More than 200 years ago, on 30 April 1760, Daniel Bernoulli (1766) read a memoir to the Royal Academy of Sciences in Paris entitled Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir (see Bradley, 1971, for a translation). In this remarkable memoir Bernoulli produced the first double decrement life table and one of the related single decrement tables, as well as deriving a mathematical model of the behaviour of smallpox in a community. This model was the forerunner of considerable developments in the mathematical theory of infectious diseases, a description of which is given in N. T. J. Bailey (1975). During the half century following Bernoulli's memoir there were a number of papers by other authors on the subject of that memoir; these, and the original memoir, seem to be little known to actuaries and are the subject of the present paper. They could have been the starting point of the actuarial development of exposed-to-risk formulae, but in fact were not.

In this paper, we examine the hedging performance of the new FTSE 100 Share Index. First, we reconstruct a back-history of the monthly and daily index values using the London Share Price Database. Then we analyse the statistical characteristics of the Index, comparing it with the FT–Actuaries All Share and FT Ordinary Indexes. Finally, we examine the sources of undiversification within the FTSE Index. We conclude that the market is likely to track the FTSE Index to within a close tolerance, and that the residual tracking error is attributable partly to sector imbalances, but more importantly, to imbalances in exposure to smaller capitalization stocks.

Our earlier paper ‘The money supply, economic management and the gilt-edged market’ (J.I.A.96, 1) introduced the money supply as being ‘of importance to investment, in general because of possible changes in the economic management in the United Kingdom, and in particular because of the possible implications for the gilt-edged market’. To a large extent these changes have come about; the Authorities are now placing greater emphasis than in 1969 on controlling the money supply and support for the gilt-edged market was withdrawn when the Bank of England published Competition and Credit Control in May 1971.

2. The present paper is more fundamental for investment. The introduction to the earlier paper referred to our main research project into the uses of the financial statistics which were first published in 1962 as a result of the Radcliffe Committee Report. Changes in the management of the economy and the gilt-edged market were side issues, however important; the present paper describes some of the interim conclusions of the main project which is still continuing.

In this paper we examine two aspects of extreme events: their calculation and their communication. In relation to calculation, there are two types of extreme events that are considered.

The first is the extent to which extreme events in two or more variables occur together. This can be gauged by using measures of tail association. Higher levels of tail association are useful for highlighting the extent to which there are concentrations of risk. We investigate the range of approaches used to measure tail association and propose a pragmatic alternative, the coefficient of finite tail dependence.

The second type of extreme event arises from combinations of losses from a series of risks that together result in total losses exceeding a particular level. This is measured using ruin lines or, in higher dimensions, planes and hyperplanes. The probability of ruin and the economic cost of ruin are considered here. In this context, it is important to consider what the term “loss” actually means, and whether it is in relation to a current set of exposures or a potential strategy.

The communication of extreme events is discussed not just in terms of the numbers that can be used, but in terms of the graphical methods that can be used to aggregate information on a range of risk combinations. This involves communicating not just the level of risk but also the importance of the risk considered.

The term ‘investment risk’ is often used loosely, and frequently confused with the notion of short-term price volatility, particularly for equity instruments. For the long-term investor, however, what is most apposite is the ability to meet future real cash flows as they become due. This paper addresses the concept of economic fundamentals of long-term investment, the objectives of long-term investors (and how these differ from those of short-term investors), the notion of real value shortfall risk, what is meant by an investor’s risk capacity (as opposed to risk appetite) and liquidity management considerations. Subsequently, some of the constraints and barriers to appropriate risk measurement and management are considered, in particular the regulatory and behavioural biases that are overlaid on fundamental asset/liability management. Various alternative approaches to measuring risk, and their appropriateness for purpose, are outlined, in the hope of further informing the discussion and thereby helping to accelerate productive change.

The last 12 years have seen the evolution of a new funding regime under the supervision of the Pensions Regulator. Over this period, there has been significant turbulence in financial markets, including record low interest rates. This paper takes a critical look at the development of funding approaches and methodologies over this period. It analyses the Pensions Regulator guidance and how scheme specific actuarial methods have emerged since the move away from the Minimum Funding Requirement in 2001 and the introduction of the Scheme Specific Funding Requirements in 2005. It asks whether these new methodologies have been successful from the perspective of members, trustees, employers and shareholders. At a time when actuarial valuation methodologies have faced considerable criticism, this paper aims to propose a pension funding methodology which is fit for purpose and also reflects the latest guidance from the Pensions Regulator on integrated risk management.

The only treatise of any value on this subject that I am acquainted with is a paper by Mr. Peter Gray, in vol. xiv of the Journal of the Institute of Actuaries. Considering the magnitude of the pecuniary interests frequently involved in transactions of the nature referred to, it is somewhat surprising that Mr. Gray's valuable labours have not, ere this, been followed up by contributions from other writers on the doctrine of Interest and Annuities. This, I think, is much to be regretted,—and must form my apology for the following attempt to add something to what Mr. Gray has given us.

The development of stochastic investment models for actuarial and investment applications has become an important area of interest to actuaries. This paper reports the application of some techniques of modern time series and econometric analysis to Australian inflation, share market and interest rate data. It considers unit roots, cointegration and state space models. Some of the results from this analysis are not reflected in the published stochastic investment models.

This paper explores the application of ERM-style techniques to pension funds. It uses the term ‘entity-wide risk management’ rather than ‘enterprise risk management’, even though both have the same acronym (‘ERM’), because many pension funds do not view themselves as business ‘enterprises’ as such. Some of the techniques that business enterprises have for managing risk (e.g. raising new capital from shareholders or branching into new business areas if existing ones have unattractive risk-reward characteristics) may not be open to many pension funds. The paper argues that the holistic approach to risk management (and governance) that is a hallmark of ERM is as appropriate to pension funds as it is to any other type of entity. This is the case whether the fund is defined benefit or defined contribution in nature, or a hybrid. It is also the case whether the ‘entity’ is deemed to be the fund itself, the sponsor or the two combined. Indeed, there are aspects of pension arrangements, such as the relationship between the fund and its sponsor, that lend added impetus to the use of ERM-style techniques in practical pension fund management.

This paper is concerned with attempts that have been made to incorporate measures of risk in the selection of financial investments. The methodology that is developed here has been primarily aimed at the evaluation of capital investment projects, but the concepts and ideas are also relevant in the financial investment field. Implicit in all this discussion is the fact that when information is obtained in order to calculate an internal rate of return or a net present value, the figures put into the calculation are estimates rather than precise and exact quantities. In some instances the estimates may be felt to be very good, whilst in other cases it may be felt that there is a wide range of possible deviations. Hence the evaluation of any project or projects should, correctly, be described not by a single criterion but by a range of possible values, some of which are judged to be more likely than others. Having said this, there now comes the need to decide upon the method by which choice is to be exercised. Should it be by the choice of the highest average value of the criterion, or by choosing some more conservative rule?

This paper examines international developments in life insurance generally accepted accounting practice (GAAP) for policy valuation and profit recognition in four major Anglo-American markets—the U.K., Australia, the U.S.A. and Canada. Each valuation method examined has its advantages and disadvantages with respect to the needs of preparers and users of the annual corporate reports of life insurance companies. The paper documents that the statutory basis and U.S. GAAP are considered to have substantive deficiencies. In contrast, the U.K. accruals method, the Australian margin on services method and Canadian GAAP have much to commend them, particularly with regard to their flexibility to accommodate valuation adjustments for unexpected events. Nevertheless, from the preparers' point of view, the systems which would have to be developed to facilitate the U.K. accruals and Australian margin on services methods would be difficult and costly to implement. Profit reporting under Canadian GAAP is also sensitive to changes in actuarial reserving assumptions. The authors conclude that, since national preferences in actuarial and accounting practices are inevitable and because the product-market structures of life insurance markets are so distinctive, international harmonisation of life office GAAP is unlikely to occur for a very long time.

The paper considers the valuation for solvency purposes of traditional long-term insurance business. It concentrates on without-profit business, and discusses the reserves that are required to protect against the contingency of sudden adverse changes in asset values (the ‘mismatching’ or ‘resilience’ test). The details of a suitable test, and a method of applying it in practice using a ‘matching rectangle’, are described. Investigations into the effectiveness of such a test, using both deterministic and stochastic methods, are followed by concluding remarks on the underlying philosophical issues raised.

Full numerical results are presented in the Appendices.

In this paper a stochastic model office offering UK-style life insurance contracts is used to demonstrate the effect on relative solvency of different investment and bonus strategies. Relative solvency is defined loosely as the probability that an individual insurer does not fall significantly out of line with the rest of the life insurance market, in terms of, for example, asset liability ratios, or payouts to with-profit policyholders. The model uses the Wilkie investment model, extended to incorporate variation between companies in equity performance.

The ‘law of mortality’ proposed by Heligman and Pollard is compared with the law of Gompertz and with English Life Table No. 14. Some new mathematical results are derived, including specific equations for the curve of deaths. Some numerical illustrations are given.

1.1. This is intended as a discussion paper or a working paper. It does present new data; but that is not its main purpose. Its aim is to stimulate thinking about possible techniques for studying mortality through morbidity, rather than to produce results. The Secretary of the International Union for the Scientific Study of Population when outlining the purpose of a new committee of the I.U.S.S.P. on “factors affecting mortality and the length of life” stated that “it is a common feeling that the traditional demographic and actuarial approach to mortality produces diminishing returns. It is also a common feeling that closer links with other biomedical disciplines, with genetics or biology, are essential if we wish as demographers to improve our contribution to the study of mortality.”(1) The formation of that Committee was a major factor prompting me to set down some views which I have long held on the subject of mortality studies.

Hogg and Klugman use the truncated Pareto distribution with probability density function

where δ≥0 is specified and α > 0 and λ > 0 are unknown parameters, to describe insurance claims. This is fitted first of all by the method of moments, using the estimators

and

where is the mean of a simple random sample, and the (biased) variance

The authors then suggest, on pp. 113–16, that these estimates be used as starting values in a Newton iteration to get the maximum likelihood estimates of the parameters, but this technique can fail as a result of convergence problems. The object of this note is to show that this has led Hogg and Klugman to underestimate seriously the area in the tail of a fitted loss distribution, and to discuss a method of circumventing this difficulty.

Although pensions have been discussed at the Institute on many occasions in recent years, the papers submitted have generally been limited to some particular aspect of the subject, such as the preservation of pension rights, the position of pension funds in the national economy, or the influence of State pensions, whereas the subject of the valuation process has been to some extent neglected.

This is the fourth, and, let it be hoped, the last, of a series of papers on mortality tables in their various aspects which the Institute has honoured me by accepting. The investigation had its origin many years ago in efforts to improve upon the then methods of constructing mortality tables from Census Returns; and, when the enquiry seemed to be nearing its conclusion, the fact became evident that the formulas which had presented themselves could be applied effectively in summation form to the graduation of tables derived from other sources. There was, however, a preliminary objection to be overcome. Summation Formulas of Graduation were under a cloud. Dr. Sprague had condemned them in unmeasured terms, and Mr. Todhunter, in the head-note to his paper (J.I.A., xxxii, 378), had accepted Dr. Sprague's authoritative judgment. Hence my first paper of the series, “On the Error introduced into Mortality Tables by Summation Formulas of Graduation”, (J.I.A., xli, 54), wherein it was attempted, and, I think, successfully, to show that that error is generally so minute as, with ordinary care, to be safely negligible in practice.

In my previous sketches upon the subjects embraced in the above title, whilst, on the one hand, I was desirous to avoid burdening them with any apparently superfluous details, on the other hand, I clearly foresaw that the task would remain of following out, in at least another article, some needful explanations or justifications. To my former readers I therefore inscribe the following notes, in the anticipation of their being found not wholly unserviceable in the view just adverted to.