Installment Insurance

In the policies studied thus far it has also been assumed that the face value of the policy (generally $1,000 or multiples of that amount) is payable at maturity in a single sum. But it has become a common practice to make provision for the payment of policies in periodic installments. Thus there are policies paid in monthly installments extending over a period of years, or in ten, fifteen or twenty yearly installments. These contracts differ in cost from those paid in a single cash sum and it is necessary to determine wherein this difference lies. Such installment contracts are of two kinds; one stating that the face value, $1,000, will be paid in a definite number of installments, and the other maturing regularly as a single-payment policy but giving the insured or his beneficiary the option of choosing the installment-payment plan. A policy which promises payment of $100 on the death of the insured and $100 per year thereafter until ten payments have been made is an example of the first; the contract in the second case would mature for $1,000 payable at once, but would allow the beneficiary to receive in lieu thereof a certain sum annually for ten years, this sum not being $100 but rather the amount which can be purchased by $1,000 in hand at maturity.

In the case of the first contract it is evident that the company is going to pay out a total of only $1,0 00, but during the ten years given the company in which to pay this sum, it will be earning interest on the funds in its possession. It must have on hand, therefore, at the time of maturity, only such funds as, with interest added, will yield $100 at each of the ten annual periods. The payments are made as follows: $100 immediately, $100 at the end of one year, $100 at the end of two years, etc., the tenth payment being made at the end of nine years. The first $100 will be paid at once upon the maturity of the contract and therefore earns no interest. A part of the funds will draw interest for one year, another part for two years, etc., the last portion drawing interest for nine years. Consequently the funds which must be available at the maturity of the contract will equal $100 plus such amounts as with interest for one year, two years, three years, etc., will respectively equal sums of $100. These amounts are the discounted values of $100 for one, two, three years, etc. The present value of these ten payments is found as follows:

If the company therefore has $878.61 on hand at the time the policy matures and continues to earn 3 percent, interest on all funds in its possession it will be able to pay the ten installments of $100 each as they come due. To determine the net single premium for a policy so paid, it is necessary to regard the policy as having a face value of $878.61, instead of $1,000. Thus, a term policy, a whole-life policy, a pure-endowment or an endowment insurance might be paid in ten installments, and the only change from the computations already made would consist in the substitution of $878.61 for $1,000 as the amount of insurance.

Where the policy matures for $1,000 but gives the further option of receiving payment in installments, it is clear that the premium must provide for $1,000 payable in a single cash sum at maturity since the insured or beneficiary may choose this option. There will be no difference therefore in the computation of the net single premium for this policy from the usual $1,000 policy. But since $878.61 only is necessary at maturity to provide ten installments of $100 each, $1,000 in hand at maturity will enable the company to pay ten installments, each greater than $100. A single proportion will show how the amount of these payments may be determined. Since $878.61 will provide installments of $100 each, $1,000 will provide installments greater than $100 in the same proportion that $1,000 is greater than $878.61. Thus, letting x equal the amount of the installment to be found, we have :

A policy maturing for $1,000 and giving the option of receiving it in ten annual installments could therefore pay $113.81 in each installment. By the principles here laid down the cost can likewise be determined for a contract paid in any number of installments, such as five, fifteen or twenty.

The contracts explained thus far have invariably involved but one life. Life-insurance companies, however, will issue policies covering risks on two or more lives, or joint-life policies as they are called. Especially in the field of partnership or corporation insurance has the joint-life policy been used in recent years. But the computation of costs on joint-life risks will carry us more deeply into actuarial science than it is desired here to enter, since the purpose of our premium analyses is merely to give an adequate idea of the risk involved in the most usual types of policies. Premium computations therefore will not be made for ordinary joint-life, last-survivor and contingent or survivorship insurances.