I came across an article last night that stated that the average annual return for the S&P 500 was 12% from 1926-2014.

The way this author arrived at 12% was to add up all the yearly returns over the last 89 years and divide them by 89. When I do that with my numbers, I get 11.99%. Going further, using that return, $100 invested at the beginning of 1926 would have grown to nearly $2.4 million by the end of 2014. The math looks like this:

^{89}

Unfortunately, that’s not reality. Let me try to explain it with a simple example.

Say an investor received the following returns over three years:

Year 1: 25%

Year 2: -50%

Year 3: 25%

The average of those three years would be 0%. Now let’s calculate how it would have looked had we invested $100 at the beginning of the first year and held it through year three:

Year 1: $125.00 ($100 + ($100 x .25)) OR $100 X (1 + .25)

Year 2: $62.50 ($125 + ($125 X -.50)) OR $125 X (1 – .50)

Year 3: $78.13 ($62.50 + ($62.50 x .25)) OR $62.50 x (1 + .25)

**ONE QUICK NOTE:** Some people have a difficult time with math and since this is a “Basics” post, I’ll spend a little extra time talking about how to think about returns. The reason we add 1 to return expressed as a decimal (1 + .25) is because 1 represents the original investment. So, we are getting our $100 back plus $25. When we have a negative return, we subtract the negative return from 1 (1 – .50) because we are losing part of our original investment. In this case we are losing half of our $125. Does that make sense?

Based on the average return we calculated earlier, the $100 should have been worth $100 at the end of the third year because our numbers showed a 0% return, but this shows that our $100 investment is only worth $78.13 (a 21.88% loss). What accounts for the difference?

The difference is the first average used was simply the arithmetic mean (average) and doesn’t take into account an actual investment. To find out our actual return, we need to use something called the geometric mean (average). To do that, we need to change our formula up a bit. First, we need to express the returns as multipliers by adding 1 to them. Like this:

Year 1: 1.25

Year 2: 0.50

Year 3: 1.25

Now, we simply multiply those factors together and raise them to the power of 1/3 (or find the 3rd root). Like this:

^{1/3}

^{.3333}

Now, in order to get this to a percentage, we have to subtract 1 from .921008, which leaves us with -.07899 or -7.899%.

We can check our answer by plugging our finding back into the following equation:

^{3}

IT WORKS!

So…

Back to the opening paragraph, which stated that the average return from the S&P 500 Index was 12% from 1926-2014. The geometric average is 10.04%. “Not much different,” you might think. The difference is huge when we are talking about 89 years. Let me show you:

^{89}

That’s nearly $1.9 MILLION less than the number we get using the average return. Which brings up an important question: why do people still talk about averages?

Extra credit: Here are a couple of useful YouTube videos I found that talk about the geometric average (or mean)…