CHAPTER XIV THE NET SINGLE PREMIUM (CONTINUED) By BRUCE D. MUDGETT The premiums computed thus far relate to contracts which embody only two kinds of risks, the risk of death and the risk of survival. These two types are sometimes referred to as insurance and endowments, since insurance as such is generally needed against premature death while endowments have the character of investments accumulated for the future. Every life-insurance contract covers one or both of these features, viz, protection against death or accumulation in case of survival. 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 $1,000, 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: Present value of $1,000 in ten installments $878.6120 If the company therefore has $878.61 on hand at the time the policy matures and continues to earn 3 per cent. 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 pureendowment 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 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.1 1 Annuities.— The remaining class of contracts to be analyzed are known as annuities. Annuities promise to pay the possessor a stated income, usually at intervals of one year during the lifetime of said person. It will be seen, therefore, that they furnish a type of investment whereby the recipient whose sole dependence is upon invested capital, can be assured of an income for life. And since the income is payable only during the life of the one person, the annuitant, a single annuity on one life does not furnish group protection, but each life must necessarily be covered by a separate contract. Annuities covering a single life are of two kinds, immediate and deferred. Immediate annuities, sometimes referred to as the ordinary life form, may be temporary, i.e. limited to a term of years, may continue for the whole of life, or may promise a certain number of payments irrespective of the question whether the recipient be living or not. The latter contracts are sometimes spoken of as guaranteed annuities or annuities with a guaranteed minimum number of payments. The cost of each of these contracts will be considered in turn. An immediate temporary annuity of $100 purchased, say, at age 70 and continuing for a period of ten years, will promise to pay the annuitant one hundred dollars one year from date 1 The computation of costs for joint-life contracts is effected by the application to the mortality table of the law of compound probabilities in determining the probability that joint-lives will fail, that they will survive, etc. The results are equally scientific with those obtained in dealing with single lives, but the development of jointlife formulæ cannot be undertaken within the scope of this book. of purchase if then living, and one hundred dollars at each anniversary of that date if still living until ten payments have been made. The cost of this contract will be the sum of money paid at the time of purchase, namely age 70, which will furnish these annual payments, and the net cost, which it is proposed here to determine, will be the amount necessary to provide merely for the payments of the sums promised to the annuitant without assessing against the contract anything for expenses. The formula used in computing net single premiums on insurances can again be used here, namely, net cost will equal the risk or probability insured against multiplied by the sum insured (the amount of the annuity) multiplied by the value of $1.00 discounted for the time the money is held. Since therefore a payment is made to the annuitant, if surviving, at the end of each year, the cost for each year must be determined separately and these sums added to obtain the total cost. The probability insured against is the probability that the annuitant will survive through the first year, through the second year, the third year, etc. It will be seen therefore that the annuity under consideration is equivalent to a series of ten pure endowments, one maturing in one year from date of purchase, one in two years, one in three years, etc., until ten have been paid. The probability that the first annuity payment will be made, if determined from the American Experience table, will equal the probability that a man aged 70 will survive one year, or expressed in the form of a fraction, 38569 36178 The $100 paid in case of survival is paid one year from the date of purchase of the annuity and therefore the net cost of the first payment will be the value of this sum discounted for one year at 3 per cent. and multiplied by the probability of survival. Thus the total operation for the first year is as follows: 36178 X 100 X .970874=$91.07 38569 nuity payment. = net cost of first an In like manner the net cost for the remaining nine payments will be found by multiplying the probability of surviving through two, three, four years, etc., by the amount of the |