# Geometric vs. Arithmetic Average Returns

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• 0:02 Calculation of Returns
• 0:34 The Arithmethic Average
• 1:57 The Geometric Average
• 4:17 Lesson Summary
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Lesson Transcript
Instructor: Ian Lord

Ian has an MBA and is a real estate investor, former health professions educator, and Air Force veteran.

The way many people calculate averages doesn't necessarily give an accurate picture of investment returns. In this lesson we will compare geometric and arithmetic averages and learn how they impact investment return math.

## Calculation of Returns

As an investor in stocks and bonds, Sam wants to know what kind of return on his investments he has made over the last five years. Specifically, Sam wants to know what his average return has been over that time period. But the exact method for calculating the true returns of an investment differs from the calculation of averages that most people think of. In the lesson ahead we will help Sam understand the difference between arithmetic and geometric averages.

## The Arithmetic Average

When Sam thinks of an average, the first thing that comes to mind is his memory of figuring out averages on his school exams. He's thinking of this kind of average: if his last five test scores were 95%, 80%, 99%, 86%, and 90%, his average would be 90%.

This is an arithmetic average, and the formula for calculating it is (a + b + c + d + e) / n where n is the number of data points in the average. So if we plug in Sam's test scores, the equation is (95 + 80 + 99 + 86 + 90) / 5 = 90.

Using the same formula for calculating Sam's returns on his investments, we might say that if Sam's returns over the last five years were -3%, 5%, 10%, -2%, and 20%, the arithmetic average would be 6%. But the problem with using this formula for investments is that it treats each data point as unique. In other words, how Sam scored on each test has nothing to do with the score on the previous test. But in investing, returns are incredibly dependent on how the investment has performed previously. This means the arithmetic average could be misleading, so let's look at what happens when we use a formula that accounts for past performance.

## The Geometric Average

The geometric average takes into account how an investment has previously performed when calculating the average return. Picture the following scenario: If an investment of \$100 has a 100% return in year one followed by a negative 50% return in year two, the arithmetic average would be 25%. However, the final sum is still \$100. Sam hasn't gained any actual return, and if anything, has lost money to inflation.

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