Chapter 7: Interest Rates and Bond Valuation
Concept Questions
1.
Treasury Bonds: Is it true that a U.S. Treasury security is risk-free?
No! A truly risk-free security is one that is devoid of any risk, i.e., a security that
has no potential for loss. Such a security would have a required rate of return
equal to:
RiskFreeRate  r *
As interest rates fluctuate, the value of a Treasury security will also fluctuate.
Using r* as the pure, risk-free rate of return and IP to represent an inflation
premium, then short-term Treasury bills (T-bills) would have a required rate of
return equal to:
TreasuryBi llRate  r *  IP
For a long-term Treasury bond (T-bond), which also has a maturity risk premium
(denoted MRP), the required rate of return would be:
TreasuryBo ndRate  r *  IP  MRP
U.S. Treasury securities are generally deemed to be default risk-free (i.e., no
credit risk). In addition, since Treasury securities are very liquid securities (i.e.,
there exists an active secondary market where investors can sell their Treasury
securities at market value very quickly).
This is unlike corporate bonds, which carry an inflation premium (IP), a maturity
risk premium (MRP), a default risk premium (DRP), and a liquidity risk premium
(LP). If we compare the rates of a 10-year T-bond with a 10-year corporate bond,
the results would be:
TreasuryBo ndRate10Year  r *  IP10Year  MRP10Year
CorporateBondRate10Year  r *  IP10Year  MRP10Year  DRP  LP
Although U.S. Treasury securities are considered to be default risk-free and carry
no liquidity risk premium, they are still subject to inflation and maturity risks.
10.
Term Structure: What is the difference between the term structure of interest
rates and the yield curve?
A ‘true’ yield curve is a graphical depiction of the term structure of interest rates.
Both yield curve and term structure of interest rates should be based on spot rates
derived from pure discount bonds (i.e., zero-coupon securities). However, what
most people call a yield curve is actually a ‘yield-to-maturity’ curve. This ‘yieldto-maturity’ curve plots yields-to-maturity for various homogeneous risk-class
securities versus these securities’ time-to-maturity. For zero-coupon securities,
the yield-to-maturity is the same as the spot rate. However, for coupon-bearing
securities the yield-to-maturity is not the same as the spot rate.
Problems
3.
Bond Prices: Staind, Inc., has 7.5 percent coupon bonds on the market that have
10 years left to maturity. The bonds make annual payments. If the YTM on these
bonds is 8.75 percent, what is the current bond price?
To find the present value of Staind, Inc.’s bond we use the following equation:
  1
1  
(1  r )t


BondValue Staind  C 

r



 
  F
 (1  r )t


 

1

1  
(1  0.0875)10  
$1,000


BondValue Staind  $75 

 $918.8889247

 (1  0.0875)10
0.0875




Using a financial calculator:
10n;8.75i;75PMT ;1,000 FV ; PV  $918.8889247
4.
Bond Yields: Ackerman Co. has 9 percent coupon bonds on the market with nine
years left to maturity. The bonds make annual payments. If the bond currently
sells for $934, what is its YTM?
If we try to use formulas and solve for r, we will be unsuccessful! Therefore, using
a financial calculator:
9n;90 PMT ;934CHSPV ;1,000 FV ; i  10.15300669%
6.
Bond Prices: Grohl Co. issued 11-year bonds 1 years ago at a coupon rate of 6.9
percent. The bonds make semiannual payments. If the YTM on these bonds is 7.4
percent, what is the current bond price?
To find the present value of Grohl’s bond we use the following equation:
 
 
1   1
   r t
  1
C    2 
BondValueGrohl  
r
2 

2








 
  F
 (1  r )t





 


 
1


1 
20 


 0.074   
  1 

2   
$69   
$1,000
BondValueGrohl 


 $965.1034873
20


0
.
074
2
 0.074 

 1 
2
2 

 






Using a financial calculator:
20n;
7.
7.4 69
i; PMT ;1,000 FV ; PV  $965.1034873
2
2
Bond Yields: Ngata Corp. issued 12-year bonds 2 years ago at a coupon rate of
8.4 percent. The bonds make semiannual payments. If the bonds currently sell for
105 percent of par value, what is the YTM?
Again, if we try to use formulas and solve for r, we will be unsuccessful!
Therefore, using a financial calculator:
84
PMT ;1,050CHSPV ;1,000 FV ; i  3.83736988%  2  7.67473975%
2
Calculating Real Rates of Return: If Treasury bills are currently paying 7
percent and the inflation rate is 3.8 percent, what is the approximate real rate of
interest? What is the exact real rate?
20n;
9.
The approximate relationship between nominal interest rates (R), real interest
rates (r*), and inflation (h) is:
R r h
or
r  Rh
r  7%  3.8%  3.2%
The Fisher equation, which slows the exact relationship between nominal interest
rates (R), real interest rates (r), and inflation (h) is:
(1  R)  (1  r )(1  h)
1 R 
r 
 1
 1 h 
 1  0.07 
r 
  1  3.082852%
 1  0.038 
13.
Using Treasury Quotes: Locate the Treasury issue in Figure 7.4 on page 216 of
the text (9th edition) maturing in November 2027. Is it a note or a bond? What is
the coupon rate? What is its bid price? What was the previous day’s asked price?
The information reported in the WSJ for this bond follows:
Maturity
Rate Mo/Yr
6.125 Nov 27
Bid
120:07
Asked
120:08
Ask
Chg
5
Asked
Yld
4.57
Since the Treasury matures in more than 10 years, it has to be a T-bond.
The coupon rate is given in the first column: 6.125%
The T-bond’s bid price is stated in percent of par value. The number after the
colon represents the number of 32nd of the par value. Therefore 120:07 means the
bond is trading at 120 7/32 of par value. If we assume the T-bond’s par value is
$1,000,000, then the T-bond’s current bid price is:
7 

120%  %   $1,000,000  (120.21875%)  $1,000,000  $1,202,187.5
32 

The previous day’s asked price is equal to today’s asked price minus the change
in the asked price. Today’s asked price is given in column 4 as 120:08, the asked
price change in column 5 as 5. Therefore, we have:
8
5 

120 %  %   $1,000,000  $1,200,937.50
32
32 

16.
Interest Rate Risk: Both Bond Sam and Bond Dave have 9 percent coupons,
make semiannual payments, and are priced at par value. Bond Sam has 3 years to
maturity, whereas Bond Dave has 20 years to maturity. If interest rates suddenly
rise by 2 percent, what is the percentage change in the price of Bond Same? Of
Bond Dave? If rates were to suddenly fall by 2 percent instead, what would the
percentage change in the price of Bond Sam be then? Of Bond Dave? What does
this problem tell you about the interest rate risk of long-term bonds?
Since both bonds currently sell at par value, their YTMs equal their coupon rate.
Therefore, the current level of interest (or required rate of return on these bonds)
must be 9%.
If interest rates suddenly increased by 2% to 11%, the value of the each bond
would be:
Using a financial calculator:
Bond Sam : 6n;5.5i;45PMT ;1,000 FV ; PV  $950.0446969
Bond Dave : 40n;5.5i;45PMT ;1,000FV ; PV  $839.5387531
The percentage change in price can be determined using the following equation:
P P 
P   1 0  100
 P0 
 $950.05  $1,000 
PSam  
 100  4.995%
$1,000


 $839.54  $1,000 
PDave  
  100  16.046%
$1,000


If interest rates (i.e., YTMs) suddenly fall by 2% to 7%, the value of the each bond
would be:
Using a financial calculator:
Bond Sam : 6n;3.5i;45PMT ;1,000FV ; PV  $1,053.28553
Bond Dave : 40n;3.5i;45PMT ;1,000FV ; PV  $1,213.550723
The percentage change in price can be determined using the following equation:
P P 
P   1 0  100
 P0 
 $1,053.29  $1,000 
PSam  
 100  5.329%
$1,000


 $1,213.55  $1,000 
PDave  
 100  21.355%
$1,000


You should notice two items about these bonds based on the information above:
(1) Holding everything else constant between these two bonds, the longer the
maturity of the bond, the greater is its price sensitivity to changes in interest
rates.
(2) For a given change in interest rates, capital gains due to a decrease in
interest rates are larger than capital losses due to an increase in interest
rates.
17.
Interest Rate Risk: Bond J is a 4 percent coupon bond. Bond K is a 12 percent
coupon bond. Both have nine years to maturity, make semiannual payments, and
have a YTM of 8 percent. If interest rates suddenly rise by 2 percent, what is the
percentage price change of these bonds? What if rates suddenly fall by 2 percent
instead? What does this problem tell you about interest rate risk of lower coupon
bonds?
Currently, both bonds have the same YTM. Therefore, their current prices are:
Bond J : 18n;4i;20PMT ;1,000FV ; PV  $746.8140605
Bond K : 18n;4i;60PMT ;1,000FV ; PV  $1,253.185939
If the YTM increases by 2% to 10%, the bond prices are:
Bond J : 18n;5i;20PMT ;1,000FV ; PV  $649.3123929
Bond K :18n;5i;60PMT ;1,000FV ; PV  $1,116.895869
The percentage change in price can be determined using the following equation:
 $649.31  $746.81 
PJ  
  100  13.0555%
$746.81


 $1,116.90  $1,253.19 
PK  
 100  10.8755%
$1,253.19


If the YTM decline by 2% to 6%, the bond prices are:
Bond J : 18n;3i;20PMT ;1,000FV ; PV  $862.4648692
Bond K : 18n;3i;60PMT ;1,000FV ; PV  $1,412.605392
The percentage change in price can be determined using the following equation:
 $862.46  $746.81 
PJ  
  100  15.4859%
$746.81


 $1,412.61  $1,253.19 
PK  
 100  12.7211%
$1,253.19


You should notice two items about these bonds based on the information above:
(1) Holding everything else constant between these two bonds, the lower the
coupon rate of the bond, the greater is its price sensitivity to changes in
interest rates.
(2) For a given change in interest rates, capital gains due to a decrease in
interest rates are larger than capital losses due to an increase in interest
rates.
20.
Accrued Interest: You purchase a bond with an invoice price of $968. The bond
has a coupon rate of 7.4 percent, and there are four months to the next semiannual
coupon date. What is the clean price of the bond?
The accrued interest is the coupon payment for the period times the fraction of the
period that has passed since the last coupon payment. Since we have a
semiannual coupon bond, the coupon payment is one-half of the annual coupon
payment payable every six months. There are five months until the next coupon
payment, so one month has passed since the last coupon payment. The accrued
interest for the bond is:
AccruedInterest 
$74 2
  $12.3333333
2 6
And we calculate the clean price using the following equation:
Clean Pr ice  Dirty Pr ice  AccruedInterest
Clean Pr ice  $968  $12.33  $955.67
23.
Using Bond Quotes: Suppose the following bond quotes for IOU Corporation
appear in the financial page of today’s newspaper. Assume the bond has a face
value of $1,000 and the current date is April 15, 2009. What is the yield to
maturity of the bond? What is the current yield? What is the yield to maturity on a
comparable U.S. Treasury issue?
Company
(Ticker)
IOU (IOU)
Coupon
7.2
Maturity
April 15, 2023
Last
Last EST Vol
Price Yield
(000)
108.96 ?? 1,827
The IOU bond has 14 years to maturity, so the bond’s required rate of return (or
YTM) would be:
Bond IOU : 28n;36PMT ;1,000FV ;1,089.60CHSPV ; i  3.1157123  2  6.2314246%
The current yield can be determined using the following equation:
CurrentYield 
$ AnnualInterestPayment
Current Pr ice
CurrentYie ld IOU 
$72
 6.60793%
$1,089.60
26.
Zero Coupon Bonds: Suppose your company needs to raise $30 million and you
want to issue 30-year bonds for this purpose. Assume the required return on your
bond issue will be 8 percent, and you’re evaluating two issue alternatives: a 8
percent annual coupon bond and a zero coupon bond. Your company’s tax rate is
35 percent.
a. How many of the coupon bonds would you need to issue to raise the $30
million? How many of the zeroes would you need to issue?
The coupon bonds have a 8% coupon which matches the 8% required return,
so they will sell at par value (assume $1,000). The number of bonds that must
be sold is the amount needed divided by the bond price, so we can use the
following equation:
NumberOfBondsIssued 
TotalFunds Re quired
Pr oceeds Re ceivedPerBond
NumberOfBondsIssued Coupon 
$30,000,000
 30,000 Bonds
$1,000 PerBond
For the zero coupon bonds, we first need to determine the price per bond:
PZero 
ParValue
(1  YTM )t
PZero 
$1,000
 $99.37733254
(1  0.08)30
Therefore, the number of zero coupon bonds needed to be issued would be:
NumberOfBondsIssued ZeroCoupon 
$30,000,000
 301,879.7067 Bonds
$99.37733254 PerBond
b. In 30 years, what will your company’s repayment be if you issue the coupon
bonds? What if you issue zeroes?
The repayment of the coupon bonds will be at par value plus the last coupon
payment times the number of bonds issued:
Re paymentCoupon  30,000  ($1,000  $80)  $32,400,000
The repayment of the zero coupon bonds will be par value times bonds issued:
Re paymentZeroCoupon  301,879.7067  $1,000  $301,979,706.70
c. Based on your answers in (a) and (b), why would you ever want to issue the
zeroes? To answer, calculate the firm’s after-tax cash flows for the first year
under the two different scenarios. Assume the IRS amortization rules apply
for the zero coupon bonds.
The total coupon payment for the coupon bonds will be the number of bonds
times the coupon payment. For the cash flow of the coupon bonds, we need to
account for the tax deductibility of the interest payments. To do this, we will
multiply the total coupon payment times one minus the tax rate:
CashFlowCoupon  (30,000 Bonds )  ($80 PerBondInt erest )  (1  0.35)  $1,560,000
Therefore, the company’s after-tax cash outflow each year for payment on this coupon
bond is $1,560,000.
For the zero coupon bonds, the first year interest payments is the difference in the price
of the zero at the end of the year and the beginning of the year. The price of the zeroes in
one year will be:
PZero 
$1,000
 $107.3275192
(1  0.08) 29
We have already determined the price of the zero coupon bonds at issue to be
$99.37733254. Therefore, the difference in price from beginning to end of
year would be:
PZeroCoupon  $107.3275192  $99.37733254  $7.95018666
This difference represented the first year’s interest deduction per bond.
Therefore, the total cash flow for the zeroes will be the number of bonds
issued times the difference in price times the tax rate. Note that we are not
multiplying by one minus the tax rate as we did with the coupon bond. This is
because the after-tax cash flow for the zero coupon bonds is a cash inflow!
Remember, zero coupon bonds pay no interest, so the issuing firm receives a
tax benefit for implied interest without having to pay out anything!! The aftertax cash inflow for the zero coupon bonds in the first year would be:
CashFlowZero  (301,880Bonds )  ($7.95018666PerBondInt erest )  (0.35)  $840,000.82
The issuing firm would be able to deduct $840,000.82 from EBIT in the first
year after issuing zero coupon bonds.