Chapter 5 Time Value of Money 1335-4 FINDING THE INTEREST RATE, I
Thus far we have used Equations 5-1 and 5-2 to! nd future and present values.
Those equations have four variables; and if we know three of the variables, we can
solve for the fourth. Thus, if we know PV, I, and N, we can solve 5-1 for FV, while
if we know FV, I, and N, we can solve 5-2 to! nd PV. That’s what we did in the pre-
ceding two sections.
Now suppose we know PV, FV, and N and we want to! nd I. For example,
suppose we know that a given bond has a cost of $100 and that it will return
$150 after 10 years. Thus, we know PV, FV, and N; and we want to! nd the rate of
return we will earn if we buy the bond. Here’s the situation:
FV! PV(1 " I)N
$150! $100(1 " I)^10
$150/$100! (1 " I)^10
1.5! (1 " I)^10
Unfortunately, we can’t factor I out to produce as simple a formula as we could
for FV and PV—we can solve for I, but it requires a bit more algebra.^4 However,
! nancial calculators and spreadsheets can! nd interest rates almost instantly.
Here’s the calculator setup:
N I/YR PV PMT FV10 –100 0 1504.14Enter N! 10, PV! $100, PMT = 0 because there are no payments until the secu-
rity matures, and FV! 150. Then when you press the I/YR key, the calculator
gives the answer, 4.14%. You would get this same answer with a spreadsheet.
SELF^ TEST What is discounting, and how is it related to compounding? How is the
future value equation (5-1) related to the present value equation (5-2)?
How does the present value of a future payment change as the time to
receipt is lengthened? as the interest rate increases?
Suppose a U.S. government bond promises to pay $2,249.73 three years
from now. If the going interest rate on three-year government bonds is 4%,
how much is the bond worth today? How would your answer change if the
bond matured in 5 years rather than 3? What if the interest rate on the
5-year bond was 6% rather than 4%? ($2,000; $1,849.11; $1,681.13)
How much would $1,000,000 due in 100 years be worth today if the dis-
count rate was 5%? if the discount rate was 20%? ($7,604.49; $0.0121)(^4) Raise the left side of the equation, the 1.5, to the power 1/N! 1/10! 0.1, getting 1.0414. That number is 1 plus
the interest rate, so the interest rate is 0.0414! 4.14%.