The purpose of this paper is to indicate some of the points which arise during the currency of transactions in reversions and life interests; the selections of appropriate bases for mortality, interest, valuation of funds and estate duties do not fall within the scope of the paper. References to strictly legal aspects are no more than incidental.

As in many other matters with which actuaries are concerned, mistakes (including errors of judgment) sometimes only become apparent many years after they were made. It follows, therefore, that no hard and fast distinctions can be drawn between the preliminaries necessary to a transaction and what comes after. For this reason I feel obliged to refer to my earlier paper entitled ‘Notes on transacting new business in reversions and life interests’ which appeared in this Journal (10, 1); it is hoped, however, that the present paper will be found to be complete in itself.

I have long wanted to give vent to the thoughts in this talk and therefore I should start by saying ‘Thank you’ for providing me with an audience. Although the broad ideas have been in my mind for a long time, when it came to the point I had considerable difficulty in deciding just what to say and particularly in choosing a suitable title. I am far from satisfied with the somewhat pretentious name with which I have christened the infant, but it does I hope convey the thought that although I shall be talking about statistics—a fact which may or may not be an attraction and was therefore better not concealed—it will not be in a detailed technical sense. Indeed one of my first duties must be to disavow, not only any desire but any considerable ability to expound on statistics in detail.

I have reached the age in life when I begin to gain much comfort from the thought that what matters is not the facts you know but the way you think. It is a comforting thought when you have forgotten most of the facts.

Life offices are now, in the main, large and well-established institutions, with assets which, at market values, would show a substantial margin over balance sheet values. This margin is known as the ‘inner reserve’ or ‘hidden reserve’ because details of it are rarely disclosed, in fact its very existence is seldom brought before the public gaze; this secrecy is specifically allowed by the Companies Act 1948.

The backing given by these large inner reserves and the advantages deriving from the continuity of the office are two of the most important factors in formulating life office policy, and in the choice of actuarial bases of calculation; and yet these factors are almost always taken for granted, a tribute perhaps to the length and strength of the offices' past history.

Any discussion of mortality, fertility or migration assumptions for population projections, or of the various technical questions that arise, must be preceded by some consideration of the nature and purpose of the work that has been undertaken. It should be asked why such calculations are necessary and what is expected from the results. Fundamentally, inquiry about the future of the population arises from a natural curiosity about man's surroundings in space and in time, and this does not seem to have been damped by past failures to foresee developments; otherwise, long-range projections would by now have ceased to be made, in view of the poor prospects of success. Beyond this, however, there are to-day a variety of practical reasons for which it is deemed necessary to estimate the future numbers of persons in given areas with such accuracy as is possible.

An important factor in any actuarial problem is the number of lives likely to be involved. It is at first surprising therefore that so little attention is paid to the question of size in our pension fund literature, although a brief consideration of the historical development of pension funds gives some indication of the reason for this situation.

Superannuation provisions were originally introduced for the benefit of employees of the Government in the Civil Service and the Armed Forces. The pensions were allowed to emerge as charges on current revenues and no attempt was made to fund the liabilities. Towards the turn of the last century superannuation provisions were introduced for teachers (consequent upon the Education Act, 1870) and railway staffs, and for the staffs of certain municipalities. The movement grew slowly, although several large firms began to make pension arrangements at this period, and among non-government employers there was a greater tendency to fund the liability for pensions.

Since my own experience of the cycle of monetary operations I shall describe is concerned with banking, it would seem natural to commence by discussing the functions of a bank. There is no legal definition of a bank, but the term is generally understood to apply to an institution which accepts from depositors money withdrawable on demand, in practice generally by cheque. The cardinal point to bear in mind is that a bank is a trustee of other people's money which, apart from deposit accounts at 14 days' notice, is all repayable on demand. The fundamental consideration is that it is customers' money, apart, of course, from the comparatively small proportion of resources represented by shareholders' capital.

A bank's first duty, therefore, is to its depositors (not to its borrowers!), and the exercise of that trust requires that any depositor shall be free to come to his bank and withdraw all or part of the funds he has entrusted to it. Hence liquidity is of prime importance and a bank must arrange its structure of assets so as to comply with this requirement. But like other commercial concerns a bank must have earning power to cover its outgoings, reward its shareholders and try to satisfy its staff! So, consistent with liquidity, a bank's ‘quick’ assets are arranged to bring in as large a return as possible subject to this overriding limitation.

In the following remarks I have attempted to approach my subject (necessarily in a somewhat random fashion) from a slightly different angle from that of Clarke's text-book. It must be understood that any opinions expressed are personal ones, and have no official authority.

All insurance has as its basic conception the pooling of risks, but whereas ordinary insurance is an individual contract, and steps are taken to ensure that the insured person is a reasonably healthy life when the policy is effected, social insurance brings under its cover the whole population of a country, or at any rate some more or less well-defined section of the population, and its concern is that the bad lives or heavy risks should gain protection equally with the healthy lives and lighter risks.

Measures of marriage based on the data for one sex only do not give a fair comparison between two communities unless the relative numbers and ages of the sexes available for marriage are the same in both. Such a similarity is unlikely to exist; in 1921, for example, at ages 15–54 the number of marriageable women was 87% of the number of marriageable men in New Zealand but the corresponding ratio in England and Wales was 119%, and this fell to 113% by 1931.

W. S. Hocking discussed the balance of the sexes in Great Britain in J.I.A. (74, 340) and showed the extent of variations in the relative numbers available for marriage in England and Wales in this Journal (10, 24). It is desirable in a number of problems to be able to make some allowance for such variations, particularly in connexion with the measurement of reproduction, depending as it does on marriage intensity.

These notes are an attempt to answer a question which is often asked and might at first sight appear to be a simple one, viz. ‘What is the tax and estate duty position where a company effects an assurance on the life of an employee (or director) and what would be the position were the man to take out the policy himself, the company providing the premiums?’ The first point to settle is— ‘What is the purpose of the policy?’—and then, related to that, ‘What class of assurance is intended?’ Besides the tax and estate duty aspects there is the question of the respective rights in the disposal of the policy proceeds which may in some cases over-ride the financial considerations.

Where alternative procedures are available the relative financial effect can, in some instances, only be assessed by someone with an intimate knowledge of the accounts of the company and the affairs of the life to be assured. It is desirable in any case that the inquiries should be dealt with through the company's accountant. It might well be asked whether such matters are not entirely the province of the professional accountant. Whatever opinion may be held about this, the fact remains that life offices are to an increasing extent consulted on technical matters involving taxation and estate duty even by members of the legal and accountancy professions.

The theory of statistics, like other scientific theories, can be divided into three fairly distinct sections; they are (1) the foundations of the theory, (2) the superstructure, and (3) the principles of application. It is, of course, possible to divide the subject in different ways; for example, the word ‘methodology’ is sometimes used to cover part of (2) and (3). But I find it helpful in thinking about the subject to make the division that I have indicated.

It is my purpose to show that actuarial science is essentially a particular application of the theory of statistics, that there is nothing very special about its superstructure and that in its application to practical affairs the universal principles apply. But I first want to make it clear that while I include the combination of compound interest functions with other actuarial functions as part of actuarial science I do not regard finance and investment as part of actuarial science proper. While treating finance and investment as a part of economics I appreciate that they represent an essential ingredient in the equipment of the practising actuary.

This paper deals with some technical and statistical questions in sickness benefit insurance.

In § 1 there is a survey of the various theoretical approaches to the calculation of net premiums, and the calculation of gross premiums is discussed in §2.

Some methods of calculating rates employ two basic functions: the frequencies of new cases of sickness and the duration of sickness. The derivation of these functions from the corresponding statistical data is exemplified in § 3, which contains some results from an account of the experience of a Swedish sickness insurance company. Finally, § 4 deals with some mathematical formulae for graduating data of this type.

The concept of a power function in statistics often proves rather difficult to absorb. Johnson & Tetley (1950) develop the theory at some length, and Forster (1947) in an admirable article in this Journal provides some extra reading which is of necessity in rather general terms. It is hoped that the problem discussed in this article may prove a useful example of the application of the power concept to a particular problem and clarify the underlying theoretical ideas.

In any statistical test two kinds of error may arise. The first kind of error occurs when the hypothesis tested, usually called the null hypothesis, is rejected when it is in fact true. The second kind of error occurs when the null hypothesis is accepted when it is in fact false. The hypothesis with which the test is concerned will in general be denoted by Hi. In our case simple hypotheses will be considered which are concerned with the value of some parameter defining the probability law of the individuals.

The contracts discussed in this note are annuities certain provided by Life Offices. A schedule for income-tax purposes dividing payments into capital and interest determines the variable net sum paid by the office from year to year to the payee under each contract; it is the effect of this arrangement on the office to which attention will be drawn.

To comply with the Assurance Companies Acts, 1909–46, and the Finance Act, 1938, section 27, the office must pay the considerations for such contracts into a fund which is separate from the life assurance and life annuity funds, and which must not relate to business effected before 1 January 1938. This fund is taxed on an interest less expenses basis (subject to certain limitations), and the full standard rate of tax is applied. However, the office is permitted to withhold tax on the interest portion of each annuity payment from the payee, and to retain this tax provided its amount is exceeded by the tax suffered on interest credited to the fund, adjusted for the relief on expenses. Profits tax will be ignored in this note; the tax position of the office as a whole has such an intimate bearing on the problem that further discussion would be outside the intended scope of the present analysis.

In reference(1), some large sample significance tests and confidence intervals for mortality rates were presented. These results appear to be valid for the usual type of insurance data. Also, for an ordinary mortality study, application of a test or confidence interval does not require much additional work. This paper presents a review of reference (1) along with a discussion of the concepts involved. More extensive tables of significance tests and confidence intervals are included in this paper.

To apply the results of (1), the data must be subdivided according to the first letter of the surname of the person insured. According to present practice, such information is not recorded for a mortality study. Since this recording requires little extra effort at the initial stage of an investigation, however, such information could easily be incorporated into future mortality studies.

The purpose of this note is to consider in a simple way the effect of double taxation relief on overseas investments, with special reference to those of Life Offices transacting business in the United Kingdom. This deserves consideration in practice, because under certain circumstances the spreading of Life Office investment in equities between equities of companies resident in the United Kingdom and companies overseas may be more profitable than confining investments to those expressed in the currency of the United Kingdom. The amount of the relief allowed is of course material in determining how profitable such spreading of investments may be. It is not intended to deal with the many additional considerations such as, for example, the possibility of changes in exchange rates, which have to be borne in mind when investing overseas.

The volume of group life and pension scheme business transacted by life offices in this country is considerable, and is increasing steadily. A great deal of work is being spent in the preparation of quotations for such schemes. Readers may, therefore, be interested in a method of reducing to a minimum an especially uninteresting part of the work arising in connexion with quotations for schemes in which the cost of pension benefits for future service or of life assurance benefits is calculated on the ‘single-premium’ basis.

In these schemes it is customary for the amount of life assurance cover provided during any Scheme Year, and the amount of pension earned in respect of that year, to depend only on the salary of the member and not on his age. The employee normally contributes an amount, also fixed in relation to his salary, to purchase part of the pension by means of a deferred annuity providing a return of premiums without interest in the event of his death. The employer in each year pays the single premium required to purchase the balance of the pension accruing in respect of that year's service (usually by a deferred annuity without return of premiums in the event of the employee's death), and also to purchase the whole of the temporary life assurance granted during that year. The employer's premiums are recalculated each year on the basis of the ages attained in that year and thus for any individual employee the cost to the employer increases from year to year. Considering the scheme as a whole, this rise is partly or wholly compensated by deaths, withdrawals and retirements, and it may be quite reasonable to expect that the cost in the future will only rise by a small amount, or even decline.