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of competition, claims are paid promptly, one prominent company, for instance, advertising that over ninety-five per cent. of its claims are paid within one day of receipt of proofs of death. The importance of this consideration lies in the fact that the company loses nearly six months' interest on the sum paid. For, if deaths occur on the average at the middle of the year and proof of death requires one week, as is likely to be the case nowadays, the claim is paid on the average at nearly the middle of the year. But by the assumption used in computing the premium the money is supposedly held until the end of the policy year. Computing premium rates at 4 per cent., this would mean a loss of $20 on a $1,000 policy. The assumption that claims are paid at the end of the year, however, is maintained in the face of this fact for two reasons: (1) because of the great amount of labor and expense involved in computing new tables based on the more correct assumption; and (2) because the mortality table, as explained in the preceding chapter, allows sufficient margin to cover this deficiency and make the position of the company perfectly safe.

Another assumption made by the companies in their rate computations is that the death rate is uniform throughout the year. Thus, if out of 100,000 persons of a certain age 600 die within one year, the assumption is that fifty die the first month, fifty the second month, and so on during the year. The fact is that the death rate is constantly decreasing up to about age 10 when it begins gradually to increase, and this increase continues at a constantly accelerating rate to the end of life. This assumption is of financial importance to the company only in case of policies paid for by premiums at intervals more frequent than one year. In the case of annual premiums, since all premiums are paid in advance, the money is on hand at any time during the year to pay insurance costs. In the case of monthly premiums, however, if only one-twelfth of the annual premium is collected in advance, but one-sixth of the total year's mortality should occur during the first month, the company will not have the

funds on hand to pay losses. This situation can occur only during the first ten years of life when the mortality rate is constantly decreasing and it necessitates special treatment in case of insurance of children under age 10. But after age 10 the mortality rate is increasing and the discrepancy between the assumption of uniform deaths and the actual situation is favorable to the company and therefore presents no dangers, for the company will now collect one-twelfth of the premium, but will experience less than one-twelfth of the year's losses during the first month.

BIBLIOGRAPHY

DAWSON, MILES M., Elements of Life Insurance, ed. 3, 19-23, 38-39.

FACKLER, EDWARD B., Notes on Life Insurance, 12-13, 51-52. MOIR, HENRY, Life Assurance Primer, chaps. 4, 5.

CHAPTER XIII

THE NET SINGLE PREMIUM

By

BRUCE D. MUDGETT

Classification of Premiums as Single and Periodic.Life-insurance policies may be purchased by a single premium, an annual premium, or a premium paid weekly, monthly, quarterly, or semi-annually. Of these the annual premium is by far the most important and may continue until the death of the policyholder or the maturity of the policy or may be limited to a definite number of years as in a twenty-payment life policy and a twenty-payment thirty-year endowment. In the twenty-payment life policy, for instance, the premiums continue for twenty years provided death does not intervene before this period has elapsed and after the twenty payments have been made the policy requires no further payments and matures whenever death occurs. Few insurance contracts, with the exception of annuities, are purchased by single premiums, although they may be so purchased and the companies will quote single premium rates for any kind of policy. Nevertheless, in taking up the subject of rate computation in life insurance it is necessary to begin with a thorough study of the single premium, inasmuch as it furnishes the method of approach in determining annual pre

miums.

Classification of Premiums as Net and Gross. The premium charged for a life-insurance contract is supposed to cover all contingencies the company is likely to meet, and these may be conveniently grouped into two classes, viz, mortality and expenses. Mortality has reference to that part of the premium which provides for the occurrence of the event

or risk insured against, while the second element covers the costs incident to the management of a company, such as salaries, rents, commissions, etc., which may be fairly charged against a particular policy. In computing premiums mortality costs are always determined first and to this mortality element is added an amount, determined by a more or less scientific method, called loading, which provides for expenses, and from these calculations is determined the premium charged the policyholder. According, therefore, as to whether the " premium" in question is "loaded" or not, it may be classed as net or gross. The net premium makes provision for mortality losses only, while the gross or "office" premium contains this element plus an addition, or a "loading," for expenses. The gross premium is the only one known to the policyholder, but before it is obtained an actuary must have ascertained the net premium. If, therefore, the gross annual premium is the ultimate object of the study of rate computation this study must begin by first determining the net single premium. From the latter, as will be shown later, the net annual premium can be found. Following this it will be possible to study the various methods of loading in order to ascertain the gross annual premium.

In the preceding chapter it was shown that the computation of premium rates on any kind of policy required information as to (1) the amount of the policy, (2) the age of the insured, (3) the mortality table to be used in measuring the risk incurred, and (4) the rate of interest assumed on funds possessed by the insurance company. In the computations that follow, risks will always be measured according to the American Experience table of mortality; the rate of interest assumed will be 3 per cent. and the face value of the policy will be $1,000 unless otherwise stated. The age of the insured will be stated in each instance.

Term Insurance.- Term insurance is the simplest type of contract issued insuring against premature death. Term policies usually run for five, ten, fifteen, or twenty years, and promise to pay the sum insured if the policyholder should die

within this period, nothing being paid if death does not occur during the designated term. Term policies are therefore a distinct type of temporary insurance. Attempts have been made to popularize a one-year term policy which is renewable from year to year at the option of the insured, thereby granting current cost insurance which is paid for at the beginning of each year, the premium furnishing protection for that year only, and a different rate being chargeable for the following year's insurance. This type of policy offers an excellent opportunity to explain the simple elements of rate-making. Suppose, therefore, that the net single premium is to be ascertained on a renewable one-year term insurance of $1,000 on a life aged 45. Immediate use will now be found for two of the assumptions used in rate-making which were mentioned in the preceding chapter, viz, that premiums are paid in advance and that matured claims are paid at the close of the policy year. Accordingly, it is required to find the amount of money which must be paid in at the beginning of the year by a policyholder in order to enable the company to return $1,000 at the close of the year in case the policy has matured. The question must now be asked: What is the risk insured against? It follows from the definition of term insurance that it is the chance of dying during the year. This will be determined by means of the mortality table. This shows that, of 74,173 persons living at age 45, 828 die during the year. Suppose now that an insurance company should issue 74,173 one-year term policies to persons aged 45. If the mortality experienced among this group coincides with the experience indicated in the mortality table there will be 828 deaths during the year. Since each of these deaths represents a liability of $1,000 to the company, and since the claims are payable at the close of the year, the company must have on hand at that time $828,000 to pay claims. But this entire amount need not have been collected from the policyholders since they were required to pay their premiums at the beginning of the year and the company was able to invest the money at interest for

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