The text of this paper, together with the abstract of the discussion held by the Faculty of Actuaries on 18 February 2002, are printed in British Actuarial Journal, 8, IV, 643-704.

1.1. Occupational pension schemes in Great Britain and in many other countries are experiencing a very difficult period. For several years now actuaries valuing funded pension schemes have been faced all too often with the unenviable task of presenting to their clients the unpopular message that their pension schemes are going to cost more in the future than they did in the past. Furthermore, the increasing pressure to revalue pensions in course of payment and to replace withdrawal benefits by some form of preserved rights is either threatening employers with intolerable funding burdens, or is causing some departure from the established practice of fully funding retirement benefits.

The publication of the tables of mortality based upon the experience, collected by the Committee representing the four principal insurance companies of France, puts us in possession of an extensive and valuable body of statistics bearing upon the mortality of assured lives and annuitants in that country, and at the same time places the Actuarial profession under a considerable obligation to the Committee of French Actuaries to whom we are indebted for the collection and publication of the data and the resulting monetary tables. As in this country, the life assurance companies on the Continent had for a long time, necessarily, to depend upon statistics of mortality based upon observations of the population generally, which, whatever the skill with which they were prepared, were not only affected by the uncertainty attaching to all such statistics, but were not in any case specially suited to represent the probable mortality, either of assured lives or annuitants. The earliest of these tables, to which any value could be attached, are those of Deparcieux and Duvillard, corresponding in point of date, approximately, to the Northampton and Carlisle Tables in England, and it was not until 1860 that the three companies, the Compagnie d'Assurances Generales, Union, and Nationale, published their annuity experience, nor until 1874 that the first of these companies published a table representing the mortality experience of their assurants.

This subject can appear difficult, but the main idea is delightfully simple. Let us consider an example. Suppose that a life office has issued policies at premiums based on a certain rate of interest, investing in Stock Exchange securities the premiums received from year to year, less claims, and that after a time the rate of interest rises permanently. The market value of the investments will fall and the office will lose if they have eventually to be sold, but will gain to the extent that future premiums can now be invested on more favourable terms. The gain is fixed but the loss will vary according to the redemption dates of the assets, since the longer the date, the greater the fall in value. Immunization consists in selecting investments such that the loss balances the gain. An alternative way of looking at it is on the basis of present values. The values of the assets on the one side of the balance sheet, and the net liabilities on the other, will both fall if interest rates rise. We want them to fall by the same amount, and as our liabilities are fixed, we must seek to achieve that result by a suitable choice of assets.

In applying this title to the following remarks, I am quite aware of the fact that a number of authorities, whose opinion is entitled to the highest respect, will at once pronounce it a misnomer, for the reason that, while all statistical returns disclose an increase in the rate of mortality from cancer, such increase is held to be only apparent, and is not real, but is due to improved methods of diagnosis. This view was propounded and ably advocated in a paper by Mr. Geo. King, F.I.A., and Dr. Newsholme, printed in the Proceedings of the Royal Society of London, vol. liv, 1893. The position taken up in that paper was that, as cancer is a disease of middle life and old age, no satisfactory conclusion could be reached, except by a distribution of the living and the recorded deaths from cancer into the various age periods. The gentlemen named proceeded with their investigation on this basis, and arrived at the conclusion that no actual increase in the death rate from cancer had taken place, but that the apparent increase was due to improved methods of diagnosis and to erroneous methods of tabulating the data.

The question is frequently asked, in what respect competition in life assurance business differs from that in any other trade or profession. It is contended, that in trade the effect of competition is to produce a better article at a lower price—to awaken new tastes and desires in the public, and so to create a market, which, notwithstanding the diminution in the price, may afford larger profits to the merchant, by the greater quantity of the commodity which he can sell or the greater frequency with which his capital may be turned. But there is nothing analogous to this in life assurance, for in truth it is nothing more than a redistribution of the capital of the public—a deduction of the portion which one man may save out of his profits in the course of a long life, for the benefit of the family of another who does not live to save it.

For the past few years, stimulating research has been carried out on ordered statistics. Linear combinations of the sample ordered values are used to provide estimates of parameters of the population from which the sample is drawn. Such statistics are termed systematic by Mosteller. They are in common use, because they provide simple solutions of parametric problems of statistical estimation. Often they are inefficient, but this drawback is sometimes more than offset by the ease with which the calculations are performed.

Mosteller drew attention to the possible use of the statistics Wr , which he called quasi-range of rth order and defined as

where x 1, x 2,…, x n are ordered observations in a sample of n from a population f(x) (x 1 < x 2 < … < xn ). The distributions of quasiranges in samples drawn from a normal population have been studied by Godwin and Cadwell.

1. In the year 1800, when the Parliaments of Great Britain and Ireland were fused by the Act of Union establishing the United Kingdom, the population of Ireland was of the order of 5 millions. By 1821 the figure had risen to 6·8 millions and in 1841 it was 8·2 millions. During the following decade the population fell by 1-6 millions to 6·6 millions. By the year 1861 it was only 5·8 millions and thereafter it continued to decline steadily, though not with such rapidity, until in 1911 a figure of 4·4 millions was reached. Owing to the disturbed state of the country in 1921, the next census was not taken until 1926, after the political change in 1922 when twenty-six of the thirty-two counties into which the country was divided were established as a separate political entity, the Irish Free State (later to become a Republic), while the remaining six were constituted as Northern Ireland and continued to form part of the United Kingdom. The population of the whole island in 1926 and again in 1951 was 4·3 millions, that is to say, it differed only slightly in 1951 from what it was forty years earlier in 1911—see Table 1. The population of Ireland has, therefore, remained virtually stationary at about 4¼ millions for almost half a century. The trend of Irish population since 1841 is in striking contrast with the trend in England and Wales for, whereas in 1841 the population of Ireland was more than one-half of that in England and Wales, today it is less than one-tenth; the Irish population has almost halved while that of England and Wales has almost trebled.

The paper discusses two important models of option pricing: the binomial model and the Black—Scholes model. It begins with a brief description of options.

Gentlemen,—I thank you for again electing me your President, —it is an honour which I highly esteem. You may rest assured that I will, to the utmost of my power, endeavour to perform the duties of the chair, I trust, to your satisfaction.

It is customary on the opening of a session for your President to say a few words by way of “Address” on some subject, of his own selection, which has more or less relation to the general business of the Society. As I had last winter the pleasure of reading a paper which occupied a good deal of my time, I have chosen a subject for which the materials are almost all at hand, but which when strung together may, in their continuity, possess an interest which will not be unwelcome to the members of an Actuarial Society. I propose to consider one or two incidents in the History and Statistics of Life Assurance.

The History of Life Assurance is exceedingly interesting. Bestowing all our attention upon the engagements of the passing hour, we are very apt to forget the arduous up-hill struggles of the many schemes of the past two centuries. If an account should ever be written, as I believe it will, of the career of those futile efforts, it should teach us to be very careful, and watchful of our position. There is in our time, it sometimes appears to me, rather an excess of that spirit of competition which I fear tends to weaken, and not to strengthen. Life Assurance is not entirely one of the exact sciences. The ordinary trader may easily, by applying one or two principles of accounting, exhibit the solvency or insolvency of his business; but it is almost impossible, except in the mass, to fathom the Liabilities of the business of Life Assurance. An ordinary debt is owing now, or a bill will certainly mature on a given date, but our debts accrue in the future, and are not individually dated. In the maturing of a multitude of them we have the utmost confidence.

The theory of ‘temporary selection’ is concerned with the variation, for fixed x, of q[x–t]+t the observed rate of mortality at age x during the t+1th year after the issue of an assurance or annuity contract. The classical view is that—apart from chance variations—q[x–t]+t increases gradually with increasing t until the effects of selection have disappeared after which time q[xx–t]+t is a constant depending on x only.

Various reasons have been suggested for the persistence of temporary selection in an observed series of values of q[xx–t]+t. The chief of these are:

(1) The continuing effects of an initial selection on the part of the assurance company or by the annuitant (Morgan, 1834);

(2) The gradual withdrawal from assurance of healthy lives (Higham, 1851); and

(3) Secular improvements in medicals election or in the self-selection of annuitants (Karup, 1903).

A working party was set up in February 1999 by the Research Committee of the Life Board to consider the impact of a low inflationary environment on the financial services and investment markets with particular emphasis on companies conducting long-term insurance business.

The working party has produced an interim report which reviews the causes of the present expectation of low inflation and discusses evidence from the past and the role of demographics. It considers issues surrounding the composition of the retail price index, appropriate economic assumptions and the impact on investment markets, including the effect of charges on net returns to savers. The position of long-term insurance business is then considered in more detail, looking at HM Treasury return consequences as well as disclosure and product design features. Finally, there are two investigations, the first looking at the effect of low inflation on the achieved profits derived from different product designs, and the second looking at the changes in ruin probability for a with-profits office associated with declining inflation.

Various numerical procedures have been developed in recent years for estimating the provisions for outstanding claims of a general insurer. (See, for example, Benjamin, 1977, 234–66.) These procedures do not in general have a theoretical statistical basis, and consequently they do not provide information about the reliability of the resulting outstanding claims estimates. An exception is the complex procedure of Reid (1978).

It is not surprising that actuaries have been daunted in their quest for an appropriate statistical model: the claim process is very complicated, particularly for ‘long tail’ business. One has to consider the incident leading to a claim, the size of the resulting claim, the time to settlement, and the distribution of payments up to the time of settlement both by amount and epoch. There are also the additional complications of inflation and investment earnings on the provisions.

In this paper, a theory is developed in which the complications of the claim process are overcome by considering the joint distribution of payments for a single claim in successive development periods. The model is distribution-free in that it does not assume specific distributions for claim size, time to settlement, or payments prior to settlement by amount and epoch. The method requires that payments which have been made be expressed in ‘constant-dollar’ terms (i.e. adjusted for past inflation).

The paper describes in some detail the statistical methods that may be used for analysing mortality data from medical and epidemiological studies, with particular reference to quantifying survival and the risk of mortality and to comparing the experience with a standard.