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On the Rationale of certain Actuarial Estimates
Published online by Cambridge University Press: 18 August 2016
Extract
Some three or four years ago, I submitted to the members of the Institute a few observations on the valuation of property held for life and in reversion, and on the due apportionment of it when questions arise between the life tenant and the owner of the fee. I was induced to bring this subject under the notice of the members from observing the great difference of opinion prevailing in regard to it generally, and especially in the discussions on the subject of church leases and of other property similarly circumstanced—in which discussions no sort of agreement appears to have been come to as to the true principles upon which the value of the interests of the several parties concerned should be estimated. In the paper in question, I endeavoured to point out some of these principles; and my object now is to enlarge somewhat on the arguments therein laid down, to show that the question at issue almost always resolves itself into one whether a property is to be bought or sold, to call attention to the vast difference in value which arises under the two conditions, and to point out the imperative character of the causes, from which this difference originates; finally, to urge the necessity of great care and circumspection in dealing with the questions submitted to us, so that all risk of confounding one of the conditions above referred to with the other may be avoided.
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- Copyright © Institute and Faculty of Actuaries 1860
References
page 310 note * See volume vi., page 61.
page 311 note * See page 18 of this volume.
page 311 note †
The formula in this case being, as is well known, 100 
. Since, however, annuities are now, in accordance with the provisions of an Act of Parliament passed some years ago, almost always paid to the day of death, a still more simple formula is applicable, in that case, for determining their market value. Thus, the sum advanced is represented by 1–p; and since the annuity for this is p+i, we have, for the value of £1 annuity, 
; since 
.
page 312 note * The formula being 100A simply, say at 3¼ per cent. Carlisle mortality.
page 312 note †
That is to say, the value found by the formula 1000. 
Carlisle mortality and interest at 6 percent, or by the formula 1000. [1–d(1 + A)], where A is at 3¼ per cent. and d at 4½ per cent.; interest in such an arrangement being actually realizable immediately, and the risk of extreme longevity thrown on others.
page 312 note ‡
Namely, 1000. 
, where r and A are at 3¼ per cent., and ϕ is an addition for expenses, &c.
page 313 note * Reversionary property consists so commonly of stock in the funds, that a few words as to the future value of it may, perhaps, be said here without irrelevancy. It is very much the practice to assume that Consols will, at any future time, most probably realize 85 per cent; and this assumption is grounded on the fact that that has been the average price for about half a century together (see chapter 2 of Mr. Griffith Davies'work). But I think the assumption is a fair one, considering that 3½ per cent. is about the true rate of interest for a security perfect in every respect, with the exception of the circumstance that the amount recoverable is always uncertain. An investment in Consols has, with this exception, all the characteristics of a perfect security. I t is as safe as anything can well be, interest is paid with entire punctuality, and the capital can be lent or withdrawn, at the will of the lender, almost at a moment's notice. The exception we have spoken of is the only defect, but it is a serious one, and demands at least half per cent. more in the rate of interest. It s i probable that Consols would always be nearer 100 per cent. if it were not for this peculiarity—that is to say, that no more than 3 per cent. per annum would usually be obtained from an investment in them. But i f 85 per cent. be a fair rate for the purposes we are speaking of, there seems no reason why the 3 per cent. stock of an Assurance Company should not be taken in its valuations at that rate, rather than at any of the rates arising from such other principles of valuation as those enumerated by Mr. Farren (see vol. 5, page 318); and we may observe, that, if such a practice be acted upon, it follows that every purchase of such stock at a higher rate than 85 per cent. must involve a loss.
page 314 note * Namely, 1000. [1–(d+p)(1+AB), where d is at 4½ per cent. interest, being receivable immediately, and AB at 3¼ per cent., its cost price; or, 1000 
, where the first quantity within the brackets is at 6 per cent., interest being supposed to be deferred and the risk of extreme longevity retained, and the second represents the actual cost of the contingent assurance as granted by a safe and respectable Company. (See paper “On the Contrivances required to render Contingent Reversionary Interests Marketable Securities,” vol. ii., p. 159.)
page 314 note †
That is to say, 1000. 
; AB and r being taken at3¼ per cent., and Φ being an arbitrary addition for expenses, &c., at the rate usually demanded by the particular Company.
page 314 note ‡
Found from the formula 
, where p is the whole life premium and d and AB are at the market rates of 5 per cent. and 3¼ per cent. respectively. (See paper at page 159, vol. ii., before referred to.)
page 315 note * We may here observe, that the market value of policies of assurance i s a good deal raised, from the circumstance of the Companies making (very properly) an allowance for their surrender. Were it not for the price which they give, it is to be inferred that these securities would not sell at so high a rate as they do. The sum assured, for example, would be regarded as an ordinary reversion for sale merely, whilst the premium would be looked upon as an annuity, the cost of which, in the market, would have to be set off against the value of the reversion; and, on these principles, it will be found that an assurance must be in force very many years before it can acquire any value at all.