# Geometric Average Return

Geometric Average Return is the compound annual rate of return achieved by holding the investment over the whole period. It is calculated by adding 1 to each sub-period return, multiplying all the resulting figures, raising the result to (1/N) where N is the number of sub-periods and then subtracting 1 or using Excel GEOMEAN function.

Geometric average return is theoretically sound measure of average return as compared to the arithmetic average return because it accounts for the order of return and the associated compounding effect. Arithmetic average return is overstated and where the returns are volatile, or we need to compare returns over an extended period, geometric average return is the preferred measure.

## Formula

We can calculate geometric return using the following formula:

$$Geometric\ Average\ Return\\=\left(\left(1+R_1\right)\times\left(1+R_2\right)\times\ldots\times\left(1+R_n\right)\right)^\frac{1}{N}-1$$

Where R1, R2 and Rn are sub-period returns for period 1, 2 and n respectively. N is the total number of sub-periods for which return is available.

Alternatively, we can also calculate it using the Excel GEOMEAN function.

## Example

Your university established its endowment with \$100 million 3 years ago. Annual return for the first 3 years was 15%, -5% and 10%. Suppose all the return results from capital gain.

The arithmetic average return in the above case is 10%:

$$Arithmetic\ Average\ Return\\=\frac{15\%+(-5\%)+10\%}{3}=10\%$$

The geometric average return in the same case is:

$$Geometric\ Average\ Return\\={((1+15\%)\times(1-5\%)\times(1+10\%))}^\frac{1}{3}-1\\=6.32\%$$

Please note that the arithmetic average return is significantly higher than the geometric return and its usage could be misleading. Let’s compare the endowment value worked based on actual return, arithmetic average return and geometric average return.

$$Endowment\ Value\ (actual)\\=100\ million\times(1+15\%)\times(1-5\%)\times(1+10\%)\\=120.18\ million$$

$$Endowment\ Value\ (based\ on\ Arithmetic\ Average\ Return)\\=100\ million\times{(1+10\%)}^3\\=133.1\ million$$

$$Endowment\ Value\ (based\ on\ Geometric\ Average\ Return)\\=100\ million\times{(1+6.32\%)}^3\\=120.18\ million$$

Please note that the geometric average return has replicated the actual growth trajectory of the endowment while the arithmetic average has overstated the endowment value. This is because the arithmetic average ignores the order of returns. It ignores the fact that the 5% decline in the second year occurs after a 15% growth in the first year.