Average return – Arithmetic vs geometric When investors analyze the past performance of their investments in stocks, bonds, etc., one of the key metrics they focus on is the average return, which is also known as the average historical return or average realized return.

Learning objectives:

• Use a series of past returns to calculate the average realized return.
• Understand the distinction between the arithmetic average return and the geometric average return.

Here are the monthly returns of a stock you bought a year back:

2.1%, —1.2%, —0.6%, 0.4%, 1.2%, 0%, —0.2%, —0.2%, 1.1%, 0.4%, —0.1%, 1.9%.

There are two popular metrics that you can use to evaluate this stock’s average monthly performance. We discuss them in turn.

Arithmetic average return

You can easily calculate the arithmetic average return over T periods (e.g., 6 days, weeks, months, years, etc.) by summing up each return observation Rt and dividing by the number of observations T:

Arithmetic average return = ∑Rt/ T

Note that is the summation factor and is telling us to add up net returns. In our example, the arithmetic average return for the last 12 months would be:

(2.1% — 1.2% — 0.6% + 0.4% + 1.2% + 0% — 0.2% — 0.2% + 1.1% + 0.4% — 0.1% + 1.9%) / 12 = 0.40%.

This tells us that the monthly return in an average month was 0.4%.

Geometric average return

The geometric average return calculations are a bit more involved compared to the arithmetic average return. To calculate the former, we need to compound monthly gross returns 1 + Rt and then take the 1/T th root of the result:

Geometric average return = [∏(1 + Rt )]1/T  1

Note that ∏ is the product factor and is telling us to compound gross returns. In our example, the geometric average return over the past 12 months would be:

[(1.021) (0.988) (0.994) (1.004) (1.012) (1.000) (0.998) (0.998) (1.011) (1.004) (0.999) (1.019)]1/12 — 1 = 0.39%.

This means that the value of your investment in the stock grew on average by 0.39% per month.

What is next?

This post is part of the series on investments. In our next post in the series, we introduce the concept of return volatility as a measure of risk. In the previous post, we showed how to perform holding period return calculations.