**Scenario 1: +10% + 10% + 10% + 10% - 20%**

**Scenario 2: +5% + 5% + 5% + 5% + 0%**

If you answered scenario 2, 5% gains followed by a flat year you would be right. $1,000 invested in the first scenario would result in $1,171.28 (+17.13%) after five years while $1,000 invested in the second scenario would result in $1,215.51 (+21.55%). This despite both scenarios averaging a 4% gain per year.

Why does this happen? The difference can be described by defining the arithmetic average versus the geometric average of the investment. The arithmetic average of our investments is found by adding up the returns for each year and dividing by the number of years. This is what we typically mean whenever we use the term average, as we did above when we said a 4% yearly average for both scenarios. The geometric average is found by adding 1 to the annual returns, multiplying them, raising that to the power of (1/number of years) and then subtracting 1. It sounds like a lot but the gist of it is that returns for one year affect future returns when we calculate geometric average but this does not take place when calculating arithmetic average, and thus geometric average is a more relevant measurement for investors.

Let's see the results of calculating both averages for our two scenarios:

**Arithmetic Averages**

Scenario 1: ( .10 + .10 + .10 + .10 - .20 ) / 5 = .04 or 4%

Scenario 2: ( .05 + .05 + .05 + .05 + .00 ) / 5 = .04 or 4%

Geometric Averages

Scenario 1: [ ( 1.10 * 1.10 * 1.10 * 1.10 * .80 ) ^ ( 1 / 5 ) ] - 1 = .0321 or 3.21%

Scenario 1: ( .10 + .10 + .10 + .10 - .20 ) / 5 = .04 or 4%

Scenario 2: ( .05 + .05 + .05 + .05 + .00 ) / 5 = .04 or 4%

Geometric Averages

Scenario 1: [ ( 1.10 * 1.10 * 1.10 * 1.10 * .80 ) ^ ( 1 / 5 ) ] - 1 = .0321 or 3.21%

**Scenario 2: [ ( 1.05 * 1.05 * 1.05 * 1.05 * 1 ) ^ ( 1 / 5 ) ] - 1 = .0398 or 3.98%**

As you can see, the arithmetic averages are the same for both scenarios while the geometric averages for scenario 2 is greater than that of scenario 1. The geometric averages reflect our total average annual returns while the arithmetic averages do not.

Let's try some more scenarios. 4 years of 15% gains followed by a loss of 40%, 4 years of 20% gains followed by a loss of 60% and 4 years of 25% gains followed by a loss of 80% all have an arithmetic average return of 4%. The same can be said for 4 years of 5% losses followed by a 40% gain, 4 years of 10% losses followed by a 60% gain, 4 years of 15% losses followed by an 80% gain and 4 years of 20% losses followed by a 100% gain. How do their total returns stack up? Let's take a look at a table (click for larger image):

Our best total returns are from those rows towards the middle. So what does this tell us as investors? It tells us that, all other things being equal, if two investments have the same average (arithmetic) returns but one has a greater volatility than the other, always choose the investment with lower volatility. Volatility is our enemy.

It also tells us to be wary of investments that have even several years of outperformance. I chose 5 years for a timeline here because that is the typical length of a business and stock market cycle. From the table above, you can see that 4 years of 25% yearly gains are completely wiped out and then some with a -80% year at the end.

If you would like to play around with the numbers and work out your own scenarios, click here to view my google docs spreadsheet which you can view, edit and save in google docs or download to excel. Good investing!

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