attended with such special dangers, that it seems to us that in the insurance of a female at this period, there is really a higher and not a lower risk. On this point Dr W. Robertson remarks:- "There are two circumstances which vitiate all attempts to deduce the true mortality of women in childbed from the materials in the RegistrarGeneral's hands. The number of mothers is unavoidably understated: for it does not include the mothers whose children were stillborn; still births not being registered. But a far more serious source of error is, that the women who die after giving birth to dead children, are in very few cases indeed registered among women dead from puerperal causes; yet puerperal deaths are in connection with still births probably 4 times as numerous as they are in connection with live births. The relative frequency of still births and live births in Holland, 1872, (I select the first good record that comes to my hand), was very nearly 1: 18, or in actual numbers 7309: 131,964. It is not likely that the still births are less frequent in this country than in Holland". In this connection we have to refer to the articles on CHILD-BIRTH, Risk of, and to the articles on ENGLISH LIFE TABLE No. 3 (Dr Farr's). The difficulty of bringing out the facts by statistics is no doubt considerable the probability of error in the ages of females in the census returns being well known; but we have such confidence in the judgment of Dr Farr, that we feel very much inclined to adopt without hesitation the opinion he has arrived at after full consideration of all the possible fallacies. He says: "The rates of female mortality are in singular accordance with those deducible from observation on males; the mortality of females being slightly higher at the ages 10 to 35 than the mortality of males at home in England." The diverse opinions on this subject are represented at considerable length in the above-named articles, and we presume the question will be further discussed under FEMALE MORTALITY, when the work reaches that length.

We need not prolong the notice of this cyclopædia any further. One of our chief aims has been to bring it prominently under the notice of medical readers, as they are very apt to remain in ignorance of such a work, being, as it were, outside of our domain, and the writer outside of our profession. We think, however, that our notice and extracts (which are all from this work, and not from the original sources) will suffice to indicate the vast amount of interesting matter which lies, in many cases at least, quite ready for us. No doubt, for certain purposes, reference to the originals would be necessary; but it is no small boon to have the materials so fully given, and their sources accurately indicated. The solid "blocks" are what we want; the time has not come for safely presenting many of the subjects here treated of in any form. We notice a somewhat pathetic remark in the preface to the first number; the work, the author says, "has cost me many sacrifices-pecuniary, social, personal, health. I shall find content in any small rewards that flow from it". We can readily believe that the sacrifices in such an undertaking must have been great. We hope that one of the small rewards that will flow from it will be that the work may find its place as a cyclopædia for reference in all our medical libraries.

JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

On the Rate of Interest yielded by Foreign Government Loans. By WILLIAM SUTTON, B.A., of the London and Provincial Law Assurance Society.

[Read before the Institute, 26 April 1875.]

FROM the fact that a very large amount of capital has been of recent years invested in loans to foreign governments, the question of sinking funds and other kindred matters has come prominently into notice; and matters which at one time were only looked upon as belonging to the theory of compound interest, are now become of great moment to thousands of persons who have invested, or purpose to invest, their money in the above-mentioned class of security. Various causes appear to have led to the large increase in the popularity of foreign loans among certain classes of investors. No doubt, the rapid growth of unemployed capital has considerably reduced the rate of interest realizable on first-class home securities, and has induced persons anxious to obtain a high rate of interest to invest their money elsewhere. From the nature of the case, however, even a fairly high rate of interest would not generally be sufficient inducement for an investor to lend money to a perfectly irresponsible foreign government, which might at any time, if it thought fit, renounce all its past financial obligations. Whether the general practice of providing for the repayment of

VOL. XIX.

G

loans has arisen from the scruples of foreign governments to accumulate a large national debt like our own, or whether, on the other hand, it has been found necessary to give an undertaking to repay the loan over a term of years, in order to make the loan at all palatable to investors, I cannot say; but the fact none the less remains, that very generally provision appears to be made, in the case of loans by foreign governments, for repayment in a shorter or longer period of time. It may be stated that, for all practical purposes, this is equivalent to borrowing money by means of terminable annuities, and undoubtedly there are many cases where the revenue of certain duties, taxes, or other dues specially hypothecated in favour of the said loan, is sufficient to provide the amount of the annual charge. Judging from appearances, however, it is found in practice that a fairly high rate of interest, with repayment sooner or later of the money invested, is not the most attractive form for investors. The form which these loans usually take is that of a moderately high rate of interest coupled with a considerable bonus on repayment, and it may naturally be concluded that this is the form most suited to the taste of investors. By bringing a loan before the public in this shape, two purposes at least are served-one is, that the actual terms on which the loan is obtained are to some extent concealed, otherwise, it is possible that the extravagant terms offered in some cases would act as a caution to intending investors, and instead of serving as a bait would have the contrary effect; and another reason is that, by offering a bonus on repayment, a slight gambling element is introduced, which is an almost irresistible attraction to great numbers of investors. The prospect of making 5 to 7 or 8 percent interest, and when your bond is drawn for repayment receiving a bonus of, say 15 to 25 or 30 percent, is undoubtedly a pleasant one, and the parties who bring out these foreign loans are well aware of the fact. Many persons, however, have recently learned that this pleasing prospect does not always become a reality. The disclosures already made before the Parliamentary Committee now sitting are of a very startling nature, and this investigation will doubtless check for a time the fondness of investors for the class of security in question. Two very important lessons have, at any rate, already been taught —namely, first, that the respectability of the firm or company acting as agent for the loan is no criterion whatever of the soundness of the loan itself; and, secondly, that an official quotation on the Stock Exchange must be taken quantum valeat, and no more. It is not intended, however, in the present paper, to enter further

into this question, but rather to take matters as they are, and discuss certain points more strictly within the actuary's province.

As has been already pointed out, there is generally a provision made for repayment with a bonus, the amount of such bonus being a fixed quantity, but the time at which it will be received being generally uncertain; and it has become an important practical question to determine the exact effect of this bonus, both from the borrower's point of view and the investor's or lender's. In the former case, as is well known, the problem is generally an extremely simple one theoretically. In the common case where the loan takes the form of a terminable annuity, or, as it is usually put, where the loan is redeemable by an accumulative sinking fund, we have to determine the actual rate of interest (x) paid by the borrower the equation

1−(1+x) ̄ 1−p=(r+s) XC

-n

the nominal amount of the loan being 1, the price of issue 1-p, so that p is the bonus received on repayment, r the nominal rate of interest, s the amount redeemed the first year, where s.(1+r)”—1

= 1.

r

There exists no means of solving exactly this equation in x, and various methods of approximation have therefore been resorted to, which it may be useful to consider on the present occasion.

It

Professor De Morgan, in a very valuable paper in vol. vi of the Journal, p. 61, "On the determination of the Rate of Interest of an Annuity", in speaking of the solution of the above equation, says, "The problem is, I doubt not, of infrequent occurrence. never occurred to myself to want more than a look at the tables would furnish; but it may be suspected that, as the requirements of business become closer, inverse problems will acquire more importance than they now possess". This prognostication has undoubtedly been fulfilled with regard to the problem we are now dealing with.

If i be a near value of a less* than the true value, and 1—(1+i)-”

=a', then we have

i

* This limitation is incorrect, as pointed out by Mr. Baden.-W.S. (See p. 91.)

This is the formula demonstrated by Baily, in his Doctrine of Interest and Annuities, and is also given in Jones, vol. i, p. 41.

This is Barrett's improvement on formula (1); and the earliest demonstration of it with which I am acquainted is given in a letter signed "M" in vol. vi, p. 56 of the Journal, where use has been made of Taylor's theorem in the Differential Calculus for the purpose. Recently it has been demonstrated in precisely the same way by Mr. Makeham, as a particular case of a more general formula, in his highly suggestive paper on "Loans repayable by Instalments"-see vol. xviii, p. 136. Still more recently another method of obtaining the formula has been given by Mr. McLauchlan -see vol. xviii, p. 297.

(3). A nearer approximation than the preceding:

This is given by the late Professor De Morgan in vol. viii, p. 67 of the Journal, but without demonstration. In Mr. McLauchlan's contribution already referred to, it is shown how the formula may be obtained; but the method employed by De Morgan, it is probable, was simpler than that used by Mr. McLauchlan.

where i is an approximate value obtained by inspection, and x=i +p; then the above gives us approximately,