## Thursday, December 1, 2011

### Calculating Returns Can Be Tricky

Calculating the total return of a collection of investments is a simple matter of adding up or averaging the returns of the individual investments, right? In reality, the right way to "add" or "average" returns depends on the context.

Suppose that investment A stayed flat for two years (0% return each year), and investment B fared much better returning 20% each year for both years. Suppose further an investor splits \$1000 between A and B in the first year, then rebalances and again splits his money between A and B in the second year:

To start:

A \$500
B \$500

After year 1:

A \$500
B \$600

The total is \$1100 for a 10% return. So we see that correct overall return is the average of 0% and 20%, namely 10%. This investor will get another 10% return in the second year and will end up with \$1210 for a total return of 21% after two years.

Note that the total return is not just 10% + 10% = 20%. In this case we have to "add" the returns using compounding: 1.10 x 1.10 - 1 = 21%.

Consider a second investor who owns only investment A for one year and then only investment B for the second year. We might think that this investor will get an average return of (0% + 20%)/2 = 10%, but he does not. This investor will be left with \$1200 for an overall return of 20%. But this doesn't work out to the 10% per year that the first investor got. The second investor's average return is 9.54% per year. When you compound this return for two years, you get his 20% total return for the two years.

This isn't just an academic exercise. Lenders use this against us all the time. If you get a monthly car loan at 6% per year, the actual rate is 0.5% per month. They call this "simple interest". In reality they calculate the monthly rate as though they were going to use simple interest (divide by 12), but then they compound it. If you multiply 1.005 by itself 12 times you get the real yearly rate that you’re paying: 6.17%. The difference isn't huge, but it's still extra money out of your pocket.

Understanding when to add returns and when to compound them is at the core of why higher portfolio volatility leads to lower long-term investment returns. Any time you hear someone talk about the average return of some type of investment you should ask what kind of average was used.