51. Apart from CDs, returns on investments are rarely the same from year to year, since they vary with prevailing interest rates. How should you calculate an “average” rate of return over several years? Consider a mutual fund that delivers 100% return one year and loses 50% the next year. Calculate just the ordinary average (the arithmetic mean) of the percentages to get . That sounds good, but check what happens to a $1000 investment: It grows to $2000, then halves back to $1000—for a 0% gain. Because the average of the percentages is not useful, the customary way used in finance to calculate the “average” return is to use the geometric mean. If the initial value of the portfolio was , and its value after years is , then the average annual rate of return is the value of that solves , or .