Method of Calculating the Reserve

Inasmuch as the legal reserve looks to future requirements, and is based on the assumption that a certain interest rate will be earned and must provide for mortality equal to that of the American Experience table, these factors must form the basis for calculating reserves on any policy. Likewise since insurance may be purchased by a single or by an annual level premium, reserves will differ according to the method of paying premiums, for in the latter case credit may be taken for premiums still due. Suppose therefore it is desired to calculate the reserves on a whole-life policy for $1,000 issued at age 45, based on the American Experience table and 3 percent, interest and paid for by a single premium. The net single premium for this policy was found in The Net Single Premium I to be $504.58493. The simplest method of showing the operation of the reserve on this policy will be to make the assumption that a company insures a group of 74,173 persons, the number living at age 45 according to the mortality table, and trace the disposition of the entire fund contributed by them, showing how the total fund paid at the start is increased year by year through interest accretions and decreased at the same time by payment of death losses occurring within the group.

According to the table, therefore, 74,173 persons will insure at age 45 and each will pay to the company $504.58493, giving the company a fund of $37,426,578.013 at the beginning of the first year of insurance. This sum is paid at the beginning of the year and, since death claims are assumed to be paid at the end of the year, will earn interest for one year before any claims for death payments will be made upon it. Three percent, of the above sum is $1,122,797.340 and this added to the original sum gives a fund of $38,549,375.343 at the close of the year. Death claims for $828,000 are now due and when paid leave a net surplus of $37,721,375.34. This latter sum represents the funds belonging to policyholders still living from among the original group, or 73,345, and if the insurance were cancelled at this time and the share of each returned to him there would be available $514.30 for each policyholder. In continuing the insurance, however, this $37,721,375.343 again earns interest and the process here described is repeated for the second year. The accompanying table, showing the net reserves on a single premium policy at age 45, traces the operation of the fund for the group until at age 96 they will all have died according to the mortality table, and in the last column shows the reserve standing to the credit of the individual policyholder for each of the fifty-one years of insurance. The table shows the total sum on hand at the beginning of each year of insurance, the amount of interest earned during the year, the total of these two amounts, the death claims paid during the year and the reserve fund remaining at the close of the year for the group as a whole and the pro-rata share belonging to each survivor. Since this policy is paid for by a single premium, this individual reserve constitutes the total sum available per policyholder for the payment of future claims and therefore must equal the net single premium at each age later than forty-five for a whole-life policy at that age. If these figures are correct the terminal reserve at age 94 (i.e. the reserve at the end of the year) will be the net single premium at age 95 and this sum increased at 3 percent, interest for one year will just equal the amount payable at the close of age 95, for the mortality table shows that the last person of the group insured will certainly have died by this time. The fact that the fund payable for the last three deaths equals $3,015.61 (see column 6) or a surplus of $15.61 above the amount of the claims accruing is due to failure to carry results to a sufficient number of decimal places in the computations for earlier years. The true terminal reserve at age 94 is $970.87 instead of $975.92 as shown in the table. This slight inaccuracy in no way affects the principle involved and does not appear in the figures for the individual reserve until age 87, in which instance a discrepancy of one cent is found.

Table 1 A, Table 1 B.

This table shows that the reserve at the close of each year of insurance is adequate for the payment of all future claims against such a policy if the assumptions as to mortality and interest are realized. The company, therefore, which holds this reserve against a single-premium whole-life policy issued at age 45 is solvent. It is not necessary, of course, to insure 74,173 persons under this identical policy to guarantee the adequacy of this reserve. But if a sufficient number of persons are insured under all policies and at all different ages to insure the operation of the law of average, the reserves so determined will be adequate for any policy.

The above policy may be issued as an ordinary life policy, payable by annual level premiums of $29.665318. The table on page 200 shows the operation of the reserve under this policy in a manner similar to the case where the premium was paid in a single sum. The assumption is made, viz, that a group of 74,173 persons is insured at the moment they enter upon their forty-fifth year of age. The main difference between the two tables arises from the methods of paying premiums. Whereas the entire contributions of the policyholders are paid at the beginning in the first case, in the latter instance the first annual installment only is paid and the company therefore does not hold so large a fund. Tracing the method of the second table more in detail, it shows the total premiums paid in at the start, interest earned during the year, the total sum on hand at the end of the year, before deducting death claims, and after the latter have been paid, and finally the individual reserve or proportionate share of each survivor in the total reserve fund at the end of the year. In the second year the fund on hand., namely the total reserve fund at the close of the previous year, is increased by the premiums paid at the beginning of the second year. This fund is then increased by interest accruing during the year and reduced by death payments at the end of the year in the same manner as in the previous instance. This process is repeated for each year of insurance until according to the mortality table all will have died, and in the last year, with three persons to pay premiums, their payments plus the total reserve fund on hand from the previous year increased during the year by interest should at -the close of age 95 just equal $3,000. Again the figures in the table miss the correct figure by a small amount, $19.73, due to the failure to carry the results to a sufficient number of decimal places.

Disregarding the slight inaccuracy as explained, the table shows the adequacy of the net annual premiums to pay death losses according to the American Experience table, providing the surplus from early premiums is preserved until needed in the later years. This surplus is represented by the figures in column 12 of the second table and is called the reserve.

See table A, See table B.

A comparison of the individual reserves for single-premium and for annual-premium policies will show a great difference between them. For instance, at the close of the fifth insurance year in the illustrations used, when the insured will have reached age 50, the single-premium reserve is $555.22 while the annual premium reserve is but $102.20, a difference of $453.02. Likewise after thirty years the two reserves are respectively $824.93 and $646.62, differing by $178.31. A simple test of the accuracy of the annual-premium reserve is possible from these figures. It was found on page 198 that a company is solvent and can pay future claims if it holds the single-premium reserve. The annual-premium reserve there-; fore being much smaller does not in itself assure solvency. But in the latter case the company will receive regular yearly premiums in the future and it is proper to take credit for these premiums in ascertaining its solvency. If, therefore, the annual premium reserve is $453.02 short of the amount necessary at the close of the fifth year of insurance to guarantee solvency, this figure must represent the present value of future premiums to be collected if $102.20 is the correct reserve. By the method of computing the present value of a life annuity due of $1.00 as shown on page 180 such values can be computed for any age such as 50, 55, 60, etc. The present value of a one-dollar annuity due at age 50, so determined, is $15.2710. If the net annual premium on the ordinary life policy at age 45 is $29.665 then the present value of future premiums receivable on this policy after age 50, or of an annuity due of $29.665, is $29.665 X 15.2710 or $453.02. This is the exact amount by which the annual-premium reserve fell short of the single-premium reserve and $102.20 is therefore correct. This justifies the definition of the legal reserve previously given, as that sum of money which with future premiums, if any, will enable the company to pay future claims.

The following table has been arranged to make comparisons similar to the one described above for different years during the term of the ordinary life policy issued at age 45 and shows the difference between the single- and the annual-premium reserves for every fifth year until age 80 as well as the present value at each selected age of the future net annual premiums still to be collected on the policy issued at age 45. Comparison of columns 4 and 7 will show that the present value of future premiums in each instance just equals the difference between the two specified reserves.

Table III.