| Age 94—living, 21 X ·52027 = | contributions | $10 92 | |
| Balance | 8 52 | ||
| $19 44 | |||
| Death-claims | 18.00 | ||
| Balance | $1 44 | ||
| Age 95—living, 3 X ·52027 = | contributions | $1 56 | |
| Balance | 1 44 | ||
| $3 00 | |||
| Death-claims | 3 00 |
One important element, however, the investment of money at interest, has been omitted in the above illustration, and will be introduced now. It has already been stated that premiums are payable at the beginning, while death-claims are due at the end, of the same year. To bring these different amounts and dates to a common basis we must determine the present value of each, at the age at which the insurance begins. Knowing, from the table, the amounts of the contributions and the number of deaths and the time at which they become due, it is to be ascertained what amount of money deposited at the beginning of the period, improved at compound interest, would be equivalent to the total of these sums.
The present value of $1 at the beginning of the year, at 412 per cent. interest, is $0·9569; that is to say, $0·9569 invested at 412 per cent. interest will amount to $1 at the end of the year. The present value of $l for two years is $0·9157, for three years $0·8763; or these amounts, improved at 412 per cent, compound interest, will become $1 by the end of these years.
Applying this principle to the class of 847 persons under consideration, and assuming each contribution and each death-claim to be $1, we need only multiply these by the present value corresponding to each year, bearing in mind that the living pay in advance, while the death-claims are due at the end of the year:
| Age. | Living. | Deaths. | |||
| 90 | 847 X $1= | $847 00 | 385 X ·9569 = | $368 41 | |
| 91 | 462 X ·9569 = | 442 10 | 246 X ·9157 = | 225 26 | |
| 92 | 216 X ·9157 = | 197 79 | 137 X ·8763 = | 120 05 | |
| 93 | 79 X ·8763 = | 69 23 | 58 X ·8386 = | 48 64 | |
| 94 | 21 X ·8386 = | 17 61 | 18 X ·8025 = | 14 44 | |
| 95 | 3 X ·8025 = | 2 41 | 3 X ·7679 = | 2 30 | |
| $1,576 14 | $779 10 | ||||
The above calculation shows the present value of the 1,628 contributions, at $1 each, to be $1,570.14, and the present value of the 847 death-claims to be $779.10; therefore, to meet these latter, the contributions need only be, instead of one dollar, 77910157614 of one dollar, or $0·49432 (49 cents). The item of interest has reduced the premium to $0·49432, when, without it, it would have been $0·52027.
