this analysis can be extended to any number of ages, and hence the equal age w does satisfy the necessary condition in Eq. (1). 77. Commutation Symbols Continued.—We have now achieved the object stated in Sec. 71 and can continue methods for numeri- cal computation of present values of joint contingent functions without fear that the number of necessary commutation symbols will become burdensome. We shall confine the attention to ordinary cases and upon not more than two lives. The necessary commutation symbols are defined thus:
Daw = V"lw1 Nw» = ^Dwu = Dv» + D«4.i,»^.i + Da+s.w^t . to the end of the table Mw = vN»w — N»+i,w+i
, (1) 104 MATHEMATICS OF LIFE INSURANCE [§7 and the present values of pure joint contingent functions becom T71 __ -"W+n,W+T»
These formulas are entirely analogous to those for a single life and the only additional labor involved is that of finding the equa age w from a table giving the force of mortality at each age through- out the mortality table plus the labor of interpolation when th( equal age involves a fractional part of a year, as it usually does As an example, consider the annuity a^ggg-. We have a2^25 = ^20 + ^5 ~ ^0,25. Here the equal age turns out to be exactly 23, and, therefore, a____ = NW 1 ^25 ^V'23.23 20,25 rT ' ~n~ ~ n——" "SO L'25 ^23,28 ^ 984,399.6 770,113.4 _ 654,942 46,556.2 ' 37,673.6 37,080 = 23.922 nearly. Hence a fund of $23,922 will support an annuity of $1000 payable at the beginning of each year to two persons of ages twenty and twenty-five until both are dead, assuming 3% per cent compound interest. Had the equal age contained a fractional part of a year, it would have been necessary to interpolate in the table of commuta- tion symbols to find the values of Nv,» and Z>«,«,. It should be remembered that a mortality table must be Make- hamized before values of commutation symbols depending on more than one life can be computed. It would seem wise to adopt a Makehamized table for all computations of present values of contingent functions, but such a plan has not been generally adopted by insurance companies except for joint functions.
§77] JOINT CONTINGENT FUNCTIONS 105 Exercises 1. Compute the net annual premium for an insurance certainly payable in ten annual instalments of $1000 each, the insured being twenty-five years of age at issue of the policy. 2. Compute the annual premium on the policy in Exercise 1, if the instalments are to be continued as long as the beneficiary shall live after the expiration of the 10 years, the beneficiary being twenty years of age at issue of the policy. 3. Compute the numerical values of A^g-gy and P^Tar 4. Compute the numerical values of the following symbols and state in words the meaning of each: a^, 130,10, |l0°3(uo, 10 "SO^o-