The geometric mean return is a return measure often calculated in the context of property investment performance analysis.

When measuring property investment performance over several periods the simple average, which is also referred to as arithmetic mean, does not provide a good measure of investment performance because it does not take into account the compounding of returns through time. In such case one needs to calculate the constant rate of return per period which when applied to the beginning value of the investment at a compounding fashion would result in the same ending value of the investment. This compounded periodic constant rate implied by the beginning and ending value of the investment is referred to as the geometric mean return.

The geometric mean return can be calculated in two ways: 1) Using periodic returns 2) using the beginning and ending value of the investment.

## Estimating the Geometric Mean Using Periodic Returns

In the first case we may know for example the return that was achieved in each year. So if the holding period is three years and the returns for each of the three years are R1, R2 and R3 then the formula for calculating the geometric mean of these returns (GMR) is the following:

GMR = [(1+R1) * (1+R2) * (1+R3)]^{1/3} – 1

Then the more generalized formula for n periods (months, quarters, or years) is:

GMR = [(1+R1) * (1+R2) * (1+R3)….(1+Rn)]^{1/n} – 1

Consider the following example of annual returns for three years Example: R1 = 10% R2 = 40% R3= -30%

Then the geometric mean annual return can be calculated as:

GMR = [(1+0.10) * (1+.4) * (1-0.3)]^{1/3} – 1

= [1.1 * 1.4 * 07]^{0.333} – 1

= [1.078]^^{0.33} – 1 = 0.0254

## Estimating Geometric Mean Returns Using the Beginning and Ending Values

The geometric mean return can also be calculated using the beginning and ending value of the investment. Let’s note the beginning value as BV and the ending value EV. Then the formula for calculating the GMR is:

GMR = (EV/BV) ^{(1/3)} -1

To demonstrate the application of the formula let’ s assume that the beginning value BV in the above example is 100. With this beginning value and given the annual returns the respective investment values at the end of each year can be calculated as:

Year 1 ending value = 100 *1.10 = 110

Year 2 ending value = 110 * 1.4 = 154

Year 3 ending value = 154 * 0.7 = 107.8

Thus the ending value EV is 107.8 and GMR can be calculated as:

GMR = (107.8 /100) ^{o.333} -1 = .0254

This example verifies that both formulas give the same result

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