This paper discusses the problem whether an office should issue with-profit deferred annuity contracts, and having concluded that with-profit contracts should be issued, it discusses a consistent system for the calculation of premium rates, the valuation of the business and the distribution of profits. Primarily, however, it describes what has been done in one particular case and refers but briefly to some other possibilities, there being no doubt that the discussion will embrace other systems of distribution. The approach in this particular case started from a large existing portfolio of non-profit group deferred annuity contracts and the system of distribution was designed to be applicable to group business. In the result, however, a system has emerged which could be applied to individual contracts and which has been extended to embrace group deferred annuity contracts carrying the option to take a cash sum at the maturity date. In the latter application it is sound under different systems of taxation.

In March 1944, while the second world war was still raging, the Royal Commission on Population was appointed ‘to examine the facts relating to the present population trends in Great Britain; to investigate the causes of these trends and to consider their probable consequences; to consider what measures, if any, should be taken in the national interest to influence the future trend of population; and to make recommendations’. This step was of twofold significance. First, it marked the recognition by the Government of the possible need to take policy decisions in the field of population—to translate the subject from the academic to the political plane. Secondly, it marked official recognition of the fact that despite a flood, during the immediate prewar years, of reports of grave foreboding by the demographers of the day (whose anxieties have since proved to have been exaggerated) the Government had insufficient information to decide whether or not there was a 'population problem' in Great Britain. That the problem does not now appear to be so pressing as was once thought does not abate in any way the necessity for observing the facts. The Royal Commission, indeed, emphasized the necessity for continuous study of the population problem which, they said, will always be changing.

We have chosen Investment Policy and Index Numbers as the title of this paper, because the origin of the Actuaries' Investment Index is to be found in a paper, The Statistical Groundwork of Investment Policy, by the late C.M. Douglas (T.F.A. 12, 173).

The main policy consideration discussed by Douglas was the extent to which offices should invest in equities. Douglas maintained that funds should only be invested in equities if greater security was thereby obtained. To establish the suitability of equities, it was necessary to study the relative variations in equity prices, debenture prices and business activity. For this purpose, index numbers were required.

From a study of index numbers covering the period 1924-28 Douglas concluded that share prices were related to the level of business activity, i.e. to the progress of the ‘trade cycle’, equity prices following the cycle directly, and debenture prices following the cycle inversely, but with a different tempo and interval. If, therefore, an investor could judge the future trend of business activity, a sound investment policy could be pursued.

In this paper we propose to review some of the developments that have taken place in the principles and practice of investment policy over the last twenty-five years, and to study the contribution to these principles that the Actuaries' Investment Index and other index numbers can provide. In addition, we propose to review the methods used in the construction and maintenance of the Actuaries' Investment Index in the light of current conditions.

The standard table of mortality most generally used in Britain at the present time for life assurance calculations is the A 1924–29 table, which was derived from the experience of assured lives during the period 1924–29. During the quarter of a century that has elapsed since that time there have been substantial changes in mortality rates and the A1924–29 table is to-day out of date. The publication of the A 1924–29 Light table, based upon the experience of certain selected offices whose mortality was lighter than average, has no doubt been helpful in providing a table which goes some way in the direction of the lower mortality rates of to-day. However, this table suffers from the disadvantage that, while the mortality rates at young ages are not low enough for current experience, at some of the older age-groups the rates are too low, with the consequence that the shape of the mortality curve does not accord with present conditions. The greatest proportionate reductions in mortality since 1924–29 have occurred at the younger ages, and there can be little doubt that a new table is needed to provide offices with an efficient and up-to-date instrument for life assurance calculations.

Many years ago it was suggested—I cannot remember by whom—that instead of graduating lx or qx or some such function it might be preferable to graduate annuity values. When I was fitting frequency curves to many kinds of statistics some fifty years ago, I noticed that the expectation of life from birth onwards looked something like the ordinates of a very skew frequency curve with a mode at age 4 or thereabouts. Partly because it seemed difficult to bring expectations of life within a definition of frequency distributions and partly because the example was neither a straightforward graduation nor helpful to the work on which I was then engaged, the notes I made were put aside and have long since been destroyed. From time to time, however, I have wondered whether life annuity values and expectations of life could be expressed roughly by one of the algebraic expressions used for frequency curves, and whether there would be advantages in working from joint expectations of life for n lives of the same age rather than for one life. Such a device, or the use of annuity values, may be taken to mean that we are using a series of values that could represent an expectation of life from a mortality table with heavier but related mortality. Approximations even if they are as rough as I had to anticipate in this case may be useful in unexpected ways, and that thought supplied me with some sort of justification for taking up the subject again and making some fresh calculations.

It is well known that a principle of uniform seniority applies to mortality tables which follow Gompertz's or Makeham's law, and, in a modified form, in the case of certain other laws, such as Makeham's second law and the double geometric law. It is natural to inquire what is the most general class of mortality laws to which such a principle applies. In its most general form the uniform seniority principle implies that the value of a joint-life annuity on m lives of different ages is equal to that of a joint-life annuity (possibly computed at a different interest rate) on k lives of equal age, in such a way that the new interest rate i′, the number of substituted lives k, and the difference between the youngest age and the substituted equal age, depend only on the various differences in age between the youngest life and the other original lives, and not on the actual age of the youngest life.