Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/189

 LIFE.] x + n respectively, arid a x+ ,, is the hypothetical annuity- value at the latter age. Mr Sprague has shown (Ass. May., 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 &quot;loading&quot; 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 ad justed as to balance each other. When the net-premium method is employed, it is im portant 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 cir cumstances 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 per cent, above the net premiums, and office B charges premiums with a margin of 30 per cent. 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 per cent, 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 &quot;loading&quot; or &quot;margin&quot; contained in its premiums, and set aside for future expenses and profits. ictsof Valuations may be made on different data as to mortality ?rent and interest, and the resulting net liability will be greater l - or less according to the nature of these. Under any given table of mortality a valuation at a low rate of interest will produce a larger net liability will require, that is to say, a higher reserve to be made by the office against its future engagements to the assured 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 accordin as 1- nV : . or &amp;lt; 1 - t.e., as or as l+a x 1+alt . (1), l+d x where the accented symbols throughout refer to one table and the unaccented symbols to the other. We have thus the means of ascertaining whether the policy-values of any table will be greater or less than, or equal to, those of another, either (1) by calculating for each table separately the ratios of the annuity-values at successive ages, and comparing the results, or (2) by cal culating at successive ages the ratios of the annuity-values 177 of one table to those of another, and observing whether these ratios decrease or increase with advancing age, or remain stationary throughout. The above relations will subsist whatever maybe the differences in the data employed, and whether or not the annuity-values by the different tables are calculated at the same rate of interest. When the same rate of interest is employed, any divergence in the ratios of the annuity-values will of necessity be due to differences in the rates of mortality. This interesting subject is investigated by Mr Meikle in a paper on Policy Life-Lines, one of the Actuarial Society s publications, and by Mr Sprague in the Assurance Magazine, vol. xxi. p. 77. The following table gives examples of the reserve values of policies for 100, calculated on the net-premium method by three different mortality tables, at a uniform rate of interest, 3 per cent. Age at Entry. Northampton. Carlisle. Institute of Actuaries 11&quot;. Duration of policy five years. 20 30 40 50 60 4-196 5-490 7-294 9-571 13-668 4-534 5-464 7-053 12-374 13-698 4-360 6-135 8-708 12-100 16-180 Duration of policy ten years. 20 30 40 50 60 8-738 11-572 15-220 19790 28-236 9-422 11-746 15-655 24-904 29-310 9-440 12-897 18-045 24-573 31-857 Duration of policy twenty years. 20 30 40 50 60 19-299 25-031 31-998 42-438 55-637 20-061 25-562 36-660 46-914 53-315 21-119 28-614 38-183 48-601 57-792 Tahlc reserve values. Something may be said here as to the data on which Data em- assurance companies make their valuations. The rates P lo y c&amp;lt;l of interest assumed by different offices may be said to 1} i. , J. . . . J olfices. range between .3 and 4 per cent., being in most cases lower than 4. It is, however, in regard to the tables of mortality that the greatest diversity exists. The North ampton Table has, for valuation purposes, been all but discarded. The Carlisle Table has so far lost its ground, since the introduction of the more recent Experience Tables, as to be now used by only a minority of the offices as the chief basis of their calculations. The different tables based on the experience of the Equitable Society, the Seventeen Offices Experience, and the English Life Tables have still some adherents, and (besides those offices which value by the &quot; hypothetical method &quot;) a few companies employ tables constructed specially for their own use. But there is an evident tendency towards the general adoption of the Institute of Actuaries (twenty offices) Tables, which have been used by a large proportion of the companies in their latest valuations. Of these, the tables chiefly employed are H M and H M(5), the latter being used by some offices in combination with the H M pure premiums, in order to eliminate as far as possible the effects of selec tion. Mr King (Ass. Mag., xix. 381 and xx. 233) and Mr Sprague (Ass. Mag., xxi. 229 and xxii. 391) have shown the construction of tables which would give in a more direct and scientific way the result that is aimed at by using the combined H M and H M(5 &amp;gt; tables. Mr King, to illustrate the results of his method, constructs a &quot;model office,&quot; assuming a uniform annual influx of new business and a rate of discontinuance of policies based on the experience XIII. -- 23