Chapter 8

Valuing Bonds

8-1.
a. The coupon payment is:

Coupon Rate  Face Value 0.055  $1000

CPN    $27.50
Number of Coupons per Year 2

b. The timeline for the cash flows for this bond is (the unit of time on this timeline is six-month periods):

0 1 2 3 60

$27.50 $27.50 $27.50 $27.50 + $1000

2
P  100/(1.055)  $89.85
8-2.
a. The maturity is 10 years.
b. (20/1000) * 2 = 4% so the coupon rate is 4%.
c. The face value is $1000.

8-3.
a. Use the following equation:

1/n
 FV 
1 YTM n   n 
 P 
1/1
 100 
1  YTM 1    YTM1  4.70%
 95.51 
1/ 2
 100 
1  YTM 1    YTM1  4.80%
 91.05 
1/ 3
 100 
1  YTM 3    YTM 3  5.00%
 86.38 
72 Berk/DeMarzo • Corporate Finance
1/ 4
 100 
1  YTM 4    YTM 4  5.20%
 81.65 
1/5
 100 
1  YTM 5    YTM 5  5.50%
 76.51 
b. The yield curve is

Zero Coupon Yield Curve

5.6
Yield to Maturity

5.4
5.2
5
4.8
4.6
0 2 4 6
M aturity (Ye ars )

c. The yield curve is upward sloping

8-4.
a. P = 100(1.055)2 = $89.85

4
b. P  100/(1.0595)  $79.36

c. 6.05%

8-5.
40 40 40  1000
$1,034.74     YTM  7.5%
a. YTM YTM 2 YTM 20
(1  ) (1  ) (1  )
 2 2 2

Using the annuity spreadsheet:

  NPER Rate PV PMT FV Excel Formula
Given: 20   -1,034.74 40 1,000  
Solve For Rate:   3.75%       =RATE(20,40,-1034.74,1000)

Therefore, YTM = 3.75% × 2 = 7.50%

40 40 40  1000
PV   L  $934.96.
b. .09 .09 2 .09 20
(1 ) (1 ) (1 )
2 2 2

Using the spreadsheet

With a 9% YTM = 4.5% per 6 months, the new price is $934.96
 Chapter 8  Valuing Bonds 73
  NPER Rate PV PMT FV Excel Formula
Given: 20 4.50%   40 1,000  
Solve For PV:     (934.96)     =PV(0.045,20,40,1000)

8-6.
C C C  1000
900   2
 5
 C  $36.26, so the coupon rate is 3.626%
 (1  .06) (1  .06) (1  .06)

We can use the annuity spreadsheet to solve for the payment:

  NPER Rate PV PMT FV Excel Formula
Given: 5 6.00% -900.00   1,000  
Solve For PMT:       36.26   =PMT(0.06,5,-900,1000)

Therefore, the coupon rate is 3.626%

8-7. Bond A trades at a discount. Bond D trades at par. Bonds B and C trade at a premium.

8-8. Bonds trading at a discount generate a return from both receiving the coupons and from receiving a face
value that exceeds the price paid for the bond. As a result, the yield to maturity of discount bonds exceeds
the coupon rate.

8-9.
a. Because the yield to maturity is less than the coupon rate, the bond is trading at a premium.

40 40 40  1000
b.  2
 14
 $1,054.60
(1  .035) (1  .035) (1  .035)

  NPER Rate PV PMT FV Excel Formula
Given: 14 3.50%   40 1,000  
Solve For PV:     (1,054.60)     =PV(0.035,14,40,1000)

8-10.
a. When it was issued, the price of the bond was

70 70  1000
P  ...  10
 $1073.60
(1  .06) (1  .06)

b. Before the first coupon payment, the price of the bond is

70 70  1000
P  70  ...  9
 $1138.02
(1  .06) (1  .06)
74 Berk/DeMarzo • Corporate Finance
c. After the first coupon payment, the price of the bond will be
70 70  1000
P ...  9
 $1068.02
(1  .06) (1  .06)

8-11.
a. First, we compute the initial price of the bond by discounting its 10 annual coupons of $6 and final face value
of $100 at the 5% yield to maturity:

  NPER Rate PV PMT FV Excel Formula

Given: 10 5.00% 6 100  
Solve For PV: (107.72) = PV(0.05,10,6,100)

Thus, the initial price of the bond = $107.72. (Note that the bond trades above par, as its coupon rate exceeds
its yield).

Next we compute the price at which the bond is sold, which is the present value of the bonds cash flows
when only 6 years remain until maturity:

  NPER Rate PV PMT FV Excel Formula

Given: 6 5.00% 6 100  
Solve For PV: (105.08) = PV(0.05,6,6,100)

Therefore, the bond was sold for a price of $105.08. The cash flows from the investment are therefore as
shown in the following timeline:

Year 0 1 2 3 4

Purchase Bond -$107.72

Receive Coupons $6 $6 $6 $6
Sell Bond $105.08
Cash Flows -$107.72 $6.00 $6.00 $6.00 $111.08

b. We can compute the IRR of the investment using the annuity spreadsheet. The PV is the purchase price, the
PMT is the coupon amount, and the FV is the sale price. The length of the investment N = 4 years. We then
calculate the IRR of investment = 5%. Because the YTM was the same at the time of purchase and sale, the
IRR of the investment matches the YTM.
  NPER Rate PV PMT FV Excel Formula
Given: 4   –107.72 6 105.08  
Solve For Rate:   5.00%       = RATE(4,6,-107.72,105.08)

8-12.
a. We can compute the price of each bond at each YTM using Eq. 8.5. For example, with a 6% YTM, the price
of bond A per $100 face value is
100
P(bond A, 6% YTM)   $41.73
1.0615
The price of bond D is
1  1  100
P(bond D, 6% YTM)  8  1   $114.72
.06  1.0610  1.0610

 Chapter 8  Valuing Bonds 75

One can also use the Excel formula to compute the price: –PV(YTM, NPER, PMT, FV).

Once we compute the price of each bond for each YTM, we can compute the % price change as

Percent change =
Price at 5% YTM  Price at 6% YTM 
Price at 6% YTM 
The results are shown in the table below:

Bond Coupon Rate Maturity Price at Price at Percentage Change

(annual payments) (years) 6% YTM 5% YTM
A 0% 15 $41.73 $48.10 15.3%
B 0% 10 $55.84 $61.39 9.9%
C 4% 15 $80.58 $89.62 11.2%
D 8% 10 $114.72 $123.17 7.4%

b. Bond A is most sensitive, because it has the longest maturity and no coupons. Bond D is the least sensitive.
Intuitively, higher coupon rates and a shorter maturity typically lower a bond’s interest rate sensitivity.

8-13.
a. Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0625 = 23.30. Return = (23.30 / 17.41)1/5 – 1 =
6.00%. I.e., since YTM is the same at purchase and sale, IRR = YTM.
b. Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0725 = 18.42. Return = (18.42 / 17.41)1/5 – 1 =
1.13%. I.e., since YTM rises, IRR < initial YTM.
c. Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0525 = 29.53. Return = (29.53 / 17.41)1/5 – 1 =
11.15%. I.e., since YTM falls, IRR > initial YTM.
d. Even without default, if you sell prior to maturity, you are exposed to the risk that the YTM may change.

CPN CPN CPN  FV 60 60  1000

8-14. P  2
 ...  N
  2
 $1032.09
1  YTM 1 (1  YTM 2 ) (1  YTM N ) (1  .04) (1  .043)

This bond trades at a premium. The coupon of the bond is greater than each of the zero coupon yields, so the
coupon will also be greater than the yield to maturity on this bond. Therefore it trades at a premium

8-15. The price of the zero-coupon bond is

FV 1000
P N
 5
 $791.03
(1  YTM N ) (1  0.048)
76 Berk/DeMarzo • Corporate Finance
8-16. The price of the bond is

CPN CPN CPN  FV 40 40 40  1000

P  2
 ...  N
  2
 3
 $986.58
1  YTM1 (1  YTM 2 ) (1  YTM N ) (1  .04) (1  .043) (1  .045)
The yield to maturity is

CPN CPN CPN  FV

P  2
 ...  N
1  YTM (1  YTM) (1  YTM)

40 40 40  1000
$986.58   2
 3
 YTM  4.488%
(1  YTM) (1  YTM) (1  YTM)

8-17. The maturity must be one year. If the maturity were longer than one year, there would be an arbitrage
opportunity

8-18. Solve the following equation:

 1 1 1 1  1000
1000  CPN     4

 (1  .04) (1  .043) (1  .045) (1  .047)  (1  .047) 4
2 3

CPN  $46.76

Therefore, the par coupon rate is 4.676%.

8-19.

a. The bond is trading at a premium because its yield to maturity is a weighted average of the yields of the zero
coupon bonds. This implied that its yield is below 5%, the coupon rate.

b. To compute the yield, first compute the price.

CPN CPN CPN  FV

P   ... 
1  YTM1 (1 YTM 2 ) 2
(1 YTM N ) N
50 50 50 50 50  1000
      $1010.05
(1  .04) (1  .043)2 (1 .045)3 (1  .047)4 (1  .048)5

The yield to maturity is:

CPN CPN CPN  FV

P   ... 
1  YTM (1  YTM) 2
(1  YTM) N
50 50  1000
1010.05   ...   YTM  4.77%
(1  YTM) (1 YTM) N

c. If the yield increased to 5.2%, the new price would be:

 Chapter 8  Valuing Bonds 77
CPN CPN CPN  FV
P   ... 
1 YTM (1  YTM)2 (1 YTM) N
50 50  1000
  ...   $991.39
(1  .052) (1 .052)N

8-20. First, figure out if the price of the coupon bond is consistent with the zero coupon yields implied by the other
securities:

1000
970.87   YTM1  3.0%
(1  YTM1 )
1000
938.95   YTM 2  3.2%
(1  YTM 2 )2
1000
904.56   YTM 3  3.4%
(1  YTM 3 )3

According to these zero coupon yields, the price of the coupon bond should be:

100 100 100  1000

   $1186.00
(1 .03) (1 .032)2 (1 .034)3

The price of the coupon bond is too low, so there is an arbitrage opportunity. To take advantage of it:

Today 1 Year 2 Years 3 Years

Buy 10 Coupon Bonds 11835.00 +1000 +1000 +11,000
Short Sell 1 One-Year Zero +970.87 1000
Short Sell 1 Two-Year Zero +938.95 1000
Short Sell 11 Three-Year Zeros +9950.16 11,000
Net Cash Flow 24.98 0 0 0

8-21. To determine whether these bonds present an arbitrage opportunity, check whether the pricing is internally
consistent. Calculate the spot rates implied by Bonds A, B and D (the zero coupon bonds), and use this to
check Bond C. (You may alternatively compute the spot rates from Bonds A, B and C, and check Bond D, or
some other combination.)

1000
934.58   YTM1  7.0%
(1  YTM1 )

1000
881.66   YTM 2  6.5%
(1  YTM 2 )2

1000
839.62   YTM 3  6.0%
(1  YTM 3 )3

Given the spot rates implied by Bonds A, B and D, the price of Bond C should be $1,105.21. Its price really
is $1,118.21, so it is overpriced by $13 per bond. YES, there is an arbitrage opportunity.
78 Berk/DeMarzo • Corporate Finance

To take advantage of this opportunity, you want to (short) Sell Bond C (since it is overpriced). To match
future cash flows, one strategy is to sell 10 Bond Cs (it is not the only effective strategy; any multiple of this
strategy is also arbitrage). This complete strategy is summarized:

Today 1 Year 2Years 3Years

Sell Bond C 11,182.10 –1,000 –1,000 –11,000
Buy Bond A –934.58 1,000 0 0
Buy Bond B –881.66 0 1,000 0
Buy 11 Bond D –9,235.82   0  0  11,000
Net Cash Flow 130.04 0 0 0

Notice that your arbitrage profit equals 10 times the mispricing on each bond (subject to rounding error).

8-22.
a. We can construct a two-year zero coupon bond using the one and two-year coupon bonds as follows:
Cash Flow in Year:
1 2 3 4
Two-year coupon bond ($1000 Face Value) 100 1,100
Less: One-year bond ($100 Face Value) (100)      
Two-year zero ($1100 Face Value) - 1,100

Now,
100 1100
Price(2-year coupon bond) =   $1115.05
1.03908 1.039082

100
Price(1-year bond) =  $98.04
1.02
By the Law of One Price:

Price(2 year zero) = Price(2 year coupon bond) – Price(One-year bond)

= 1115.05 – 98.04 = $1017.01

Given this price per $1100 face value, the YTM for the 2-year zero is (Eq. 8.3)

1/ 2
 1100 
YTM(2)    1  4.000%
 1017.01

b. We already know YTM(1) = 2%, YTM(2) = 4%. We can construct a 3-year zero as follows:

Cash Flow in Year:

1 2 3 4
Three-year coupon bond ($1000 face value) 60 60 1,060
Less: one-year zero ($60 face value) (60)
Less: two-year zero ($60 face value) - (60)    
Three-year zero ($1060 face value) - - 1,060
 Chapter 8  Valuing Bonds 79

Now,
60 60 1060
Price(3-year coupon bond) =  2
  $1004.29
1.0584 1.0584 1.05843
By the Law of One Price:

Price(3-year zero) = Price(3-year coupon bond) – Price(One-year zero) – Price(Two-year zero)

= Price(3-year coupon bond) – PV(coupons in years 1 and 2)
= 1004.29 – 60 / 1.02 – 60 / 1.042 = $889.99

Solving for the YTM:

1/ 3
 1060 
YTM(3)    1  6.000%
 889.99 

Finally, we can do the same for the 4-year zero:

Cash Flow in Year:

1 2 3 4
Four-year coupon bond ($1000 face value) 120 120 120 1,120
Less: one-year zero ($120 face value) (120)
Less: two-year zero ($120 face value) — (120)
Less: three-year zero ($120 face value) — — (120)  
Four-year zero ($1120 face value) — — — 1,120

Now,
120 120 120 1120
Price(4-year coupon bond) =     $1216.50
1.05783 1.05783 1.05783 1.057834
2 3

By the Law of One Price:

Price(4-year zero) = Price(4-year coupon bond) – PV(coupons in years 1–3)

= 1216.50 – 120 / 1.02 – 120 / 1.042 – 120 / 1.063 = $887.15
Solving for the YTM:
1/4
 1120 
YTM(4)    1  6.000%
 887.15 
Thus, we have computed the zero coupon yield curve as

7%

6%

5%
Yield to Maturity

4%

3%

2%

1%

0%
0 1 2 3 4
Year
80 Berk/DeMarzo • Corporate Finance
8-23. The yield to maturity of a corporate bond is based on the promised payments of the bond. But there is some
chance the corporation will default and pay less. Thus, the bond’s expected return is typically less than its
YTM.

Corporate bonds have credit risk, the risk that the borrower will default and not pay all specified payments.
As a result, investors pay less for bonds with credit risk than they would for an otherwise identical default-
free bond. Because the YTM for a bond is calculated using the promised cash flows, the yields of bonds with
credit risk will be higher than that of otherwise identical default-free bonds. However, the YTM of a
defaultable bond is always higher than the expected return of investing in the bond because it is calculated
using the promised cash flows rather than the expected cash flows.

8-24.
a. The price of this bond will be

100
P  96.899
1 .032

b. The credit spread on AAA-rated corporate bonds is 0.032 – 0.031 = 0.1%

c. The credit spread on B-rated corporate bonds is 0.049 – 0.031 = 1.8%
d. The credit spread increases as the bond rating falls, because lower rated bonds are riskier.

8-25.
a. When originally issued, the price of the bonds was

70 70  1000
P  ...   $1065.29
(1 0.065) (1 .065)30

b. If the bond is downgraded, its price will fall to

70 70  1000
P  ...   $1012.53
(1 0.069) (1 .069)30

8-26.

a. The price will be

65 65  1000
P  ...   $1008.36
(1 .063) (1  .063)5

$10,000,000
b. Each bond will raise $1008.36, so the firm must issue:  9917.13  9918 bonds .
$1008.36
This will correspond to a principle amount of 9918  $1000  $9,918,000 .

c. For the bonds to sell at par, the coupon must equal the yield. Since the coupon is 6.5%, the yield must also
be 6.5%, or A-rated.
 Chapter 8  Valuing Bonds 81
d. First, compute the yield on these bonds:

65 65  1000
959.54   ...   YTM  7.5%
(1  YTM) (1 YTM)5

Given a yield of 7.5%, it is likely these bonds are BB rated. Yes, BB-rated bonds are junk bonds.

8-27.
35 35  1000
a. P  ...  10
 $1,021.06  102.1%
(1  .0325) (1  .0325)
35 35  1000
b. P  ...  10
 $951.58  95.2%
(1  .041) (1  .041)

c. 0. 17

Appendix

A.1. From Eq 8A.2,

(1  YTM 2 )2 1.0552
f2  1  1  7.02%
(1 YTM1 ) 1.04

A.2. From Eq 8A.2,

(1  YTM 3 )3 1.0553
f3   1   1  5.50%
(1  YTM 2 )2 1.0552
When the yield curve is flat (spot rates are equal), the forward rate is equal to the spot rate.

A.3. From Eq 8A.2,

(1  YTM 5 )5 1.0455
f5   1   1  2.52%
(1 YTM 4 )4 1.0504
When the yield curve is flat (spot rates are equal), the forward rate is equal to the spot rate.

A.4. Call this rate f1,5. If we invest for one-year at YTM1, and then for the 4 years from year 1 to 5 at rate f1,5,
after five years we would earn

1 YTM11 f1,54

with no risk. No arbitrage means this must equal that amount we would earn investing at the current five
year spot rate:

1 YTM11 f1,54  1 YTM55

(1  YTM 5 )5 1.0455
Therefore, (1  f1,5 )4    1.19825
1  YTM1 1.04
and so: f1,5  1.198251/4  1  4.625%
82 Berk/DeMarzo • Corporate Finance
A.5. We can invest for 3 years with risk by investing for one year at 5%, and then locking in a rate of 4% for the
second year and 3% for the third year. The return from this strategy must equal the return from investing in a
3 year zero coupon bond (see Eq 8A.3):

1 YTM33 1.051.041.031.12476

Therefore: YTM3  1.1247613 1  3.997