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The differences in future value from investing at these different rates of return are small for short compounding periods (such as 1 year) but become larger as the compounding period is extended. For instance, with a 40-year time horizon, the future value of investing in stocks, at an average return of 12.4%, is more than 12 times larger than the future value of investing in treasury bonds at an average return of 5.2% and more than 25 times the future value of investing in treasury bills at an average return of 3.6%.

### III. The Frequency of Discounting and Compounding

The frequency of compounding affects both the future and present values of cash flows. In the examples above, the cash flows were assumed to be discounted and compounded annually — i.e., interest payments and income were computed at the end of each year, based on the balance at the beginning of the year. In some cases, however, the interest may be computed more frequently, such as on a monthly or semi-annual basis. In these cases, the present and future values may be very different from those computed on an annual basis; the stated interest rate, on an annual basis, can deviate significantly from the effective or true interest rate. The effective interest rate can be computed as follows

T-^f-f- .. T , , n * i. Stated Annual Interest Rate % Effective Interest Rate = 11 +-1 -1

n where n = number of compoundin g periods during the year (2=semiannual; 12=monthly) For instance, a 10% annual interest rate, if there is semiannual compounding, works out to an effective interest rate of

Effective Interest Rate = 1.052 - 1 = .10125 or 10.25%

As compounding becomes continuous, the effective interest rate can be computed as follows

Effective Interest Rate = expr - 1

where exp = exponential function r = stated annual interest rate Table 2 provides the effective rates as a function of the compounding frequency.

Table 2: Effect of Compounding Frequency on Effective Interest Rates

Frequency

Rate

t

Annual

Rate

Annual

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