ferent inputs. See Ch 02 Tool Kit.xlson the textbook’s web site that accompanies
this text for spreadsheet models of the topics covered in this chapter.
TVM calculations generally involve equations that have four variables, and if you
knowthreeofthevalues,you(oryourcalculator)cansolveforthefourth.
If you know the cash flows and the PV (or FV) of a cash flow stream, you can de-
termine the interest rate.For example, in the Figure 2-3 illustration, if you were
given the information that a loan called for 3 payments of $1,000 each, and that
the loan had a value today of PV $2,775.10, then you could find the interest rate
that caused the sum of the PVs of the payments to equal $2,775.10. Since we are
dealing with an annuity, you could proceed as follows:
With a financial calculator, enter N 3, PV 2775.10, PMT 1000, FV
0, and then press the I key to find I 4%.
Thus far in this section we have assumed that payments are made, and interest
is earned, annually. However, many contracts call for more frequent payments;
for example, mortgage and auto loans call for monthly payments, and most
bonds pay interest semiannually. Similarly, most banks compute interest daily.
When compounding occurs more frequently than once a year, this fact must be
recognized. We can use the Figure 2-3 example to illustrate semiannual com-
pounding. First, recognize that the 4 percent stated rate is a nominal rate that
must be converted to a periodic rate, and the number of years must be con-
verted to periods:
iPERStated rate/Periods per year 4%/2 2%.
Periods Years Periods per year 3 2 6.
The periodic rate and number of periods would be used for calculations and shown
on time lines.
If the $1,000 per-year payments were actually payable as $500 each 6 months,
you would simply redraw Figure 2-3 to show 6 payments of $500 each, but you
would also use a periodic interest rateof 4%/2 2% for determining the PV or
FV of the payments.
If we are comparing the costs of loans that require payments more than once a
year, or the rates of return on investments that pay interest more frequently, then
the comparisons should be based on equivalent(or effective) rates of return using
this formula:
For semiannual compounding, the effective annual rate is 4.04 percent:
.
The general equation for finding the future value for any number of compounding
periods per year is:
where
iNomquoted interest rate.
m number of compounding periods per year.
n number of years.
FVnPVa 1
iNom
m
b
mn
,
a 1
0.04
2
b
2
1.0(1.02)^2 1.01.04041.00.04044.04%
Effective annual rateEAR (or EFF%)a 1
iNom
m
b
m
1.0.
Summary 93
Time Value of Money 91
