Almost a quarter of a century ago—in 1958 to be precise—one of my predecessors devoted the main part of his Presidential Address to a discussion of state and occupational pensions. He spoke at a time when price inflation was 2¾%, growth in the national income had been averaging 2% p.a. in real terms and the number out of work was half a million. The redemption yield on long-dated Government stock was about 5¼% and the dividend yield on the Actuaries' Investment Index of Ordinary Shares was 5¾%—we did not yet have the reverse yield gap that recognizes the endemic nature of inflation.

1.1 The price of a particular ordinary share represents an equilibrium position between the views of those participants in the market who wish to buy and those who wish to sell. Most participants, and certainly all institutional investors, have access to a vast amount of background information, and share prices adjust continuously as these participants revise their buying or selling prices in the light of new information or in the light of a changed interpretation of existing information. If an explicit price model can be developed solely from the principle that prices are in equilibrium once all participants in the market have acted on their interpretation of the information available to them, this model will be of considerable assistance in the management of ordinary share portfolios.

1.2 This paper describes the construction and application of such a price model and discusses the optimal extent to which mathematical and statistical methods can be employed in portfolio management. A recent paper by Clarkson (1980) describes the stage of development that had been reached at the end of 1978; the present paper expands on the underlying concept of market equilibrium and on the practical implications for the management of institutional portfolios of ordinary shares.

Let E be an event whose probability of occurrence depends on some continuous variable x, P(E|x) = qx

For example, E may be death and x age, E may be incidence of lung cancer and x amount of smoking, or E may be reconviction of a parolee with x previous criminal convictions (with suitable definitions of the underlying time interval for the occurrence of E). Given observations on n individuals with characteristic x and the incidence of E, it is desired to estimate the function qx.

The simplest case is when the data are grouped—suppose x occurs in nx cases of which E occurs sx times. Then the elementary crude estimate is

This paper describes a simple, non-parametric method of graduating observed rates or probabilities of the form The technique has been used for smoothing data sets arising in medicine and criminology (Copas(1)(2)) and is extended here to an actuarial example and the results compared with more traditional approaches.

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.

For most people, statistical forecasting means modelling time series using the methods described by Box and Jenkins (1970). A Box and Jenkins model requires a large number of known data points before it can be properly chosen and, since it cannot be changed without again observing another batch of input, is only of use if the data conforms in the future to the chosen model. This means that this style of forecasting is essentially rigid, unadaptable and of limited use in practice. This paper sets out, with the aid of the examples, the essentials and some applications of Bayesian forecasting as developed by Harrison and Stevens (1976).

The paper is concerned initially with the admissibility of particular claim payment models. Admissibility is restricted by the requirement that, if claims escalation is excluded, the amount of outstanding claims at any epoch of development of a generation of claims should be independent of the future speed of finalization.

This simple requirement represents an invariance condition. The problem of admissibility of models can therefore be approached in the manner of invariance problems in physics. Standard techniques of variational calculus are used.

1. In the three years since projections of occupational pension scheme membership and expenditure were made for the Wilson Committee more data about schemes has become available, such as that from the Government Actuary's survey of Occupational Pension Schemes 1979 (the G.A.'s Survey), so the occasion of the publication of the Government Actuary's National Insurance Fund Long Term Financial Estimate (the Quinquennial Review or Q.R.) has been taken to revise and extend the Wilson Committee projections, using the starting date and earnings and prices assumptions of the Q.R.