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example, the value of a policy for 1 with bonus additions B is (1+B)A, +, .-P(r-I-a, +, .). But the general principles of calculation are the same in all cases. The present value of the whole sums undertaken to be paid by the office is ascertained on the one hand, and on the other hand the present value of the premiums to be received in future from the insured. The difference between these (due provision being made for expenses and contingencies, as afterwards explained) represents the “net liability ” of the office. Otherwise the net liability is arrived at by calculating separately the value of each policy by an adaptation of one or other of the above formulae. In either case, an adjustment of the annuity-values is made, in order to adapt these to the actual conditions of a valuation, when the next premiums on the various policies are not actually due, but are to become due at various intervals throughout the succeeding year.

So far in regard to the provision for payment of the sums conthe policies, with their additions. We now come to the provision for future expenses, and for contingencies not tained in

7"""s""' embraced in the ordinary calculations. In what is called °" the “ net-premium ” method of valuation, this provision

P°"”s' is made by throwing off the whole “ loading " in estimating

the value of the premiums to be received. That is to say, the premiums valued, in order to be set off against the value of the sums engaged to be paid by the office, are not the whole premiums Ne” actually receivable, but the net or pure premiums derived I m from the table employed in the va uation. The practical "ZZ "d effect of this is that the amount brought out as the net m ° ' liability of the office is sufficient, together with the net premium portion of its future receipts from policyholders, to meet the sums assured under its policies as they mature, thus leaving free the remaining portion-the margin or loading-of each year's premium income to meet expenses and any extra demands. When the margin thus left proves more than sufficient for those purposes, as under ordinary circumstances it always ought to do, the excess falls year by year into the surplus funds of the office, to be dealt with as profit at the next periodical investigation.

There appears to be a decided preference among insurance companies for the net-premium method as that which on the whole is best suited for valuing the liabilities of an office trans-N°3'”"° acting a profitable business at a moderate rate of expense, values' and making investigations with a view to ascertaining the amount of surplus divisible among its constituents. In certain circumstances it may be advisable to depart from a strict application of the characteristic feature of that method, but it must always be borne in mind that any encroachment made upon the “ margin ” in valuing the premiums is, so far, an anticipation of future profits. Any such encroachment is indeed inadmissible, unless the margin is at least more than sufficient to provide for future expenses, and in any case care must be taken to guard against what are called “ negative values.” These arise when the valuation of the future premiums is greater than the valuation of the sums engaged to be paid by the office, or when in the expression (P, +, ,-P,)(I +a, +, ,) the value of P, is increased so as to be greater than that of PH... It is evident that any valuation which includes “ negative values " must be misleading, as policies are thereby treated as assets instead of liabilities, and such fictitious assets may at any time be cut off by the assured electing to drop their policies. In recognition of the fact that a large proportion of the first year's premiums is in most offices absorbed by the expense of obtaining new business, it has been proposed by some actuaries to treat the first premium in each case as applicable entirely to the risk and expenses of the first year. At a period of valuation the policies are to be dealt with as if effected a year after their actual date, and at the increased age then attained.

Another modification of the net-premium method has been advocated for valuing policies entitled to bonus additions. It con” sists in estimating the value of future bonuses (at an “;':;;;I assumed rate) in addition to that of the sum assured and method existing bonuses, and valuing on the other hand so much of the office premiums as would have been required to provide the sum assured and bonuses at the time of effecting the insurance. This tends to secure, to some extent, the maintenance of a tolerably steady rate of bonus.

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 o ce premiums are made the basis o 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' == if (1 -v), and the policies are valued according to the formula

V n, '; = (P';+n "' Plx)(l +U'g+'»),

where P', and P', +.. are the office premiums at ages x and x~{-n respective lg, and a', +, . is the hypothetical annuity-value at the latter age. Mr prague 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 adwste as to balance each other.

hen 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 va uation), 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 jgjef 'if interest will produce a larger net liability-will datim 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', >, =, or <, .V,

according as

1 a' I a

1 'l' z;(»n>Y:yor < I 'l' Z-tn

I+t1x I-l-as

1 +az+n I +Gl¢+"

1.. . s — = —1

e a 1, az >, ,0r< I, a, z . . (1),

1+a'¢ r+a'., ..

o a — = -L-I

S I+az>' 'Or < I+a.+.. (2>

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 calculating at successive ages the ratios of the annuity-values 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 may be 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.

A prevailing fallacy in the popular mind, which has grown out of the practice of net valuations, is the inference that the average technical reserve represents the value of the Fa, ,, , cy of individual policy. Each risk is properly assumed at single its probable or average value at the time. But from |;g;¢;J'w that moment its circumstances are constantly changing in directions then unforeseen, and the expectation that such changes will occur is the motive for' insuring. To treat' them

singly as unchanged in value at any later time is as illogical as