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at entry for an assurance without extra risk), unless V'x'n=V
xin, the liability of the Office is at once altered, and our calculation of the necessary reserve upset. And if, as I have shown to be probably the case, the reduction to Px is greater than is warrantable, then not only is the amount held by the Society to discharge its contract altered, but changed in the direction contrary to the interests of the Office. It will require a greater sum to discharge its obligations, so that unless in the unlikely case of V'zin being equal to Vein, in any Office wholly taking off extra on the removal of special risk, the policyholder has at any time the option of upsetting the calculations of the Company as to the reserve it ought to keep, and either diminishing, or much more probably increasing, the liability of the Office in a manner not provided for in its periodical valuations. No doubt it is possible, by substituting P, for the future premium in the foregoing equations, to find such a value for V'x'n as will permit of the reduction being given. But here, again, we labour under the disadvantage that if this be done in all contracts having this option, the liability under policies not then
Multiplying out and arranging we have tred,
2(1.x+27 +1x+372 +....)
) (=vletlá x+1) +v22x+20 x
+v4(lx+37 +13+972 + .) =v?lz+2d +2) +031c+30
#28(lz+an+lz+32+ ...)(=volz+za*c+3) +
..:) +0w-213+w-2a +2:0–21276-19
(=ww-212+w-2a'w-2) +w-1 13+w-101
From the foregoing expression then it will be seen that it is practically certain that Vxlı must exceed V'x11: For though in an individual instance the annuity at an older age may be greater than that at a younger, yet this excess can never be so much, even when all possible cases of it are taken into account, as to counterbalance the facts that in the left member of the foregoing inequality we have one more term than in the right member, and that also a'z is constant in the left, while in the right it is replaced by the diminishing series of annuities a'z+1, 6'x+2, &c. Since then we know that Vx11 must be greater than V' therefore
1 + 2x+1 1+ax
1+ ax 1 + Ox+1
Substituting (x+1) for x, we have
1+ar 1 + a2+2 à
V.x/2>V'x122 1+2x+2 and so on with other ages. Therefore it may be asserted that Vxın>V'xın.
> i+d's+1, and à fortiori
1 + 0.4+1
1 + Alex
exercising this privilege will be erroneously estimated; and, in fact, it is only by importing into our calculations the additional element of the chance of the option being exercised that we can have the values of such contracts correctly arrived at. It appears, then, that as in the other form of extra, where the life assured was "rated up,” so in this form, where this option is granted, the ordinary method of valuation leaves it quite unsettled what amount must be reserved to meet the liability incurred: in the former, from the want of mortality tables; in the latter, from the option of reduction to normal entry rate ;-a most unsatisfactory state of things, and a matter of no small moment to Companies issuing such policies in considerable numbers.
I have now examined some of the principal phases presented by the two modes of imposing extra premium, but though it will make somewhat longer a paper already sufficiently lengthy, it will be well, before concluding, to glance at a very interesting method of charging extra in use by some Offices.
In a few Societies, the life assured, if accepted at an annual extra of f, is at liberty to pay the ordinary premium, it being stipulated that in the event of death occurring in the first year, a deduction of ef will be made from the sum assured (e being the average after-lifetime of an ordinary life, aged x), and in consecutive years thereafter (e-1)f, (e-2)f, &c., are respectively deducted, according to the number of years from entry at which the claim arises. Is this mode of treating extra, or the ordinary one of charging f, year after year, likely to be the more profitable to the Office ?
Let, as before, dashes signify mortality at under-average rates. By this method the life assured gives the Society an assurance of ef, diminishing f per annum.
This equals lz i'z+
l'E while by the more usual method of charging extra, the life assured gives the Society an annuity-due of f, that is
ve + fela ao
1 l'ate-1 ye-lt
xte qeļ te axe T's
Divide each by f, and we have the two series-
velt ľz and
ľ, vt Multiplying out the first series, and transposing from the one to the other, we have the expression for the ordinary form of charging extra,
(l'e 1+2 vt
I and the expression for the method we are now considering,
v+02 + ....ve-1 +ve. That is, the ordinary form gives an under-average life annuity due + an annuity on an under-average life for e years, and the other gives an annuity certain for a number of years equal to the expectation of an average life. Which is greater ?
If the probabilities of an under-average life were those of an average one, then, as in e years, since at very few ages at which lete
lath Ixt? assurances take place does fall under ), it follows that lx
le To &c., would form a diminishing scries, of which the value of the smallest term was not less than , and we should thus have
I, ly greater than v + v2 + ....ve, and, à fortiori, the whole of the first series greater than the second; therefore, to a life which has been erroneously classed as under-average, the new method of charging extra is much the more advantageous.
But in the case when the life really involves extra risk, since
is less than Pote, it may possibly happen that the chance of
le the under-average life attaining the expectation age of an average life is less than 1, and thus we cannot say whether the new or the old method will be the more advantageous for the Office, as it altogether depends on the probabilities of invalid life reaching the
Were even the number alive at the age X +
as great as half those at entry age, a supposition probable enough even in assurances at extra risk, the first scries rould be
of which, since the coefficients of the powers of v up to vi in the bracket are greater than the corresponding coefficients in the latter series, it is probable, taking also into account the term edz that the first expression as a whole is greater than the seco That is, even when in a number of years, equal only to half the expectation of an average life, there are no more than ?? survivors of the l’x living at age x, it would be more advantageous to the Office to charge extra during life than by the method we are now considering. When the probabilities of death become very great in the first few years, this way of meeting the extra risk may give more favourable results to the Society than the more usual one. In such a case, the age which the probability of reaching is less than must be very much less than that which represents the average age at death of ordinary assurers. It is very unlikely, however, that in many cases the chances of death in the first few years of assurance will be so great as to cause the value of an annuity certain for a period equal to the expectation of an average life, to exceel twice the value of an annuity of the same number of terms on an invalid life by 1+the value of the deferred annuity; and so, though it is impossible to speak with absolute certainty, it is exceedingly probable that the ordinary system of charging extra during life is more profitable for the Office than the method now being considered, and since in each case the assumption is made that only the value of the extra risk incurred is charged, it is likely that this new system will fail to repay the Society for the increased liability undertaken.
In my treatment of the subject of extra premium, I have altogether neglected the question of loading. I have done so, partly because its introduction would very much complicate the elucidation of these features I have more particularly desired to exhibit; and partly because I believe that the subject of the loading on the risk premium ought to be considered in conjunction with the method adopted in dividing surplus-a field of enquiry much too wide for the scope of the present paper.
I have now examined some of the leading phases in which extra premium presents itself. Whatever may be thought of my analysis, I shall not have written altogether fruitlessly if I succeed in calling the attention of the Institute to the subject. When other branches of the science of life insurance are rapidly advancing, this seems in very much the same position as it was when the actuarial profession was yet in its infancy-anything but a satisfactory state of matters, when we consider the practical importance of the subject, and how intimately connected it may be with the solvency of a Life Office.
The Influence of Occupation upon Health, as shown by the Mortality
experienced. By FRANCIS G. P. NEISON, F.S.S.
[Read before the Institute, 29th April 1872.] THE influence that occupation has upon health is a subject that has hitherto attracted but little attention, for though in the works of one or two writers on vital statistics it has been slightly alluded to, no work has yet appeared which, while founded on direct observation, has been devoted to the treatment of this subject.
Perhaps also it was not thought that the results would be such as to much affect any of those questions in which the duration of life enters into account, and which concern so vitally the welfare of the numerous Life Offices and Benefit Societies of this kingdom; and that, therefore, the investigation would not repay its labour. When, however, the results now submitted are considered, the importance of the question will be fully revealed, and it will be perceived that the influence of occupation is one that essentially concerns all those Societies in which many of like or different trades are associated together for the purpose of securing contingent benefits.
A brief description of the data on which the results presented are founded, may be now given.
The Census of 1861 afforded a ready means of ascertaining the number living of any given occupation in that year; and the Supplement to the 25th Annual Report of the Registrar-General gave the occupations of all males that died in the years 1860, 1861.
There was thus the means of ascertaining the mortality experienced by various occupations in those years; and Dr. Farr, the able Assistant of the Registrar-General, availed himself of the opportunity, and gave, in the Supplement above referred to, some interesting results for one or two occupations.