# Average yearly return

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## Average yearly return

Average yearly return, average annual return, -- there is a trick to calculating it: finding the "geometric mean," which is not properly calculated by merely summing up the yearly positive and negative percentages and dividing the sum by the number of years, as the novice supposes.

For example, suppose a mutual fund company has told you its fund has gained 40% one year and only lost 30% the next. Have they earned 5% on average? Only if you erroneously figure the mean like this: 40%-30 = 10; 10/2 = 5%. In reality, your \$100 rose to \$140 after one year then fell by 30% the next: 140(.7)= \$98. So, in fact, you lost two dollars overall.

The formula for the average yearly return as a geometric mean: AYR = (1 + first year's % return)(1 + second year's % return)(1 + third year's % return)and so on = X. Then take the square root of X for two years of returns, the cube root of X for three years of returns, and so on = Y. Then Y - 1 = AYR. In the above example: AYR = (1.4)(.7) = .98; the square root of .98 = .9899 - 1 = -1%. Thus, instead of gaining 5% per year on average, the fund lost 1% per year on average. A shortcut: avoid the last two steps by just looking at X, say, .98, and seeing what portion of your original investment would be left, which is often all you really need to know to get a sense of the true outcome of the yearly returns over the given period.

• As investors use the term, the return an investment provides over a period of time, expressed as a time-weighted annual percentage. Sources of returns can include dividends, returns of capital and capital appreciation. The rate of annual return is measured against the principal amount of the investment and represents a geometric mean rather than a simple arithmetic mean.
• Annual return is the de facto method for comparing the performance of investments with liquidity, which includes stocks, bonds, funds, commodities, and some types of derivatives.

US mutual funds report the average annual compounded rates of return for 1-year, 5-year, and 10-year periods as the "average annual total return" for each fund. The following formula is used.

• P (1+T)n = ERV

Where:

• P = a hypothetical initial payment of \$1,000.
• T = average annual total return.
• n = number of years.

ERV = ending redeemable value of a hypothetical \$1,000 payment made at the beginning of the 1-, 5-, or 10-year periods at the end of the 1-, 5-, or 10-year periods (or fractional portion). [1] [2]