An essentially different method is employed by some offices, and is not without the support of actuaries whose judgment is entitled to every respect. It has been called the “hypothetical method.” By it the office premiums are made the basis of valuation. Hypothetical annuity-values, smaller than those which would be employed in the net-premium method, are deduced from the office premiums by means of the relation P′ = 1/(1 + a′) − (1 − v) and the policies are valued according to the formula
nV′x = (P′x+n − P′x) (1 + a′x+n),
where P′x and P′x+n are the office premiums at ages x and x+n respectively, and a′x+n is the hypothetical annuity-value at the latter age. Mr Sprague has shown (Ass. Mag. xi. 90) that the policy-values obtained by this method will be greater or less than, or equal to, those of the net-premium method according as the “loading” is a constant percentage of the net premium or an equal addition to it at all ages, or of an intermediate character, its elements being so adjusted as to balance each other.
When the net-premium method is employed, it is important that the office premiums be not altogether left out of view, otherwise an imperfect idea will be formed as to the results of the valuation. Suppose two offices, in circumstances as nearly as possible similar, estimate their liabilities by the net-premium method upon the same data, but office A charges premiums which contain a margin of 20% above the net premiums, and office B charges premiums with a margin of 30%. Then, in so far as regards their net liabilities (always supposing the sum set aside in each case to be that required by the valuation), the reserves of those offices will be of equal strength, and if nothing further were taken into account they might be supposed to stand in the same financial position. But it is obvious that office B, which has a margin of income 50% greater than that of office A, is so much better able to bear any unusual strain in addition to the ordinary expenditure, and is likely to realize a larger surplus on its transactions. Hence it appears that in order to obtain an adequate view of the financial position of any office it is necessary to consider, not only the basis upon which its reserves are calculated, but also the proportion of “loading” or “margin” contained in its premiums, and set aside for future expenses and profits.
Valuations may be made on different data as to mortality and interest, and the resulting net liability will be greater or less according to the nature of these. Under any given table of mortality a valuation at a low rate of Effects of different data. interest will produce a larger net liability—will require a higher reserve to be made by the office against its future engagements to the insured—than a valuation at a higher rate. The effect of different assumptions in regard to the rates of mortality cannot be expressed in similar terms. A table of mortality showing a high death-rate, and requiring consequently large assurance premiums, does not necessarily produce large reserve values. The contrary, indeed, may be the case, as with the Northampton Table, which requires larger premiums than the more modern tables, but gives on the whole smaller reserve values. The amount of the net liability depends, not on the absolute magnitude of the rates of mortality indicated by the table, but on the ratio in which these increase from age to age.
If the values deduced by the net-premium method from any two tables be compared, it will be seen that
V′x >, =, or < nVx
according as
| 1 − | 1 + a′x+n | >, =, or < 1 − | 1 + ax+n |
| 1 + a′x | 1 + ax |
i.e. as