1  Introduction

Professional investment practitioners routinely report expected geometric return as a summary measure of the prospective performance of asset classes and investment portfolios. Expected geometric return has intuitive appeal because its historical counterpart, the geometric average, provides an annualised measure of the proportional change in wealth that actually occurred over a past time series, as if there had been no volatility in return. However, as a prospective measure, expected geometric return has limited value and often the expected annual (or arithmetic) return is a more relevant statistic for modelling and analysis. Despite this, the distinction between expected arithmetic return and expected geometric return is not well understood, both in respect of individual asset classes and in respect of portfolios.[1] This confusion persists even though it is explained routinely in finance textbooks and other reference sources.[2]

This paper addresses these issues in the context of the financial projections and capital contribution calculations for the New Zealand Superannuation Fund.[3] Section Two provides an introduction to the issues by explaining the distinction between expected arithmetic return and expected geometric return. I show that, although average geometric return can be a useful measure of historical return, it is of only limited relevance in future-orientated analyses, where the expected arithmetic return is invariably more relevant. Section Three examines measures of portfolio return and illustrates that the portfolio geometric return is not (and is greater than) a weighted average of the geometric returns of the underlying asset classes. Indeed, it is possible for the portfolio geometric return to be greater than that of all of its constituent asset classes. Section Four then turns to measurement of the expected value of the stock of an investment portfolio, such as the New Zealand Superannuation Fund. Although the intuitive approach would be to compound the expected geometric return over time, this understates the expected portfolio size, whereas compounding the expected arithmetic return provides the correct result. Section Five shows that the calculation of the required contributions to the New Zealand Superannuation Fund also relies on the expected arithmetic return on the Fund’s investment portfolio, not the expected geometric return. Finally, Section Six provides some concluding remarks.

In order to derive some of the mathematical results presented below, some assumptions are required about the time-series behaviour of returns. The usual assumptions are that returns are stationary (E[r_t] is constant), homoscedastic (Var[r_t] is constant and finite) and serially independent. The purpose here is not to defend these assumptions and not all of them are necessary to derive most of the results presented here. They are clearly not entirely descriptive of reality: expectations change, volatility can vary over time, and some serial dependence can be detected in some returns series. Nonetheless, these are standard assumptions that are adopted in financial analysis and they produce rigorous results. These assumptions can be relaxed without changing the general tenor of the results presented in this paper, but it would be at the cost of unnecessary added complexity in the calculations.[4]