Solving Typical FE Problems
All of the formulas used in making time value calculations are based on effective interest rates. Therefore, whenever the interest rate that is provided is a nominal rate, it is necessary to convert it to an effective interest rate. As shown below, an effective interest rate, i, can be calculated for any time period longer than the compounding period.
The most common way that nominal interest rates are stated is in the form 'x% per year compounded y' where x = interest rate and y = compounding period. An example is 18% per year compounded monthly. When interest rates are stated this way, the simplest effective rate to get is the one over the compounding period because all that is required is a simple division. For example, from the interest rate of 18% per year compounded monthly, a monthly interest rate of 1.5% is obtained (i.e., 18% per year/12 compounding periods per year) and this is an effective rate because it is the rate per compounding period. To get an effective rate for any period longer than the compounding period use the effective interest rate formula.
This effective interest rate formula can be solved for r or r/m as needed to determine a nominal interest rate from an effective rate.
For continuous compounding, the effective rate formula is the mathematical limit as m increases without bounds, and the formula reduces to i = er - 1.
Example #9: For an interest rate of 12% per year compounded quarterly, the effective interest rate per year is closest to:
- 4%
- 12%
- 12.55%
- 12.68%
Solution: An effective interest rate per year is sought. Therefore, r must be expressed per year and m is the number of times interest is compounded per year.
Example #10: For an interest rate of 2% per month, the effective semiannual rate is closest to:
- 11.55%
- 12%
- 12.62%
- 26.82%
Solution: In this example, the i on the left-hand side of the effective interest rate equation will have units of semiannual periods. Therefore, the r must have units of semiannual periods (i.e., 12% per six months) and m must be the number of times interest is compounded per semiannual period, 6 in this example.
The types of calculations used to obtain effective interest rates are summarized in Table 3.
| Table 3- Summary of Calculations Involved in Finding Effective Rates | ||
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| Interest Statement | To Find i for Compounding Period | To Find i for any Period Longer than Compounding Period |
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| i = 1% per month | i is already expressed over compounding period | Use effective interest rate equation |
| i = 12% per year compounded quarterly | Divide 12% by 4 | Use effective interestrate equation |
| i = nominal 16% per year compounded semiannually | Divide 16% by 2 | Use effective interest rate equation |
| i = effective 14% per year compounded monthly | Use effective interest rate equation and solve for r/m | For effective i values other than yearly, solve for r in effective interest rate equation and then proceed as in previous two examples |
