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Geometric return definition and tutorial

Deeply understanding the three methods for calculating return can be a career differentiator. In terms of difficulty, this one is right in the middle.
  1. Define - Define geometric return.
  2. Calculation - See how it is calculated and interpreted.
  3. Context - Use it in a sentence.
  4. Video - See the video and transcript.
  5. Quiz - Test your knowledge.
Paul Alan Davis, CFA, August 26, 2016
Updated: December 16, 2018
Geometric return takes compounding into consideration and that requires exponents. Keep reading.

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Geometric Return

Beginner

Geometric return is one of three methods for calculating return over multiple time periods. It is commonly used in backward-looking performance reviews because it takes compounding into consideration. There are two versions: 'total geometric return' and 'average geometric return'. This is time-certain, meaning, the order of returns does matter.

Synonyms: geometric mean, geometric average, compounded rate of return


How it is Calculated

As an example, let's say an investment had a compounded return over two periods at 5% each period. The total geometric return formula in Excel would read =((1+0.05)^2)-1 or 10.25%.

In math and statistics, a distinction is often made between discrete and continuous data. Log return is the more theoretical continuous version. In the practical world, however, most people think of returns broken into discrete periods instead.

  • Arithmetic return - One period, not-compounded, discrete.
  • Geometric return - Multi-period, compounded, discrete.
  • Logarithmic return - Infinite-periods, compounded, continuous.

So geometric returns is a compounded version where you add one to the return, take that to the power of the number of periods, then at the end subtract one. See below for the average geometric return.

Because it compounds returns over time it is most resembles an investor's actual performance, so it is used in backward-looking performance contexts.

In a Sentence

Leo:  Professor, I'm soooo looking forward to your lecture on geometric return .
Doc:  [silently] It's 4:25. What was he doing 5-minutes ago?

Video

This video can be accessed in a new window or App here , at the YouTube Channel or within this page below.

Geometric return definition for investment modeling (4:54)

Video Script

The script includes two sections where we visualize and demonstrate the calculation of geometric return.

Visualize

We're sitting in Excel and this is a snippet from our boot camp course.

We'll simplify this to two-periods but it applies to 3, and beyond. Here we have two versions of geometric return.

The geometric total return, sometimes called geometric return, involves adding 1 to each return, mulitplying them together, then subtracing 1.

And for the average, often called the geometric mean, take that total to the power of one over the number of observations. That's what this caret symbol means, here we'd raise it to the 1/2 power. I put the total here in double parentheses because that's similar to the logic in Excel.

Demonstrate

Let's now demonstrate the 'total' and 'average' calculations using two hypothetical stocks, over three periods.

ABC had returns of 10%, followed by -11%, then another positive 10%. XYZ had returns of 100%, followed by -50%, then it was flat.

Total geometric return is calculated by adding 1 to each return. You could multiply them together manually, like this, and subtract one, or use the =PRODUCT function, with the range of returns in parentheses, minus 1.

We have observations here. And for the geometric average take that same product, raise it to the power of 1 over 3, then subtract 1, like this. Or you could use a cell reference to the cell containing the number of observations.

And to review, you would choose the geometric method for backward-looking perforamnce reviews because it measures the compounding of returns, and I put an example here showing the change in dollars, starting with $1,000 so you can follow the logic. Notice this, and this.

The sentence earlier about time-certain is meaningful now. Let me show you why. Arithmetic returns, are more forward-looking, when you don't know the order of returns, unlike here. It is more reflective of the estimates for future periods. Take a moment to think about this.

Using the same data, but using arithmetic returns, look at the differences. 2.5% versus 3%, 0% versus 16.67%. To see why follow the link at the end.

Quiz

Click box for answer.

Which return calculation method is best suited for forward-looking risk estimation. | Arithmetic or Geometric?

Arithmetic, because the order of returns doesn't matter

Questions or Comments?

Still unclear on geometric return? Leave a question in the comments section on YouTube. Also, see a tutorial page and video on Stock return calculation methods from the Quant 101 Series on YouTube. There we go over when to use arithmetic, geometric and log returns.

Related Terms

Our trained humans found other terms in the category return math you may find helpful.


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