CHAPTER VIII INCREASING INSURANCES 60. Increasing Insurance.—Hitherto we have considered only standard life insurance policies, in which the benefit is either a fixed sum payable at the end of the year in which the insured dies or, in endowment policies, at the end of a term of years. Clearly, however, the benefit may vary in any way desired. The simplest mode of varying the benefit is to increase the insur- ance by a fixed amount each year. Thus we may have an insur- ance of k dollars increasing by h dollars each year, so that the benefit is k, k + h, k + 2h, k + 3h, . . . dollars in case of death during the first, second, third, fourth, . . . years. If the insured dies during the nth policy year, the total benefit is k + (n — l)h dollars. The net single premium chargeable for such an insurance is AA,+A(iA,+i,|A,+i,|A,+ . . . T^IA,), since the additional insurance consists of the sum of insurances of A dollars deferred for 1,2,3, . . . , w — x years. The most interesting case is that for which k = h = 1, that is, an insurance beginning at $1 and increasing by $1 each year thereafter. The symbol for the present value of such an insurance is (7A),, where x indicates the age of the insured at the issue of the policy. Hence (ZA), = A, + iA, + sJA, + . . . + ^|A,, (1) or, in commutation symbols, (7A). = ^M-+M^+M^+ -+M») L/x We shall now call the numerator of the fraction on the right Rx, so that R, = Af, + Af,+i + Af,+2 + . . . + Mu (3) 81 82 MATHEMATICS OF LIFE INSURANCE [§61 is a new commutation symbol obtained by a cumulative addition of the Mi's just as the Nx's were formed from the Di's (Table III). Thus, Eq. (2) becomes (^). = § (4) For example, ITA\ ^2° 397,283.63 „ ., , (7A)20 = ^o = -4W2- = 8-53 Dearly- Therefore, the payment of a net single premium of $8.53 will secure an insurance of $1, increasing by $1 each year as long as the insured lives. The net annual premium is _ _ (7A), R, . N. R,,., "'-"ar^-^r; (5) where the Greek letter v is used to indicate that the insurance is not standard, or ordinary. Thus, T2o = R^ = 0.404. tVzo An annual payment of $404 net will secure an insurance of $1000 increasing by $1000 each year as long as a person of age twenty lives. Exercises 1. Check the values of R, from age eighty to the end of Table III. 2. Compute the numerical value of (lA),, and state in words the meaning of the result. 3. Compute the net annual premium for an insurance of 81000 increasing by $1000 each year throughout life at age thirty-five. 61. Intercepted Increasing Insurance.—The symbol jn(ZA). means the present value of an insurance to (a:) beginning at $1 and increasing by $1 for n years and then remaining constant through- out the rest of life. Hence, |n(ZA), = A, + ijA, + slA. + . . . + »-ijA. _ (M. + Af,+i + M^ + . . . + M^,.i)
D.
(R, — R,+n) D, '
(1) §61] INCREASING INSURANCES 83 and 1^ = (RS 'N^- (2) At age twenty, i (Rip — Rap) nioni-i i 110^20 = ——,r——— == 0.12014 nearly. * 20 A net annual premium of $120.14 at age twenty will secure an insurance of $1000 increasing by $1000 each year for 10 years and then remaining constant for the rest of life. Should the insured die at any time after the expiration of 9 years, the full benefit will be $10,000. Such an insurance is not altogether impracticable, assuming that, in general, incomes increase for a period of years after service begins. By the symbol \n(IA)x:T\ (On) we shall mean the present value of an insurance of $1 increasing by $1 for n years and then remaining constant for t — n years and then terminating. Hence, |»(7A),:7J = |lA, + l|,-lA,, + 2|(-2A, + . . . n-lll-n+lA, ^ (Af. - M^,) + (M,^ - M,+.) + . . . + (M,^_i -M^)
(R, — j?»+n — nMi^t)
D,
D. Or, more simply, |,(7A)^ = 1»(7A), - n,|A. _ (Rx — JZi+n —n Mx+i).
(3)
D,
If ( =- n we have
I (JA\ — - (-RI — Rx+n — nM,+n) ,.» \»(.lA)x:n\ — ——————_————————, (4) and the net annual premium for n years is | - - - (R- — R'+n — nM^+n) inTX'nl ~ ——(^ - N^)—— <5) Thus, |i.(7A)2o=^ = (^-^0-10^30) ^ Q ^4^ (.-(Vso — A1 so) The payment of a net annual premium of $40.44 for 10 years at age twenty will very nearly pay the death claims as they fall due for a term insurance of $1000 increasing by $1000 for 10 years and then ceasing. The following table exhibits the history of the policy. 84
MATHEMATICS OF LIFE INSURANCE
Note that the valuation symbol kx increases in value for each additional $1000 of insurance.
Exercises 1. Compute the numerical values of | s(.IA)is and ^(IA),,:^ and state in words the meanings of the results. 2. Compute the net annual premium for a 10-year term insurance of $1000 increasing by $1000 for 5 years and then remaining constant for the remainder of the term at age twenty-three. Construct a table for this policy like the one in this section. 62.Return-premium Policies.—Among the many ways the benefit side of an insurance contract may be varied, perhaps the most important is that secured by a return-premium policy. By this is meant an insurance of $1, say, plus all premiums paid prior to the death of the insured. In a limited-payment policy the premiums may be returned in the form of an endowment, provided the insured survives the premium-paying period. Or the premiums may be returned in case of death or survival. We shall consider only a few of the many possible forms of return- premium policies. 1. To determine the net annual premium for an ordinary whole-life insurance of $1 with the return of all net annual premiums paid prior to the death of the insured. Let ir represent the net annual premium required. The net single premium for the proposed insurance is, then, A, + 7r(ZA),,
§62] INCREASING INSURANCES 85 since A, is the present value of the insurance of $1 and Tr(,IA)x is the present value of an insurance of TT dollars, increasing by TT dollars as long as (a;) lives. The net annual premium is, therefore ,.d^M, y; Solving this equation for IT, we have '-.—N:-?.^:- ») Thus, at age twenty, TT = Mto = 13,267.32 Nio - Rw 984,399.6 - 397,283.63 ~ -0226- The full benefit at death for an insurance of $1000 of this character is $1000 plus $22.60 X the number of premiums paid. To make return-premium insurance feasible, the office premium must be returned and not the net premium. 2. To determine the annual office premium far an ordinary whole- life insurance with the return of all office premiums paid prior to death, the loading being k per cent of the net premium plus c dollars per $1000 of ordinary insurance. Let TT be the net annual premium. Then -'-(l+^+i^O is the annual office premium per dollar of ordinary insurance. Hence "A^(M) a, and, therefore, -'-^(l+t'+i^-(A4^(-")0 +*)+,„, (2) Or, clearing of fractions, ^'a, = [A, + 7r'(7A),l(l + k) + ^ . a,. (3) The left member of this equation represents the payment side of the contract, and the right member the benefit side. Solving Eq. (3) for TT', we get _/ ^ A,(l + k) + ca.,/1000 _ M,(l + k) + cN^/lOOO a. - (7A),(1 + k) ~ N,-R,(l+~k)—— W 86 MATHEMATICS OF LIFE INSURANCE [§62 It is clear, from Eq. (4), that 1 + k must be much less than N^/Rx. With a heavy percentage loading, the denominator in Eq. (4) becomes small, the numerator large, and the office pre- mium correspondingly large. If 1 + k is equal to (or greater than) Nz/Rx, the proposed insurance is impossible. 3. Todetermine the annual office premium for an endowment insurance of $1 for n years with return of all office premiums in case the insured survives the endowment 'period, the loading being the same as in the previous problem. Here the net annual premium is \nA, + nE, + mr'nE, _ M, - M^ + (1 + nv')D^n /,, " - —————^————— - ^ _ ^ W since the proposed insurance is a standard endowment insurance plus an endowment of all office premiums in case of survival. The office premium is , _ [Mx - M^n + (1 + nT')D^n](l + k) _c_ , T - ————————N, - N,+n 1000 ' ; Solving for TT', we have , _ (M, - M^n + D^n)(l +k)+ c(N, + JV,+n)/1000 /y T ~ N^ - N^n - nD^n(l + k) - ; Here, again, 1 + k must be less than (Nx — A^+nVwOrt+n ir order that the proposed insurance be possible. 4. The net annual premium for an n-payment whole-life insurance of $1 with return of all office premiums paid if death occur within th> premium-paying period or in case of survival of this period i indicated by the expression ^+n.(JA)^+?<JE.
n x
To illustrate costs and net reserves on a return-premium policy consider a term insurance of $1000 issued to a person of age twent: with return of all office premiums paid in case of death during th term or in case of survival of the term. The term is 10 years am the office premium charged is $60 per year. The net annua' premium is then given by the expression IOOO(MM - Mm) + 60(fi2o - Rw - lOMso) + GOODsa NM — Nsa §62]
INCREASINGINSURANCE
87
On substituting the numerical values of the commutation symbols involved, we. find the net annual premium is $57.767. With a net annual premium of $57.77, therefore, we should expect to come out at the end of the tenth year with a trifle more than enough to pay the $600 promised in the contract.
Unless an insurance company can place a great many such policies, it would not be justified in advertising them or instructing their agents to bring them before the public.
Exercises 1. Compute the net annual premium for an insurance of $1000 with return of all net premiums paid prior to death at age thirty-five. 2. With the same policy as in Exercise 1, compute the net annual premium and the annual office premium when all office premiums paid prior to death are returned, the loading being 25 per cent of the net premium plus $2 per thousand of ordinary insurance. 3. Construct a table like the one in this section for a 10-year term insurance of S1000 with return of annual office premiums in case of survival only, the loading being the same aa in Exercise 2, and the age thirty-five. 4. Interpret in words the following symbols: