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.

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.

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.

An allusion made by Mr. Jellicoe to this subject, in a paper read before the Institute of Actuaries on the 28th November, has induced me to place on record the conclusions at which I had arrived after a somewhat recent consideration of the question.

It appears to me that the buyer of a policy by way of investment, in considering the price at which he should purchase, should come to his decision entirely independent of what might be the Office value. The two values have no connexion whatever one with another. In the one case the purchaser is about to make an investment; in the other, the Office proposes to redeem a liability.

The remainder form of Newton's formula of interpolation with divided differences has been known for at least 37 years; yet actuaries have taken hardly any notice of the complete formula which ought to be one of their principal tools. It will, therefore, not be superfluous to begin by reproducing the very elementary proof from first principles.

1.1. For a number of years now it has been a commonplace amongst actuaries actively involved in unit-linked business that it was time someone wrote a paper dealing adequately with the subject of unit pricing. To those not actively involved with the subject the absence of such a paper may seem unimportant, but the fact is that the price-making process is central to the whole conduct of linked life business. It exerts a major influence upon the profitability and solvency of the office, and upon the relative interests of different classes and generations of policyholders and shareholders. In this respect, it is analogous to the bonus policy of a with profits office.

1 Whichever way you look at it, pensions are a problem. The Policy Studies Institute has expressed concern at the increased burden of State pensions fifty years from now when the number of pensioners increases in relation to the number of contributors. The Institute for Fiscal Studies too is anxious for the same reason and makes the criticism that insufficient consideration was given to the long-term cost when the State scheme was extended in 1978 so as to provide an additional earnings-related pension. The report of the Scott Committee has reduced but not wholly eliminated controversy on the extra value of index-linked public service pensions compared with the pensions commonly provided in the private sector. Employers, on whom the financial burden finally rests, are hard pressed to finance the accrued pension rights of their employees in final salary pension schemes and are unable to make the additional contribution necessary to preserve the values of the pensions and deferred pensions of former employees.

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.

The Institute of Actuaries has always been ready to welcome contributions from members able to give information regarding the mortality found to prevail among special classes of lives, and among lives resident in other parts of the world than Great Britain. The authors of the present paper having had occasion to look closely into the subject of West Indian mortality, chiefly in connection with the financial affairs of the Barbados Mutual Life Assurance Society, have therefore willingly responded to an invitation to lay before the members of the Institute some account of the statistics which they have been able to gather, and of the conclusions at which they have arrived. They desire, in the first place, to express their thanks to the directors of the Barbados Mutual Society for permitting them to publish the results of the society's mortality experience, and to Mr. Spencer C. Thomson, the manager of the Standard Life Office, for some valuable statistics relating to the experience of that office regarding West Indian mortality.

Few who are engaged in the practice of Life Assurance will deny that the prevailing restrictions imposed on, and charges made for foreign travelling and residence are unsatisfactory to all parties concerned. A considerable and increasing portion of the public whose pursuits or inclinations, especially in early life, take them to different parts of the world, are deterred from effecting life assurances on this account. The value of policies as securities is diminished owing to the uncertainty of the extra premiums which may at any time be imposed, and the risk of the assurances being forfeited altogether by causes over which mortgagees have no control. And actuaries feel that this subject is the opprobrium of their profession; for as in general no sufficient data exist whereby the risks incurred may be measured, the extra charges are therefore arbitrary and unscientific, for the most part little better than random guesses made on no intelligible principles. For these reasons it is hoped that any suggestions on the subject, however imperfect, may not be unacceptable, and may be the means of eliciting some useful discussion.

The origin of savings connected with human life was not a particularly happy one in France. A certain Neapolitan named Lorenzo Tonti submitted to Mazarin, the Finance Minister of ‘Sun King’ Louis XIV, a savings plan, the profits from which would help to replenish the sadly depleted royal coffers of this grandiose monarch. The plan was called a ‘Tontine’ after its originator and provided for a lump sum to be divided amongst the survivors of those who had contributed after a period of say 10 to 15 years. The ‘sales psychology’ was good as the individual is normally of the opinion that he will outlive his contemporaries. Unfortunately the results were disastrous as Parliament refused to ratify the royal edict of 1653 and Mr Tonti was clapped in the Bastille.

British life offices transacting life assurance outside the United Kingdom have until now confined themselves almost entirely to the countries of the British Commonwealth—because among the countries with stable currencies the members of the Commonwealth have in the main been the ones with which we have had the largest numbers of other business contacts. Another influence has been the absence of a language barrier. Before the 1914–18 War some offices transacted business on the Continent of Europe but thereafter most of them pulled out; today as far as I know only one British office transacts life assurance in France, one in Germany and none in Italy.

One might be forgiven therefore for thinking that actuaries handling life office affairs need not concern themselves with life business on the Continent of Europe—particularly as, at the beginning of last year, the negotiations for Britain's entry into the European Common Market broke down.

The subject of valuation is one of the most important, and at the same time most difficult, of all the problems faced by actuaries.

The traditional approach to the subject has centred on discussion of the relative merits of the net premium and bonus reserve methods of valuation, and on the inherent conflict between valuation as a means of demonstrating solvency and valuation as a tool for ensuring a smooth emergence of surplus in accordance with the need to declare uniform reversionary bonuses. This approach is covered in the well-known papers by Redington (2) and Skerman (3) (4) and forms a significant part of the course of reading for examination subject 8 (formerly B2).

Bonus reserve methods are, typically, ‘active’ methods of valuation in which all relevant parameters, including expenses and future bonuses, are given values close to their expected values. The premium valued is the office premium, and assets are usually valued at market value or by discounting expected investment income. Because actual conditions are always changing, an active valuation method tends to lead to frequent changes of basis and this introduces an element of volatility to the results which can, in consequence, be difficult to interpret. Nevertheless, because of the emphasis on using realistic values for all parameters, it is commonly believed that bonus reserve methods are particularly appropriate for investigations into solvency, especially when external conditions are changing rapidly, and in determining appraisal values.