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 (nVx) 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 Ax + n its equivalent Px + n(1 + ax + n) this expression becomes
nVx = (Px+n − Px) (l + ax+n)
(2);