By JLP | June 6, 2006
Take a look at the total returns for the S&P 500 Index for the last 20 years:
Over the last 20 years, the S&P 500 Index had an average return of 13.17% (.1317 as a decimal) per year. So, if you had invested $10,000 in 1986, it would have been worth $118,748 at the end of 2005. The math looks like this:
Is this how much your account would have really been worth? NOPE!
Here’s what your account would have looked like through the years:
According to those numbers, the account was worth $95,421 at the end of 2005. That’s a far cry from the $118,748 we got from using the average return. So, how do we calculate the Geometric Mean (also called the average annualized rate of return)? If you use Microsoft Excel, it is simple to calculate if you use their function (located in the “Insert” menu). You can also calculate it yourself relatively easily.
If you look at the chart above, you will see the year listed, the total return received that year and a column titled “Factor.” To get the factor, you simply add 1 to the return expressed as a decimal. For example:
A return of 18.67% is .1876 as a decimal, for a factor of 1.1867 (1 + .1867 = 1.1867)
For a year with a negative return, the calculation looks like this:
A return of -3.11% is -.0311 as a decimal, for a factor of .9689 (1 + (-.0311) = .9689)
So, if you had $10,000 at the start of one year and received a return of 18.67%, you would have $11,867 at the end of the year (10,000 X 1.1867). On the other hand, if you had $10,000 at the beginning of the year and received a return of -3.11%, you would have $9,689 at the end of the year, or only 96.89% of the amount you had at the beginning ($10,000 X .9689 = $9.689).
Now, to find your Geometric Average, you simply multiply all the factors together and raise them to the 1/n power (where n is the number of periods—usually years—for which you are making the calculation) . Got that? It’s not as confusing as it sounds. Using the numbers from the table above, it looks like this:
Subtract 1 from the answer above and you get the geometric average of .119392 or 11.9392%.
Now, to check and see if our answer is right, test it out with this equation:
Holy cow! It works!