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The sum

Hence the sum assured during the first year is 1000-678-067= 321.933; and the amount for each succeeding year is found by adding 678·067 = 10=67-807 to that for the year preceding. assured during the eleventh year is thus £1000; and it remains at this amount during the rest of life.

The deduction at the outset seems here somewhat heavy; but it rapidly diminishes, and vanishes at the end of ten years. Were the term extended to twenty years the deduction at the outset would be only 365.053, and the assurance would consequently commence at 634.947.

I must defer till another opportunity the development of the schemes here shadowed forth. I will now merely mention, that they find their practical applications in cases in which it is arranged that a party who has been "rated up," instead of paying additional premium, shall be subjected to a temporary abatement of assurance.

I am, Sir,

Your most obedient servant, London, 21 Oct. 1872.

P. GRAY.

ON THE RELATION BETWEEN THE VALUE OF A POLICY

AND THE RATE OF INTEREST, To the Editor of the Journal of the Institute of Actuaries. SIR,—In the paper on “Extra Premium,” by Mr. J. R. Macfadyen, in the current volume of the Journal, that gentleman has given, in a footnote on p. 89, a demonstration intended to show that “in any given case it is practically certain that the value of a policy by a higher rate of interest must always be less than by a lower." Having, sometime ago, myself arrived at a similar result to Mr. Macfadyen's by a rather different process, I venture to send it you, with the hope that it may be of interest to some of your readers. We have, by a well-known formula,

V=1-(1-1)(1-V3+1)......(1-V3+x-1); consequently, it will be sufficient to consider how the value of a policy one year old is affected by increasing or diminishing the rate of interest at which it is calculated.

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do;

that is,

Hence, if az is greater than Ax+1, Q3+2, ......i.e., if the value of an annuity on x's life is greater than the value of an annuity on any life older than X,

da then

a(1+a) > 0 dy

is a positive quantity: in other words, as v increases so do

1 also does V. But as v

,

1+ lower the rate of interest the greater is the value of a policy of one year's standing on x's life.

Similarly, if ax+1 is greater than Qx+2, 2x+3......, Vx+1 increases in value as the rate of interest diminishes; and generally, if the values of annuities on the lives x, x+1, 2+2,..... form a continually decreasing series, the value of a policy of one year's standing taken out at any age from a upwards is greater the less the rate of interest. Whence it follows from (1) that „Vx increases as the rate of interest decreases.

I am, Sir,

Your obedient servant, 18 Lincoln's Inn Fields,

W. SUTTON. 26 August 1872.

ERRATUM. The last line on p. 4 of this volume is misplaced and should be carried over so as to be the last line on p. 8.

JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

On Reversionary Life Interests as Securities for Loans. By T. B. SPRAGUE, M.A., Vice-President of the Institute of Actuaries.

[Read before the Institute, 25 November 1872.] Of all the securities proposed to insurance companies, reversionary life interests are the most troublesome to deal with in the ordinary way of mortgage ; and the objections to so dealing with them have been considered by many actuaries so serious, that they have laid down the rule that the only safe way of making an advance on the security of a reversionary life interest is by way of reversionary charge. The transaction, they think, should be in the nature of a sale rather than a mortgage, a portion of the reversionary life interest being sold in consideration of the present advance.

When a borrower applies for a loan on a reversionary life interest, he generally expects to be called on to effect an insurance of about the same amount as would be required to secure a loan on an immediate life interest; and it is difficult to make him understand why a greatly larger insurance is necessary to protect the lender.

The reason becomes obvious enough when we consider what remedy a lender has in the event of the borrower failing to pay his interest and premiums. In that case, there appear to be three possible courses open to the lender. First, he may, by agreement with the borrower, allow the premiums and

VOL. XVII.

interest to accumulate at compound interest; or, secondly, he may sell the reversionary life interest; or, thirdly, foreclose. Suppose the amount of the insurance that has been effected to be such as would be required in the case of an advance on the security of a life interest in possession, or to exceed the sum lent by about 20 per-cent. Then, if the first of the above courses is adopted, and the interest and premiums are allowed to accumulate at compound interest, it is clear that in a very few years, (probably within three years, under the circumstances supposed) the accumulated amount of the loan will exceed the insurance, and a further insurance will become necessary to protect the lender from loss by the death of the borrower. Should the borrower be still in good health, no difficulty arises. If, however, he has fallen into bad health, and his life is only insurable at a greatly increased premium, the value of his reversionary life interest will be much reduced; and unless the sum originally lent was but a small fraction of the value, the lender may probably find that the security is worth less than he has lent upon it. But it may happen that the borrower's life has become wholly uninsurable. If it were then certain that he would die within a short term of years, the difficulty would be met by increasing to a moderate extent the original amount of the insurance; but, as is well known to all persons familiar with life insurance business, a person whose life is practically uninsurable may nevertheless live for ten or even twenty years; and if the borrower's life in the case supposed should be extended in this way, the accumulations at compound interest would make his debt very greatly exceed the amount of the insurance, so that the lender would receive on the death of the borrower only a small part of the sum due to him. It appears then that, in this case, in order to protect the lender completely, the full insurance that will be ultimately required should be effected (or arranged for) at the outset when the loan is originally granted. Secondly, suppose that the lender attempts to sell the reversionary life interest; then since the purchaser will in no case receive any income until the death of the life tenant, and will receive nothing at all if the reversioner should die before the life tenant, he will of course require to have an insurance on the reversioner's life sufficient to cover the probable amount of the accumulations at compound interest of his purchase money and the premiums which he may pay before he comes into possession of the reversionary life interest. Here the same difficulty meets us; for, under the circumstances supposed, a new insurance will have to be effected when the reversionary interest is put up for sale, and it may happen that the reversioner's life has become uninsurable; in which case the reversionary life interest will be perfectly unsaleable. Lastly, if the lender forecloses, he practically becomes the purchaser himself, and the same remarks apply.

From these considerations we conclude that, in order to make a loan in the ordinary way of mortgage on the security of a reversionary life interest, an insurance on the reversioner's life must be either effected or arranged for at the outset, of sufficient amount to render the reversionary life interest practically saleable, at such a price as to return the lender the amount of his advance with all arrears of interest and premiums and legal costs. Suppose, for example, that the life tenant is 60 and the reversioner 30, then a reversionary annuity of £1000 will be worth £4276, and the policy necessary to protect a purchaser fully will be £14,286 (see the tables appended to my paper “On the Valuation of Reversionary Life Interests” vol. xiv, pp. 432, 3). The annual premium on this policy will be £319. 138.; and the very largest sum that could be lent on the security of the annuity would be £3500. In this case, then, the amount of the insurance exceeds four times that of the advance, even when we take the smallest margin consistent with safety. Taking the interest at 5 per-cent, the annual sum which the borrower is required to pay is £495, or more than 14 per-cent on the advance. If the loan is to be allowed to accumulate at compound interest, of course the amount that could be advanced is greatly reduced. Thus, for example, if in the above case the interest and premiums are to accumulate at 5 per-cent compound interest for 5 years, the sum to be lent must be so fixed that the accumulated amount of the debt at the end of the five years

shall be so much less than the then value of the reversionary annuity that there is no doubt, in the event of that annuity being sold, the lender will receive the full amount of the debt and costs. The value of the reversionary annuity of £1000 and the policy of £14,286, at the end of the five years will be very nearly the same as that of a similar annuity on a life of 30 expectant on the death of 65, which the table referred to above gives as £5227. The accumulated amount of the debt at the end of five years should therefore not be more than about £4700; whence it follows that the original loan should not be more than £2229, so that the insurance is no less than 6.4 times the sum lent.

In all the cases we have considered, it is to be observed that the large policy is not required at the outset of the transaction, but is

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