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4
The Basics of Return, Compounding, and Annualization

One can visualize the annualized return in a chart that shows the account balances over time. The chart below shows the actual account balances for the 5-year and the 6-year investments as solid lines. The account balances of the fixed rate accounts that replicate the respective investments are shown as dotted lines. The latter are straight lines because the scale of the vertical axis is logarithmic. Their slope indicates the annualized return of the respective investement.

Calculating the annualized return is not hard. Equation (5) tells us that the ending value E of an investment of B dollars for n years at an annual rate of a is given by

When calculating the annualized return, we are given the beginning balance B, the ending balance E, and the number of years n (possibly a fractional number). Solving this equation for the unknown annualized return a is simple pre-calculus math: divide both sides of the equation above by B, take both sides to the power 1/n, and subtract 1 on both sides:

It is obvious from Equation (2) that the expression E/B in the above is equal to 1 + r, where r is the return of the investment for the full n years. Plugging that into Equation (7) above gives us a way to calculate the annual rate of return a from the ordinary return r:

Applying this to the five year investment with the 30.696% return and the six year investment with the 35.6007% return, we obtain the annual rates of return 5.5% and 5.2% that we mentioned earlier: