# Geometric Average Return

Geometric Average Return is the average rate of return on an investment which is held for multiple periods such that any income is compounded. In other words, the geometric average return incorporate the compounding nature of an investment.

Geometric average return is a better measure of average return than the arithmetic average return because it accounts for the order of return and the associated compounding effect. Arithmetic average return overstates an investment's performance where the returns are volatile. When we need to compare returns over an extended period, geometric average return is the preferred measure.

## Formula

Geometric average return can be calculated using the following formula:

Geometric Average Return

= ((1 + R_{1}) × (1 + R_{2}) × ... × (1 +R_{n}))^{(1/n)} - 1

Where,

*R _{1}*,

*R*and

_{2}*R*are sub-period returns for period 1, 2 and n, respectively, and

_{n}*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 = | 15% + (-5%) + 10% | = 10% |

3% |

The geometric average return in the same case is just 6.32%:

Geometric Average Return

= ((1 + 15%) × (1 + (− 5%)) × (1 + 10%))^{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 us compare the endowment value worked based on actual return, arithmetic average return, and geometric average return.

Endowment Value (Actual)

= $100 million × (1 + 15%) × (1 − 5%) × ( 1 + 10%)

= $120.18 million

Endowment Value (AAR)

= $100 million × (1 + 10%)^{3}

= $133.1 million

Endowment Value (GAR)

= $100 million × (1 + 6.32%)^{3}

= $120.18 million

It is evident 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.

Written by Obaidullah Jan, ACA, CFA and last modified on