The text of this paper, together with the abstract of the discussion held by the Faculty of Actuaries on 17 February 2003, are printed in British Actuarial Journal, 9, IV, 725-786.

The interest with which I have perused Mr. Gray's very able paper in the last number of the Journal, has led me to give some consideration to the general theory of varying Annuities; and as the results of the investigation appear to me somewhat striking, I am induced to think that a brief paper on the subject may not be unacceptable to that section of the scientific world to which this Journal more particularly addresses itself.

My first impressions on considering the magnitude and pretensions of the investigation which I now present to the Institute were that it would perhaps be more suitable for a communication to the Journal, or that at any rate it would scarcely provide sufficient matter to occupy the attention of the meeting for a whole evening. This impression was strengthened when I considered the importance and wide scope of the researches which have been presented as food for our reflection at many of the meetings held in recent years. But, on further consideration, it appeared the subject I had chosen was one likely to give rise to useful discussion. Also I feel that, from a good number of the members of the Institute, at any rate, on whose time and attention the cares of business are continually making greater and greater demands, I may hope for forgiveness if I place before them a somewhat lighter paper than usual.

The title of the paper may suggest to readers that the authors are about to enter into a highly moralistic debate comparing those persons who take the ‘evil weed’ against those that lead ‘clean wholesome lives’. Whilst this is obviously not the case, the inference of competition is totally relevant. As the U.K. life insurance market moves slowly but surely to a smoker/non-smoker division, new dimensions of competition could open up between those offices who adopt a differential system and those who do not. One purpose of this paper is to comment on this feature.

The main purpose of the paper, however, is to highlight the financial and underwriting problems involved in this type of market which represents a highly significant departure from previous practice. Many readers will have already debated these problems within their offices and perhaps with their reassurer. However, to our knowledge, no discussion has yet taken place across the British actuarial profession as a whole, and we believe it highly desirable that such debate takes place. This paper, therefore, should be seen as offering nothing new or conceptually original, but as merely the catalyst for discussion.

At the present time the subject of legislation in reference to Insurance Companies is attracting much attention in Germany as well as England. Until quite recently the regulation of insurance matters in Germany was under the sole control of the several States, as is still the case in the United States of America: the former German Confederation possessing neither the organization nor the authority to interfere in the matter. As a consequence of this state of affairs, we are informed that a company organized in one of the German States was regarded in the others as a foreign company, and all companies, whether German or foreign, which desired to extend their business over the whole of Germany, were compelled to put themselves in communication with more than thirty different Governments, each of which had its own special regulations. Some of the larger States had passed laws on the subject, but the provisions of them differed very widely from each other. In the remaining States, each Government imposed on the companies such conditions as it thought proper. This state of things is evidently very unfavourable for Insurance Companies, which require—more than any other undertaking—a wide field for their successful development. Under the old political organization of Germany, a uniform system of legislation, though long and ardently desired, was quite unattainable. But after the events of the year 1866 had led to the formation of the present North German Confederation, which gives a common legislation to at least three-fourths of Germany, the hopes of reformers revived. The 4th article of the Constitution of the Confederation expressly provides that legislation upon insurance shall belong to the province of the Confederation.

The assumed rate of future mortality improvement has increased over three recent sets of the United Kingdom's national population projections. This optimism has not been so marked in countries which share ancestors with the U.K. population. New Zealand is one such country that provides a data-rich case example in which to investigate the portability of mortality trends.

This paper compares mortality trends in New Zealand with those in England & Wales. Both countries seem to have a ‘golden cohort’ which enjoys faster improving mortality than people born before or after. The birth of the golden cohort in England & Wales coincided with cohort life expectancy there catching up with New Zealand's.

We show that first generation migrants from the U.K. have better mortality than New Zealand born residents likely to have British ancestry. The advantage lasts into older ages, decades after migration. We hypothesise that migrants from the U.K.'s golden cohort brought with them an early life mortality improvement advantage, and additionally benefited from the healthier environment of New Zealand at middle to older ages. Further, given the recent strong mortality improvement in New Zealand, the U.K.'s assumptions for future mortality look relatively optimistic.

If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavour to persuade me that he was larger than the elephant, I might by possibility be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be unable wholly to reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others—I should feel quite at my ease.

The paper sets out the method required to be followed when estimating reserves for a Company or a Lloyd's Syndicate which has accepted reinsurance treaties that have given rise to catastrophe losses, sufficiently large to upset the normal development pattern and to affect the gross account quite differently from the net account. The losses may be caused by single factors such as aircraft crashes or oil rig disasters, or by the aggregation of claims resulting from a windstorm or an earthquake. The paper discusses two possible approaches to estimation of the gross losses; via exposure totals or via statistical modelling techniques.

A considerable number of investigations on mortality in times past has already been published, and without scarcely any exception they all go to show the same thing, that the mortality now-a-days is much smaller than it used to be. Some of these results are, however, only found by a sort of induction, in considering the conditions of life and the state of medical science in past centuries, and most of the statistical facts do not go farther back than to 1750; for more remote periods, the statistical information which has been gathered is scanty, untrustworthy, and dispersed.

I therefore undertook the task of investigating a distinct class of society during as long a period as possible, hoping thus to be able to confirm or correct our notions on the variations of mortality. The materials were taken from an elaborate biographical work by the Rev. S. V. Wiberg, published 1870—71, with a supplement from 1879.

I confess that the general impression of the results was that of a disappointment. In fact, the materials appeared so incorrect, that distinct numerical conclusions were not admissible. Still, I have thought it of interest to publish the chief results, hoping that, by-and-by, more correct materials may be at hand, and that the readers will consider the results here laid down as merely preliminary.

The two studies described below were made as part of a large-scale programme of official research into a series of topics relating to fertility, mortality and migration. The outcome of all this research (including that recorded below) will be published by the Government in due course but the manner and place of publication have not yet been decided. Although the present paper is announced under joint authorship, it is not a fully integrated effort: Part 1 was devised and written by the first-named contributor and Part 2 by the second. Nevertheless, their preparation took place over the same period of time and during that period steady contact was maintained between the authors by correspondence and meetings, and thus the work proceeded in parallel.

The mode of using Barrett's method has now something of farrago about it: a little discussion may be useful, were it only to awaken attention to the confusion arising from changes made without adherence to the recognised laws of perspicuity.

The farrago to which I have alluded arises out of a struggle between the annuity, as commonly understood, and what I shall continue to call the annuity due. According to the technical meaning of the word annuity, the payment begins to grow at the moment of the grant: thus, a yearly annuity for 12 years, deferred 20 years, begins action at the end of 20 years, and grows the first payment in the 21st year. I shall vary the phrase used in my “Essay on Probabilities” (Cab. Cycl.) by allowing the words “annuity due for x years” to signify an annuity of x payments, the first immediately. Thus an annuity for 12 years, now granted, will in a year become an annuity due, also for 12 years.

1.1. This paper is written at a time when, I believe, the greatest challenge facing actuaries engaged in pension work is to communicate.

1.2. We must communicate with Trustees, with members, with employers, with Government, with accountants, with the O.P.B., with the S.F.O. and, through the media, with public opinion: we must also communicate with each other.

An investigation into the effects of family and personal history upon assurance risks has long been desired by Actuaries, but hitherto in this country the material at their disposal has not been sufficient in volume to admit of analytical treatment. The only attempt the Institute as a body has made to deal with this subject is in the old Institute experience, which contains the records of 11,146 rated-up lives, in which all kinds of risks are combined without any attempt at classification.

In the hope that this long-felt want might, to some extent, be met, an investigation was projected into the experience of a large number of lives assured in the Ordinary Branch of the Prudential Assurance Company whose family or personal history showed some marked peculiarity, by means of which they could be separated into more or less well-defined classes. The actual work of compiling the necessary data was commenced in 1903, and by the courteous permission and public-spirited action of the Directors we have now the pleasure of placing the main results of that investigation before the Institute. No claim is made in this Paper either to exhaustive analysis or to complete solution of the numerous and important problems that arise in practice, but it is thought that, at least, the results may form a useful aid to the Actuary in arriving at those conclusions which necessarily have to be formed—mainly from a priori reasoning—when risks come before him for adjudication.

In the course of an investigation that was recently being made into the rate of mortality prevailing among a body of assured lives subject to extra risks, it occurred to me that it would be advisable. if possible, to procure some standard of comparison in addition to the ordinary mortality tables at present in use, it being apparent that the rate of mortality in question would depart too markedly from the existing tables for them to serve as any criterion. Several modes of forming hypothetical tables exhibiting features similar to those characterizing the actual observations—namely, an increased rate of mortality, especially heavy at the earlier ages—suggested themselves to me; but the difficulty of fixing the degree in which they should differ from the table on which they were to be based, and the arbitrary nature of the assumptions it was necessary to make, were obstacles I was unable to remove in a satisfactory manner. Finally, influenced, among other considerations, by the practice in vogue with Life Assurance companies of the present day, I selected the basis explained in the following article, and deduced a table of mortality from the HM 3 per-cent whole-life pure-premium increased at each age by the net addition of 01.