# Average Return Calculation: Folks Foolishly Fool Themselves

Way back in the summer, Chris Sells wrote The “Average Return” Myth. His commentary can be quite an eye-opener for some people. I know this concept he discussed surprised me when I first encountered it many years before I came across his post. I'm reproducing a big chunk of his explanation here.

Assume that you have \$1000 to invest. After the first year, you get a 25% return so of course you invest it again the following year. At the end of the second year you end up losing 15%. What's the average rate of return over the two years? You probably think that it's 5%, that is, (25% + (-15%))/2. Let's do the math for 25% and -15%:

* Using simple interest, after 1 year, \$1000 + \$1000 * 25% = \$1250
* After 2 years, \$1250 + \$1250 * -15% = \$1062.50

Here we're using Interest = Principle * Rate * Time calculation for yearly aka simple interest (I = PRT and Time is 1 year). Taking the numbers the other way, i.e. -15% the first year and 25% the next year, yields the same result:

* After 1 year, \$1000 + \$1000 * -15% = \$850
* After 2 years, \$850 + \$850 * 25% = \$1062.50

In fact, the result is the same no matter in which order that the rates come or how many there are:

Table 1: From Good to Bad
0 0 \$1,000.00
1 25% \$1,250.00
2 15% \$1,437.50
3 5% \$1,509.38
4 -5% \$1,433.91
5 -15% \$1,218.82

Table 2: Starting Bad to Good
0 0 \$1,000.00
1 -15% \$ 850.00
2 -5% \$ 807.50
3 5% \$ 847.88
4 15% \$ 975.06
5 25% \$1,218.82

Table 3: A Mixed Bag