We may now state this general proposition : When the net premiums have been correctly computed and calcula tions are made on the same assumptions as to interest and mortality, as in computing the premiums, the "retrospec tive" and the "prospective" methods will bring out the same reserve, which means that there will have been saved and accumulated out of past premiums just enough, so that this sum, together with net premiums payable in future, accumulated at the expected rate of interest, will enable the future claims to be met as they fall due. In other words, the accumulations at interest from past net premiums which form the "unearned premium" re serve balance the discounted deficiencies of future net pre miums, which form the "re-insurance" reserve.
99 If the net premiums are not adequate, the two methods will not produce the same results. The "retrospective" method will bring out a lower reserve and the "prospec tive" method a higher reserve than if the net premiums were precisely adequate; in such case, the latter will be the reserve, required under the actual circumstances, to enable the claims to be met. Likewise, if the net pre miums employed were redundant (i. e., larger than neces sary) the "retrospective" method would bring out a higher reserve and the "prospective" method a lower reserve than if with precisely adequate net premiums; but, again, the latter will be the correct reserve under the actual circumstances. It is the custom, however, to treat all in excess of the usual net premium as surplus and value on the basis of the usual net premiums; but this would not be proper if no part of the future net premiums could be used otherwise than to pay death claims, unless the pur pose be to over-state the reserve required.
Suppose it were desired to find the reserve at the end of six months, instead of at the end of one year, for the too, 000 insurances at age to, paid for by aggregate net single premiums of $1,273,151. This sum would have accumu lated at 4 per cent. to $1,298,614 by the end of six months. Death claims to the amount of $338,000 would have been incurred; but, as these are not deemed to be payable until the end of the year, only $338,000 1.02 = $331,372 is set aside to meet them. This leaves $1,298,614 $331, 372 = $967,242 as the reserve at the end of six months, by the retrospective method.
By the prospective method we have $338,000 of claims to accrue in the next six months, payable at the end of that time, and $674,000 of claims to accrue during the year following, payable at the end of that year, or eighteen months from this date. The present values of these re spective sums are : $338,000 ÷ 1.02 = $331,372, and 96 $674,000 ÷ X I.02)= ÷ = $635, 370, and the sum of these is $966,742, the aggregate re serve by the prospective method. These reserves thus
approximately computed, are $5oo apart, owing to the assumption that the accumulated and discounted values for six months at 4 per cent. interest, annually com pounded, are found by multiplying and dividing by 1.02 respectively, which is not quite accurate.
An approximation to the correct reserve at the end of six months may be made as follows : Assume that changes in the reserve, by death claims diminishing the fund and interest increasing it, take place uniformly throughout the year. Then, since, we start the year with $1,273,151 and close it with $648,077, which we have found to be the aggregate reserve at the end of the first year, in six months the reserve will have diminished by half of ($1,273,151 $648,077), i. e., by half of $625,074 = $312,537; that is, the reserve will be $1,273,151 less $312,537 = $960,614. This approximation would be ac curate if the claims for the first six months were paid at the end thereof; for, in that case, we would have the accu mulation $1,298,614 less $338,000 = $960,614. As, not withstanding the assumption in computing net premiums, claims are usually paid very soon after the deaths take place, this is usually taken as the reserve at the end of six months; and the initial reserve, that is, the fund on hand at the beginning of the year, modified by the proportionate change, whether it be an increase or decrease, which it undergoes during the year, is considered to be the reserve at a given time during the year. Thus the reserve at the end of six months is called the mean reserve; that is, the mean between the initial reserve and the terminal reserve, between the reserve at the beginning of the year and at its end. It may be computed, in the case we have been considering, in the following manner : ($1,273,151 -I 97 When premiums are paid annually in advance, the last terminal reserve, plus the premium due at the beginning of the year, constitutes the initial reserve. Take the case of the too,000 insurances of $1,000 each for two years from age 10, with annual premiums. At the end of the first six months the reserve is the sum of the premiums paid which alone constitute the initial reserve the first year and the terminal reserve, the sum divided by two, $651,21I + $1,269) ± 2 = $652,480 ± 2 = $326,240. At the end of a year and six months it will be : The terminal reserve plus the premiums then paid that is, taken to gether, the initial reserve plus zero, because nothing is reserved at the end of the second year, since the insurance then expires, the whole divided by two, ($1,269 --I- $646, 808)± 2 = $648,077 ± 2= $324,038.50.