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.

Annual premiums, therefore, are annuities but they differ in three important respects from the annuities thus far considered. (1) In the first place they are annuities paid by the insured to the company, while regular annuities are paid by the company to the insured. To be sure both annual premiums and annuities are based on the life of the same person, viz, the insured, but this does not affect the principle involved. (2) Annuities, moreover, were found to be purchased, ordinarily, by a single premium, i.e. a single cash sum. If annual premiums are analogous to annuities, how, then, are annual premiums purchased? Or, to state the proposition directly, in what way does the company return value received for the annual premiums it collects? Obviously, not by a cash sum to the insured upon the issue of the policy. Bather it pays for them with the policy which promises cash upon the happening of some future event and this future promise of money has a present value which can be expressed in money. This "present value" is comparable to the cash payment for annuities.

(3) A third and fundamental difference between annual premiums and annuities exists with reference to the time when they respectively begin. It will be remembered that the cost of an immediate life annuity is computed on the assumption that the first payment of annual income is received one year from the date of issue of the contract. But it is impossible to issue a life-insurance policy, allowing the premium to be paid on any such basis. The law of contracts requires the payment of a consideration as a necessary preliminary to the creation of the contract and the policy states that it is issued "in consideration of the payment of $_____ and a like amount annually thereafter". Hence the first annual premium is always payable when the policy is issued, and not one year later, as is the case with annuities. The series of annual premiums is, therefore, equal to the usual annuity plus one payment made immediately. The distinction between the two is expressed by calling the annual premium a life annuity due. Life annuities due are not sold as annuity contracts and the purpose of this term is to have a convenient expression to describe an annual premium in terms of an annuity. The problem stated on page 175 may now be restated in the following terms: The net annual level premium will be a life annuity due equivalent to the net single premium.