## Tuesday, July 24, 2012

### "AVERAGE RETURN ON INVESTMENT" IS MISLEADING

If you add the percentage gains and losses over a series of time periods and then average them to get some “average” return on your investment, ROI, it is known that the result can be quite misleading.

Say you make 60% gain (G) on your investment (I) this year and take a 40% loss (L) next year. You might think your average return on the investment was (60%-40%)/2yr=+10%/yr. Unfortunately, it was really a 4% loss, -2% per year, a substantial difference.

Let’s say you have invested \$1000. The first year you gained 60%, making your total \$1600. The next year you lost 40% of the \$1600, making your total \$960. Though your “average” return on investment was (60%-40%)/2=10%, you actually lost money, (\$960-\$1000)=-\$40, a 4% loss, or 2% per year on your original \$1000 investment.

The lesson: you cannot simply average the yearly percentage returns on investment to get a reliable estimate of how well the investment is performing.

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A little mathematics shows that it does not matter whether the loss was in the first year or the second. At the end of two years, you have your investment of \$1000 times (one plus the fractional gain, G) times (1 minus the fractional loss, L):

\$1000(1 + G)(1 - L) = \$1000(1.60)(0.60) = \$960.

For those who are curious about the mathematics, the general formula for this two-period case

(1 + G)(1 - L) = (1 + G - L - G*L)

and the product G*L is why the true outcome is different from the naïve average, (G-L)/2. If both periods showed gains, G and G’, then the equation uses +G’ rather than -L, and the correct answer for the characteristic gain per year is again different from the naïve average, (G+G’)/2.

Averaging over more than two periods produces a similar outcome, more difficult to express succinctly, but the message is the same, the naïve averaging of the ROI values incorrectly assesses the return on investment at the end of the period being analyzed.

FOOTNOTE:

The proper calculation of a characteristic or effective yearly return on investment requires determining the ratio of the final value of the account to its initial (investment) value and then finding the internal rate of return r, which when used in the formula (1+r)^n, where n is the number of periods (^ means exponentiation, raising to a power), gives that correct ratio. In our example 0.96=(1+r)^2, which can be solved as r = -0.02, a 2% loss per year.