6 Conclusion
Expected geometric return is routinely reported as a summary measure of the prospective performance of asset classes and investment portfolios. It has intuitive appeal because its historical counterpart, the geometric average, provides a useful descriptive measure of the annualised proportional change in wealth that actually occurred over a past time series, as if there had been no volatility in return. However, for applications that involve future projections or other prospective analyses, expected geometric return has limited value and often the expected annual arithmetic return is a more relevant statistic for modelling and analysis. Despite this, the distinction between expected annual arithmetic return and expected geometric return is not well understood, both in respect of individual asset classes and in respect of portfolios. This confusion persists even though it is explained routinely in finance textbooks and other reference sources.
Even the supposedly straightforward calculation of weighted average portfolio return becomes somewhat complicated, and can produce counterintuitive results, if the focus of reporting is expected geometric return. Simply calculating the portfolio expected geometric return for a particular time horizon as being the weighted average of the expected geometric returns of each asset class for that time horizon will understate the expected portfolio geometric return. The weighted average calculation should be carried out starting with the expected annual arithmetic returns of the individual asset classes. The true expected portfolio geometric return will be at the upper end of (and could possibly exceed) the spread of individual asset class expected geometric returns.
The issues are also interpreted in the context of the analysis underlying the New Zealand Superannuation Fund. Projections of the expected size of the Fund should be based on compounding the expected arithmetic return over time, not the geometric return. Similarly, calculation of the capital contributions the Crown is required to make to the Fund is a function of the expected arithmetic return on the Fund, not of the expected geometric return.
