This paper must be forward-looking. To dwell on the past, and even the present, would fill many pages to little purpose. The author is convinced after reviewing many files of documents that the European Common Market would not have gone forward if its founders had allowed themselves to be sidetracked by narrow interests. This perhaps sounds ominous to experts in particular fields, such as actuaries, for it is not always a happy position when political motives may overrule, or cut short, technical considerations. But the concept of the Common Market is a political one and this is becoming more apparent as the years roll on. The collective European interest will prevail but that does not mean that British philosophy and practice cannot make a distinguished contribution to European thought in the field of insurance, particularly life assurance. In practical terms, it is clearly not possible to argue for the adoption of British practice in its entirety. But I do believe that unless the profession takes every opportunity to argue the merits of our own ways of providing life assurance in the private sector, we may well live to regret the omission.

In 1970 the Association Royale des Actuaires Belges celebrated the 75th anniversary of its foundation. To mark the occasion, the Association had in the previous year invited papers to be submitted on two subjects, one of which was ‘Determination of the heterogeneity of the classes of risks’, and the paper which follows this introductory note was one of those submitted on this subject. The paper deals exclusively with motor insurance, in which the study of heterogeneity must be regarded as being of major importance, and discusses both the measurement of heterogeneity and the relationship between heterogeneity and methods of experience rating such as the no claim discount (NCD) systems used in the United Kingdom.

The aim of this paper is to review some of the theoretical and practical questions associated with experience rating in motor insurance, with particular reference to the no claim discount (NCD) systems widely used in the United Kingdom. We shall describe some work carried out in one U.K. insurance office which has in force about 650,000 private motor car policies each covering a single vehicle.

The Caribbean is usually taken to include the number of widely scattered islands in the Caribbean Sea, as well as four neighbouring mainland territories which, for historical reasons, are closely associated with one or other of the island groups. The islands comprise: three Republics—Cuba, Haiti, and the Dominican Republic; three former British Colonies which have become independent countries within the Commonwealth during the last ten years, Jamaica (1962), Trinidad and Tobago (1962), and Barbados (1966), and a number of other islands which continue to have some level of dependent relationship with one or other of the ‘metropolitan’ countries including Britain (the Windward and Leeward Islands, the British Virgin Islands and the Bahamas); France (Martinique and Guadeloupe), the Netherlands (Curaçao, Aruba, Bonaire), and the United States of America(Puerto Rico and the U.S. Virgin Islands). The Mainland Territories include the independent Commonwealth country Guyana (formerly British Guiana and independent since 1966), French Guiana, Surinam (Dutch Guiana), and British Honduras.

This paper investigates the recalculation from its base date of the Financial Times Index of Industrial Ordinary Shares (hereafter called the FT Index) using the ordinary arithmetic mean of the values of its constituents, rather than the geometric mean that is actually used in its calculation. The value of this arithmetic version of the FT Index represents that of a portfolio invested equally in each of the original constituents at the base date. The behaviour of the recalculated index may therefore be described in portfolio terms.

The arithmeticized index has been calculated at each end-year since the base date of the FT Index (1 July 1935). By the end of 1970 its value was 38% higher than that of the (geometric) FT Index. The percentage by which the new index exceeded the original was found to have increased approximately linearly with time; a least-squares linear fit over the whole period indicated that the percentage excess of the arithmetic version over the original index had increased on average by 0·8% a year. A separate fit for the period from end-1960 onwards shows that this rate had increased to 2·7% per year, and even over the period since 1950, it has been 1·1% per year. Use of these relationships gives a rough adjustment of the FT Index for portfolio comparison purposes.

We also calculated a series of arithmetic indices with base dates at the end of each year since the original base date of the FT Index. In the majority of calendar years since its foundation, the FT Index has not fallen far behind the corresponding arithmetic version over the twelve month period. However, over periods of more than a year, the geometric and arithmetic indices can show dramatically different results, particularly if the percentage change in the index value is considered. It is clear that longer-term changes in the value of the FT Index cannot be interpreted as the changes that would have occurred in the value of a portfolio equally invested in the thirty constituent shares at the beginning of the period.

None of these findings affects the usefulness of the FT Index for its original purpose, as a sensitive daily indicator of changes in the industrial ordinary share market. It is purely its long-run use as a portfolio standard that is misleading. For this purpose the more recently instituted arithmetic indices are likely to be useful in the future. Meanwhile, our arithmeticized FT Index values may be useful to correct the downward drift imposed on the FT Index by its geometric construction.

In his paper of 1941, Seal included details of some experiments he performed in an attempt to estimate the appropriate number of degrees of freedom for the chi-square goodness-of-fit test of a summation formula graduation. These results are referred to by Tetley and by Benjamin and Haycocks in their textbooks when they mention the difficulty of determining the number of degrees of freedom or mean chi-square value.