RATE-MAKING PURE ENDOWMENTS AND LIFE ANNUITIES. It would be possible now to pass directly to the single premium rates for other forms of insurance; but, since, as will be seen, the method of computing annual premiums cannot be explained without first explaining how to com pute pure endowments and annuities, these fall first to be considered. After this, single and annual premiums can be taken up together.
A pure endowment is a promise to pay a sum of money at the end of a given period, to a person, if then surviving, the purchase money to be forfeited in case he dies during the period. The net single premium for this that is, the premium that would need to be collected in one sum in advance, without adding anything for expenses and con tingencies is found as follows : Taking age 10 and the Actuaries' Table again and as suming that funds will earn 4 per cent. interest until dis bursed, let us consider the case of 100,000 pure endow ments of $1,000 each, payable in 10 years to each survivor of that period. According to the table there will be 93,268 surviving at age 20 out of the 100,000 setting out from age 10Therefore, $93,268,000 would be required to pay each survivor $1,000; and, if it were not for interest, each of the original 100,000 would need to deposit $932.68 in order to make up a fund large enough to pay $1,000 to each survivor.
But $i accumulates at 4 per cent., annually com pounded, to $1.4802 in 10 years. Therefore, if the 100,000 persons altogether deposit in advance, the sum of $93,268,000 $1.4802 = $63,010,000 nearly, they will ac cumulate enough to amount, with interest, to $93,268,000 at the end of 10 years and thus to pay all the endowments. So that, if each deposits $630.10, in advance, instead of $932.68, the accumulation will be sufficient. Therefore, $630.10 is the net single premium at age 10 for a pure endowment of $1,000, due in 10 years.
The process we have followed is self-explanatory. It is equivalent, however, to this : Multiply the probability of surviving (93,268 .2.- mo,000) by the discounted value
of $1,000 ($1,0oo ÷. 1.4802). And, generally, for the net single premium for a pure endowment of $1, this would be : Multiply the probability of surviving the en dowment period by the value of $1 certain, due at the end of the endowment period, discounted to the beginning of the same.
The probability of surviving is always a fraction, of which the "number living" in the mortality table at the original age of the endowment holders is the denominator and the "number living" at the age attained at the end of the endowment period is the numerator.
If the promise is to pay $1 at the end of each year, in case a certain person survives, this evidently is equivalent to a series of pure endowments; that is, a pure endowment of $1 due at the end of one year, plus a pure endowment of $1 due at the end of two years, etc. If it is to continue to be paid at the end of each year until the person dies, the last possible payment is at age 95 according to the Ameri can Experience Table, and at age 99 according to the Actuaries' Table. That series of pure endowments is called a life annuity. If $1 is also to be paid at the outset, it is called a life annuity due, or sometimes, less accu rately, an immediate life annuity. Technically, the latter term means an annuity with the first payment at the end of the first of the intervals at the end of which the payments are to be made, as the first year, first quarter, etc.
The net single premium for a life annuity of $1 at age 10 then, is the sum of the single premiums for a pure en dowment of $i in one year, a pure endowment of $i in two years, etc., up to'and including a pure endowment of $i in 85 (1. e., 95-10) years, if by the American Experience Table, and 89 (i. e., 99-Jo) years, if by the Actuaries' Table. And the single premium for a life annuity due is just $i more than this; since, in addition to these pure endowments, there is $1, also, to be paid at once.