# When to use geometric vs. arithmetic mean? Why is the former better for percentages?

While computing the average price of a stock or index over a period of time, I have the choice between using an arithmetic mean or a geometric mean. Which should I use and when?

I am reading a book on Trading Systems by Kaufman and he says:

The geometric mean has advantages in application to economics and prices. A classic example compares a tenfold rise in price from 100 to 1000 to a fall to one tenth from 100 to 10. An arithmetic mean of the 2 values 10 and 1000 is 505, while the geometric mean gives

`G = (10 × 1000)^(1/2) = 100`

and shows the relative distribution of prices as a function of comparable growth. Due to this property, the geometric mean is the best choice when averaging ratios that can be either fractions or percentages.

I am not able to understand what he means by that last part (from "relative distribution" onward.) Could somebody please explain?

• On re-reading, the explanation cited doesn't reflect reality. The math is correct, but a more realistic +/-50% is exaggeration enough. +/-20% is still zero average but -2%/yr geometric, which I suppose should be a decent number. Mar 13 '14 at 2:26

Simple. Say in 2012 you were up 50% (brilliant) but then in 2013 you were down 50% (sorry). i.e. if you started with \$1000, you were up to \$1500, then down to \$750. You lost \$250 overall.

If you were to compute the mean of the percentages using each method, then:

• Arithmetic mean: The average of +50% and -50% (really 150% and 50% of each period's initial value) is zero, not up or down.

• Geometric mean: `1.5 * .5 = .75`, i.e. you are down 25% over 2 years, or about 13.4% per year.

It should be clear the Geometric makes more sense in such a case.

• Given the massive edits, it should be clear Joe should not answer questions on his iPhone at red lights. Thank you, Chris. Mar 12 '14 at 22:27
• Thanks. But while calculating GM in your example, why you did not take the square root of 1.5*.5. Sorry, but can you elaborate more on your answer Mar 13 '14 at 13:47
• First, I did. I edited as I was off a tiny bit. SQR of .75 is .866. Or the -13.4% per year. Better? Mar 13 '14 at 13:52
• Thanks. Is there a case where using AM to compute average returns will be better than GM. Or is it always better to use GM? Mar 13 '14 at 13:56
• AM is a misleading number. A financial celebrity is known to cite an average expected return of 12%. But the GM is more like 10%, and over the long term, it's not about a pedantic quibbling over words, but a real difference in the returns one would see. Mar 13 '14 at 14:08

JoeTaxpayer nailed it.

Here's another way to look at it: Generally, we invest in something, then might leave it there for a few years, then take it out, but don't touch it in between. In that case, to get the final amount X(N), we need to take the initial amount, then multiply by growth in the first year, then multiply by growth in the second year, etc.

So, for three years, we have:

X(3) = X(0) * G(1) * G(2) * G(3) = X(0) * "average annual growth" ^ 3

So, here, we see that we want the average annual growth to the power three equal to the product of the annual growth rates, thus, geometric mean:

geometric mean = (G(1) * G(2) * G(3)) ^ (1/3)

On the other hand, consider a situation where I have three investments X,Y,Z over one year. Now I have, after one year:

X(1)+Y(1)+Z(1) = X(0)*G(1,X) + Y(0)*G(1,Y) + Z(0)*G(1,Z) = ( X(0)+Y(0)+Z(0) ) * "average annual growth"

Now, in this case, if we assume X(0) = Y(0) = Z(0) = 1, i.e. I put equal amounts in each, we see that the average annual growth rate we want in this case is the arithmetic mean:

arithmetic mean = (G(1,X) + G(1,Y) + G(1,Z)) / 3

(if we had unequal amounts at the beginning, it would be a weighted average).

TL;DR:

• if we're interested in ONE investment over MANY years, we need to multiply -> use the geometric mean.
• if we're interested in MANY investments over ONE year, we need to add -> use the arithmetic mean.
• if we're interested in MANY investments over MANY years.... it gets complicated :-)
• Thanks. But what is wrong with using GM for many investments in one year? It seems the choice of the average is personal preference here. Apr 4 '14 at 20:30
• If I put 100\$ into stock X, and 100\$ into stock Y, and X stays at 100\$, and Y goes to 144\$, I made 44\$ on a 200\$ investment, or 22%. Now, 1.22 is the arithmetic mean of 1.00 and 1.44, while the geometric mean is 1.20. So, to be maybe a bit more precise, the arithmetic mean is appropriate for parallel investments, while the geometric mean is appropriate for successive investments (that are re-invested).
– Fab
Apr 4 '14 at 21:04