2  Expected geometric return and expected arithmetic return (continued)

For a given distribution of annual expected return and volatility, the geometric mean is smaller the longer the time period (N) that is being examined. This is illustrated in Figure 1 using the expected annual return on equities of 12.7% and standard deviation of 20.2% reported by Ibbotson Associates (Ibbotson Associates 2002). The expected annual return stays at 12.7%, regardless the time horizon. However, the expected geometric return starts at 12.7% if the time horizon is one year then declines so that for a twenty-year time horizon, the expected geometric return is about 11%.

Figure 1 – Arithmetic and Geometric Measures of Expected Return

Stocks are more volatile than bonds, so most of this difference is also reflected in the risk premium. From the same source, the risk premium of stocks over bonds measured on an arithmetic basis was 7.0%, but the difference between the geometric averages was only 5.4%.[7]

There also is a frequently used approximation of the geometric mean. This is calculated as the expected annual arithmetic mean minus half its variance. This understates the true geometric mean for all time horizons, and it is especially wrong for shorter time horizons. This incorrect geometric approximation is also illustrated in the example in Figure 1.

Another central measure of returns is the median. Under lognormality, the median geometric return is a constant that does not decline over time, and it is equal to the median arithmetic return.[8] It is also illustrated in Figure 1. It is the asymptote that the expected geometric return converges to as the time horizon is expanded out infinitely.

Understanding the distinction between geometric and arithmetic return is important because both metrics are used by commentators discussing issues of investment returns, such as the equity risk premium, and there is scope for confusion about which is relevant in any particular situation. Recognising this, some authors report their analyses on both bases (for example, Cornell 1999, Lally and Marsden 2002). Prospective (that is, future-orientated) applications, such as the capital asset pricing model[9] and the assessment of the required capital contribution to the New Zealand Superannuation Fund, require an unbiased estimate of the expected annual return.[10] As shown above, the expected geometric return over any period greater than one year will understate the expected annual return, while the expected arithmetic return provides the appropriate measure for this purpose.

A further complication with using any measure of prospective analysis is that the expected values (and other parameters of the return distribution) are not known with certainty and so must be estimated. However, Blume (1974) shows that an arithmetic average provides an unbiased and consistent estimate of the expected annual return, while the geometric average provides a downward biased estimate and it has a larger sample variance than the arithmetic average. In the related case of estimating discount factors for present value calculations, Cooper (1996) shows that both arithmetic and geometric averages provide downward biased estimates of the discount factor, and that the arithmetic average is least biased. This holds even if returns are serially correlated.