The authors, Hodges and Schaefer, of the preceding paper [2], taking up where my own article [3] left off, have contributed to a better understanding of the geometric mean index of stock price relatives. Their basic point is that, if in any practical situation a portfolio were managed according to a policy of periodic reallocation, the wealth relative of the portfolio would not be approximated by the geometric index. This is demonstrated through simulation, using randomly generated price sequences as well as empirical data. In addition, they have presented a verbal characterization of the hypothetical portfolio policy whose wealth relative is measured by the mth-order power mean of price relatives discussed in my paper. This policy, as I had stated, is not an intuitively simple one like “maintain equal dollar amounts at all times” or “buy and hold.”

The literature of Index numbers contains much discussion of the relative merits of geometric and arithmetic averages of prices and quantities. The controversy on this subject dates from the middle of the nineteenth century and is fully described by Crowe [2], In recent times both types of averages have been applied to security price relatives to measure the performance of groups of securities over time. The purpose of this paper is to demonstrate the properties of these security indexes and to show the relationships between them and an index based upon a more general type of average called the power mean. The concluding section of this paper contains the proof of an interesting and important limit property which provides the conceptual link between geometric and arithmetic security indexes.