Deferred Annuities

Immediate life annuities are purchased by persons of advanced age, and contemplate the payment of benefits at periodic intervals following the date of issue. It is necessary, therefore, that the person considering investment in such a contract shall have accumulated the fund with which to make such purchase. This fund is presumably created from savings over the productive period of a man's lifetime. The experience of probate courts leads to the conclusion, however, that most men dying after age 60 leave little or no capital accumulated. Realizing this and knowing how easy it is to forget the future some men are interested in an annuity contract that will furnish an income during old age, as do the contracts just described, but which can be purchased by annual sums laid aside during their productive years; in other words a contract that will enable them to create this fund by annual payments, say, between ages 40 and 70, which fund can then be returned to them as an annuity after age 70. The deferred annuity offers this opportunity. It bears a close resemblance to the old-age pensions now operated by a number of governments and private corporations under which plans money is accumulated year by year in small amounts either from the wages of the pensioners, or is donated by the employer or the state and is paid periodically during the lifetime of the pensioner after he attains a stated age.

The deferred annuity is the only type of single-life annuity sold by insurance companies which can be purchased by an annual premium. In theory, of course, it is possible to pay for such a contract by a single premium paid at the date of purchase of the contract but in practice such is not ordinarily done. It is necessary, however, in this instance, as in the computations previously made, to compute the net single premium before determining the net annual premium.

If, therefore, it is desired to find the net single premium payable at age 40 which will purchase the right to receive a life annuity of $100 beginning at age 70, there are two possible ways of approaching the problem. In the first place it may be asked, what is the amount of money that must have been accumulated by the company by the time the annuity begins ? This is equivalent to asking how much money must be on hand at age 70 to furnish $100 annually during life, the first payment to be made when the annuitant reaches, age 70. The problem at this point is, therefore, identical with that of the immediate life annuity just discussed, with the single exception that here the first $100 payment is made at age 70 while in the former case the first payment was made at age 71. If therefore the insurance company has on hand at the time the annuitant becomes 70 years of age the amount of money necessary to purchase an immediate life annuity the first payment being at age 71 plus an additional $100 for the payment made on arriving at age 70, or., taking the figures from our previous computations, $666.55 + $100.00 or $766.55, this amount may be considered as the net cost at age 70 of a life annuity the first payment of which is made at that age.

It is now necessary to determine how much must be paid to the insurance company by the purchaser who takes such a contract when 40 years of age. The cost of this contract is ordinarily computed on the assumption that the single premium paid at age 40, or the annual premium paid from ages 40 to 70 is a sum laid aside for use after age 70, the purchaser relinquishing any right to his contributions in case he fails to survive to that age. By this means he is able in case of survival to share proportionately in all funds relinquished by other annuitants who failed to live to age 70. Clearly the chance that a man aged 40 will collect any portion of his annuity is the chance that he will survive this period. In other words it may be stated that the period of deferment is a pure-endowment period.

It is now possible to state the problem in actuarial terms. In case of survival from age 40 to age 70 the annuitant must have standing to his credit the then present value of the whole-life annuity payments beginning at age 70. This amount was found to be $766.55. The amount payable at age 40 which will furnish this sum if living at age 70 will be the present value of this sum discounted for thirty years at the assumed interest rate and multiplied by the probability of surviving the thirty-year period of deferment, viz: 38569/78106 x 766.55 X .411987 = $155.94734

The problem of computing the net single premium for the 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 :

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 old-age 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 $325. This return premium feature would, of course, cost an extra premium beyond what was necessary to purchase the deferred annuity by itself.