The aim of this paper is to illustrate the background of Industrial Assurance. The paper therefore contains as little technical detail as possible. Industrial Assurance is fundamentally similar to Ordinary Assurance, but the organization and administration of Industrial business varies from that of Ordinary business partly as a result of special legislation applicable to Industrial Assurance and partly as a result of the frequent collection of premiums at the home of the policyholder.

The technical and legal side of the business has been dealt with in various papers given before the Institute and the Students' Society. Little has been written of a descriptive character, and it is thought that an appreciation of the nature and day-to-day conduct of the business might help the student to get the technical side more easily into perspective. A problem in Industrial Assurance cannot be tackled from the point of view of Ordinary Assurance.

I Shall not try in this paper to give a complete picture of the actuary in Canada, nor of the Canadian life insurance business. I have neither the space nor qualifications to attempt such a task.

I shall, however, try to bring out the differences in these fields which would be most noticeable to a United Kingdom trained actuary coming to live and work in Canada. I have, I think, advantages in attempting to do this. I have not been in Canada for so long that I have forgotten the contrasts which struck me most forcibly, but on the other hand, after almost six years, I can hope to be a reasonably accurate guide and in some cases at least suggest an explanation for the differences.

References to the rate of interest are frequent. In reality, of course, there is no single rate of interest but only a pattern of yields on individual assets or classes of assets. Nevertheless, it is useful as a first approximation to treat the complex of interest rates as a whole, thus bringing into sharper focus the forces that are common to the determination of the yield of all fixed-interest securities, with the exception of the risk of default. It is in this sense that the term ‘the rate of interest’ is used throughout this paper; consideration of the risk of default has been omitted as a variable that can easily be assimilated into the argument at any stage. The paper aims at making a broad survey of the most important forces that influence the rate of interest in a modern industrial economy where capital is for the most part in private hands. Where the assumption that we can regard interest rates as one united family departs too far from reality the more important qualifications that are needed are indicated very briefly.

When invited to prepare a paper to the Students' Society I had in mind selecting a topic from each of the various branches to illustrate the types of problems that are inherent in non-life business. However, as my ideas took positive shape, I found myself drawn into a more active study of certain aspects which have received little attention in this country, although some considerable theoretical and a limited amount of practical work has been done on the Continent. Instead, therefore, of giving a general description of various features I have largely limited myself to techniques which could be used in practical application and thus endeavoured to bring into relief some problems which indicate the almost unlimited scope for practical and theoretical investigations.

2. Initially the treatment is simplified; I hope that I have succeeded in picking out the spots where intuition rather than formal proof has been my guide, but if I have not I hope my critics will give me latitude in this exploration into largely unexplored territory.

The present is a particularly appropriate time to embark on a comparison of the demographic situation in the United States of America and that in England and Wales. P. K. Whelpton's study, Cohort Fertility: Native White Women in the United States, and the Report on the Family Census of 1946 conducted under the auspices of the Royal Commission on Population, have recently been published. These two works, one American and the other British, deserve careful study. This paper is designed mainly to make a broad comparison of the populations of the two countries and to provide a background for readers of Whelpton's book and of the Family Census Report.

The author wishes to acknowledge the ready assistance he received, on all occasions, from officials of the U.S. National Office of Vital Statistics and of the U.S. Bureau of the Census.

In the opening sentence of his paper on the valuation of industrial assurance policies given to the Institute in February 1937, W. J. Fulford said that his subject was essentially a practical one. Certainly the underlying actuarial principles are in no way different from those of the ordinary branch. In fact, the use of continuous functions gives the various valuation formulae a deceptive appearance of simplicity. There are none of the adjustments to allow for incidence of premium income, the formulae for which are the bane of the student grappling with the valuation of ordinary branch policies. The practical aspects, however, are very different. Reference to the published accounts of the Ordinary Offices shows that the new business, even for the largest, amounts only to a few hundred policies each week. In contrast in one large Industrial Office no fewer than 6000 new proposals are received each week while the total ‘offs’ in 1952 amounted to nearly 400,000. Since the Industrial Offices have a pre-disposition to hold their annual general meetings in late March or early April the actuary responsible for the annual valuation is faced with a formidable task if his figures are to relate to business on the books the previous 31st December, bearing in mind that the final results with recommendations have to be considered by the Board, decisions taken, and accounts published all before the Annual Meeting.

In several recent papers given to the Institute and Faculty of Actuaries attention has been focused on options in both the assets and the liabilities of a life office. The warnings that have been given about the concealed danger of options and the desirability which has been stressed of making adequate, if not severe, charges therefor are, it is hoped, sufficient justification for introducing a paper confined to the subject of annuity and cash options. The paper is limited to the development of the theory and to supplying tables to assist in the calculation of option costs in practice; questions of policy are considered to be beyond the scope of the paper.

Chiefly, perhaps, owing to the increase in the average size of our industrial or other organizations and to the increased facilities for carrying out the work by punched-card and other office machinery, there has of recent years been an increase in the country generally in the use of a number of organizational or management techniques of an analytical or statistical character. The tendency is more marked in America than in Britain, but even here we find an increased use of such techniques as ‘Market Research’, ‘Sales Forecast’, ‘Cost Analysis’, etc.

The more this type of statistical analysis finds favour, the greater will be the demand for increased statistical data relating to the economy of the country or of its social structure. Mr John Ryan, Vice-Chairman of the Metal Box Company Ltd, made this the main subject of his Alfred Watson Memorial Lecture to the Institute of Actuaries on 24 March 1952.

The type of contract which will be considered is a renewable oneyear term assurance on a group of lives which provides, in addition to a capital sum on the death of any life, that if at the end of the year the total claims in the year are less than some fraction, k, of the premiums paid, then a fraction, l, of the difference will be refunded.

Suppose that P is the net and P′ the office premium for n lives assured for a sum of 1 each and that q(r) is the probability that exactly r deaths will occur during the year, then

where c is the integral part of kP′. Since equals the expected deaths, μ say, the equation may be written

where and

The following note was written in 1949 as a supplement to the official course of reading for students preparing to take Part IA of the Institute's examinations and is issued in duplicated form to students by the Actuarial Tuition Service. The editors feel that the note may be of interest to many who, like themselves, did not have the opportunity of reading it in the course of their studies; those more recent students who have read the note may like to refresh their memories.

Mr Perks has kindly given permission for publication, and wishes to express his thanks to H. W. Haycocks and G. V. Bayley for the helpful criticisms and suggestions that he received from them.

A brief, introductory talk on a subject, such as this, will naturally be expected to start with a definition. In this, I must frankly admit to some difficulty, as I do not know of any really satisfactory definition of operational research. This, however, is nothing to be abashed at. It is usually easy enough to be clear about a field of study when you are in the centre of it; the boundaries, however, are commonly diffuse—and it is with boundaries that definitions are concerned. This difficulty of precise definition can be found even with quite well-established fields such as physics or, dare I suggest, the actuarial profession.

I shall therefore be content to quote a phrase which is commonly applied to operational research, really more a description than a definition, and shall then try to expand and explain the salient points incorporated in it.

A tool that is missing from the standard equipment of the actuarial student is the technique of maximizing, or minimizing, a function of several variables which are subject to certain constraints. The conception of an extreme value of a function, subject to constraints, is, we feel, as important as the mathematical technique of evaluating such a value in a particular instance.

We shall first deal with the technique and its mathematical development and then demonstrate some applications. It is as if we were demonstrating a new car, how it works and how to drive it. The first example is simple and is in the nature of a trial run to show how the car handles. Succeeding examples go further and we hope that the actuarial countryside into which we go will prove as interesting as the mechanical means of getting there. To many the vehicle itself will be familiar, but perhaps some of our byroads may lead to unfamiliar places or to familiar places by unfamiliar routes.

Lidstone has shown that the solution to the theoretical problem of valuing a life interest is essentially indeterminate. For a uniform life interest this is generally avoided by requiring of the formula that at the end of any policy-year the purchaser will have received sufficient income to discharge all premiums to date on the supporting policy, and to provide the predetermined interest on his total outlay.

Consider a life interest to be valued of 1 per annum, and let P be the office rate of premium on the whole-life non-profit policy required to return to the purchaser his initial outlay. In the evolution of a formula three unknown quantities are involved:

G, the gross purchase price (including expenses);

T, the total initial outlay;

S, the sum assured.

It is hoped that this article, which must be anonymous, will convey to students the nature of practical problems which had to be faced in dealing with one particular trust fund. In order that the transaction may not be readily identifiable, figures and dates have been altered, but not so as to affect the story materially.

In dealing with reversionary interests, the attention of the student is usually confined to the type of interest derived under personalty trusts, and he may not always be aware of the interesting problems that can arise in dealing with trusts to which the Settled Land Acts apply. Accordingly, in this article, it is proposed to describe some of the interesting points which lay behind one particular purchase of a reversionary interest in Settled Land.