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Corporate Finance, 12th Edition
51
that sells for
d par values of
Government
000 = .12.
Dupon divided
237
CHAPTER 8
Interest Rates and Bond Valuation
Second, the bond offers $80 per year for 10 years. The present value of this annuity
stream is:
to maturity
ally issued
ty declines
from
present
When
Annuity present value = $80 x (1 – 1/1.080)/.08
= $80 X (1 - 1/2.1589)/.08
= $80 X 6.7101
= $536.81
We add the values for the two parts together to get the bond's value:
Bond value = $463.19 + 536.81 = $1,000
This bond sells for exactly its face value. This is not a coincidence. The going interest
rate in the market is 8 percent. Considered as an interest-only loan, what interest rate
does this bond have? With an $80 coupon, this bond pays exactly 8 percent interest only
when it sells for $1,000.
To illustrate what happens as interest rates change, suppose that a year has gone
by. The Xanth bond now has nine years to maturity. If the interest rate in the market
has risen to 10 percent, what will the bond be worth? To find out, we repeat the
present value calculations with 9 years instead of 10, and a 10 percent yield instead
of an 8 percent yield. First, the present value of the $1,000 paid in 9 years at 10
percent is:
y the
Tket
he
Present value = $1,000/1.10º = $1,000/2.3579 = $424.10
Second, the bond now offers $80 per year for 9 years. The present value of this annuity
stream at 10 percent is
Annuity present value = $80 x (1 – 1/1.10%)/.10
= $80 x (1 – 1/2.3579)/.10
= $80 X 5.7590
= $460.72
We add the values for the two parts together to get the bond's value:
Bond value = $424.10 + 460.72 = $884.82
A good bond site
to visit is: www
.bloomberg.com
/markets/rates
-bonds, which has
loads of useful
information.
Therefore, the bond should sell for about $885. In the vernacular, we say that this bond,
with its 8 percent coupon, is priced to yield 10 percent at $885.
The Xanth Co. bond now sells for less than its $1,000 face value. Why? The
market interest rate is 10 percent. Considered as an interest-only loan of $1,000,
this bond pays only 8 percent, its coupon rate. Because the bond pays less than the
going rate, investors are willing to lend only something less than the $1,000 prom-
ised repayment. Because the bond sells for less than face value, it is said to be a
discount bond.
The only way to get the interest rate up to 10 percent is to lower the price to less
than $1,000 so that the purchaser, in effect, has a built-in gain. For the Xanth bond, the
price of $885 is $115 less than the face value, so an investor who purchased and kept
the bond would get $80 per year and would have a $115 gain at maturity as well. This
gain compensates the lender for the below-market coupon rate.
Another way to see why the bond is discounted by $115 is to note that the $80
coupon is $20 below the coupon on a newly issued par value bond, based on current
market conditions. The bond would be worth $1,000 only if it had a coupon of $100 per
©
Black
Stop
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(el)
Corporate Financial Policies
52
PART II Valuation and Capital Budgeting
238
year. In a sense, an investor who buys and keeps the bond gives up $20 per year for nine
my
the
years. At 10 percent, this annuity stream is worth:
Annuity present value = $20 x (1 -- 1/1.10°)/.10
= $20 X 5.7590
= $115.18
pr
This is the amount of the discount.
Online bond
calculators are
available at gpi
.fidelity.com
/ftgw/interfaces
Ipycl; interest
rate information is
available at money
.cnn.com/data/
bonds and www
.bankrate.com
of rising by 2 percent? As you might guess, the bond would sell for more than $1,000.
What would the Xanth bond sell for if interest rates had dropped by 2 percent instead
This case is the opposite of a discount bond. The Xanth bond now has a coupon
rate of 8 percent when the market rate is only 6 percent. Investors are willing to pay a
premium to get this extra coupon amount. In this case, the relevant discount rate is 6
percent and there are nine years remaining. The present value of the $1,000 face amount is:
Such a bond is said to sell at a premium and is called a premium bond.
Present value = $1,000/1.06' = $1,000/1.6895 = $591.90
The present value of the coupon stream is:
Annuity present value = $80 x (1 – 1/1.06°)/.06
= $80 x (1 – 1/1.6895)/.06
= $80 X 6.8017
= $544.14
We add the values for the two parts together to get the bond's value:
Bond value = $591.90 + 544.14 = $1,136.03
The total bond value is therefore about $136 in excess of par value. Once again, we can
verify this amount by noting that the coupon is now $20 too high, based on current market
conditions. The present value of $20 per year for nine years at 6 percent is:
Annuity present value = $20 x (1 – 1/1.06%)/.06
= $20 X 6.8017
= $136.03
Gre
crea
Learn more
about bonds at
investorguide.
Copyright 2019 by Mo
Printed in the United S
Act of 1976, no part of
means, or stored in a
publisher
This is as we calculated.
Based on our examples, we can now write the general expression for the value of a
bond. If a bond has (1) a face value of F paid at maturity, (2) a coupon of C paid per
period, (3) t periods to maturity, and (4) a yield of R per period, its value is:
Bond value = C [1 – 1/(1 + R)']/R + F/(1 + R)'
Present value of the Present value of
Bond value
+
(8.1)
coupons
the face amount
This McGraw-Hill Crea
the instructor of this co
materials. Instructors
EXAMPLE
8.1
Semiannual Coupons In practice, bonds issued in the United States usually make coupon
payments twice a year. So, if an ordinary bond has a coupon rate of 14 percent, the owner will
receive a total of $140 per year, but this $140 will come in two payments of $70 each.
Suppose the yield to maturity on our bond is quoted at 16 percent. Bond yields are quoted
as annual percentage rates (APRs); the quoted rate is equal to the actual rate per period
ISBN-13: 978130740
ISBN-10: 130740801/
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"x6"
oto Paper
Plain Paper
8.5" x 11"
CL
Stop
BK
53
Corporate Finance, 12th Edition
es up $20 per year for nine
CHAPTER 8
01/.10
239
Interest Rates and Bond Valuation
ped by 2 percent instead
Il for more than $1,000.
bond.
ond now has a coupon
Ors are willing to pay a
vant discount rate is 6
$1.000 face amount is:
multiplied by the number of periods. With a 16 percent quoted yield and semiannual payments,
the true yield is 8 percent per six months. If our bond matures in seven years, what is the bond's
price? What is the effective annual yield on this bond?
Based on our discussion, we know the bond will sell at a discount because it has a coupon
rate of 7 percent every six months when the market requires 8 percent every six months. So,
if our answer exceeds $1,000, we know that we have made a mistake.
To get the exact price, we first calculate the present value of the bond's face value of
$1,000 paid in seven years. This 7-year period has 14 periods of six months each. At 8 percent
per period, the value is:
Present value $1,000/1.0814 $1,000/2.9372 = $340.46
The coupons can be viewed as a 14-period annuity of $70 per period. At an 8 percent discount
rate, the present value of such an annuity is:
Annuity present value = $70 x (1 - 1/1.084)/.08
$70 x (1 - 3405)/.08
$70 X 8.2442
$577.10
5591.90
06
".06
The total present value is the bond's price:
Bond value = $340.46 + 577.10 = $917.56
To calculate the effective yield on this bond, note that 8 percent every six months is equiv.
alent to:
Effective annual rate = (1 + .08)2 - 1 = .1664, or 16.64%
The effective yield, therefore, is 16.64 percent.
nce again, we can
on current market
it is:
Learn more
about bonds at
investorguide.com
the value of a
of C paid per
As we have illustrated in this section, bond prices and interest rates always move
in opposite directions. When interest rates rise, a bond's value, like any other present
value, declines. Similarly, when interest rates fall, bond values rise. Even if the bor-
rower is certain to make all payments, there is still risk in owning a bond. We discuss
this next.
is:
(8.1)
INTEREST RATE RISK
The risk that arises for bond owners from fluctuating interest rates is called interest rate
risk. How much interest rate risk a bond has depends on how sensitive its price is to
interest rate changes. This sensitivity directly depends on two things: the time to maturity
and the coupon rate. As we will see momentarily, you should keep the following in mind
when looking at a bond:
1. All other things being equal, the longer the time to maturity, the greater the inter-
est rate risk.
2. All other things being equal, the lower the coupon rate, the greater the interest rate risk.
ake coupon
e owner will
ch.
are quoted
Der period
We illustrate the first of these two points in Figure 8.2. As shown, we compute and
plot prices under different interest rate scenarios for 10 percent coupon bonds with
maturities of 1 year and 30 years. Notice how the slope of the line connecting the prices
is much steeper for the 30-year maturity than it is for the 1-year maturity. This steepness
Corporate Financial Policies
54
COU
in
PART II Valuation and Capital Budgeting
240
ex
m
2,000
Figure 8.2
Interest Rate Risk
and Time to Maturity
$1,768.62
is
h
1
30-year bond
1-year bond
$916.67
1,500
Bond value ($)
$1,04762
$502.11
1,000
500
20
5
10
15
Interest rate (%)
Value of a Bond with a 10 Percent Coupon Rate for Different Interest Rates and Maturities
Interest Rate
5%
10
Time to Maturity
1 Year
30 Years
$1,047.62 $1,768.62
1,000.00
1,000.00
956.52 671.70
916.67
502.11
15
20
ME
Graw
NI
cre
Copyright 2019 by
Printed in the Unite
Act of 1976, no pa
means, or stored /
publisher
tells us that a relatively small change in interest rates will lead to a substantial change
in the bond's value. In comparison, the l-year bond's price is relatively insensitive to
interest rate changes.
Intuitively
, shorter-term bonds have less interest rate sensitivity because the $1.000
face amount is received so quickly. The present value of this amount isn't greatly affected
by a small change in interest rates if the amount is received in, say, one year. However,
even a small change in the interest rate, once compounded for, say, 30 years, can have a
significant effect on present value. As a result, the present value of the face amount will
be much more volatile with a longer-term bond.
The other thing to know about interest rate risk is that, like many things in finance
and economics, it increases at a decreasing rate. A 10-year bond has much greater interest
rate risk than a 1-year bond has. However, a 30-year bond has only slightly greater interest
rate risk than a 10-year bond.
This McGraw-Hill
the Instructor oft
materials. Instruc
ISBN-13: 9781
The reason that bonds with lower coupons have greater interest rate risk is essen-
tially the same. As we discussed earlier, the value of a bond depends on the present
value of both its coupons and its face amount. If two bonds with different coupon
rates have the same maturity, the value of the lower-coupon bond is proportionately
more dependent on the face amount to be received at maturity. As a result, its value
will fluctuate more as interest rates change. Put another way, the bond with the higher
ISBN-10: 1307
Current mar
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55
Corporate Finance, 12th Edition
Q
CHAPTER 8 Interest Rates and Bond Valuation
241
coupon has a larger cash flow early in its life, so its value is less sensitive to changes
in the discount rate.
Bonds are rarely issued with maturities longer than 30 years, though there are
exceptions. In the 1990s, Walt Disney issued "Sleeping Beauty" bonds with a 100-year
maturity. Similarly, BellSouth, Coca-Cola, and Dutch banking giant ABN AMRO all
issued bonds with 100-year maturities. These companies evidently wanted to lock in the
historically low interest rates for a long time. The current record holder for corporations
looks to be Republic National Bank, which sold bonds with 1,000 years to maturity.
And low interest rates in recent years have led to really long-term bonds. For example,
in July 2017, Japanese telecommunications company SoftBank issued $4.5 billion in
perpetual bonds.
We can illustrate the effect of interest rate risk using the 100-year BellSouth issue.
The following table provides some basic information on this issue, along with its prices
on December 31, 1995; May 6, 2008; and February 1, 2018.
Percentage
Change
in Price
1995-2008
Percentage
Change
in Price
2008-2018
Coupon
Rate
Maturity
Price on
12/31/95
Price on
5/6/08
Price on
2/1/18
2095
7.00%
$1,000
$1,008.40
+.84%
$1,164.21
+15.45%
Several things emerge from this table. First, interest rates apparently fell between Decem-
ber 31, 1995, and May 6, 2008 (Why?). The bond first gained .84 percent and then gained
an additional 15.45 percent.
FINDING THE YIELD TO MATURITY: MORE TRIAL AND ERROR
Frequently, we will know a bond's price, coupon rate, and maturity date, but not its yield
to maturity. Suppose we are interested in a 6-year, 8 percent coupon bond. A broker quotes
a price of $955.14. What is the yield on this bond?
We've seen that the price of a bond can be written as the sum of its annuity and
lump-sum components. Knowing that there is an $80 coupon for six years and a $1,000
face value, we can say that the price is:
$955.14 = $80 x [1 - 1/(1 + R)°1/R + $1,000/(1 + R)
where R is the unknown discount rate, or yield to maturity. We have one equation here
and one unknown, but we cannot solve for R directly without using a financial calculator
or a spreadsheet application. Instead, we must use trial and error.
We can speed up the trial-and-error process by using what we know about bond
prices and yields. In this case, the bond has an $80 coupon and is selling at a dis-
count. We know that the yield is greater than 8 percent. If we calculate the price at
10 percent:
Bond value = $80 x (1 - 1/1.10%)/10 + $1,000/1.109
= $80 x 4.3553 + $1,000/1.7716
= $912.89
Current market
rates are available
at www.bankrate
.com
At 10 percent, the value is lower than the actual price, so 10 percent is too high. The
true yield must be somewhere between 8 and 10 percent. At this point, it's "plug and
chug" to find the answer. You would probably want to try 9 percent next. If you did, you
would see that this is in fact the bond's yield to maturity.

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