**THE CUMULATIVE GAIN** was 21% in the first example given in the previous section. The annualized gain, however, was 10%. This is known as a geometric average: It’s the amount you would have to earn each year to achieve the cumulative result. Note that this is different from simply dividing the cumulative gain by the number of years—the so-called arithmetic average. The latter wouldn’t give you the annualized return, because simple averaging doesn’t take into account the effects of compounding.

Suppose that, over the next five years, you earned annual returns of 10%, -10%, 5%, 0% and 15%. Your cumulative gain would be 19.5%, which you can find by performing this calculation:

1.1 x 0.9 x 1.05 x 1 x 1.15 = 1.195

To turn this into an annualized (or geometric) return, you would need the help of a financial calculator or a spreadsheet. If you had that handy, you’d discover that the annualized return over the five years is 3.6%.

By contrast, the arithmetic average—which you find by simply adding up the performance each year and dividing by the number of years—would be 3.9. Unless you earn the same return every year, the geometric average will always be less than the arithmetic average. In fact, if there’s a wide gap between the geometric and arithmetic average, that’s a sign that a portfolio is highly volatile. This volatility isn’t good for efficient investment compounding.

Next: Rule of 72

Previous: How Money Compounds

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