1.l. The aim of this paper is to examine a number of issues that arise in decision-making. In particular it seeks to explore the relevance to decision-making of the commonly made polarization as between the so-called objective and subjective approaches to probability. The paper is in three parts. The first concerns utility, and the second various aspects of probability. The third and final part looks at the interplay between these two concepts. Whilst a number of the comments and illustrations refer specifically to the insurance industry, the concepts discussed are broadly set in the wider context of business and commerce as a whole. It is hoped that the paper will thereby provide a basis for discussion on a topic that has acquired increasing importance in recent years.

During the last session of the Institute of Actuaries, I had the honour to submit a paper, the purpose of which was to devise a method of adjustment as even and correct as that of Mr. Woolhouse, and more easy in application. This was to be done by combining two summations of numbers increasing by third differences; the summations being so chosen as to introduce small factors only into the formula, and to admit of its being worked in columnar shape. (Journal, vol. xxiii, page 335.)

Thereupon, Mr. G. F. Hardy showed that Mr. Woolhouse's formula can be used in columnar shape; and Mr. Ackland, improving on Mr. Hardy's arrangement, contributed a table in which an adjustment of the HM decrement was so calculated. (Journal, xxiii, 354.)

This important and unexpected result of the reading of my paper rendered it of little value, except that it supplied a short formula which may be convenient for use where a rough graduation only is required. Any degree of smoothness can be obtained by repeating the process; but this is tedious, and I would now furnish a formula derived in the same manner, and yielding a satisfactory adjustment by a single operation.

It is quite common for stocks (usually loan stocks) to be issued by companies and purchased by investors, which include among their terms an option to convert at some future date into other stocks (usually ordinary shares) on specific terms. Such options obviously have value, but I am unaware of any simple published method for estimating the value of such options in practical cases. The purpose of this note is to suggest a formula, based upon easily understood estimates of the future market prices of the stocks in question, which may be applied readily in practice to estimate the value of such options.

The results of the investigation from 1945 to 1959 were presented in a paper to the Institute by R. E. Hayward and B. C. Lucena (J.I.A. 91, 286) and readers are referred to this for further detail on practical aspects of the investigation. This further report owes its existence to all those originally involved as well as those specifically involved in the follow-up. Professor J. M. Malins and Dr M. G. FitzGerald have continued to give us the help needed to obtain the required data from the hospital records. Those of our Society particularly involved in producing this follow-up report include B. C. Lucena, R. Betteridge, A. Roberts and M. Willett.

Nearly twenty-two years have elapsed since the masterly essay of our President gave to the institute an opportunity for comprehensive examination of the British friendly society system. During the intervening period, phases of the subject have been occasionally before the Institute, but the period has been one of great activity amongst the friendly societies, and developments of interest to actuaries culminating, it may be suggested, in the co-operation of the State and the societies proposed by the Chancellor of the Exchequer, have emerged. In the circumstances it has seemed to me that the Institute might again survey this wide field of provident effort with interest to its members and profit to the community.

I propose accordingly to examine the more important groups of societies, paying particular regard in respect of the leading affiliated orders to the machinery by which it is sought to equip the constituent parts for the effective recognition of their duties; thence passing to some questions of special interest to ourselves. For the purpose of connected description it will be necessary to mention particular organizations, but I trust that nothing of what I have written may seem to depart from the excellent custom of neutrality which governs our debates. I venture indeed to say that so far from being unduly sensitive, all the societies which are striving for improvement (and I think this may be said of most of them) are receptive towards the stimulus of informed criticism.

Some apology is needed, perhaps, for proposing to rediscuss a subject which has been handled so often, and by so many distinguished actuaries. But I am probably addressing gentlemen who have not had occasion to become familiar with all that has been written or said on the subject, and others to whom it may usefully be presented from a fresh point of view; and there are some important practical questions too on which the profession seem scarcely yet agreed, which will admit of further consideration. I shall not attempt to notice the various papers which from time to time have been contributed by actuaries towards the solution of the questions I am to consider. For many of the views I am to express I am necessarily indebted to those who have preceded me; and the aim of this paper is less to introduce what is new, than to enforce, and illustrate if possible, sound and correct opinions. I am anxious to set the subject in a light which will make it intelligible to others than actuaries, and to discuss principles rather than formulas.

On the occasion of the bicentenary of the foundation of scientific life assurance, it was thought fitting that a work should be prepared in which the development of some aspect of the subject might be discussed. The authors undertook some research on the origins and disposal of surplus and the results are set out in a book entitled Surplus in British Life Assurance—Actuarial Control over its Emergence and Distribution during 200 Years. The following pages convey some of the impressions formed in the course of its preparation. Much of the later part of the paper consists, in fact, of extracts from the text of the book. The introductory sections present a short outline of the book as a whole.

Several writers, notably Messrs. Higham and Sprague, have treated of the Mortality Experience of Life Offices, basing their investigations upon various data; and it may be thought that there is no necessity to pursue the subject further. The question is, however, one of great interest and importance, both in its relation to abstract science, and in its bearing on the actuarial management of companies; and as it possesses collateral branches which have hitherto received but little attention, it occurred to me that my time would not be wasted in attempting to elicit new facts, and illustrate old ones, from the abundant materials now at our disposal.

It will no doubt be in the recollection of many of our readers, that some few years ago attempts were made by the Council of the Institute to collect, from the Offices willing to contribute the results of their experience, such information connected with the extra risks undertaken by them as might afford the means of determining, if not exactly, at least with some approach to accuracy, the proper rates of premium to meet them. The Council made application to all the existing Companies; but did not succeed, unfortunately, in obtaining returns from more than fifty-two of them. These fifty-two Companies, however, were at the pains to supply as many as 6,154 cases—each of them of a special nature, as regarded unusual hazard to life arising from residence in foreign climates, or from particular pursuits; but so infinitely varied in their conditions and circumstances were they, that it became immediately obvious that considerable difficulty would arise as to grouping them into classes which should be sufficiently well defined to yield results of a useful and practical character. If, on the one hand, the conditions of the risk were kept within limits sufficiently restricted, the numbers in each class were all but insignificant; and if, on the other, attempts were made to remedy this last inconvenience, the groups comprised risks which were altogether heterogeneous.