UNEARNED PREMIUM OR RE-INSURANCE RESERVES. If no interest were counted upon, if mortality were precisely according to the mortality table, and if losses were payable at the instant of death, a company which had insured too,000 lives at age to for $1,000 each, just six months ago, receiving net premiums in the aggregate of $676,00o, ought to have on hand just $338,00o. For, on the basis of uniform deaths throughout the year of age, 338 would already have died out of the 1oo,000 per sons insured, and 338 more would die in the ensuing six months.
This sum, then, would be the aggregate "unearned pre mium" reserve for these insurances, because it is what would remain of the original net premiums, after the in surance already furnished had been paid for. It would also be the aggregate "re-insurance" reserve because another company, counting upon experiencing mortality precisely as per the table employed, could afford to accept this sum as a re-insurance premium and assume the liabil ity under the policies for the ensuing six months.
Because, by looking at the past receipts and disburse ments calculated as per the original assumptions, we may find the value or reserve, we call that method the retro spective method of valuation, i. e., of computing the re serve; and, because by looking at the future requirements, on the same assumptions, we may also find it, we call that the prospective method.
The "retrospective" method, setting out with the net premiums actually received, will produce the reserve actu 93 ally required, only in case the net premiums are precisely sufficient according to the mortality table and rate of in terest assumed. The real test of the sufficiency of a re serve is the "prospective" method, as we shall see here after. Thus, if the net premiums actually collected had been $600,000 there would remain at the end of six months only $600,000 $338,000 = $262,000, which would not be enough to meet the $338,000 of claims that would accrue in the next six months; the actual reserve would then be $262,000, but $338,000 the required re serve.
Suppose that the losses were not really payable until the end of the year, in such case no claims would have been paid and the whole $676,000 would be in hand. We should still call $338,000 the "unearned premium" or "re insurance" reserve, however, and the remaining $338,000 would be offset by a liability of $338,000 for death-claims, already accrued but not paid.
Suppose, now, that instead of an insurance for one year and of assuming that no interest is earned, we consider 100,000 term insurances for two years from age 1o, of $1,000 each, paid for by aggregate net single premiums of $1,273,151, and that the funds earn 4 per cent. per annum until disbursed; and it is desired to ascertain the "unearned premium" or "re-insurance" reserve at the end of the first year. By the "retrospective" method, first, at the end of the first year, the aggregate net single pre miums amount, with one year's interest at 4 per cent., to $1,324,077, out of which $676,000 is immediately payable because of deaths the first year, leaving the sum of $648, 077 as the reserve to provide for paying the claims of the second year i. e., to cover 99,324 insurances of $1,000 each, for one year from age H.
By the "prospective method" we have: $674,000 will be due at the end of one year to pay the death claims of 94 the 99,324 insured for $1,000 each at age 11. Its value at the beginning of the year is $674,000 1.04 = $648, 077, 1. e., the same amount.
Suppose, however, that the 100,000 insurances for two years for $1,000 each from age 10 have been paid for by annual premiums in advance, aggregating $651,211. At the end of the first year the premiums paid at the begin ning of that year, improved at interest, would amount to $677,269, out of which $676,000 is immediately payable for first year's death losses, leaving $1,269. There now remain 99,324 persons living to pay the second year's pre mium, which is for each person $651,211 ÷ 100,000 $6.51211, which makes $646,8o8 for the 99,324 persons. We have found that, on the basis of no premium payable at the beginning of the second year, the reserve should be $648,077 in order to meet the $674,00o death losses pay able at the end of that year; but premiums amounting to $646,808 are now payable, leaving $648,077 $646,808 = $1,269 to be supplied by the reserve. This sum, then, is the aggregate reserve, by both the "retrospective" and the "prospective" methods.