2  Expected geometric return and expected arithmetic return

Measures of expected value provide essential information when preparing projections of the behaviour of financial investments into the future. However, there are two measures of return over time. The average of an observed set of returns can be measured either arithmetically, by summing the percentage return for each year then dividing by the number of years, or geometrically, by compounding the annual returns and putting this to the power of the inverse of the number of years.

The difference between arithmetic and geometric historical averages can be seen from a simple numerical example. Suppose returns in two years are +40%, followed by -40%. Starting with $100, we have $140 after the first year. In the second year, we lose 40% ($56), giving an ending stock of $84. The arithmetic average return is zero (being the simple average of +40% and -40%). The geometric average return is -8.3% (being ). In other words, the same ending wealth could have been achieved with a constant compounded annual return in both years of -8.3% ($91.70 after year 1, then $84 after year 2).

If returns are not constant and the time period of measurement is greater than one year, the geometric average will always be less than the arithmetic average.[5] The difference will be greater the longer the time period and the greater the volatility. As a purely descriptive measure of historical return, the geometric average provides an annualised measure of the proportional change in wealth that actually occurred over the time horizon being examined, as if the wealth grew at a constant rate of return.

Like historical averages, the expected value of future returns can also be specified either in terms of an annual arithmetic mean or in terms of a geometric mean that is measured over a specified time horizon. This is illustrated in the following example. Suppose there is a 50% probability of a+40% return per year and a 50% probability of -40% per year. Starting with $100, the probability tree of possible outcomes over three years is shown in Table 1. After one year, the wealth level is either $140 or $60, giving an expected wealth level of $100 and expected return is 0%. After two years, there are three possible wealth levels and the expected wealth is still $100. However, the expected geometric return is -4.2% per year over two years and, measuring over a three year time period, the expected geometric return declines further to -5.6% per year. The expected total wealth at any time in the future is calculated by compounding the initial wealth ($100) by the expected annual arithmetic return. In this case, that is 0%, so the expected wealth stayed at $100 over the three years. The expected annual return does not change with the time horizon. However, the expected geometric return does. It declines as the time horizon increases and it is not a particularly meaningful measure of the expected growth of wealth over time.

This example illustrates how the expected geometric return declines as the time horizon over which it is measured increases. Therefore, if a measure of long-term expected geometric return is reported, it will be relevant only to the specific time horizon to which it relates. It will understate expected geometric returns over shorter periods. In addition, the expected growth in wealth over time is obtained by compounding the expected annual arithmetic return. Compounding the expected geometric return will understate the expected growth in wealth. This is discussed further in Section Four. Conversely, when discounting back to present value, using the expected geometric return for the discount factor will overstate the present value, while using the expected annual arithmetic return will give the correct result.

The above example assumes that returns follow a discrete binomial distribution. It is more common in financial analysis to think of returns as following a continuous distribution. If returns are assumed to follow a lognormal distribution and are serially independent,[6] then the exact relationship between the arithmetic mean (E[r]) and the locus of the geometric mean (E[g_N]) over time (N) is:

(1)