The two leading formulæ at present in use for interpolation by central differences are due to Newton‡ (Methodus Differentialis, Prop. III), though often called by the name of Stirling, who republished them about 20 years later.

The introduction of high-speed electronic computers has opened the way for analyses of motor insurance risk statistics on a scale not previously possible. We may regard the purpose of the analyses which form the subject of this note as being to test the validity of the existing rating structure by finding out how the claims experience varies from one rating cell to another within that structure; and, more generally, to explore the possibility of devising a more efficient rating structure by defining new levels within the existing factors, or by using new factors in addition to, or in replacement of, those already used.

To achieve this purpose it is appropriate to consider on the one hand a detailed investigation of the experience over a limited period, analogous to that carried out when constructing a new standard life table and, on the other hand, a system for keeping the emerging experience under more or less continuous review. For this second aspect it is natural to think in terms of making comparisons between the actual experience and that expected according to the standard table.

I have heard that a former distinguished President of this Institute once publicly stated that “Doctors rush in where Actuaries fear to tread.” It is also said that “at forty a man is either a physician or a fool”: but the presidential saying seems to suggest that a man may be both. With such a warning ringing in my ears, I yet venture to read before this learned Institute a paper on a thorny subject. I am doing so, first because that subject can be dealt with only by the united labour of actuaries and doctors; and second, because I am sure of the consideration and forbearance of the Fellows in any matters wherein I travel with halting gait.

This paper is partly descriptive, partly analytical, and covers the following aspects:

a description of the market in general;

a brief description of the organization of Lloyd's;

currencies and exchange rates;

calculation of reserves;

taxation;

underwriting; and

accumulation of risk.

§§ 1 and 2 and the earlier paragraphs of § 3 convey information regarding the market that will be well known to actuaries practising in this field and may be passed over without loss. They are added in an attempt to convey a picture of the type of operation that results within an office concentrating on reinsurance business, as against looking at reinsurance in its peripheral effect on a direct writing office.

Mr. John Finlaison's method of graduation (Journal, vol. xxi, page 50) deals with an irregularity, which, for convenience, may be assumed to be unity, in the following manner:—It diminishes the irregularity to a small fraction on its first appearance at one end of the formula; conducts it along the formula by easy stages, as over a double inclined plane; and dismisses it gently at the other end. The method has the advantage of simplicity in working and check, but it is correct to first differences only.

Mr. Woolhouse's method is correct to third differences, and the irregularity glides over a curve; but the application of the formula requires sustained attention.

The object of this paper is to combine the facility of the one method with the smoothness and correctness of the other.

Let S be the sum of nine numbers, u 0 to u 8, increasing by third differences,

the central term,

whence .

This paper sets out a framework, based on option pricing theory, that can be used to assess the value of deferred unrealised capital gains tax. In the U.K. and Australia, capital gains tax is paid on realisation of assets and the basis for determining the tax allows for inflation indexation of the cost base of the asset. Capital gains tax payments under these circumstances are shown to resemble those of a complex option. A number of theoretical approaches to the valuation of this option are discussed in the paper.

In this paper the reader is asked to watch the heuristic development of an idea and its implications.

The subject is the way in which the working of a life office is unaffected by changes in the market rate of interest. For this purpose all other questions such as mortality, expenses, etc., will be ignored as far as possible. The idea is that actuarially a life office is qualitatively different from the sum of its individual policies; that many of the difficulties in discussing this subject occur because basic actuarial concepts and techniques, especially

(i) the idea of an average rate of interest, and

(ii) the method of discounting into a present value,

which have been designed primarily to deal with the problems of individual policies, are inappropriate to the large-scale problems of a life office, e.g. valuation, investment and bonus declarations.

The heuristic development is that of an alternative theory and technique; these do not, for instance, make use of (i) and (ii) above.

The use of classical bonus–malus systems entails very high maluses and other problems which, during recent years, have been criticised by actuaries. To avoid these problems, new bonus–malus models have been developed. For instance, it is well known that the use of an exponential loss function reduces the differences between overcharges and undercharges, solving the problem of high maluses. In order to measure the sensitivity of the exponential bonus–malus system, and according to robust Bayesian analysis, we first model the structure function by specifying a subclass of the generalised moments class. We then examine the range of relativities for each prior. Finally, we illustrate our method with a numerical example based on real data.

In 1947 the company with which the author is associated decided to institute an enquiry into the mortality of impaired lives. From the first the venture was both medical and actuarial, and a leading part in the planning of the investigation was undertaken by the company's Principal Medical Officer, T. W. Preston, M.D., who devised an impairment coding scheme and has since continued to supervise the coding of policies qualifying for inclusion in the data. While the present paper describes methods used in the statistical analysis of the results, the author wishes to place on record the major role played by Dr Preston in the carrying out of the investigation. The work has, in fact, involved the combination of medical and actuarial skills, and without the participation of the medical profession no actuarial investigation into the mortality of impaired lives would be practicable.

The underlying idea of this paper is the valuation of the sum assured (S) in groups of t years by multiplying the total sum by a tabulated function α and by a tabulated β.

This paper focuses on sections of the insurance industry which are not often widely discussed. If catastrophe insurances are to be transacted successfully, a large number of different technical skills and experiences must be combined. Actuaries in the U.K. are still relatively little employed on the problems of general insurance, and are rarely involved in the complexities of these particular sections of it. It must be recognized that there is only limited scope for the application of actuarial skills in this field. However, there may be some advantage to be gained by mixing an intelligent statistical opinion with other specialized technical or practical ones when some catastrophe problems are being discussed. Before expressing a view on any topic, it is, of course, essential to learn about the background and existing conditions.

The concept of catastrophe is considered in section 2 of this paper. Potential catastrophe situations can be classified into two groups, natural hazards and man-made hazards. Each of the sections 3–9 deals with one of the principal natural hazards, giving an outline of the causes of catastrophic events, the type of losses to be expected, and the form of insurance cover likely to be available in vulnerable areas. Section 10 describes the New Zealand state scheme, which provides cover in the event of extraordinary losses from nearly all elemental perils, and war. Sections 11–16 each deal with a different type of situation where a man-made catastrophe can occur, and the insurance provisions available in each case.