continuation of the calculation would show that the difference between simple and compound interest grows greater with time, so that, for example, at the fortieth year the difference between the two interests for $100 is $220.10; that is, $110 at compound interest would amount to $480.10, while at simple interest it would amount to $260. But the above illustration does not represent the actual facts in regard to the finances of an insurance company, because each year it receives from the policyholder another premium which is added to the previously received premiums and their interest accumulation. This, then, constitutes a new principal upon which interest is earned. If $100 is paid each year and invested at 4 per cent on the plan that each year's payment with its interest accumulation becomes a new principal, with the new annual payment of $100 added to it, then the true results of compound interest as it operates in insurance calculations are seen. The $100 payments made in this manner amount at the close of the fourth year to $446.13, whereas the interest on an original and single payment of $100 with its interest accumulations compounded amounts only to $116.98. If I be invested for a year at a rate i, the amount at the end of the year is (1 + i) and if any other sum, P, is invested at the same rate, the amount at the end of the year would be P(1 + i); likewise the amount in n years will be P(1 + i)". If this accumulated amount is expressed by the letter S, the equation for the calculation of the sum for any year is : S = P(1 + i)" and from this formula any one of four unknown quantities may be calculated, the remaining three being given. For example, suppose $382.88 will be paid in five years for $300 paid now. At what rate of interest can this be done? The rate of interest which any sum doubles itself may be calculated. In this case P = 1 and consequently S=2. Therefore log 2-log I n = log (1 + i) By using the natural logarithm to the base I and neglecting the small value of i in the formula, the general rule from this formula for finding the time in which money doubles itself is divide 69 by the rate per cent. A company must not only know that it can accumulate funds at a certain rate, but also what sums it must collect upon which interest is to be accumulated. That is to say, knowing from the mortality table that a certain number of deaths will occur each year and knowing, therefore, that the total payments to be made are the sums named in the policies of those dying, it must know what sums invested at a certain rate of interest will amount to the total face value of the claims made. The application of this principle will be explained when the subject of premiums and premium calculation is considered. We thus see that the science of life insurance is based upon calculations involving mortality statistics and compound interest earnings. It is a combination of the theory of probabilities with certain principles of finance. REFERENCES Pearson, Karl. Chances of Death, Vol. I, Chap. I. Grammar of Science, Chap. IV. Jevons, Stanley W. Principles of Science, Chap. X. Bowley, A. L. Elements of Statistics, Part II, pp. 261-355. VII. Yale Readings in Insurance, Vol. I, Chaps. I, IV, Insurance Guide and Handbook, Fifth Edition, Chap. IX. Insurance Science and Economics. Frederick G. Hoffman, CHAPTER III MORTALITY TABLES If the reader seeks to understand the principles upon which life insurance is based as well as the practical application of these principles in the actual conduct of the business, the significance of the mortality table must be understood. The mortality table is the expression of the theory of probabilities as applied to life insurance. Mortality Table Defined. A mortality table is a table which shows the number of persons remaining alive at each age out of a given number and also the number dying during each year of age. It is "the instrument by means of which are measured the probabilities of living and dying." The table does not show the actual individual experience of the group at each age, but an average with the deviations reduced. The number upon which the experience is based, usually 100,000, is called the radix of the table. A table may commence at any age, but usually begins at 10 with. the upper limit at 100 years. The sources of the data from which such tables are constructed are usually either general population statistics or statistics of insured lives. We therefore have the two chief classes of tables, the general or population tables and the select tables of mortality. Manifestly the early tables were of the first kind. Mortality Table as a Life Table. The table may be considered either a Mortality or a Life Table, although such tables are usually called Mortality Tables. That is, if a large number of children could be placed under observation throughout life, and the number dying each year could be tabulated, there would be remaining each year the number living; or on the other hand there could be tabulated the number living out of the group at the beginning of each year, thus accounting for the number dying. Such a method of constructing a Life or Mortality Table is manifestly impossible and therefore other methods to be described later must be used. Life Insurance is based upon the principle that the number of deaths occurring each year among a large group follows a definite law of average and is not dependent upon chance. This law forms the basis of calculating the sums which are to be paid to the policyholders or their beneficiaries. Development of Mortality Tables. The duration of life was long a subject for speculation, and the ability to predict with accuracy its duration with respect to large groups of individuals represents one of the greatest accomplishments of the human intellect. There were many stages in the collection of facts about life, but even after a large number of such facts had been collected, it was a long time before these facts could be interpreted and applied in the form of a mortality table. There were many early estimates and speculations of the duration of life and the causes of that duration. It was an early custom to record the ages at death, and many early efforts were made to discover a law limiting |