The development of a ‘law of mortality’, a mathematical expression for the graduation of the age pattern of mortality, has been of interest since the development of the first life tables by John Graunt (1662) and Edmund Halley (1693). Although Abraham De Moivre proposed a very simple law as early as 1725 the best known early contribution is probably that of Benjamin Gompertz (1825). Since World War II mathematical formulae have been used to graduate sections of the English Life Tables, as well as assured lives mortality, and pensioner and annuitant mortality. Reviews of attempts at finding the ‘law of mortality’ have been given by Elston and Benjamin and Haycocks.

This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a ‘best estimate’; reporting the variability of claims reserves; and use in dynamic financial analysis models.

Most writers on the subject of life annuities have had occasion to lament the paucity of tables available for the performance of calculations involving two or more lives. The late Mr. David Jones has done much to supply this deficiency by the publication of complete sets of tables for two lives at various rates of interest; but, beyond this, it is extremely improbable that, under the present system, any considerable progress will be made, owing to the multiplicity of the different combinations when three or more lives are concerned, and the consequent magnitude of the task involved in the construction of complete sets of tables for such cases.

The original purpose of this paper—undertaken at request but not unwillingly—was to review the principles and practice of life-office valuations in the light of modern conditions. It was difficult, however, to deal satisfactorily with the principles of valuation in vacuo without reference to more fundamental principles. As a consequence the paper has become more ambitious in its scope than originally intended—and has threatened to run away with itself. The reader will perhaps be less disappointed if he is warned in advance that he is to be taken on a ramble through the actuarial countryside and that any interest lies in the journey rather than the destination.

This paper addresses the problem of longevity risk — the risk of uncertain aggregate mortality — and discusses the ways in which life assurers, annuity providers and pension plans can manage their exposure to this risk. In particular, it focuses on how they can use mortality-linked securities and over-the-counter contracts — some existing and others still hypothetical — to manage their longevity risk exposures. It provides a detailed analysis of two such securities — the Swiss Re mortality bond issued in December 2003 and the EIB/BNP longevity bond announced in November 2004. It then looks at the universe of hypothetical mortality-linked securities — other forms of longevity bonds, swaps, futures and options — and investigates their potential uses. It also addresses implementation issues, and draws lessons from the experiences of other derivative contracts. Particular attention is paid to the issues involved with the construction and use of mortality indices, the management of the associated credit risks, and possible barriers to the development of markets for these securities. It suggests that these implementation difficulties are essentially teething problems that will be resolved over time, and so leave the way open to the development of flourishing markets in a brand new class of securities.

In these days of continuous change in mortality, with the necessity of forecasting in many of our operations, a paper concerned with the graduation of mortality statistics may appear to be academic, the more so if attention is restricted to the fitting of mathematical curves, with or without any attempt to advance that elusive and, as some think, delusive conception, the law of mortality. Nevertheless, it is felt that if progress is to be made in any branch of science, theoretical research and experiment are essential, as also is the publication of the results, unless the work is completely abortive. It has become the fashion to speak of graduation as though its function were restricted to the provision of a practical instrument for monetary calculations; as Todhunter is reputed to have said, the tabulated values of lx are merely successive terms in a commutation column. Notwithstanding this, most of us retain, consciously or unconsciously, a feeling that, underlying all the roughnesses in our data referable to errors of observation and an ever-changing environment, there may be an inherent mathematical system of law and order, which if it could but be discovered would give such insight into the meaning of the unadjusted figures that a considerable advance would be made in the practical application of our science.

In this paper the ‘Wilkie investment model’ is discussed, updated and extended. The original model covered price inflation, share dividends, share dividend yields (and hence share prices) and long-term interest rates, and was based on data for the United Kingdom from 1919 to 1982, taken at annual intervals. The additional aspects now covered include: the extension of the data period to 1994 (with omission of the period from 1919 to 1923); the inclusion of models for a wages (earnings) index, short-term interest rates, property rentals and yields (and hence property prices) and yields on index-linked stock; consideration of data for observations more frequently than yearly, in particular monthly data; extension of the U.K. model to certain other countries; introduction of a model for currency exchange rates; extension of certain aspects of the model to a larger number of other countries; and consideration of more elaborate forms of time-series modelling, in particular cointegrated models and ARCH models.

1.1. The purpose of this paper is to present to the actuarial profession a stochastic investment model which can be used for simulations of “possible futures” extending for many years ahead. The ideas were first developed for the Maturity Guarantees Working Party (MGWP) whose report was published in 1980. The ideas were further developed in my own paper “Indexing Long Term Financial Contracts” (1981). However, these two papers restricted themselves to a consideration of ordinary shares and of inflation respectively, whereas in this paper I shall present what seems to me to be the minimum model that might be used to describe the total investments of a life office or pension fund.

In the following pages I shall have frequent occasion to avail myself of a term which the progress of the analysis of life contingencies has rendered indispensable, but which is not found in any of the standard elementary works in that science. I think, therefore, that I cannot better commence this paper than by an attempt to give an explanation of the expression “force of mortality,” sufficiently ample to obviate any difficulties which might otherwise be experienced on this score.

The purpose of the report is to achieve a greater understanding of the United Kingdom ‘cohort effect’ by exploring research in other fields and analysing population mortality data by cause of death in more detail. The ‘cohort effect’ in this context is the observed phenomenon that people born in the U.K. between 1925 and 1945 (centred on the generation born in 1931) have experienced more rapid improvement in mortality than generations born either side of this period.

In a Continuous Mortality Investigation (CMI) Bureau working paper published in 2002, a similar trend was noted in the mortality experience of male pensioners and males with life assurance policies. The CMI Bureau investigation showed peak rates of improvement for the cohort born in 1926. Interim projection bases for future mortality experience were produced as a result of the study. The projections made various assumptions about the extent to which the observed cohort effect would continue to shape the pattern of future mortality improvement.

This report suggests that it is highly likely that the cohort effect has been caused by a number of different factors in combination. Prevalence of smoking from one generation to the next has certainly been one such factor. Furthermore, an analysis of patterns of cigarette smoking suggests that there is a degree of inevitability in some element of likely future improvement, especially for mortality at older ages from conditions strongly linked to smoking.

However, trends in heart disease and breast cancer mortality suggest that smoking is not the only factor. The differences between lung cancer and heart disease trends by year of birth are especially interesting. The report shows that there are two ‘sub-cohorts’ of the 1925 to 1945 cohort: an earlier group where the improvements may be largely due to smoking; and a later one where other factors, such as diet in early life, may have played a greater role.

Historic patterns of smoking behaviour by socio-economic class provide an explanation for the five-year difference in the year of birth showing the fastest improvements, i.e. the difference between 1926 for the CMI Bureau investigation and 1931 for the general population. It is also notable that the second ‘sub-cohort’ of high improvement, applying to people born in the early 1940s, can be seen in both population and CMI experience.

A case study examining Japanese mortality experience shows that strong cohort trends can be projected well into old age. This does not provide proof that the U.K. cohort effect will do the same. However, it does counter arguments that year of birth effects will inevitably wear off with age. It is especially interesting given recent epidemiological research linking early life experience with markers of ageing.

There are a number of reasons to believe that the U.K. cohort effect will have an enduring impact on rates of mortality improvement in future decades. These include historical patterns of smoking behaviour and the impact of early life experience on health in later life. There appears to be little evidence to support the idea that the width of the generation experiencing rapid improvement will reduce with time.

This paper presents a statistical model underlying the chain-ladder technique. This is related to other statistical approaches to the chain-ladder technique which have been presented previously. The statistical model is cast in the form of a generalised linear model, and a quasi-likelihood approach is used. It is shown that this enables the method to process negative incremental claims. It is suggested that the chain-ladder technique represents a very narrow view of the possible range of models.

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.

The main object of this paper is to propound and discuss a new indifference rule for the prior probabilities in the theory of inverse probability. Being invariant in form on transformation, this new rule avoids the mathematical inconsistencies associated with the classical rule of ‘uniform distribution of ignorance’ and yields results which, particularly in certain critical extreme cases, do not appear to be unreasonable. Such a rule is, of course, a postulate and is not susceptible of proof; its object is to enable inverse probability to operate as a unified principle upon which methods may be devised of allowing a set of statistics to tell their complete and unbiased story about the parameters of the distribution law of the population from which they have been drawn, without the introduction of any knowledge beyond and extraneous to the statistics themselves. The forms appropriate for the prior probabilities in certain other circumstances are also discussed, including the important case where the unknown parameter is a probability, or proportion, for which it is desired to allow for prior bias. Before proceeding to the main purpose of the paper, however, it is convenient to provide some background to the subject. Reference is first made to certain modern writers to indicate how the problem with which inverse probability is concerned occupies a central place in the foundations of scientific method and in modern philosophy. In quoting from these writers I am not to be taken as suggesting that they necessarily support the inverse probability approach to the problem. The next section of the paper contains some brief comments on the direct statistical methods which have been devised in recent times to side-step induction and inverse probability, and this is followed by a few remarks on the various definitions of probability.

Increasingly, modern business and investment management techniques are founded on approaches to measurement of profit and risk developed by financial economists. This paper begins by analysing corporate pension provision from the perspective of such financial theory. The results of this analysis are then reconciled with the sometimes contradictory messages from traditional actuarial valuation approaches and the alternative market-based valuation paradigm is introduced. The paper then proposes a successful blueprint for this mark-to-market valuation discipline and considers whether and how it can be applied to pension schemes both in theory and in practice. It is asserted that adoption of this market based approach appears now to be essential in many of the most critical areas of actuarial advice in the field of defined benefit corporate pension provision and that the principles can in addition be used to establish more efficient and transparent methodologies in areas which have traditionally relied on subjective or arbitrary methods. We extend the hope that the insights gained from financial theory can be used to level the playing field between defined benefit and defined contribution arrangements from both corporate and member perspectives.

In the course of work undertaken as members of the Executive Committee of the Continuous Mortality Investigation Bureau in the preparation of graduated tables of mortality for the experiences of 1979–82, we have had occasion to make use of and develop a number of statistical techniques with which actuaries may not be familiar, and which are not fully discussed in the current textbook by Benjamin & Pollard (1980), though some of them have been referred to in previous papers by the CMI Committee (1974, 1976). We therefore felt that it would be useful to the profession if we were to present these methods comprehensively in one paper. We do this with the permission of the other members of the CMI Committee, who do not, however, take responsibility for what follows, whether good or bad.

Deaths and exposures by individual calendar year and individual years of age for the U.K. male assured lives experience over the recent past are comprehensively modelled using generalised linear modelling techniques. Our principal objective is to develop a model which incorporates both the age variation in mortality and the underlying time trends in the mortality rates. The approach has considerable advantages over ad hoc methods of fitting parametric models to represent the age variation in mortality and then separately attempting to represent the time trends in the parameters of these models. The approach advocated can be seen as an extension to the conventional parametric graduation techniques used by the CMI Bureau to represent trends in mortality.

This paper provides a detailed overview of variable annuities. Consideration is given first to the definition of the term variable annuity. Common terminology used in the variable annuity market is introduced. The current state of the United Kingdom and other international markets is described. Then, by reference to a simplified product, an analysis of customer outcomes, pricing, reserving, risk management and hedging is carried out. The paper ends with a description of current U.K. pensions legislation and how it potentially constrains product development.

Dynamic financial analysis (DFA) is a technique which uses Monte Carlo simulation to investigate the evolution over time of financial models of funds, complex liabilities and entire firms. Although of increasing popularity, the drawback of DFA is the dearth of systematic methods for optimising model parameters for strategic financial planning. This paper introduces strategic DFA which employs the only recently mature technology of dynamic stochastic optimisation to fill this gap. The new approach is described in terms of an illustrative case study of a joint university/industry project to create a decision support system for strategic asset liability management involving global asset classes and defined contribution pension plans. Although the application of the system described in the paper is to fund design and risk management, the approach and techniques described here are much more broadly applicable to strategic financial planning problems; for example, to insurance and reinsurance firms, to risk capital allocation in financial institutions and trading firms and to corporate investment and business development involving real options. As well as describing the mathematical and statistical models used in the case study, the paper treats econometric estimation, asset return and liability scenario generation, model specification and optimisation, system evaluation and historical backtesting. Throughout the system visualisation plays an important role.

Late-life mortality patterns are of crucial interest to actuaries assessing longevity risk. One important explanatory variable is year of birth. We present the results of various analyses demonstrating this, including a statistical model which lends weight to the importance of year-of-birth effects in both population and insured data. We further find that a model based on age and year of birth fits United Kingdom mortality data better than a model based on age and period, suggesting that cohort effects are more significant than period effects. The financial implications of these cohort effects are considerable for portfolios with long-term longevity exposure, such as annuities written by insurance companies and defined benefit pension schemes.

1.1. This paper describes a consistent and justifiable means of establishing adequate claims provisions in General Insurance. The topic has created widespread interest amongst actuaries, accountants and regulatory authorities. The issue of adequate provisions is of utmost importance to policyholders, whose justifiable claims must be paid, insurance companies who must be able to satisfy shareholders and make proper assessments of premiums, and regulatory authorities who must be satisfied that adequate provision has been made for all liabilities.