applied to the assumed population of 100,000 at age 10, in the manner herewith explained, and the result was the American Experience table of mortality.

Kinds of Tables and Important Tables Used in the United States. There is an important classification of tables of three kinds dependent on the data used in their calculation. They are known as select, ultimate, and aggregate tables. These terms have reference to the question whether the data used have been affected by medical selection. It is a wellknown fact that lives which have been newly examined by an insurance company and have passed the medical tests required before becoming policyholders show a much lower rate of mortality than lives not so examined. The number of deaths occurring, for example, among 10,000 policyholders aged 40 who have just passed the medical examination will be fewer than among 10,000 aged 40 who were insured at age 30, and have been policyholders for ten years. So it is important for a company in estimating the probable mortality to know whether it has a large number of newly selected lives. An unusually low mortality is to be expected among the risks of a new company, but such a record in the first few years furnishes no basis for assuming that the low mortality will continue.

Since newly selected lives, therefore, furnish a lower mortality it is generally considered the safer plan for a company to compute premium rates on the basis of the mortality among risks with whom the benefits of fresh medical selection have passed. A select mortality table is based on data of freshly selected lives only; an ultimate table excludes this early data, usually the first five years following entry, and is based on the ultimate mortality of insured lives. Aggregate tables include all the mortality data, the early years following entry as well as the later.

The tables most used in the United States to-day by insurance companies are three. The Actuaries', or Seventeen Offices table, was calculated from the experience of seventeen British life-insurance companies and was introduced into the

United States by Elizur Wright as the standard for the valuation of policies in Massachusetts. This table has at the present time been largely supplanted by the American Experience table, the one found on page 132. The latter table was published in 1868 by Sheppard Homans and was calculated from the mortality experience of the Mutual Life Insurance Company of New York. Most premium rates for American companies are to-day computed with this table as the basis. It is what was described heretofore as an ultimate table.

The National Fraternal Congress table was derived from the experience of two American fraternal orders and was first published in 1898. It has been adopted by the National Fraternal Congress and by a number of states as a standard for the computation of premiums and the valuation of policies among the fraternal societies.

Application of the Theory of Probabilities to the Mortality Table. The statement was made earlier in this chapter that risk in life insurance is measured by the application of the laws of probability to the mortality table. Now that these laws are understood and the mortality table has been explained, a few simple illustrations may be used to show this application. Suppose it is desired to insure a man aged 35 against death within one year, within two years, or within five years. It is necessary to know the probability of death within one, two, or five years from age 35. This probability, according to the laws heretofore explained, will be determined according to the mortality table and will be a fraction of which the denominator equals the number living at age 35 and the numerator will be the number who have died during the one, two, or five years, respectively, following that age. According to the table, 81,822 persons are living at age 35, and 732 die before the end of the year. Hence the probability of death in one year is 132. During the two years following the stated age there are 732 + 737 deaths, or a total of 1,469. The probability of dying within two years is therefore 1469 Likewise the total number of deaths within five

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years is 732 737 + 742 + 749 756 or 3,716, and the probability of dying within five years is thus 3716

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Probabilities of survival can also be expressed by the table. The chance of living one year following age 35 will be a fraction of which the denominator is 81,822 and the numerator will be the number who have lived one year following the specified age. This is the number who are living beginning age 36, or 81,090, and the probability of survival for one year is therefore 81999. These illustrations furnish an opportunity for a proof of the law of certainty. The chance of living one year following age 35 is 81829 and the chance of dying within the same period is 732 The sum of these two fractions equals 1822 or 1, which is certainty, and certainty represents the sum of all separate probabilities, in this case two, the probability of death and the probability of survival. In like manner many more instructive examples of the application of these laws to the mortality table could be made, but they need not be carried further at this point, for the subject will be fully covered in the chapters on "Net Premiums."

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BIBLIOGRAPHY

DAWSON, MILES M., Elements of Life Insurance, ed. 3, 24-37. FACKLER, EDWARD B., Notes on Life Insurance, chaps. 2, 5. GEPHART, W. F., Principles of Insurance, chaps. 2, 3. MOIR, HENRY, Life Assurance Primer, chaps. 3, 6. WICKENS, C. H., "On the methods of ascertaining the rates of mortality amongst the general population of a country, district or town, or amongst different classes of such population, by means of returns of population, births, deaths and migration." Journal of the Institute of Actuaries, xliii, 23–84. (Probably the best complete statement of the subject in the English language.)

CHAPTER XII

FUNDAMENTAL PRINCIPLES UNDERLYING RATE-MAKING

By

BRUCE D. MUDGETT

To compute premium rates in life insurance the following facts must be known: (1) the age of the insured; (2) the kind of policy to be issued and its face value; (3) the mortality table to be used in measuring the incurred risk; and (4) the maximum rate of interest which the company is willing to guarantee on funds in its possession. For example, if a contract is issued promising to pay the holder $1,000 should death occur within the following twelve months, and if the chance of death within one year is measured by the American Experience table of mortality and it is further known that the person to be insured is forty years of age, all the facts are at hand for determining the amount of money to be contributed by him in order to cover the risk. At age 40 the table shows that his chances of dying are 9,794 in 1,000,000, or, expressed as a decimal, .009794. This decimal multiplied by 1,000 represents the amount of money the insured must pay to receive the protection promised, if it is assumed. that the money is put away and no use made of it until needed to pay losses. While the illustration is exceedingly simple and makes no attempt to bring out many of the complicated factors found in a more complete analysis of rate-making, it contains the essential features of any rate computation, viz, the determination of the risk covered and the amount payable in case the risk occurs. But before a fuller analysis can be undertaken it is necessary to explain certain peculiarities of life insurance which differentiate it from insurance of other hazards and which are fundamental to any discussion of rate-making.

Certain arbitrary rules used in rate computations must also be stated. To this twofold task the present chapter is devoted.

Features Peculiar to Life Insurance.- ·Protection and investment. While most kinds of insurance contracts have a single purpose, namely, the assumption of a particular risk, the great majority of life-insurance policies embody a twofold purpose by combining insurance with investment. Every policy which contains an endowment feature, i.e. which creates a fund available upon survival for a stated period, is to that extent an investment, and the increase of this investment fund constantly minimizes the insurance element. For instance, a policy issued ten years ago and having an endowment fund to its credit at the time of the insured's death equal to $500 will pay this $500 and in addition $500 more out of the "insurance fund." In other words, by the growth of the "investment fund" the insurance element of the policy is constantly decreased, While this fact is clearly apparent in the case of an endowment policy, it is not so evident in the so-called "ordinary life" policy. But there is no difference in principle, for the ordinary life policy accumulates a reserve which eventually wipes out the insurance. As is often stated, an ordinary life policy based on the American Experience table of mortality matures as an endowment at age 96. This difference between life insurance and fire insurance, for instance, is fundamental, for the loss in fire insurance is measured by the total risk of burning, whereas in life insurance it is always equal to the total risk involved less the reserve fund.

The hazard of death. Closely associated with this reserve factor in the life-insurance contract is the nature of the hazard or risk insured against. Fire insurance may again be called upon for a contrast. In fire insurance, the risk is loss by fire; and fire may or may not occur. The premium therefore need only provide against the possibility that fire occur within the term of the policy, and there is always the chance that the property may never burn. But not so