the two contrary operating causes do not quite destr they suffice in many cases to reduce the balance to small that it may with safety be neglected. In a letter which I addressed in January 1868, to th Journal, and which appeared in the 14th volume, I h point out that the hypothesis of deterioration involved i tion of a given addition to the age, was by no mea adapted to all cases, and that some reform in the methods usually adopted in dealing with extra risks The subject, I am glad to see, has recently been tak point where I left it, by Mr. Macfadyen, who, in before the Institute early in the present year, als necessity of a more scientific mode of dealing with Having been the first to broach the subject, it was me to find that Mr. Macfadyen's paper was favourably the leading members of the Institute, a circumstance cates that the question is now ripe for a further ad path of improvement. As Mr. Macfadyen's mode of subject differs in some respects from mine, I propose he the views which further consideration has led me to matter. As I have already explained, the hypothesis of an the age involves the assumption that the extra risk increases year by year as the life gets older; or, in that the aggregate extra risk is distributed in an ind gression. Now, it is extremely probable that such may with many forms of deterioration; to which, therefore thesis in question would be perfectly applicable. But probable, I think, that in other cases the additional supposed to distribute itself more uniformly over the e of life, or indeed may even form a decreasing progressi to say, we may easily suppose a case where the unfavour is such as to constitute a considerable addition to the but which, if the person should survive a certain numb would disappear, or at least become of much less Cases of these two last-mentioned descriptions are evide well met by the endowment assurance plan as the for would seem at first sight that in dealing with them ponding 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 of that period, but if death should take place during year, a small sum is deducted from the amount assur nineteenth year, twice that sum is deducted; if in th year, three times, and so on. These deductions are su sufficient to compensate the Office for the non-payment premiums required. It will thus be seen that the scheme really consists an increasing insurance, at the rate of premium re uniform assurance equal to the maximum amount; an of increase be so regulated that the premium required t increasing assurance, calculated for the increased age shall be equal to the premium for the uniform assuranc for the actual age, the Society gets the full benefit premium, without any actual additional payment by the p For the purpose of investigating the practical worl scheme, I have calculated the following rates of premi to secure an increasing assurance, commencing at diff and progressing by equal annual increments during 25 it reaches a maximum of £100. The table used in the is the Carlisle 4 per-cent. Now, comparing these premiums with those requ uniform assurance of £100, I find that the three terms column correspond to the ordinary premiums for the ag and 44, respectively; the three terms of the second colu ages 13, 28, and 43; the three terms of the third colu ages 12, 27, and 42; and, finally, the three terms of t column to the ages 11, 26, and 41. Of course it is no to imply that the correspondence in any case is exact difference is always within a fraction of a year. From thi 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. 16s., 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. 20 years for ages between 30 and 50. 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 say, 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 repre sented hu addition of e civen number of years to the are—that on supposed to be a decreasing extra risk-then the deduced by the preceding rules will be in favour of th it is in the early years that the deduction from the su greatest. Now, it will be remembered that precisely t this is the case with the endowment assurance pla have seen is strictly applicable only to the hypot increasing extra risk. The new method, which may the probationary plan, has therefore the advantag universally applicable, and may safely be used in case endowment assurance plan would not adequately meet th b The law of the distribution of risk (i.e. of the fo tality) is an important element in problems dependin average duration of life. The probability of living ove term is, however, in no way affected by it,-the lat function only of the aggregate sum of the force of mortalit during the given term. Thus, if Kr denote the aggrega mortality from age 0 to x,—that is to say, if K2=f shall have po=-K, where e is the base of the Neperia 1 dlr logarithms; for Tx dx =μx lx =p» .. −logo=dữ, an ―log +log =—log K =logo, the logari Neperian. POSTSCRIPT.-In Mr. Macfadyen's paper, above refer stated that in my letter of January 1868, I have tal granted that a person having been subjected to Extra I residence in a foreign climate) will be in exactly the sam health, after return to Europe, as if the extra risk had incurred. With reference to this remark, I have to obs in the letter in question I have not advanced, either by in or otherwise, any opinion whatever on the subject. The my investigation was to expose the error of the present sy on abstract grounds, but upon the bases of its own ass The question raised by Mr. Macfadyen is no doubt ani one, but as it admits of a very obvious and simple mode ment, upon the principles I have advocated, and as it i affected the truth of my conclusions, I should certainly thought it necessary to complicate my subject by advertin |