1. This paper originated with a sense of irritation at the way in which the graduation of age rates of mortality has become more and more complicated by the apparent determination to assume some underlying law of mortality, a law which could not possibly be deduced from observations in the past or present man-made conditions of dangerous living. It then developed into a consideration of graduation from a less frequently adopted point of view, i.e. a look at the distribution of lengths of life (the so-called curve of deaths). This led inevitably to a review of the long-pursued controversy as to whether or not there is a predetermined biological maximum to the human lifespan under ideal conditions and beyond that to a review of the biological evidence for this maximum and the possibility of extension by environmental or medical intervention. Finally, this led to the likely results of such environmental and medical advances and to some experiments in mortality projection.

1.1 On coming into pensions consultancy after many years in a life office I looked for an indication of current thinking on the valuation of funded pension schemes. Official guidance by the profession is confined to the 1969 booklet from the Institute and Faculty; the memorandum on professional conduct concerns itself with independence and how we present ourselves. The Institute's examination textbooks pass over the main objectives lightly and concentrate on detail. While there have been papers on many aspects of pensions, they all seem to skirt the fundamental principles of valuing final salary schemes.

1.2 This paper attempts to go some way to filling the gap. It provides an opportunity to review and discuss current practice. I have found it particularly illuminating to compare the way in which practice between the United Kingdom and North America diverges.

Various numerical procedures have been developed in recent years for estimating the provisions for outstanding claims of a general insurer. (See, for example, Benjamin, 1977, 234–66.) These procedures do not in general have a theoretical statistical basis, and consequently they do not provide information about the reliability of the resulting outstanding claims estimates. An exception is the complex procedure of Reid (1978).

It is not surprising that actuaries have been daunted in their quest for an appropriate statistical model: the claim process is very complicated, particularly for ‘long tail’ business. One has to consider the incident leading to a claim, the size of the resulting claim, the time to settlement, and the distribution of payments up to the time of settlement both by amount and epoch. There are also the additional complications of inflation and investment earnings on the provisions.

In this paper, a theory is developed in which the complications of the claim process are overcome by considering the joint distribution of payments for a single claim in successive development periods. The model is distribution-free in that it does not assume specific distributions for claim size, time to settlement, or payments prior to settlement by amount and epoch. The method requires that payments which have been made be expressed in ‘constant-dollar’ terms (i.e. adjusted for past inflation).

We will be considering a productive investment project or financial security which yields a sequence of cash flows, positive or negative, over time. Let a1 (dollars) be the cash flow from the project at time t, where t takes the values 0, 1, 2,…, n. Given the known cash flows at from the project, and a known market rate of interest, i per period, at which money may be borrowed or invested, a common procedure is to accept the project if its present value P is greater than zero, where

An internal rate of return of the project is defined to be a solution of the equation

in (− 1, ∞), if, of course, one exists.

Suppose we have a finite number N of observed frequencies [a(i)], and suppose further that we have a ‘reasonable belief’ that the (truncated) Poisson distribution could be used to describe those frequencies. The parameter μ (which represents both the mean and the variance of the entire unit Poisson distribution) and a suitable scaling-factor H would, in some way, be estimated from the data. In the normal course of events, the chi-squared test would probably be used to test for ‘goodness of fit’, and the number of ‘degrees of freedom’ would reflect the extent to which μ and H had been directly estimated from the data. Depending upon the degree of significance considered appropriate, the test would be used to decide whether or not a ‘reasonable fit’ had been obtained, and whether or not a particular formally-defined hypothesis should be accepted or rejected.

In the above note I suggested that the improved accuracy of E.L.T. No. 13 as compared with E.L.T. No. 12 was largely because E.L.T. No. 13 was based on census and death registrations at which the date of birth had been asked for instead of age. Since writing that note I have come across interesting evidence that more accurate results are obtained when date of birth is asked for at a census than when age is required.