# 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
0 0 \$1,000.00
1 25% \$1,250.00
2 -15% \$1,062.50
3 5% \$1,115.63
4 15% \$1,282.97
5 -5% \$1,218.82

Surprised? It's unintuitive that no matter how the rates vary over time, it doesn't matter if they come first, last or in between. Most people expect that large losses up front swamp later gains or that early gains do make up for late losses. In reality, the change of the underlying principle amount evens things out, e.g. a smaller percentage drop later is against a larger principle if there have been early gains.

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