What Is the Geometric Mean?
Let me explain the geometric mean to you directly: it's the average of a set of products, and people like analysts and portfolio managers use it all the time to figure out investment or portfolio performance.
From a technical standpoint, the geometric mean is the nth root of the product of n numbers. You have to use it when you're dealing with percentages that come from values, whereas the standard arithmetic mean is for the values themselves.
Key Takeaways
Here's what you need to know: the geometric mean averages a set of values by using the products of the terms. It's especially right for series with serial correlation, like investment portfolios. Most financial returns are correlated, such as bond yields, stock returns, and market risk premiums. For numbers that fluctuate a lot, the geometric mean gives a much more accurate true return by factoring in year-over-year compounding, which smooths out the average.
Understanding the Geometric Mean
You might hear the geometric mean called the compounded annual growth rate or time-weighted rate of return—it's the average return rate for a set of values, calculated from the products of the terms. What does that mean for you? It multiplies the values and raises them to the 1/nth power.
This is a key tool for portfolio performance for several reasons, mainly because it handles compounding effects. Take simple numbers like 2 and 8: multiply them to get 16, then take the square root, and you end up with 4. With more numbers, you'll want a calculator or software.
The big advantage is you don't need to know the actual investment amounts; it just looks at the returns for a fair comparison across periods. Remember, the geometric mean is always a bit smaller than the arithmetic mean, which is just a simple average.
Formula and Calculation With Example
The formula for the geometric mean is μ_geometric = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) - 1, where R1 to Rn are the returns of an asset or other observations.
Suppose your portfolio had these returns over five years: 5% in year one, 3% in year two, 6% in year three, 2% in year four, and 4% in year five. Plug them in: [(1 + 0.05)(1 + 0.03)(1 + 0.06)(1 + 0.02)(1 + 0.04)]^(1/5) - 1. That multiplies to about 1.2161, raised to 0.2 is roughly 1.0399, minus 1 gives 0.0399 or 3.99%. That's slightly less than the arithmetic mean of 4%.
You can also calculate this in a spreadsheet like Google Sheets using the GEOMEAN function. Set up your returns as 1.05, 1.03, 1.06, 1.02, 1.04 in cells, then enter =GEOMEAN(range) in an empty cell.
Keep in mind, the longer your time horizon, the more important compounding is, so stick with the geometric mean.
What Is the Geometric Mean of N Terms?
For n terms, it's the nth root of their product, where n is the number of terms.
Can You Calculate the Geometric Mean With Negative Values?
No, you can't directly—geometric means don't work with negative numbers. Convert them to proportions first; for a -3% return, use 1 - 0.03 = 0.97.
How Do You Find the Geometric Mean Between Two Numbers?
Multiply the two numbers and take the square root of that product.
The Bottom Line
In math, the geometric mean shows the central tendency of numbers through their product and root. In investing, it's a metric that figures portfolio returns with compounding in mind. Use it to see how your portfolio is doing and if you need to make changes.
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