How to Calculate Geometric Average Return for Investments

Investment returns vary from year to year, making it difficult to assess overall performance using simple averages. The geometric average return provides a more accurate measure by accounting for compounding over time, which is especially useful for evaluating long-term investments.

Understanding how to calculate this metric helps investors compare different options and assess historical performance effectively.

Formula and Key Variables

The geometric average return accounts for compounding, unlike a simple average, which merely sums returns and divides by the number of periods. The formula is:

Rg = [(1 + R1) × (1 + R2) × … × (1 + Rn)]^(1/n) – 1

where Rg is the geometric average return, Ri represents the return for each period, and n is the total number of periods. The multiplication of each period’s return plus one reflects compounding, while taking the nth root ensures the result is an annualized rate.

This method is particularly useful when returns fluctuate. For example, if an investment gains 20% one year but loses 10% the next, the simple average suggests a 5% return per year. However, this does not accurately reflect the impact of the loss on total value. The geometric method corrects this by factoring in how each year’s return influences the next.

Steps to Calculate for Multiple Periods

First, convert each return into its growth factor by adding 1. For example, an 8% return becomes 1.08, while a 5% loss is 0.95.

Next, multiply all growth factors to find total compounded growth. If an investment has returns of 15%, -10%, and 20% over three years, the product of their growth factors is:

1.15 × 0.90 × 1.20 = 1.242

Then, take the nth root, where n is the number of periods:

(1.242)^(1/3) = 1.075

Finally, subtract 1 and convert back to percentage form:

1.075 – 1 = 7.5%

This process ensures the return reflects compounding rather than just averaging yearly percentages.

Example Calculation

Consider an investor whose stock investment sees annual returns of 12%, -8%, and 15% over three years. First, convert these percentages into growth factors:

1.12, 0.92, and 1.15

Multiply them together:

1.12 × 0.92 × 1.15 = 1.183

Take the cube root since there are three years:

(1.183)^(1/3) = 1.0576

Subtracting 1 and converting back to percentage form gives an annualized geometric return of approximately 5.76%.

Interpreting the Outcome

A significant difference between geometric and arithmetic averages indicates volatility. The larger the gap, the more pronounced the fluctuations. This is important for risk assessment, as higher volatility increases the likelihood that an investor’s actual compounded return will be lower than the simple average suggests.

For example, two funds may report an identical 8% arithmetic average return, but if one has a geometric return of 6% while the other is 7.5%, the latter has exhibited more stable growth. This insight helps investors prioritize assets that maximize long-term compounded returns while minimizing extreme fluctuations.

In taxable accounts, geometric returns also affect effective tax rates on capital gains. Frequent fluctuations can trigger higher realized gains and increased tax liabilities. Understanding this effect allows investors to adopt tax-efficient strategies, such as holding assets longer to qualify for lower long-term capital gains rates or using tax-loss harvesting to offset gains.