CHAPTER V
CONTINGENT FUNCTIONS
LIFE INSURANCE POLICIES. NET PREMIUMS
33. Whole-life Insurance. Net Single Premium.—An insurance on the life of (x) covering the complement of life at age x is called a whole-life insurance; and the present value of $1 payable at the end of the year in which (x) dies is called the net single premium for an insurance of $1 on the life of (x).
If a large number of persons, each of age x, were each to pay this net single premium, the fund so obtained at compound interest, compounded annually, would just suffice to pay the estate of each person of the group $1, at the end of the year in which that person died, until all were dead.
The symbol Az is used to denote the net single premium for an insurance of $1 on the life of (x).
The probability that (x) will die during the nth year, that is, that he will live n — 1 years and die the next year, is n-ipz — "pz (Sec. 22, Eq. (5)). Hence the mathematical expectation of having to pay the insurance of $1 at the end of the nth year is (Sec. 27),
—
"pz).
and the total expectation, covering the complement of life at age x, is,
Az = ~jvn(n-lpz — "pz) = Zze"„-1ps — Ev""pz (1) = UP"-1 n-1pz — Z v" "pz = Paz — az, (Sec. 31, Eqs. (3)).
Replacing az and az by their values in commutation symbols (Sec. 31, Eq. (6)), we have,
Az = (vNz — Nz,}.1) (2)
vz
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46 MATHEMATICS OF LIFE INSURANCE [§33
The numerator of the fraction in the right-hand member can be computed for each age x and for various rates of interest and tabuonce for all. This is the commutation symbol Mx; that is,
ltlx = vNx — N„+1. (3)
We have, then, finally
Ax = x. (4)
We may proceed as follows, without the use of mathematical expectation. The number of persons dying during the nth year from the present is denoted by and there will be necessary now a fund denoted by v.. dx+n_1 to pay the death claims at the end of the nth year. Hence each of the lz persons alive now must contribute vnd~+n_1/lx to make up the required sum. Since this must be done for each year throughout the complement of life at age x, the total sum required now is
A. = zvndlzn 1.
Multiplying both numerator and denominator in the right-hand member by vx, we see that the denominator becomes D. and the
numerator, Zvx+ndx+n_i. If we agree to put
Cx = vx+'dx (6)
then vs+ndx+„_1 = Cs+~_1 and Eq. (5) becomes
A. _ 'Cx+n-1 = (Cx + Cx+l + Cx+2 + . + Cw), (7)
Dr Dx
The symbol Cx is thus a new commutation symbol whose value for each age x is easily computed. From Eq. (4) above we see
that Mx = ZCx+n-1 = Cx + C. +1 + Cx+2 + . + (8)
Cr is not often tabulated, but if it were, the values of the Mx's could be found by cumulative addition of the Cx's from the last age in the table to the first age, just as the Nx's were found from the D.'s.