1. This note incorporates a large part of the evidence given to the Scott Committee by the Government Actuary's Department, with additional material by myself as appropriate.

The Pay Comparison system

2. The actuarial calculations in question arise from job comparability exercises, when the value of the total remuneration package of specified jobs is being compared. In recent years, assessments have been made in the following circumstances:

(i) Pay comparison studies for the Non-industrial Civil Service.

(ii) Pay comparison studies for the Industrial Civil Service.

(iii) Adjustments are made by the Review Bodies, who recommend salaries for the Armed Forces, Doctors and Dentists and Top Salaries. The Review Bodies consider various pension comparisons, but are not obliged to give reasons for their salary recommendations.

(iv) Advice to the former Standing Committee on Pay Comparability (Clegg) for staff groups referred to them.

(v) Compensatory additions to pay are quoted for a number of bodies whose staff are paid by reference to Civil Service and other public service rates, but who are in other pension schemes.

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.

This paper addresses certain aspects of management practice. As such, it is unusual if not unique in the catalogue of topics discussed by the Institute, and it is instructive to examine briefly why this may be.

Firstly, management and the skills associated with its successful practice are not specifically actuarial concerns. There are numerous platforms from which one may advance theories of management practice and, because management skills are of general application, those platforms are generally broader than the technical disciplines which draw us together as members of one profession.

Secondly, the opportunities for and the consequences of good management are necessarily imprecise, relating as they do to the enhancement of ventures of all kinds through the intelligent use of that most indispensable resource available to managers: the individuals on whom they can call to execute the consequences of their decision-making. The degree of imprecision is so great that the practice of management is frequently termed an art rather than a science. The net of actuarial science has a mesh of a shape to catch a number of philosophies (such as the concept of Equity) which are not purely value-based—but the philosophy of human-resource management has never been one of these, escaping from the sweep of our professional scrutiny to be caught (if at all) in the trawl of the behavioural sciences. It is certainly possible to succeed in the written examinations leading to an actuarial qualification without any consideration of the aptitudes, skills and resources needed for good management.

The paper describes in some detail the statistical methods that may be used for analysing mortality data from medical and epidemiological studies, with particular reference to quantifying survival and the risk of mortality and to comparing the experience with a standard.

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).

In my 1981 paper ‘The flock and the sheep’ (J.I.A., 108,361) I explained that I had been writing a number of historical essays and that the paper was based on those which were about distribution of surplus. Some of the remaining essays fall naturally into a set under the general heading of valuation and it is these which have been combined into the present essay. It also includes something which has been on my conscience for a number of years—a critical examination in the light of history of my 1952 paper on the principles of valuation (J.I.A., 78, 286).

The complete expectation of life at birth ė0 is frequently used as a measure of the level of mortality of a population. It is also used for assessing trends in mortality and trends in mortality differentials. Although the relationship between mortality and expectation of life is essentially reciprocal, the exact connexion is rather more complicated, and becomes important when, for example, trends in differentials are analysed.

In this paper, the relationship between mortality and expectation of life is explored in some detail, and formulae are developed for analysing the effects of mortality changes on expectation of life, and trends in mortality differentials on ė0 differentials.

Unlike Keyfitz (1977), who concentrates on the proportional change in ė0 corresponding to equal proportional changes in mortality at all ages, we study the relationship between absolute changes in mortality, generally different at different ages, and the corresponding absolute change in ė0.

It is demonstrated that two populations may experience diminishing mortality differentials and at the same time widening ė0 differentials. Numerical examples are given using Australian data over the periods 1921–71 and 1971–79.

Although the formulae in the paper relate solely to the expectation of life at birth, the methods and formulae are readily adapted to expectations of life at other ages and indeed, temporary expectations.

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.

1.1. This paper explores the possibility that a company which sells unit-linked policies with death and maturity benefit guarantees may follow an investment strategy which reduces the risks associated with them. It originated when the author was requested by the Institute to study papers on the subject by P. P. Boyle, M. J. Brennan and E. S. Schwartz of the University of British Columbia in Canada and also a paper presented to the Society of Actuaries in Ireland by J. C. Fagan.

In an earlier introductory paper (J.I.A.107, 513), Balzer and Benjamin (1980) presented a model of an insurance system with delayed profit/loss sharing feedback from insurer to insured. The system was considered from a general control and dynamic systems theory viewpoint. Certain fundamental concepts were introduced and the dynamic responses of cash flow f(k) and accumulated cash flow fa(k) to an isolated group of unpredicted claims presented. This paper takes the analysis further in a number of important ways.

The general structure and details of the mode1 of the insurance system can be found in the earlier paper.

1. The English Life Tables Nos. 8 to 13 are based on the population figures by age obtained at the successive censuses of England and Wales from 1911 to 1971 and the deaths recorded in the three calendar years centred on the census year. Over this 60-year period there have been changes in the information sought regarding age at census and death registration and improvements in the exposed-to-risk formula; these are summarized in Table 1.

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.