deferred annuity in question can be approached in a different way. It consists of dealing with each separate annual income payment by itself instead of obtaining the combined value at age 70 of all these payments and then discounting this value in one operation to its value at age 40. By considering each annuity payment separately it is possible to find the amount of money to be paid as a single premium at age 40 which will furnish a payment at age 70 if living, another at age 71 if living and so on until according to the mortality table the annuitant will have surely died.

Thus if $100 is to be paid at age 70, if surviving, its cost or present value at age 40 will be equal to the present value of $100 discounted for thirty years and multiplied by the probability of surviving to age 70. In like manner the present value at age 40 of the second annuity of $100 will equal $100 discounted for thirty-one years and multiplied by the probability of surviving from age 40 to age 71. This process will be continued to the end of the mortality table and the net single premium for the deferred annuity will be equal to the total sum of these present values. The computations are shown herewith:

× 100 ×.325226= 7.895181

Total $155.935608=Net Single Premium.

The total obtained equals the net single premium for the annuity purchased at age 40 with benefits deferred until age 70. Comparison of this result with that found by the first method used will show that they are identical. For analytical purposes the former method has an advantage over the latter in bringing out in a more striking manner the pure

endowment nature of the period of deferment from age 45 to age 70 wherein the insured loses all in case of death before age 70.

Of course, a deferred annuity can be computed on a different basis to eliminate the speculative element whereby all accumulations are lost through death before age 70. The oldage pensions issued by governments and private corporations sometimes include a proviso that in case of death or withdrawal before the first annuity is paid, the insured may receive a return of all his individual contributions with interest compounded at a nominal rate. Likewise the old line companies arrive at a somewhat similar result by attaching a provision that in case of prior death the insured shall have returned to him all the premiums paid in, without interest. Thus, if a particular annuity such as is here considered were costing $15 a year between ages 40 and 70 and the insured died after having paid fifteen premiums his estate would receive fifteen times $15 or $225. This return premium feature would, of course, cost an extra premium beyond what was necessary to purchase the deferred annuity by itself.

BIBLIOGRAPHY

The bibliography on Premium Computation is deferred to the end of the chapter on The Net Level Premium inasmuch as the bibliography quoted does not analyze separately the net single from the net level premium.

CHAPTER XV

THE NET LEVEL PREMIUM

By

BRUCE D. MUDGETT

The Level, or Periodic, Premium System.— Insurance policies may be purchased by a single cash sum or by periodic payments made weekly, monthly, quarterly, semi-annually, or annually. The method of computing the net single premium has been described in Chapters XIII and XIV. Therein it was explained that policies are ordinarily purchased by annual or periodic premiums but that the determination of the latter is possible only after the single premium has been ascertained. It requires but a brief comparison to show why most insured persons choose the annual- rather than the single-premium method of paying for insurance. The net single premium on a $1,000 whole-life policy issued at age 35 (American Experience 3 per cent. basis) is $419.88 while the net annual level premium is only $21.08. Two reasons favor the choice of the latter method of payment. In the first place most persons insure to protect an income the continuation of which during their lifetime enables them to assume certain definite family or business responsibilities, the cessation of which income by death would leave these obligations unfulfilled. It is a man's earning power which enables him safely to marry or to engage in business, for the majority. of people do not obtain their capital by inheritance. It is from current income, therefore, that insurance premiums must ordinarily be paid. If the protection of a $4,000 income requires $10,000 of insurance, this amount on the single-premium plan for whole-life insurance at age 35 would cost $4,198.80 while on the annual-premium plan it would

mean an outlay of $210.80 per year. The former sum is clearly impossible of payment from a single year's income, while the latter would occasion no special hardship.

A second reason for the choice of annual- rather than singlepremium payments for life insurance lies in the reduced cost of a policy purchased by the former in case of early death. If the insured in the above illustration should die within one year after the issue of his policy this insurance would cost him $4,198.80 under the one plan and but $210.80 under the other. This difference cannot be lightly overlooked. It will require the payment of twenty annual premiums before the amount paid in will equal the single premium and therefore the annual plan of premium payments is the cheaper to the policyholder whenever death occurs before the twentieth year of insurance is begun. There is a corresponding disadvantage in the annual-premium plan if the insured lives beyond the payment of his twentieth premium for he will then pay more than would have been the case with the single premium. In other words among the policyholders of an insurance company for everyone who pays in less than the amount of the single premium there must be someone who pays correspondingly more than that amount.

Analogy Between Periodic Premiums and Annuities.— If a policyholder is given the choice of paying for his insurance by a single or an annual premium the amount of the latter must be determined on such a basis that in a large group of policyholders the company will receive the same amount of money under the one plan as under the other. Since, .therefore, the manner of computing the net single premium is known, the problem in hand at this point will be solved by finding a net annual premium mathematically equivalent to the net single premium. In order to do this it is necessary to inquire into the circumstances affecting the payment of annual premiums. They are paid regularly during the life of some person, generally the insured, or for a limited number of years, but always cease upon his or her death. This is the definition of an annuity, as stated in the previous chapter.