SummaryThis paper investigates the recalculation from its base date of theFinancial TimesIndex of Industrial Ordinary Shares (hereafter called theFTIndex) 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 theFTIndex 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 theFTIndex (1 July 1935). By the end of 1970 its value was 38% higher than that of the (geometric)FTIndex. 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 theFTIndex 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 theFTIndex. In the majority of calendar years since its foundation, theFTIndex 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 theFTIndex 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 theFTIndex 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 arithmeticizedFTIndex values may be useful to correct the downward drift imposed on theFTIndex by its geometric construction.
Unification Theories: Means and Generalized Euler Formulas
