Arithmetic vs. Geometric Mean: Key Differences in Financial Returns

Arithmetic Mean vs. Geometric Mean: An Overview

Measuring financial portfolio performance accurately can be challenging, as different methods can yield varying results. One of the most reliable approaches is the geometric mean, which accounts for the compounding of returns over time.

Unlike the arithmetic mean, which only averages periodic returns, the geometric mean reflects how investment growth accumulates from period to period, making it a more realistic measure of long-term performance.

This article will explore the differences between the two methods and their applications in evaluating portfolio returns, bond yields, and market risk premiums.

Understanding the Arithmetic Mean in Finance

An arithmetic mean is the sum of a series of numbers divided by the number of items in that series. The formula for the arithmetic mean is simple and is very commonly used to find an average for a dataset. It is best used in calculations involving items that, while the same type, have no relationship with each other.

In finance and investing, one might use the arithmetic mean to get an idea of the average earnings estimate for a series of estimates issued by a number of analysts covering a stock. Simply add up the various estimates and divide by the number of estimates.

Or, the arithmetic mean could be used to determine a moving average for a stock price. A moving average is helpful for traders and investors because, when calculated and plotted over time, it smooths out a long series of price movements to present a big picture of a price trend. Market participants can also chart long-term points of support and resistance with a moving average.

Arithmetic Mean Formula Explained

 $\begin{aligned} &A = \frac{1}{n} \sum_{i =1}^n a_i = \frac{a_1 + a_2 + \dotso + a_n}{n} \\ &\textbf{where:} \\ &a_1, a_2, \dotso, a_n=\text{Portfolio returns for period } n \\ &n=\text{Number of periods} \\ \end{aligned}$​A=n1​i=1∑n​ai​=na1​+a2​+…+an​​where:a1​,a2​,…,an​=Portfolio returns for period nn=Number of periods​

Step-by-Step Guide to Calculating the Arithmetic Mean

To calculate a 14-day moving average for a stock, simply add up its closing price for the past 14 days and then divide that sum by 14. As an example, take ABC stock. Its closing prices and the resulting figure for the moving average are shown below.

22 + 20 + 18 + 19 + 24 + 25 + 27 + 28 + 30 + 29 + 30 + 32 + 31 + 29 = 364

364 ÷ 14 = 26

The moving average for the past 14 days of closing prices is 26.

Exploring the Geometric Mean in Investments

The geometric mean for a series of numbers is calculated by taking the product of these numbers and raising it to the inverse of the length of the series. The geometric mean is best used to calculate the average of a series of data where each item has some relationship to the others. That’s because the formula takes into account serial correlation.

This sort of relationship is useful when comparing portfolio returns, bond yields, and total returns on equities. Earnings and compounding represent that correlation. They affect the return for each succeeding period measured. Geometric mean accounts for that impact.

The geometric mean is considered to provide a more accurate idea of average return than a mean calculated simply by dividing a sum of items in a dataset by the number of items.

Breaking Down the Geometric Mean Formula

 $\begin{aligned} &\left( \prod_{i = 1}^n x_i \right)^{\frac{1}{n}} = \sqrt[n]{x_1 x_2 \dots x_n} \\ &\textbf{where:} \\ &x_1, x_2, \dots = \text{Portfolio returns for each period} \\ &n = \text{Number of periods} \\ \end{aligned}$​(i=1∏n​xi​)n1​=nx1​x2​…xn​​where:x1​,x2​,⋯=Portfolio returns for each periodn=Number of periods​

Calculating Geometric Mean: A Step-by-Step Approach

To calculate the geometric mean, we add one to each number (to avoid any problems with negative percentages). Then, multiply all the numbers together and raise their product to the power of one divided by the count of the numbers in the series. Then, we subtract one from the result.

The calculation looks like this:

$\begin{aligned} &[ ( 1 + \text{R}_1) \times (1 + \text{R}_2) \times (1 + \text{R}_3) \dotso \times (1 + \text{R}_n) ]^{\frac {1}{n} } - 1 \\ &\textbf{where:} \\ &\text{R} = \text{Return} \\ &n = \text{Count of the numbers in the series} \\ \end{aligned}$​[(1+R1​)×(1+R2​)×(1+R3​)…×(1+Rn​)]n1​−1where:R=Returnn=Count of the numbers in the series​

The formula appears complex, but it’s not so difficult. Suppose you have invested your savings in the financial markets for five years. If your portfolio returns each year were 90%, 10%, 20%, 30%, and -90%, your average return would be the following:

$\begin{aligned} &(1.9 \times 1.1 \times 1.2 \times 1.3 \times 0.1)^{\frac{1}{5}} -1 \\ \end{aligned}$​(1.9×1.1×1.2×1.3×0.1)51​−1​

The result is an average annual return of -20.08%.

Key Differences Between Arithmetic and Geometric Means

Arithmetic Mean

We used an arithmetic mean for a moving average because the closing prices have no correlation. One closing price may be higher or lower than the next, but there’s no intrinsic relationship.

However, the arithmetic mean is not an appropriate method for calculating an average where the data exhibit serial correlation, or have some relationship to each other.

Consider investment returns and take the example used above for the geometric mean. If your portfolio returns for each of five years were 90%, 10%, 20%, 30%, and -90%, what would your average return be during this period using the calculation for the arithmetic mean?

That average return would be 12%. At first glance, that appears to be impressive. But that’s not entirely accurate.

Geometric Mean

As shown previously, at -20.08%, the geometric mean provides a return that’s a lot worse than the 12% arithmetic mean. But it is the result that represents reality in this case.

Annual investment returns over the years have an impact on each other. If you lose a substantial amount of money in a particular year, you have that much less capital with which to invest and generate returns in the following years.

So, for a more accurate measure of your average annual return over time, it’s more appropriate to use the calculation for geometric mean.

The Bottom Line

For investors who wish to study their portfolio performance over a number of periods—e.g., years—the calculation for the geometric mean can provide a more accurate picture of return compared to that provided by the arithmetic mean.

That’s because the geometric mean formula takes into account earnings and compounding growth from one year to the next. The formula for the arithmetic mean does not.