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## Average vs. Geometric Average

**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:

**$10,000 X (1 + .1317)**

^{20}**$10,000 X 1.1317**

^{20}**$10,000 X 11.88748**

**$118,748**

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:

**(1.1867 X 1.0525 X …X 1.0491)**

^{1/20}**9.542119**

^{.05}**1.119392**

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:

**$10,000 X (1+.119392)**

^{20}**$10,000 X (1.119392)**

^{20}**$10,000 X 9.5421**

**$95,421**

Holy cow! It works!

**Topics:** Financial Math Basics | 18 Comments »

June 6th, 2006 at 4:22 pm

Great calcs, I’m saving this for future reference!

June 7th, 2006 at 12:38 am

Excellent post! This is good stuff, makes me take a different view on some of these mutual funds.

June 7th, 2006 at 2:04 am

That is fantastic JLP! I have wondered about this for quite a while now trying to accurately present my returns. For all of my two years of data the average is 12.39% while the geometric average is 12.37%. Not a big difference yet, but time will exacerbate the discrepancy. Thanks for the lesson.

June 7th, 2006 at 10:00 am

Trip,

Thanks! I love doing these kinds of posts! It’s too bad that more people don’t like to read them!

June 7th, 2006 at 4:42 pm

Holy crumb! I must start using excel to its full capabilities so I can do stuff like this. Either way, the rate of return is yet another reason to stay in it for the long term.

June 7th, 2006 at 8:50 pm

Average vs. Geometric AverageArticle on how the standard average calculation can produce misleading historical return numbers, and how to use the Geometric Mean to correct for this.

June 7th, 2006 at 11:51 pm

I made the same arithemtic vs. geometric average when I tried to tackle the Rule of 72 as a convenience calculation. Numbers do deceive and you gave a great example! and I love reading these posts!

June 11th, 2006 at 6:57 pm

Weekly Roundup – 06/09/06Here’s a quick look at a few things from the MoneyBlogNetwork (and beyond) that caught my eye over the past week…

MightyBargainHunter asks if one should pay down their mortgage or sock extra savings away in their ING Direct account (click…

June 12th, 2006 at 9:50 am

Average vs. Geometric AverageBe sure you are using geometric means when reporting your returns. This article tells you how and why it is important.

September 19th, 2006 at 3:02 pm

Ah so good to have found this. was trying to work out geometric average of Net income growth with negitve numbers and was not working. Using your method I got the answer and my sanity.

Great work!!!!!!!!!!!!

January 23rd, 2007 at 3:00 pm

excelent analysis. you have to use georeturns!!!!

March 13th, 2007 at 1:16 am

Very kind of you. I then got a clear idea of what they are.

Thank you for all you have done.

January 28th, 2008 at 12:46 am

[...] Now, this graphic shows you how 2007’s returns impacted the annualized rate of return (also called the geometric average) for the indexes: [...]

April 23rd, 2008 at 11:06 am

Thank u, now it all make a whole lot of sense….

June 19th, 2008 at 9:00 pm

Geometric method is by far superior than arithmetic average!

April 20th, 2009 at 2:35 pm

Thanks for the great explanation of geometric vs arithmetic average!

It makes sense though when you stop to think about it… every year you multiply your returns by the relevant factor, since this will give the increase (or decrease) in your returns achieved over that period. When multiplying however you are not simply multiplying the amount you initially put in but you are also multiplying the factors from previous years and hence a factor from a particular year influences all the factors that came before it, so you simply can not use the arithmetic average. The factors are not independent! I hope this makes sense, I tried my best…it’s still a great article though so thanks!

November 20th, 2009 at 2:56 pm

Just remember, the geometric mean only informs you about the historical return. It is not a good predictor of next year’s return and therefore should not be used to compare across asset classes. For more information, see:

Blume, Marshall, 1974, Unbiased estimators of longrun expected rates of return

January 22nd, 2010 at 12:27 pm

thanks for this helped me out a lot