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has elapsed since that time there have been substantial changes in mortality rates and the A1924–29 table is to-day out of date.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.