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term of life. But the other portion of the premium, namely, that required to provide for the sum assured in the event of the proposer surviving the given age, is evidently diminished by the assumed increase of mortality; and thus it happens that although the two contrary operating causes do not quite destroy each other, they suffice in many cases to reduce the balance to an amount so small that it may with safety be neglected.
In a letter which I addressed in January 1868, to the editor of the Journal, and which appeared in the 14th volume, I had occasion to point out that the hypothesis of deterioration involved in the assumption of a given addition to the age, was by no means necessarily adapted to all cases, and that some reform in the very primitive methods usually adopted in dealing with extra risks was needed. The subject, I am glad to see, has recently been taken up, at the point where I left it, by Mr. Macfadyen, who, in a paper read before the Institute early in the present year, also urged the necessity of a more scientific mode of dealing with these cases. Having been the first to broach the subject, it was gratifying to me to find that Mr. Macfadyen's paper was favourably received by the leading members of the Institute, a circumstance which indicates that the question is now ripe for a further advance in the path of improvement. As Mr. Macfadyen's mode of treating the subject differs in some respects from mine, I propose here to explain the views which further consideration has led me to take of the matter.
As I have already explained, the hypothesis of an addition to the age involves the assumption that the extra risk of mortality increases year by year as the life gets older; or, in other words, that the aggregate extra risk is distributed in an increasing progression. Now, it is extremely probable that such may be the case with many forms of deterioration; to which, therefore, the hypothesis in question would be perfectly applicable. But it is equally probable, I think, that in other cases the additional risk
be supposed to distribute itself more uniformly over the entire period of life, or indeed may even form a decreasing progression; that is to say, we may easily suppose a case where the unfavourable feature is such as to constitute a considerable addition to the present risk, but which, if the person should survive a certain number of years, would disappear, or at least become of much less importance. Cases of these two last-mentioned descriptions are evidently not so well met by the endowment assurance plan as the former, and it would seem at first sight that in dealing with them we have no option but to insist upon the payment of an extra premium corresponding to the extent of the deterioration.
The following little table, extracted from my letter of January 1868, before referred to, affords a good illustration of the view just explained. It shows the amount of extra premium for different descriptions of assurance required upon the assumption of a constant or uniformly distributed extra rate of mortality of about 2 per-cent per annum, with a profit loading of 30 per-cent.
From these examples it will be seen that, quite contrary to the former case, the extra premium is greater for a short term assurance than for one effected for the whole term of life; while for an endowment assurance it is nearly as great. If results so different are produced by changing an increasing into a uniform distribution of extra risk, it is evident that the substitution of a decreasing distribution would present a contrast still more striking.
In the discussion, however, which followed the reading of Mr. Macfadyen’s paper, I find a plan described by two or three members, which seems to me likely to afford a satisfactory substitute for the endowment assurance plan, and which, indeed, becomes more advantageous to the Society precisely in those cases where the latter plan fails to afford the requisite compensation. The only question is whether the method suggested is likely to meet with the acceptance of the public, a point upon which the testimony of the gentlemen who described the process (and who spoke from experience of its actual working) is entirely favourable. I am further induced to look hopefully upon the practicability of the scheme, by the fact that some very energetic and successful assurance agents have specially urged it upon my attention, as one likely to remove a serious difficulty experienced by them in canvassing for business.
The plan in question consists in making a certain deduction from the amount assured for each year that the actual duration of the life falls short of a given period of time. Thus, supposing 20 years to be the probationary period fixed upon, the sum assured will be paid in full if death should take place after the expiration of that period, but if death should take place during the twentieth year, a small sum is deducted from the amount assured ; if in the nineteenth year, twice that sum is deducted; if in the eighteenth year, three times, and so on. These deductions are supposed to be sufficient to compensate the Office for the non-payment of the extra premiums required.
It will thus be seen that the scheme really consists in granting an increasing insurance, at the rate of premium required for a uniform assurance equal to the maximum amount; and if the rate of increase be so regulated that the premium required to secure the increasing assurance, calculated for the increased age of the life, shall be equal to the premium for the uniform assurance calculated for the actual age, the Society gets the full benefit of the extra premium, without any actual additional payment by the policyholder.
For the purpose of investigating the practical working of the scheme, I have calculated the following rates of premium required to secure an increasing assurance, commencing at different sums, and progressing by equal annual increments during 25 years, when it reaches a maximum of £100. The table used in the calculation is the Carlisle 4 per-cent.
Now, comparing these premiums with those required for a uniform assurance of £100, I find that the three terms of the first column correspond to the ordinary premiums for the ages 14, 29, and 44, respectively; the three terms of the second column to the ages 13, 28, and 43; the three terms of the third column to the ages 12, 27, and 42; and, finally, the three terms of the fourth column to the ages 11, 26, and 41. Of course it is not intended to imply that the correspondence in any case is exact; but the difference is always within a fraction of a year. From this remark
able coincidence, I deduce the very simple and convenient practical rule :
When the proportion which the first year's assurance bears to the maximum is
As Three to Six the diminution of the amount at ( Six years
Three to Seven risk, during the first twenty-five) Seven years
to the age.
We have thus the means of determining very readily the rate of deduction corresponding to any given addition to the age. Thus, suppose that the medical officer recommends an addition, say, of 10 years to the actual age,--the first year's assurance will be to the maximum as three to ten—that is to say, £30 for every £100 assured ; and the difference (£70) divided by 25, namely, £2. 168., is the annual rate of deduction for every year which the actual duration of the life falls short of 25 years.
The term of 25 years may, perhaps, be considered too long for ages over 44, as it would then exceed the average duration or expectation of life. I have therefore deduced a similar rule for the terms of 20 and 15 years respectively. The general rule thus extended is as follows :-add 5 years to the probationary term and divide the result by 10. The ratio which the quotient bears to the number of years added to the age, is the ratio which the first year's assurance bears to the maximum. Thus, suppose 5 years to be added to the age, and the probationary term to be 15 years. Twenty divided by 10, or 2, and 5 express the required proportions —that is to say, the first year's assurance should be ths of the maximum amount, or £40 for every £100 assured. The difference, £60, divided by 15, gives £4 for the yearly deduction in this case.
In general it will be found convenient to adopt the following scale in fixing the probationary term :
25 years for ages under 40.
15 years for ages between 40 and 60. As, however, the rule fails when the number of years added is small and the probationary term long, the latter should never exceed five times the integer next less than the former. That is to
the first year's assurance should never be more than one-half of the maximum.
To conclude, it will be observed that these equivalents are deduced upon the supposition that the deterioration is truly represented by an addition of a given number of years to the age—that
is to say, upon the supposition of an extra risk comparatively small at first, but increasing with the age of the life assured. It is evident, therefore, that if the aggregate extra risk on the other hand be distributed equally at all ages--and still more, if it be supposed to be a decreasing extra risk~then the equivalents deduced by the preceding rules will be in favour of the office--for it is in the early years that the deduction from the sum at risk is greatest. Now, it will be remembered that precisely the reverse of this is the case with the endowment assurance plan, which we have seen is strictly applicable only to the hypothesis of an increasing extra risk. The new method, which may be designated the probationary plan, has therefore the advantage of being universally applicable, and may safely be used in cases where the endowment assurance plan would not adequately meet the extra risk.
The law of the distribution of risk (i.e. of the force of mortality) is an important element in problems depending upon the average duration of life. The probability of living over any given term is, however, in no way affected by it,--the latter being a function only of the aggregate sum of the force of mortality prevailing during the given term. Thus, if K, denote the aggregate force of mortality from age 0 to x,—that is to say, if Ko=Mm.xdx,—we shall have xpo=e-Kx, where e is the base of the Neperian system of
1 dl logarithms; for = Mo .. - log l»=SMzdx, and founda=
To dx = lo
– log lx+log =log .. Ko= -log xpo, the logarithms being Neperian.
POSTSCRIPT.-In Mr. Macfadyen’s paper, above referred to, it is stated that in my letter of January 1868, I have taken it for granted that a person having been subjected to Extra Risk (from residence in a foreign climate) will be in exactly the same state of health, after return to Europe, as if the extra risk had not been incurred. With reference to this remark, I have to observe, that in the letter in question I have not advanced, either by implication or otherwise, any opinion whatever on the subject. The object of my investigation was to expose the error of the present system, not on abstract grounds, but upon the bases of its own assumptions. The question raised by Mr. Macfadyen is no doubt an important one, but as it admits of a very obvious and simple mode of treatment, upon the principles I have advocated, and as it in no way affected the truth of my conclusions, I should certainly not have thought it necessary to complicate my subject by adverting to it.