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CHAPTER 4 Time Value of Money 167
Another type of number-of-periods problem involves finding the number of
periods associated with an annuity. Occasionally we wish to find the unknown
life, n, of an annuity, PMT, that is intended to achieve a specific objective, such as
repaying a loan of a given amount, PVA n, with a stated interest rate, i.
EXAMPLE Bill Smart can borrow $25,000 at an 11% annual interest rate; equal, annual
end-of-year payments of $4,800 are required. He wishes to determine how long it
will take to fully repay the loan. In other words, he wishes to determine how
many years, n, it will take to repay the $25,000, 11% loan, PVA n, if the payments
of $4,800, PMT, are made at the end of each year.
Input Function
4800 PMT
25000
11
Solution
8.15
PV
I
CPT
N
Table Use Substituting PVA n$25,000 and PMT $4,800 into Equation 4.24
and rearranging the equation to solve PVIFA 11%,n yrs, we get
PVAn $25,000
PVIFA11%,n yrs 5.208 (4.27)
PMT $4,800
The number of periods for an 11% interest rate associated with the annuity
factor closest to 5.208 in Table A–4 is 8 years. Therefore, the number of periods
necessary to repay the loan fully is approximately (to the nearest year)
8 years.
Calculator Use (Note: Most calculators require either the PV or the PMT value
to be input as a negative number in order to calculate an unknown number of
periods. That approach is used here.) Using the inputs shown at the left, you will
find the number of periods to be 8.15, which is consistent with the value found
using Table A–4.
Spreadsheet Use The number of years to pay off the loan also can be calculated
as shown on the following Excel spreadsheet.