As the premium payments extend over only five years, the expense contributions must all be paid during that time and are most conveniently made by a uniform addition to the net premium. The present value of the amounts in column 3 is $60.0819, and the equivalent annuity for five years is $12.9769. This amount, received for five consecutive years, will put the company in funds to pay current expenses and leave a reserve of $42.6981 at the beginning of the sixth year, which, as we have seen in the analysis of the single-premium policy, is the sum required for future expenses on the paid up basis. In like manner we find that the 10-year annuity equivalent to the present value of the annual contributions in the case of an annual-payment policy is $5.534, thus: ------------+----------+-----------+--------+---------+ | 1 P. Ct. | 20 P. Ct. | | | | on | on | Total. | Initial | | Reserve. | Cost. | | Fund. | ------------+----------+-----------+--------+---------+ 1st year | $.8234 | $1.3514 |$2.1748 |$ 5.5340 | 2d " | 1.6473 | 1.2478 | 2.8951 | 9.0275 | 3d " | 2.5096 | 1.1388 | 3.6484 | 11.9116 | 4th " | 3.4124 | 1.0210 | 4.4334 | 14.1277 | 5th " | 4.3572 | .8916 | 5.2488 | 15.6161 | 6th " | 5.3479 | .7534 | 6.1013 | 16.3160 | 7th " | 6.3853 | .5966 | 6.9819 | 16.1572 | 8th " | 7.4726 | .4270 | 7.8996 | 15.0763 | 9th " | 8.6127 | .2418 | 8.8545 | 12.9977 | 10th " | 9.8086 | .0344 | 9.8430 | 9.8430 | ------------+----------+-----------+--------+---------+ The present value of the ten yearly expense items given in the "total" column above is $46.6812, which is equal to a ten-year annuity of $5.534. The several premiums stand now as follows: ENDOWMENT: $1,000, AGE 30, PAYABLE AT DEATH OR 40 Net Prem.[2] Margin. Total. At single premium. $687.228 $71.6394 $758.8674 At five premiums. 150.615 12.9769 163.5939 At annual premiums. 84.172 5.5340 89.7060 [Footnote 2: Thirty American offices. Discount from middle of year, Vx-½ or (M x 1.01961) / Dx.] By the actuaries' rate we have, with the customary loading for expense: Single premium: $721.66 (loaded, $34.36). Five premiums, $188.70 (loaded $37.78). Annual premium, $105.65 (loaded $21.11).