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RATE-MAKING-RETURN PREMIUM A variation from the usual forms of life insurance is a policy which promises the payment of a sum equal to a part or of all the premiums paid, together with the face of the policy, at death. Usually this promise is limited to 10, 15 or 20 years, but policies have been issued with all premiums returnable at death, whenever occurring. This proposal is at first a little bewildering, but all that is neces sary in order to grasp it is to observe that it is merely an increasing life insurance with a level premium.

Let us first consider how we should go to work to find the net single premium for an increasing insurance of \$1 in event of death the first year, \$2 in event of death the second year, etc., and let us again make use of age 90 and make the term 10years, in which case it will also be for life.

\$2,995.04 Dividing by 1,319 we have \$2,995.04 ÷ 1,319 = \$2.27 as the net single premium for an insurance at age 90 of \$i in event of death the first year, \$2 in event of death the second year, etc., to the end of life. The net annual pre mium is found by dividing this by \$2.485, the present value of a life annuity due of \$1 at age 9o; that is, \$2.27 ÷ 2.485 = \$.913. For an increasing insurance of \$io this would be \$9.13 per annum, and in order to return a premium of \$1o, this amount, \$9.13, must be paid as a net extra premium each year.

will be equal to P' plus this extra premium.

In other words, to arrive at the entire premium, divide the premium before the extra is added, by \$1, less the net annual premium for an increasing insurance of \$1 for the desired term.

The foregoing is more algebraic in form than is desired usually for this book; but the subject awakens so much curiosity and even incredulity in the minds of many who are interested in life insurance that, since it is next to impossible to explain the matter otherwise, it is thus pre sented.

When the extra premium is itself loaded for expenses and contingencies, the analysis of the process is yet more complex and will not be attempted here. It proceeds, however, upon very similar lines.

When only half or some other portion of the premium is to be returned, the sole alteration is that the original premium, not involving the return of premiums, is divided by \$1 less the half or other portion of the net annual pre mium for an increasing insurance of \$1 for the desired term, instead of less all of it.

When only the premiums paid in the last 10 years of a 20-year deferred dividend period are to be returned, for instance, the net annual premium payable for 20 years for an insurance of \$i beginning in io years and increasing each year by \$i for 10 years, is found as follows : The net single premium is a net single premium for a pure endow ment in xo years, equal to the net single premium at an age io years higher for an increasing insurance for 10 years; divide this by the present value of a 20-year term annuity due.