The cash ´¬‚ows can be equal, in which case CFt PMT constant, so we
have an annuity, or the CFt can vary from period to period. Now suppose
we know the current price of the asset, which is by de´¬ünition the PV of the
cash ´¬‚ows, and the expected cash ´¬‚ows themselves, and we want to ´¬ünd the
rate of return on the asset, or its yield. The assetÔÇ™s expected rate of return,
or yield, is de´¬üned to be the value of i that solves Equation 28-6. For exam-
ple, suppose you plan to ´¬ünance a car that costs $22,000, and the dealer
offers a 5-year payment plan that requires $2,000 down and payments of
For some purposes it is appropriate to use instantaneous, or continuous, compounding. Equation 28-5 can be used for
any number of compounding intervals, m, per year: m 1 (annual compounding), m 2 (semiannual compounding),
m 12 (monthly compounding), m 365 (daily compounding), m 365(24) 8,760 (hourly compounding), and
even more frequently. As m approaches in´¬ünity, the compounding interval approaches zero, and the compounding
becomes instantaneous, or continuous. The EAR for continuous compounding is given by the following equation:
EARcontinuous 1.0. (28-5a)
Here e is approximately equal to 2.71828; most calculators have a special key for e. The EAR of a 7 percent investment
compounded continuously is e0.07 1.0 0.0725 7.25%.
Equation 28-5a can also be used to ´¬ünd the future value of a payment invested for n years at a rate inom compounded
Similarly, the present value of a future sum to be received in n periods discounted at a nominal rate of inom compounded
continuously is found using this equation:
PV FVn(e ). (28-2a)
28-6 Basic Financial Tools: A Review
$415.17 per month for 60 months. Since you would be ´¬ünancing $20,000
over 60 months, the monthly interest rate on the loan is the value of i that
solves Equation 28-4a:
$20,000 415.17(PVIFAi,60). (28-4a)
This equation would be dif´¬ücult to solve by hand, but it is easy with a ´¬ünan-
cial calculator. Enter PV 20000, PMT 415.17, FV 0, and N 60,
0.75, or 3Ôü„4 percent a month.
and press CPT and then the I key to ´¬ünd I
The nominal annual rate is 12(0.0075) 0.09 9%, and the effective
annual rate is (1.0075) 1 9.38%.
Complex Time Value Problems
The present and future value equations can be combined to ´¬ünd the answers
to more complicated problems. For example, suppose you want to know
how much you must save each month to retire in 40 years. After retiring,
you plan to withdraw $100,000 per year for 20 years, with the ´¬ürst with-
drawal coming one year after retirement. You will put away money at the
end of each month, and you expect to earn 9 percent on your investments.
Here is a diagram of the problem:
retirement Years after retirement
0 1 2 480 1 2 20
100,000 100,000 100,000
PMT PMT PMT
on this date
The solution requires two steps. First, you must ´¬ünd the amount needed to
fund the 20 retirement payments of $100,000. This amount is found as fol-
lows: PV $100,000(PVIFA9%,20) $100,000(9.12855) $912,855.
Therefore, you must accumulate $912,855 by the 40(12) 480 month if
you are to make the 20 withdrawals after retirement. Each of the 480
deposits will earn 9%/12 0.75% per month. The second step is to ´¬ünd
your required monthly payment. This involves ´¬ünding the PMT stream that
grows to the required future value, and it is found as follows:
FV $912,855 PMT(FVIFA0.75%, 480) PMT(4,681.32).
$912,855/4,681.32 $195.00 per month.7
The problem could also be solved with a ´¬ünancial calculator. First, to cal-
culate the present value of the 20 payments of $100,000 at an annual inter-
est rate of 9 percent, enter N 20, I 9, PMT 100000, and FV 0, and
then press CPT and then the PV key to ´¬ünd PV $912,855. Second, clear
For simplicity we have assumed that your retirement funds will earn 9 percent compounded monthly during the accu-
mulation phase and 9 percent compounded annually during the withdrawal phase. It is probably more realistic to
assume that your earnings will be compounded the same during both phases. In that case you would use the effective
annual rate for 9 percent compounded monthly, which is 9.38 percent, as the annual interest rate during the withdrawal
phase. This larger effective rate would reduce the amount you need to save by the time you retire to $888,620, and your
monthly contribution would be $189.82.
Note also that it is relatively easy to solve problems such as this with a spreadsheet program such as Excel.
Moreover, with a spreadsheet model, you could make systematic changes in variables such as the interest rate earned,
years to retirement, years after retirement, and the like, and determine how sensitive the monthly payments are to
changes in these variables. This is especially useful to ´¬ünancial planners.
the calculator and then enter FV 912,855, N 480, PV 0, and I
9/12, and then press CPT and then the PMT key to ´¬ünd PMT 195.00,
which is the monthly investment required to accumulate a balance of
$912,855 in 40 years. Note that the $912,855 is both a present value and a
future value in this problem. It is a present value when ´¬ünding the accumu-
lated amount required to provide the post-retirement withdrawals, but it is
a future value when used to ´¬ünd the pre-retirement payments.
What is compounding? How is compounding related to discounting?
Explain how financial calculators can be used to solve present value and future
What is the difference between an ordinary annuity and an annuity due?
Why is semiannual compounding better than annual compounding from a
saverÔÇ™s standpoint? What about from a borrowerÔÇ™s standpoint?
Define the terms ÔÇťeffective (or equivalent) annual rate,ÔÇŁ ÔÇťnominal interest rate,ÔÇŁ
and ÔÇťperiodic interest rate.ÔÇŁ
How would one construct an amortization schedule?
Finding the value of a bond is a straightforward application of the dis-
counted cash ´¬‚ow process. A bondÔÇ™s par value is its stated face value, which
is the amount the issuer must pay to the bondholder at maturity. We assume
a par value of $1,000 in all our examples, but it is possible to have any value
that is a multiple of $1,000. The coupon payment is the periodic interest
payment that the bond provides. Usually these payments are made every six
months (semiannual payments), and the total annual payment as a percent-
age of the par value is called the coupon interest rate. For example, a 15-
year, 8 percent coupon, $1,000 par value bond with semiannual payments
calls for a $40 payment each six months, or $80 per year, for 15 years, plus
a ´¬ünal principal repayment of $1,000 at maturity. A time line can be used to
diagram the payment stream:
0.5 year 0 1 2 3 28 29 30
Year 0 0.5 1.0 1.5 14 14.5 15
$40 $40 $40 $40 $40 $1,040
The last payment of $1,040 consists of a $40 coupon plus the $1,000 par
The price of a bond is the present value of its payments, discounted at the
current market interest rate appropriate for the bond, given its risk, matu-
rity, and other characteristics. The semiannual coupons constitute an annuity,
and the ´¬ünal principal payment is a lump sum, so the price of the bond can
be found as the present value of an annuity plus the present value of a lump
sum. If the market interest rate for the bond is 10 percent, or 5 percent per
six months, then the value of the bond is8
By convention, annual coupon rates and market interest rates on bonds are quoted at two times their six-month rates.
So, a $1,000 bond with a $45 semiannual coupon payment has a nominal annual coupon rate of 9 percent.
28-8 Basic Financial Tools: A Review
rd)t (1 rd)N
t 1 (1
t 1 1.05
This value can be calculated with a ´¬ünancial calculator. There are 30 six-
month periods, so enter N 2(15) 30, I 10/2 5, PMT 40, and FV
1000, and then press CPT and then the PV key to ´¬ünd PV 846.28.
Therefore, you should be willing to pay $846.28 to buy the bond. This bond
is selling at a discountÔÇ”its price is less than its par value. This probably
occurs because interest rates increased since the bond was issued. If the mar-
ket interest rate fell to 6 percent, then the price of the bond would increase to
t 1 1.03
Thus, the bond would trade at a premium, or above par. If the interest rate
were exactly equal to the coupon rate, then the bond would trade at $1,000,
or at par. In general,
1. If the going interest rate is greater than the coupon rate, the price of
the bond will be less than par.
2. If the going interest rate is less than the coupon rate, the price will be
greater than par.
3. If the going interest rate is equal to the coupon rate, the bond will sell
at its par value.
Yield to Maturity
We used Equation 28-7 to ´¬ünd the price of a bond given market interest
rates. We can also use it to ´¬ünd the interest rate given the bondÔÇ™s price. This
interest rate is called the yield to maturity, and it is de´¬üned as the discount
rate that sets the present value of all of the cash ´¬‚ows until maturity equal
to the bondÔÇ™s current price. Financial calculators are used to make this cal-
culation. For example, suppose our 15-year, 8 percent coupon, semiannual
payment, $1,000 par value bond is selling for $925. Because the price is less
than par, the yield on the bond must be greater than 8 percent, but by how
much? Enter N 30, PV 925, PMT 40, and FV 1000, and then
press CPT and then the I key to ´¬ünd I 4.458. This is the rate per 6 months,
so the annual yield is 2(4.458) 8.916%.9
Price Sensitivity to Changes in Interest Rates
Equation 28-7 also shows that a bondÔÇ™s price depends on the market inter-
est rate used to discount cash ´¬‚ows. Therefore, ´¬‚uctuations in interest rates
give rise to changes in bond prices, and this sensitivity is called interest rate
risk. To illustrate, consider an 8 percent, $1,000 par value U.S. Treasury bond
with a 30-year maturity and semiannual payments when the going market
By convention, a bond with a yield of 4.458% per six months is said to have a nominal annual yield of 2(4.458%)
8.916%. Also, for most calculators PV must be opposite in sign to both PMT and FV in order to calculate the yield. How-
ever this convention may differ across calculators, so check your user manual for your particular calculator to make sure.
interest rate is 8 percent. Since the coupon rate is equal to the interest rate,
the bond sells at par, or for $1,000. If market interest rates increase, then the
price of the bond will fall. If the rate increases to 10 percent, then the new
price will be $810.71, so the bond will have fallen by 18.9 percent. Notice
that this decline has nothing to do with the riskiness of the bondÔÇ™s coupon
paymentsÔÇ”even the prices of risk-free U.S. Treasury bonds fall if interest
The percentage price change in response to a change in interest rates
depends on (1) the maturity of the bond and (2) its coupon rate. Other
things held constant, a longer-term bond will experience larger price changes
than a shorter-term bond, and a lower coupon bond will have a larger
change than a higher coupon bond. To illustrate, if our 8 percent Treasury
bond had a 5-year rather than a 30-year maturity, then the new price after
an interest rate increase from 8 percent to 10 percent would be $922.78, a
7.7 percent decrease versus the 18.9 percent decrease for the 30-year bond.
Because the shorter-term bond has less price risk, this bond is said to have
less interest rate risk than the longer-term bond.
A bondÔÇ™s interest rate risk also depends on its couponÔÇ”the lower the
coupon, other things held constant, the greater the interest rate risk. To illus-
trate, consider the example of a zero coupon bond, or simply a ÔÇťzero.ÔÇŁ It
pays no coupons, but it sells at a discount and provides its entire return at
maturity. A $1,000 par value, 30-year zero in an 8 percent annual rate mar-
ket will sell for $99.38. If the interest rate increases by 2 percentage points,
to 10 percent, then the bondÔÇ™s price will drop to $57.31, or by 42.3 percent!
The zero drops so sharply because distant cash ´¬‚ows are impacted more
heavily by higher interest rates than near-term cash ´¬‚ows, and all of the
zeroÔÇ™s cash ´¬‚ows occur at the end of its 30-year life.
These price changes are summarized below:
Initial Interest Rate
Bond Price @ 8% Price @ 10% Percent Change Risk
30-year, 8% coupon $1,000.00 $810.71 18.9% Signi´¬ücant
5-year, 8% coupon 1,000.00 922.78 7.7 Rather low
30-year zero 99.38 57.31 42.3 Very high