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RATE-MAKING SOME MISCONCEPTIONS AND MYTHS. Premiums and Values Not Based on Expectancy: Every now and then some one conceives the idea, either that the net single premium for a whole life insurance is equal to the sum insured, discounted for the average num ber of years the insured may expect to live, which is called the expectancy or expectation of life, or that the present value of a life annuity is equal to the present value of an annuity certain for the expectancy. These ideas once had earnest advocates; the plan of valuing annuities, introduced by Ulpian and adopted by the Roman courts, was based upon this theory.

The expectation of life is the aggregate number of years that, according to the mortality table, all the persons in a large group, setting out from a given age, survive, divided by the number in the original group. Thus, those who die the first year live a half-year on the average; those who die the second year live a year and a half, and so on to the end of the table. Add all these years together and divide by the original number and you have the average number of future years of lifetime.

Or, since the number living at the end of one year have each survived a year or that number of years altogether, the number living at the end of the second year have each survived another year or that number of years more alto gether, and so on, we may obtain the number of complete years survived in the aggregate by all, by adding together the number who survive one year, the number who sur vive two years, etc.; and we may get the average number $4 of complete years survived by each, or what is called the curtate expectation, by dividing this total by the original number of lives. Since all survive on the average half the year in which they die, add half a year to the curtate ex pectation and the sum is the complete expectation.

Suppose each of the original group was in receipt of a life annuity of $i, the dollars payable at the end of the first year would be equal to the number living at the end of that year; the dollars payable at the end of the second year would be equal to the number then living, and so on.

The total number of dollars paid out in annuities would be equal to the number living at the end of one year, added to the number living at the end of two years, and so on. The average number of dollars paid each person in an nuities would be this total, divided by the original number of lives. In other words, the average number of dollars paid would be precisely equal to the number of years in the curtate expectation; and half the money paid to the entire group will have been paid before the end of the expectation and half will be payable after ward. That is, the present value of a life annuity would be precisely equal to that of an annuity certain for the curtate expectation, if in both cases no interest earnings were taken into account.

But when interest is taken account of, the dollars of annuity payable at the end of the years late in life are discounted, to bring them to their present value in a much larger proportion, than the dollars payable earlier in life. Therefore, the half, payable after the end of the expec tancy, will be discounted much more, compared with the same discounted for the term of the expectancy, than the discount on half, payable before the end of the expectancy, is less than the discount for the term of the expectancy. But the value would be exactly equal to that of an annuity certain for the term of the expectancy, only in case the values of these two halves continued to be equal as when no interest is earned. On this account, the present value of an annuity certain for the term of the curtate expecta tion is greater than the present value of the life annuity. The excess, also, is greater, the larger the rate of interest; and is greater, also, the younger the age.

The net single premium for a whole life insurance is larger than the present value of the face of the policy discounted for the term of the complete expectation of life, if the insurance is payable at the moment of death. In no case can the net single premium be accurately found by reference to the expectation.

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