I have heard that a former distinguished President of this Institute once publicly stated that “Doctors rush in where Actuaries fear to tread.” It is also said that “at forty a man is either a physician or a fool”: but the presidential saying seems to suggest that a man may be both. With such a warning ringing in my ears, I yet venture to read before this learned Institute a paper on a thorny subject. I am doing so, first because that subject can be dealt with only by the united labour of actuaries and doctors; and second, because I am sure of the consideration and forbearance of the Fellows in any matters wherein I travel with halting gait.

Corporate bonds and privately placed loans constitute a large proportion of the portfolios of several financial institutions. When bonds of a corporation are purchased in the market or loans are granted by some other arrangement, the lender evaluates the risk relating to the investment in question. Determination of the level of risk is important, among other things, for facilitating comparison between different investments, setting the terms of loan agreements and making a rational decision concerning the acceptance or rejection of a loan proposal. Since evaluation of risk is very important for rational investment decision-making, the nature of risk in bond investment is examined briefly in the next section. In the subsequent sections, the prevailing techniques of evaluation of risk will be reviewed and the cash flow simulation model presented.

The two leading formulæ at present in use for interpolation by central differences are due to Newton‡ (Methodus Differentialis, Prop. III), though often called by the name of Stirling, who republished them about 20 years later.

It may well seem an act of no small temerity to attempt a fresh paper on a subject so often and so ably treated in essays read before this Institute as that of the net premium method of valuation. I am fully conscious of this, and the more so that I cannot profess to be able to add anything appreciable to what has already been said, either expressly or by implication, upon the matter. It will be admitted, however, that the practical importance of the subject, its vital bearing upon the conduct of the business of life assurance, can scarcely be exaggerated. Mr. J. Hill Williams, commenting upon Mr. A. H. Bailey's paper on “The Pure Premium Method of Valuation”, in March 1878 (J.I.A., xxi,, 126), said “The subject of this paper is one upon “which we can never have too much writing or too much “discussion, for it is the very life of our science, and upon the “views which we take of this most important question depends “the safety of many of the greatest financial institutions in the “country”; and with the spirit of such remarks we shall probably all agree. That being so, I think I am right in saying further that we have had no express discussion of this topic at our Institute for several years past. It has, doubtless, on many occasions been referred to—as a thing so intimately intertwined with our daily practice could hardly help being—in connection with other matters primarily before us; but it has not been the actual and avowed theme for consideration.

In a short paper, which I read before the Second International Actuarial Congress, held in these rooms in May 1898, I gave the solution of some problems which frequently arise out of the Rules of Pension Funds and Friendly Societies. That paper is printed in the Transactions of the Congress, published by the Institute, page 860. I was unfortunate in the selection of the table which I used for illustrating the formulas, as there was a printer's error in the value of the most important term. That, however, did not affect the formulas themselves. Mr. McGowan, in a letter to the Insurance Record of 2 February 1900, said that my remark on page 861 (of the Transactions), that “it will be P years before the annuity payments equal the price given, does not seem quite correct in the case where P is an integer plus a fraction”, and I explained, in a letter which appeared the following week, that the correction, which would be the value of the assurance of the fraction of the annuity for the fraction of the year in which the fraction would occur, is so insignificant that it might be ignored.

In the following essay we shall attempt the solution of a problem which has hitherto remained unsolved, though it is one of the most important problems connected with the science of life contingencies.

[Dr. Wittstein then explains the notation he uses; but we have substituted for it the usual notation, with which all our readers will be familiar; and he states that the Table B, on page 167, may serve as a specimen of a Mortality Table.]

Mortality Tables are based upon observations. We do not now propose to discuss the mode of collecting these observations. Upon this subject the author's treatise on Mathematical Statistics (Hanover, 1867), may be consulted. By whatever method the observations are made, the data resulting from them are always put in the form given in columns 1–3 of Table A, page 166. This table contains the data which Brune published in the year 1847 from the experience of the Prussian Widows' Fund, and from which he constructed his Mortality Table for Males (Table B); and we select this table in illustration of our subject, simply because it is readily accessible.

The text of the Presidential Address is printed in this part of British Actuarial Journal on pages 1-33.

It is now some ten years since it was accepted, cautiously at first, that computers might be of some value in performing ordinary clerical procedures. Since that time, they have been gradually installed at an everincreasing rate in offices of all kinds, including insurance offices. Our own Institute was, I believe, the first body to publish tables—the a(55) tables—which had been prepared by means of a computer, In some respects it is a pity that the insurance world in this country has not been more to the fore in utilizing this new equipment although, from the number of insurance people who attend any meeting connected with computers, it is not due to any lack of interest in new methods but rather to a cautious attitude concerning making use of equipment until it has been fully proved by actual field service. This is in contrast to the position in U.S.A. where the insurance offices have been well to the front in development of office use of computers; indeed, an insurance office there was the first non-governmental business user and that office now has four very large machines in full-time use. As a result of this we were fortunate to have had the very instructive surveys published by our sister body, the Society of Actuaries.

Many years having elapsed since the Eagle Insurance Company first undertook to issue policies of assurance upon unsound lives, and some doubts having arisen as to the way in which the funds of the Society had been affected by such transactions, the Directors decided that a thorough investigation should be made into the business connected therewith, with the view of obtaining not only a record of the past, but also, if possible, of establishing a guide for the future, and I now have the pleasure, with their permission, of making public the results of that investigation.

Mr. John Finlaison's method of graduation (Journal, vol. xxi, page 50) deals with an irregularity, which, for convenience, may be assumed to be unity, in the following manner:—It diminishes the irregularity to a small fraction on its first appearance at one end of the formula; conducts it along the formula by easy stages, as over a double inclined plane; and dismisses it gently at the other end. The method has the advantage of simplicity in working and check, but it is correct to first differences only.

Mr. Woolhouse's method is correct to third differences, and the irregularity glides over a curve; but the application of the formula requires sustained attention.

The object of this paper is to combine the facility of the one method with the smoothness and correctness of the other.

Let S be the sum of nine numbers, u 0 to u 8, increasing by third differences,

the central term,

whence .

This paper is partly descriptive, partly analytical, and covers the following aspects:

a description of the market in general;

a brief description of the organization of Lloyd's;

currencies and exchange rates;

calculation of reserves;

taxation;

underwriting; and

accumulation of risk.

§§ 1 and 2 and the earlier paragraphs of § 3 convey information regarding the market that will be well known to actuaries practising in this field and may be passed over without loss. They are added in an attempt to convey a picture of the type of operation that results within an office concentrating on reinsurance business, as against looking at reinsurance in its peripheral effect on a direct writing office.