LIFE INSURANCE 431 it. Recurring to the actuaries' table, it will be noticed that the natural premium to insure $1,000 for one year, at the age of 99, is simply the present value of $1,000 certainly due in one year. It is assumed that the insured, at whatever age he entered, will certainly die in that year, if he should live to enter it. Hence, so far as the calculation is concerned, a whole- life policy, payable at death whenever that event may occur, is identical with an endow- ment policy payable at 100 or on previous death. Hence, in converting the natural pre- miums of a whole-life policy, under this table, into a level annual premium, we are doing the same thing as commuting the natural premiums of an endowment insurance payable at 100 or previous death. But if the table had assumed that human life terminates at 40 instead of 100, the mortality of the ages previous to 39 being just the same, then the natural premium of age 39 would be the same as 99, viz. (at 4 per cent.), $961 54 per $1,000; and an endowment insurance policy, at whatever age entered, payable at 40 or previous death, would be iden- tical, under the assumption, with a whole-life policy. Hence the method of commuting must be the same, whether life is supposed to stop at 99 or at 40. The patience of the reader however will be least taxed by selecting a policy only long enough fairly to illustrate the mode of operation. Let the age of entry be 32, and life terminate at 40. Let the natural pre- miums from the table, with that of the new assumption of no life beyond 40, be placed against the ages in column A. In column B place the present values of $1 payable certain- ly when the premiums are due, discounting at 4 per cent, compound interest. In column place, in decimal form, the fractions expressing the chance of the insured being alive to pay each premium when due. AGE. A. B. C. D. E. 82 8-41x1 xl = 8-41 I'OOOO 83 8-58 x -9615 x -9913= 8-18 -9531 84 8-75 x -9246 x -9824= 7'95 -9083 35 8-93 x -8890 x -9735= 7'78 -8654 86 9-12 x -8548 x -9644= 7'52 -8244 87 9-81 x -8219 x -9553= 7'81 -7852 88 9-53 x -7903 x -9460= 7'13 -7477 39 961-54 x -7599 x -9367=684-41 -7118 738-64 6-Ti The first premium, being paid in advance, is a certainty, which is expressed by a unit. The chance of the person being alive to pay the second premium is expressed by the ratio of those living at 33 to those living at 32 (see actuaries' table), or ^='9913. So the chance of the person being alive at 34 to pay the third premium is expressed by &^ = '9 82 4; and so on. Now the present value of any fu- ture payment can only be such part of its present value as discounted at the assumed rate of interest, as those living to that age are of those living at the start. For example, the 495 VOL. x. 28 $961 54 which is payable at the end of seven years would be worth only -7599x961-54= $730 67, if it were payable certainly. But as there are only 9,367 chances out of 10,000 that it will be paid at all, it is really worth only -9367 x 730-67=$684 41. The values of the anterior payments in column D are ascer- tained in the same way, and their sum, $738 64, is the single premium equivalent, if paid in advance, to all the natural ones. To ascertain the equal annual premium equivalent to this single one, we must first find the value of one dollar payable annually during the term, if the person is alive to pay it. This is done simply by substituting unity for the natural premi- ums in column A and placing the products in E ; or, to be more particular, the first dollar is payable certainly in advance, and we set that down undiminished in column E. The pres- ent value of the dollar payable in one year is given in column B as '9615, and the chance of its being paid in column as -9913. Hence its value is -9615x-9913=-9531 in column E. And in the same way the factors in B and produce all the present values in E, the sum of which, $6*7959, is the present value at the start of $1 payable annually, subject to the chance of the person being alive to pay it. By rule of three, as this present value, $6 '795 9, is to the equivalent payment of $1 a year, so is the present value of all the natural premiums, $738 64, to the equivalent level annual pre- mium, $108 69. This may be tedious, but it is plain, and it is absolutely all there is in com- muting the natural premiums of the scale into the level net premiums of practice. No mat- ter what is the length of the policy's term, each possible year of it must be treated sepa- rately, as above, in commuting directly from the original scale of living and dying. The security for the fulfilment of the contract and persistence of the payments, in other words against the deterioration of the average vital- ity, which arises from the commutation of the natural premiums, has already been remarked. A still more important thing is its effect on the risks assumed by the company. A contract to insure a given sum for life, on the payment yearly in advance of the natural premiums, is a contract to carry a series of risks of ever in- creasing magnitude. The equivalent level pre- mium has the effect of throwing a portion of those risks from the start, and growing larger and larger to the end, on the insured party himself ; and in all cases of endowment insu- rance, the more so the shorter the term. It in fact converts what was wholly insurance into two complementary processes of insurance and self-insurance, the former (unless the policy extends beyond the age of 75) a constantly de- creasing and the latter an increasing series. The non-recognition of this important fact in the conduct of ordinary whole-life (that is, longest possible) endowment insurance has constituted that most serious defect in British life insurance which Dr. Farr has labored so
