### The Geometric Average (also called the geometric mean)

 The arithmetic average -- which is the plain old regular average that we're used to seeing -- is off with rates of return. Although usually it's only a little off. Here's why: Suppose a stock had a return of 150% in year 1, and a return of  negative 100% in year 2. What is the average return? If you use the arithmetic average you get: (150% + -100%) / 2 = 25%. But this is far off, because you didn't really get a 25% per year return over 2 years. If you invested \$100, then at the end of year one you gained 150% of that, so your balance grew to \$100 + \$150 = \$250. But then in year 2, you started out with \$250, but you lost 100% of that with the negative 100% year 2 return. So at the end of year 2, you have zero dollars.You invested \$100, and two years later you had 0, so really your average return was negative100%, not what the arithmetic average gives you, which is positive 25%. Whenever there is variance in the returns from year to year, the arithmetic average overstates the true average return. The more variance there is, the more it overstates it. What does give you the true average return is something called the geometric average.Now, a typical very well diversified stock portfolio has significant variance in it's returns from year to year, but typically not giant variance, so the arithmetic average return is not that different from the correct geometric average return. For a well diversified stock portfolio the historical arithmetic average return is about 12%, while the true average return, the geometric one, is about 10.5%.How does the geometric average (also called the geometric mean) work? Suppose your stock portfolio earned a 10% return in year 1 and a 20% return in year 2. What happened? If you invested \$100, then at the end of year 1 it grows to \$100 x (1 + 10%) = \$100 x (1.10) = \$110. So, we start year 2 with \$110, and that year goes very well, and we earn 20%, so our balance grows to \$110 x 1.20 = \$132.What is the geometric average return? It turns out that it's 14.89% (I'll talk about the formula for how to calculate this soon) . Now, let's look at what would have happened if we had earned that geometric average return every year: We start out investing \$100, and we get a return of 14.89% in the first year, so we end the first year with a balance of \$100 x 1.1489 = \$114.89. Then, we start year 2 with \$114.89, and again we get a return of 14.89%, so we end year 2 with a balance of \$114.89 x 1.1489 = \$132. So, as you can see the geometric average is perfectly accurate. It is the return where, if we got it each year, we would end up with the exact same amount of money at the end as we actually did. We end up with the same \$132. It doesn't overstate what we get, and it doesn't understate what we get. 14.89% each year ends up making the same exact amount of money as a 10% return the first year and a 20% return the second year (except for some rounding error. The actual geometric average return here is 14.891252931...%).So, the true average return, the geometric one, is (approximately) 14.89%, but the arithmetic average is (10% + 20%) / 2 = 15%. So, the arithmetic average overstates the true average return, as I earlier said it would, but not by that much because in this case the annual returns don't vary that much from year to year, also as I said earlier.Ok, now how do you calculate the geometric average? It's not that hard; there's a formula that's not that long, and Excel has a function that will do it for you. The formula is: [(1+Ret1)(1+Ret2)...(1+Retn)](1/n) – 1 . But don't worry if you have an aversion to math; your mutual fund provider, or other finance people, media, and other sources will calculate geometric averages for you. It's not necessary to conduct great personal finance that you be able to calculate it yourself.
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