With the appearance of the Report of the Royal Commission on Population and several volumes of ancillary papers, it may seem that the potentialities for fruitful statistical analysis of recent marriage and fertility data in this country have been exhausted, and that little fresh work can usefully be done on them until the extent of the experience has been enlarged by the passage of time.

There is, however, a wide area for statistical investigation that appears not yet to have been completely charted, in spite of the penetrating studies made by those who worked on behalf of the Royal Commission. The available data offer scope for the formulation of new techniques of analysis and presentation, and for examination by approaches different from those made by the Commission's demographers, which had, in the main, the particular purpose of ascertaining whether or not the people of this country are replacing themselves.

We may approach our subject by considering how we are to define probability. This at once raises a number of questions which may be classified under a few broad headings in the following way. First, is the definition to be subjective? By a subjective definition we shall mean in this paper one that admits the possibility that two persons in the same state of relevant knowledge may reasonably hold different views as to the probability of a given event. Secondly, is probability to be only a theoretical concept, or must our definition be such that a probability can be determined by practical experiments and applied to practical problems? Thirdly, are we to embrace all types of event or only a restricted class, e.g. those that occur or can be imagined to occur on a large number of occasions? Fourthly, shall we define probability as a number or are we to admit the existence of a non-numerical probability? And if we stipulate that it is a number, must it be a rational one?

Only a small proportion of examination candidates have practical experience of reversions and life interests, and the purpose of this paper is to indicate some of the practical points which arise in transacting this class of business; the selection of appropriate bases for mortality, interest, valuation of funds and Estate Duties do not fall within the scope of this paper. Reference to strictly legal aspects are no more than incidental.

Since 1945 there has been, on the basis of new business figures, a marked revival in reversionary business defined as covering purchased reversions and life interests, and loans secured thereon. On balance-sheet values the reversionary business of British life offices in force at 31 December 1948 was rather more than £15m. There was approximately the same volume at 31 December 1943, but at 31 December 1938 there was about £2m. more in force. Over the 10-year period loans have decreased by about £3½m. to only little more than £5m., but purchases have increased by about £1½m. to more than £10m. These figures can give no more than an approximate picture of market value through the different methods of arriving at balance-sheet values, but they serve to show that reversions and life interests continue to form an important part of the assets of certain British life offices.

With these words Sir George Maddex, in his Presidential Address, directed our attention to the present system of taxation of annuity funds. M. E. Ogborn (1), on 27 October 1947, touched on the subject of uneven incidence of tax upon differing funds, and some uneasiness about the consequences was manifest in the discussion on that paper.

Much has been written in recent years upon the general subject of taxation of annuities. S. J. Rowland and F. H. Wales (2), A. H. Shrewsbury (3), and M. E. Ogborn (1) have fully described and commented upon the present method. Relatively little attention, however, has been devoted to the effects of the system upon annuity funds, probably because it is only recently that we have become aware of its full implications. There is some need therefore to review the consequences and to investigate more closely the actuarial problems which follow.

It is proposed here to analyse the effects of the system and, proceeding from some simple basic principles, to point to a method of solution of the actuarial problems connected with valuation, investment and the terms for new business and withdrawals.

The object of this note is to provide an introduction to methods used in time-series analysis, together with a general appraisal of their usefulness. The treatment is purely expository and, of necessity, rather condensed. Nothing more than a broad survey of the subject is attempted; those interested in the technical details of theory and application should consult the references given at the end.

The term time series is applied to any series of observed values of some character when the order of the items in the series, as well as their actual magnitudes, is of importance. Examples of such series are common in statistical practice, and the following may be cited in illustration: economic series such as weekly, monthly or annual production figures, prices or index numbers; series of results of routine inspection and testing of successive batches of manufactured product; continuous blood-pressure records; results of psychological fatigue tests expressed as the rate of working on some standard task in successive 5 sec. intervals. In all these examples the factor which determines the order of the observations is time. The study of ordered series of observations is entitled ‘Time Series’ because in a large proportion of applications time is the ordering factor. Nevertheless, it is not necessary that the series be ordered according to time.

Actuaries are closely interested in the economic state of the nation, both as ‘scientific financiers’ and as citizens of a democracy, but this interest involves them in a dilemma: although it is possible to obtain only a hazy picture of the whole, even by devoting all of one's time to the study of economic affairs, yet for the majority this subject can be only a sideline. Each actuary must therefore make his own decisions, first as to how much time he can spare, and second as to which aspects of the nation's economy he might most profitably study. These two decisions will depend on the particular actuary's interests and employment, and on the state of the nation's economy at the time.

This paper discusses one selection of economic data, comprehensive enough to give a reliable impression of the nation's economy, yet short enough to permit frequent examination.

Although the subject of probability is of such vital importance to insurance, no great detailed study was given to it until the sixteenth century. This was nearly two thousand years after the world's first insurance venture, that of Antimenes on behalf of Greek slave-owners against loss of their slaves (the early Greeks had also a good system of marine insurance). As in many fields of study, the early students of probability met with a great deal of opposition, which must have extended well into the nineteenth century, judging from the evidence of de Morgan's book of 1838.

The actual system of book-keeping used by an office will depend on many factors, such as the volume of business transacted, the departments into which the office is divided and the scope of their work, and the information required to be produced. This paper deals mainly with premium income accounts, as the production of the revenue account figure for premiums is more complex than that of other revenue items, and as more departments in an office are concerned with premiums in one connexion or another than with the other revenue items. The object of the paper is, therefore, to show the relationship between the requirements of various departments, i.e. the accountants, the valuation (Actuarial) department, the departments concerned with issuing policies, with collecting renewal premiums and with settling claims and surrenders, etc., and the different aspects peculiar to each. For instance, the accountants require premiums to be classified according to funds, e.g. Life, Annuity or Capital Redemption, whereas the department concerned with the collection of premiums requires them to be classified according to the method of renewal, e.g. monthly or non-monthly, or terms of non-forfeiture. The actual books of account will not be described in great detail but only the principal objects that are required of each book, which would be ruled to suit the workings of the office; and the specimen headings given in the Appendices to the paper are merely to illustrate the system of book-keeping being described, which relates to an Ordinary Life Office.

When I was invited to address the Society on some legal subject of my own choice, it was natural perhaps that my thoughts should go back to my first appearance before the Society some 25 years ago. My subject then was ‘Children's Deferred Assurances’, which was appropriate from a personal point of view—I had just become a proud father! My present subject is ‘Estate Duty’, and I am not asking anyone for a q factor to show me how appropriate it may be from the same standpoint!

It is bold for anyone outside the Estate Duty Office to speak on estate duty; but I have deliberately chosen a dull and uninteresting subject because of its recent added importance to all who are concerned with life policies and annuities. The maximum rate of duty under the Finance Act, 1894, which first imposed it, was 8%. The maximum rate to-day is 80 %. Actuaries and lawyers must clearly do something about it—if they can—for otherwise there will be still less of estates left to be frittered away among the beneficiaries!

A Great deal of ink has been spilt during recent yearsonthe subject of the trend of the birth-rate and whether the population of this country will be maintained or will decline. Relatively little attention has, however, been paid to one of the factors that has a great deal to do with the course of births, viz. the supply of potential parents of both sexes. It is sometimes assumed that the position in this respect remains ‘normal’, whereas rather startling changes have occurred in recent years in the size of the reservoir of unmarried adults from which marriages are drawn, and it may be concluded from these that a considerable fall in the number of marriages, to be followed by a consequential fall in the number of births, is almost inevitable during the next few years.

The following communication is not an original paper, but the summary of an article published by the author in collaboration with H. Zimmermann in the Bulletin de l'Association des Actuaires Suisses, L, pt. 2 (1950). As the method of reserve calculation by grouping described below assumes a fundamentally new point of view and combines substantial advantages with great simplicity, it may, however, interest those actuaries who have no access to the original paper.

As is well known the individual valuation of every single policy, even for a medium-sized portfolio, is a long and arduous task. Therefore different methods for the valuation of policies grouped in certain ways have been in use for a long time. In this respect it is possible to distinguish methods giving the exact total of the individual reserves (e.g. the so-called Altenburger or Karup or valuation constant method) and those leading to approximate results (e.g. the Lidstone or Z-method). But most group methods have the evident disadvantage of a grouping which is in other respects inappropriate for the classification of the individual policy within the portfolio. Thus, in the valuation constant method, groups of equal age attained have to be formed, and the Z-method requires groups of the same remaining term. Yet one would much prefer a grouping which necessitates no sorting of the policies contrary to their natural grouping. Thus the putting together of all policies with the same year of entry for the purpose of collective reserve calculation is in some way a natural manner of grouping, and this was the procedure used in the t-method formerly proposed by the author.

In a recent pension fund valuation, which involved splitting the fund between two groups of members in proportion to the net liabilities, the writer found that he had to calculate a large number of annuity values for a variety of ages between 60 and 80 at several rates of interest with guarantee periods ranging from 1 to 10 years. In view of the nature of the valuation a broad adjustment was not good enough, and the problem arose of how to reduce the calculations to a minimum and yet retain an acceptable degree of accuracy in the approximation. It is thought that an account of the method which was evolved may be of interest and of use to readers of the Journal.

The gross and net reproduction rates differ only on account of the mortality factor and the difference, at present, is quite appreciable. If we are investigating the possibilities of increased reproductivity we would naturally study the causes of the difference between these rates and the likelihood of that difference reducing and thus increasing the reproductivity. In this paper we will determine the extent to which this difference is caused by mortality in different age groups and mortality due to different causes.

In a paper by the present writer (J.I.A. Vol. LXXV, p. 151) the rates of mortality for Australia for different ages and different causes of death were determined. The same material will be used in the present analysis where, assuming constant (1938) fertility and using 1935 mortality, we shall determine the difference between the gross and net reproduction rates and the extent to which this difference results from mortality at different ages and from various causes of death. It should be emphasized here that the many arguments raised against any analysis based on causes of death (transfers between causes, variations in diagnosis, selecting a single cause from multiple causes, etc.) do not apply with much force to the causes of death at the ages that are, in this problem, of most interest.

Heavy taxation by Estate Duty and Income Tax has recently led to a serious consideration of the avoidance of this onerous liability by means of gifts made by a donor in his lifetime and, in the case of a settled fund, of an exchange of benefits where the Beneficiaries can come together and bring the trusts to an end.

If, in the case of a Settlement, Life Tenant and Reversioner can by mutual agreement arrange to divide the Fund between them, the Life Tenant may benefit considerably by realizing a sum in excess of the market value of his interest and using it to provide as much or more spendable money than before. He may also promote a smaller liability to Sur-Tax. The Reversioner may succeed in securing a cash sum larger than he could expect at a future date on the death of the Life Tenant reduced as it would be by Estate Duty at the aggregable rate applicable to the Life Tenant.

From time to time members of actuarial staffs are faced with the valuation of some class of business which, when carried out in full, is very tedious or which represents such a limited liability that a detailed valuation is somewhat out of place. Many methods of approximate valuation have been devised and remain hidden in the records of individual offices—amongst them those relating to Joint Life and Survivorship Annuities. The method described in the following pages was devised to enable an accurate valuation of that business to be carried out in about two hours each year, instead of having to rely on interpolated values for the years between full individual valuation at quinquennial intervals.