# Subject: Analysis - Computing Compound Return

Last-Revised: 18 Feb 2007
Contributed-By: Paul Randolph (paulr22 at juno dot com), Chris Lott (contact me)

This article discusses how to compute the effective annual percentage rate earned by a single investment after a number of years have passed. A related concept called "average annual return" is frequently seen when reading about mutual funds but is computed very differently; it is discussed briefly at the end of this article. Yet another related concept called "internal rate of return" is used to calculate the percentage rate earned by an investment made as a series of purchases, such as monthly investments in a mutual fund; also see the article on that topic elsewhere in this FAQ.

To calculate the compound return on an investment, first figure out the factor by which the original investment multiplied, which is sometimes known as the total return. For example, if \$1,000 became \$3,200 in 10 years, then the multiplying factor (the total return) is 3,200/1,000 or 3.2. Next, take the 10th root of 3.2 (the multiplying factor) and you get a compound return of 1.1233498. (If you have forgotten your algebra, here's a quick reminder - just compute 3.2 raised to the 1/10 power.) The fractional part of this value, .1233, is known as the annualized return. To check that this works, note that 1.1233498 raised to the 10th power equals 3.2.

Here is another way of saying the same thing. This calculation assumes that all gains are reinvested, so the following formula applies:

```TR = (1 + AR) ^ YR
```

where TR is total return (present value/initial value), AR is the annualized return (above that was .1233), and YR is years. The symbol '^' is used to denote exponentiation (e.g., 2 ^ 3 = 8).

To calculate annualized return, the following formula applies:

```AR = (TR ^ (1/YR)) - 1
```

For example, a total return (multiplying factor) of 9.5 over 20 years yields an annualized return of 0.1191 (11.91%). To think of this in percentages, a 950% gain includes your initial investment of 100% (by definition) plus a gain of 850%.

For those of you using spreadsheets such as Excel, you would use the following formula to compute AR for the example discussed above.

```   = TR ^ (1 / YR) - 1
```

where TR = 9.5 and YR = 20. If you want to be creative and have AR recalculated every time you open your file, you can substitute something like the following for YR:

```( (*cell* - TODAY() ) / 365)
```

Of course you will have to replace '*cell*' by the appropriate address of the cell that contains the date on which you bought the security.

Don't confuse a compound return with something called an average annual return, which is a simple arithmetic mean (also see the FAQ article on this topic). That method simply adds the annual rates and divides by the number of years. For example, 5% one year and 10% the next year, average is 7.5% over those two years.

Let's compare the two methods with a contrived example. You invest \$100. After one year, you have \$200, which means in that first year, the investment returned 100%. At the end of the second year, you have \$100, which means in that second year, the investment lost 50%. (In short, you're back where you started.) Do the calculations for the compound return and you'll get 0%. Calculate the average annual return and you get 25%. So this contrived example yields a big difference. However, common scenarios yield less striking differences, and the average annual return is a useful approximation.

Here's the one thing to remember from this article. When you read an investment company's statements about their "average return", you should check carefully just exactly what they calculated.

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