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The Basics of Return, Compounding, and Annualization

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What Does This Article Do for Me?

Explains the meaning and the basic mathematics of return, compounding, and annualization. Gives an outlook on more advanced topics in financial performance measurement. The reader should be mathematically inclined. However, no more than a high school math education is required.

The Return of a Single Investment

Suppose you invest an initial amount of money, and you look at it again after some period of time. The return of the investment for that time period is the percentage change of your money. It can thus easily be calculated from the begin and end values of the investment:

If we use fractional returns rather than percentage returns, the equation becomes

Conversely, if we're given the begin value of the investment and the return, it is easy to calculate the end value:

If we use the letters r for return, B for begin value, and E for end value, we thus have the following fundamental equation:

Periodic Returns

If we look at the return periodically, like every month or every year, then we can use Equation (2) above iteratively. At the end of the first period, the value of the investment is

where r1 is the return for the first period. At the end of the second period, the value has become

where r2 is the return for the second period. Continuing in this manner, we see that after n periods, the value of the investment is