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.

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.

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 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.

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.

The object of this note is to suggest a method, believed by the writer to be new, whereby once a pension fund contribution has been calculated the revised contribution occasioned by a small change in the rate of interest can be approximately determined with great ease and rapidity.

Let 100cx be the percentage contribution for entry age x required to support a pension per annum equal to 100k % of pensionable salary for each year of service. Assume for simplicity that normal retirements and early retirements (if any) can together be sufficiently allowed for by the use of a single average retirement age (M).

The purpose of this note is to establish, by finite difference methods, some of the properties of two probability distributions: (1) the distribution of the sum of n independent variates, for each of which every value in the interval (0, 1) is equally probable; and (2) the distribution of the sum of n independent variates, each of which must take one of f equally probable values, 0, 1, 2, …, f–1. For brevity we shall refer to these distributions as (1) the continuous case and (2) the discrete case. The continuous case is important because it gives a solution to a problem that often arises in practical computing: ‘If n numbers, which are all “rounded-off’ at the same [decimal] place, are added together, what is the chance that the total will be the correct (rounded-off) total, or will be 1, 2, etc., different in the last place from this correct (rounded-off) total?’ (Forster, 1947).

Laplace (1776) gave a proof of the continuous case; and later (1812) gave a proof of the discrete case, from which, by a limiting process, he derived the result for the continuous case. The result for the discrete case had, however, been published much earlier by de Moivre (1711), and proofs had been published by de Montmort (1713) and de Moivre (1730). Laplace's proofs are long and difficult to follow; but, as far as I can discover, no later alternative proofs were put forward until Rietz (1924) gave a much simpler proof of the continuous case. Irwin (1927) used characteristic functions to prove the continuous case; and Hall (1927) gave a remarkable geometrical proof of the continuous case and derived a general expression for its moments.

In his paper (J.S.S. Vol. VI, p. 172) Dr Wishart showed that the variance-ratio test can be very easily used with the aid of the appropriate tables. It may not have been obvious to the reader of that paper that, unlike most distribution functions, the one in question can be evaluated by elementary methods and that a simple formula can be established to give the required probability independently of any table. It is not suggested that the use of the formula to be demonstrated in this note is preferable to the ordinary employment of tables. It may be remarked, however, that in certain cases where a probability has to be directly calculated the formula has a big advantage, since the tables give only the values of F (or z) corresponding to a few specific ‘probability points’ and interpolation between these must be very unsatisfactory, as is obvious from a glance at Table 3 of Dr Wishart's paper.