Page:EB1911 - Volume 14.djvu/700

lower on the average in the 19th century than in the 18th, but was lower through the middle decades of the 18th century than through those of the 19th. On the whole, it seems clear that the accumulation of wealth in itself has no necessary tendency to diminish the productiveness of capital; that this productiveness, on the general average, has not materially varied in many generations; but that the promise and expectation of productiveness which prompt the demand for its use depend upon the activity of enterprise, growing out of the prevailing spirit of hope; upon the rapidity with which new inventions are made, industries extended, and floating or loanable capital expended in permanent works. These conditions are subject to fluctuations extending through considerable periods, so that for a number of years the rate may be higher, and then for a similar series of years lower than the normal rate, determined by average productiveness, but always tending to return to this normal rate, as the tide-swept surface of the ocean to its normal level.

While the excess of the average yield of capital in America, above that of the older nations, is diminished as the facilities of transfer and exchange increase, there is no reason to conclude that it will disappear for generations to come. It seems, therefore, that the general assumption of 3% for the valuation of British offices, and that of 31/2% which is becoming the accepted standard for the companies of the United States, should command unquestioned confidence.

The business of life insurance being founded on well-ascertained natural laws, and on principles of finance which in their broad aspect are of the simplest description, there exists no necessity for frequent close scrutiny of the affairs of an insurance office, in so far as the maintenance of a mere Assets and reserve. standard of solvency is concerned. We have seen that the premiums charged for insurances are based on certain assumptions in regard to (1) the rate of mortality to be experienced, (2) the rate of interest to be earned by the office on its funds, and (3) the proportion of the premiums to be absorbed in expenses and in providing against unforeseen contingencies. If these assumptions are reasonably safe, an insurance office proceeding upon them may be confidently regarded as solvent so long as there is no conspicuously unfavourable deviation from what has been anticipated and provided for, and so long as the funds are not impaired by imprudent investments or otherwise. The ascertainment and division of profits, however, require that the affairs should be looked into periodically; but the fluctuations to which the surplus funds are liable within limited periods of time are generally regarded as furnishing a sufficient reason why such investigations should not take place too frequently. Accordingly in most offices the division of profits takes place only at stated intervals of years—usually five or seven years—when a complete survey is taken of the whole engagements present and future, and of the funds available to meet these. The mode in which the liability of an office under its current policies is estimated requires explanation.

All statistical observations on the duration of human life point to the conclusion that, after the period of extreme youth is past, the death-rate among any given body of persons increases gradually with advancing age. If, therefore, insurance premiums were annually adjusted according to the chances of death corresponding to the current age of the insured, their amount would be at first smaller, but ultimately larger, than the uniform annual payment required to insure a given sum whenever death may occur. This is illustrated by the following figures, calculated from the HM mortality table at 3% interest. In column 2 is the uniform annual premium at age thirty for a whole-term insurance of £100. In column 3 are shown the premiums which would be required at the successive ages stated in column 1 to insure £100 in the event of death taking place within a year. Column 4 shows the differences between the figures in column 2 and those in column 3.

From this table it appears that if a number of persons effect, at the age of thirty, whole-term insurances on their lives by annual premiums which are to remain of uniform amount during the subsistence of the insurances, each of them pays for the first year £1.130 more than is required for the risk of that year. The second year the premiums are each £1.111 in excess of that year’s risk. The third year the excess Is only £1.093, and so it diminishes from year to year. By the time the individuals who survive have reached the age of fifty-four, their uniform annual premiums are no longer sufficient for the risk of the following year; and this annual deficiency goes on increasing until at the extreme age in the table it amounts to £95.207, the difference between the uniform annual premium (£1.880) and the present value (£97.087) of £100 certain to be paid at the end of a year. Now, since the uniform annual premiums are just sufficient to provide for the ultimate payment of the sums insured, it is obvious that the deficiencies of later years must be made up by the excess of the earlier payments; and, in order that the insurance office may be in a position to meet its engagements, these surplus payments must be kept in hand and accumulated at interest until they are required for the purpose indicated. It is, in effect, the accumulated excess here spoken of which constitutes the measure of the company’s liability under its policies, or the sum which it ought to have in hand to be able to meet its engagements. In the individual case this sum is usually called the “reserve value” of a policy.

Age, 30 + n. (1)	P30. (2)	|1A30+n. (3)	P30−|1A30+n. (4)
30	£1.880	£.750	+£1.130
31	1.880	.769	+⁠1.111
32	1.880	.787	+ 1.093
..	..	..	..
..	..	..	..
..	..	..	..
53	1.880	1.806	+ .074
54	1.880	1.916	− .036
55	1.880	2.042	− .162
..	..	..	..
..	..	..	..
..	..	..	..
95	1.880	61.848	−59.968
96	1.880	79.265	−77.385
97	1.880	97.087	−95.207

In another view the reserve value of a policy is the difference between the present value of the engagement undertaken by the office and the present value of the premiums to be paid in future by the insured. This view may be regarded as the counterpart of the other. For practical purposes it is to be preferred as it is independent of the variations of past experience, and requires only that a rate of mortality and a rate of interest be assumed for the future

According to it, the reserve value (_nV_x) of a policy for the sum of 1, effected at age x, and which has been in force for n years—the (n + 1)th premium being just due and unpaid—may be expressed thus, in symbols with which we have already become familiar.

nVx = Ax+n − Px(1 + ax+n)

(1).

If we substitute for A_x+n its equivalent P_x+n(1 + a_x+n) this expression becomes

nVx = (Px+n − Px) (1 + ax+n)

(2);

whence we see that the sum to be reserved under a policy after any number of years arises from the difference between the premium actually payable and the premium which would be required to assure the life afresh at the increased age attained. By substituting for P_x+n and P_x their equivalents 1/1 + a_x+n − (1 − v) and 1/1 + a_x − (1 − v), we obtain another useful form of the expression,

Vx = 1 − 1 + ax+n/1 + ax

(3)

= ax − ax+n/1 + ax

(4).

The preceding formulae indicate clearly the nature of the calculations by which an insurance office is able to ascertain the amount of funds which ought to be kept in hand to provide for the liabilities to the assured. In cases other than whole-term insurances by uniform annual Net liability. premiums, the formulae are subject to appropriate modifications. When there are bonus additions to the sums insured, the value of these must be added, so that by the foregoing formula (1), for