In situations where the interest compounding period is something other than annual, the effective interest (Section 2.5) can be used in time value of money calculations. Consider the following scenario where an initial investment of $65,000 is made in an account that provides a nominal interest rate of 4%, compounded monthly, for a period of 10 years. Determine the future sum of this investment.
The expression required to calculate this future value is given by,
F = P(F/P, i, n)
There are two options for the variables 'i' and 'n', depending on which interest rate you would like to use.
Option 1: Using the nominal interest rate
When using the nominal interest rate, the compounding period is taken into account by modifying the interest rate to be consistent with each compounding period. In this case, there are 10(12) = 120 compounding periods. In each compounding period, the interest is 0.04/12. Therefore, the future sum can be found by,
F = P(F/P,i,n) = $65,000(F/P, 0.04/12, 120) = $96,904
Option 2: Using the effective interest rate
The effective interest rate reflects the annual rate taking into account the compounding period. In this case, since compounding is monthly, the effective interest rate is,
i_eff = (F/P, 0.04/12, 12) - 1 = 0.0407415429
Expanding the interest rate to several decimal places eliminates the effect of round-off error when using this value. Now that the effective interest rate is known, the future sum can b found by,
F = P*(F/P,i,n) = $65,000(F/P, 0.0407415429, 10) = $96,904
Summary
This example demonstrates that there are really two different ways to solve an engineering economy problem where the compounding period is less than one year. Either approach is valid. In order to ensure that the proper result is obtained, just remember to make sure that the interest rate used in the calculation of the interest factor is consistent with the time period used. If you have selected the effective interest rate, then carry as many decimal places as you can when computing the interest factor to avoid round-off error.