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Rate-Making Pure Endowments and Life Annuities

The method of finding the net single premium for a life annuity of $i may, then, be given this general form : Multiply the probability of surviving one year by the dis counted value of $1, due in one year; multiply the prob ability of surviving two years by the discounted value of $1, due in two years; and so on, including the number of years which is equal to the highest age in the mortality table less the present age; then add together these prod ucts. If the net single premium is for a life annuity due, add $1.

If the life annuity is payable for a certain number of years only, as Jo, for instance, then, of course, the process stops at to years and the sum of the first 10 years' pure endowments only is taken; this is the net single premium.

If it is a life annuity due of $1 for a certain number of years only, then take the sum of the pure endowments for one year less, and add $1. Thus a life annuity due of $1 for to years is a life annuity for 9 years, plus an immedi ate payment of $1.

All net single premiums are also known as "present values;" as, for instance, the present value of an insur ance, an endowment or annuity. They are also sometimes called "values," but that is likely to cause confusion, as the same word is used for "reserves." To find, for example, the net single premium or the present value of a life annuity of $1 from age 9o, Actu aries' Table and 4 per cent., assume 1,319 such annuities at that age, involving payments and values as follows: Dividing by 1,319, the number of annuitants at the out outset, age 9o, we have $1.485 as the net single premium or present value of a life annuity of $1 from age 9o.

If it were a life annuity due, the present value would be increased by the $1 immediately payable, and would be $2.485.

The foregoing mode of calculation can also be em ployed to illustrate the computation of a life annuity due limited to five years, thus : Dividing by 1,319, the number of annuitants at the out set, age 9o, we have $1.398 as the present value of life

annuity of $1 for four years at age 9o. Adding $1 to this we have the present value of a life annuity of $1 due for five years at age 90, $2.398.

Annual premiums are paid in advance each year if the insured is surviving and, therefore, are life annuities due. Thus, if a pure endowment of $1,000 were issued at age 90 due in five years, and to be paid for by five annual premiums in advance these premiums would constitute a life annuity due from age 90 for the amount of the pre mium. This annuity due must be equivalent in present value to the net single premium for the pure endowment; that is to say, the annual premium must be for as many dollars as the present value of a life annuity due of $1, at age 90 for five years, is contained in the net single premium for the pure endowment.

We have seen that if 1,319 of these pure endowments were issued at age 9o, for $1 each, $89 would need to be paid to the 89 survivors at the end of five years; the value of which, discounted five years, is $73.1515. If the amount payable to each were $1,000 this would be $73,151.5o,which, divided by 1,319, gives $55.46 as the net single premium at age 90 for a pure endowment of $1,000, payable at the end of five years.

Remembering that the annual premium is a life annuity due, at age 9o, for five years, we have seen that each $1 of it has a present value of $2.398. But the present value of the whole of it must be equal to $55.46, the net single premium for the endowment, because both are equivalent to the same thing, viz. : the value of the endowment itself; so that the net annual premium is equal to $55.46 ÷ 2.398, that is, $23.13.

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