BD5 SM06 - Corporate Finance Spring Semester Textbook explanation. PDF

Preview

Summary

Corporate Finance Spring Semester Textbook explanation....

Description

Chapter 6

Valuing Bonds 6-1.

A 15-year bond with a face value of $1000 has a coupon rate of 4.5%, with semiannual payments. a.

What is the coupon payment for this bond?

b. Draw the cash flows for the bond on a timeline. a.

The coupon payment for this bond is:

CPN = b.

6-2.

Timeline: 0

0.045× $ 1,000 Coupon Rate × Face Value =$ 22.50 = 2 Number of Coupons per Year 1

2

30

$22.50

$22.50

$22.50 + $1,000

Assume that a bond will make payments every six months as shown on the following timeline (using six-month periods):

0

a.

1

2

3

$30

$30

$30

20

$30 + $1,000

What is the maturity of the bond (in years)?

b. What is the coupon rate (in percent)? c.

What is the face value?

a.

The maturity is 15 years (2 payments per year).

b.

($30/$1,000) × 2 = 6%, so the coupon rate is 6%.

c.

The face value is $1,000.

1

©2017 Pearson Education, Inc.

2

Berk/DeMarzo, Corporate Finance, Fourth Edition

6-3.

The following table summarizes prices of various default-free, zero-coupon bonds (expressed as a percentage of face value):

Maturity (years)

Price (per $100 face value)

a.

1

2

3

4

5

$96.21

$91.83

$87.16

$82.51

$77.38

Compute the yield to maturity for each bond.

b. Plot the zero-coupon yield curve (for the first five years). c.

Is the yield curve upward sloping, downward sloping, or flat?

a.

Use the following equation with FV = $100, P = price per $100 face value, and n = maturity (years) from the table,

( )

1+YTM n =

FV n P

1 n

Price (P) $96.21

Maturity (n) 1

$91.83

2

$87.16

3

$82.51

4

$77.38

5

YTM

( ( ( ( (

$ 100 $ 96.21 $ 100 $ 91.83 $ 100 $ 87.16 $ 100 $ 82.51 $ 100 $ 77.38

) −1=0.0394=3.94 % ) −1=0.0435=4.35 % ) −1= 0.0469=4.69 % ) −1=0.0492= 4.92% ) −1=0.0526=5.26 % 1 1 1 2

1 3 1 4

1 5

b.

©2017 Pearson Education, Inc.

Chapter 6/Valuing Bonds 3

c. 6-4.

The yield curve is upward sloping.

Suppose the current zero-coupon yield curve for risk-free bonds is as follows:

Maturity (years)

YTM

a.

1

2

3

4

5

4.28%

4.75%

4.89%

5.20%

5.45%

What is the price per $100 face value of a three-year, zero-coupon, risk-free bond?

b. What is the price per $100 face value of a four-year, zero-coupon, risk-free bond? c.

What is the risk-free interest rate for a three-year maturity?

P=

a.

6-5.

b.

P=

c.

4.89%

$ 100 =$ 86.66 3 ( 1.0489 )

$ 100 =$ 81.65 4 ( 1.0520 )

In the Global Financial Crisis box in Section 6.1, Bloomberg.com reported that the three-month Treasury bill sold for a price of $100.002556 per $100 face value. What is the yield to maturity of this bond, expressed as an EAR? 4

100    100.002556   1  0.01022%  

6-6.

Suppose a 10-year, $1000 bond with a 7% coupon rate and semiannual coupons is trading for a price of $1181.64. a.

What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?

b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?

a.

$ 1,181.64=

(

$ 35+ $ 1,000 $ 35 $ 35 +… + 20 2 1 YTM YTM YTM 1+ 1+ 1+ 2 2 2

) (

)

(

)

With 7% YTM = 3.5% per 6 months, the new price is $1,181.64 Solving using financial calculator

Given Solve for I/Y

N 20

I/Y

PV -1,181.64

PMT 35

FV 1,000

2.35

Therefore, the YTM is 2.35% x 2 = 4.7% b.

P=$ 35 ×

(

)

$ 1,000 1 1 + 1− =$ 455.2777758 +$ 414.6428597=$ 869.92 20 0.045 1.045 1.04520

Solving using financial calculator

©2017 Pearson Education, Inc.

4

Berk/DeMarzo, Corporate Finance, Fourth Edition

N 20

Given Solve for PV

6-7.

I/Y 4.5

PV

PMT 35

FV 1,000

-869.92

Suppose a five-year, $1000 bond with annual coupons has a price of $1050 and a yield to maturity of 6%. What is the bond’s coupon rate?

$ 1,050=C ×

$ 1,050−

(

)

$ 1,000 1 1 + 1− 5 0.06 ( 1.06 )5 1.06

(

1 $ 1,000 1 =C × 1− 5 0.06 1.065 (1.06 )

)

$ 1,000 5 ( 1.06) 302.7418271 =$ 71.87 = C= 4.212363786 1 1 1− 0.06 1.065 $ 1,050−

(

)

Thus, the coupon rate is 7.19% 6-8.

The prices of several bonds with face values of $1000 are summarized in the following table: Bond

A

B

C

D

Price

$936.57

$1095.48

$1170.97

$1000.00

For each bond, state whether it trades at a discount, at par, or at a premium. Bond A trades at a discount. Bond D trades at par. Bonds B and C trade at a premium. 6-9.

Explain why the yield of a bond that trades at a discount exceeds the bond’s coupon rate. Bonds trading at a discount generate a return both from 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.

6-10.

Suppose a seven-year, $1000 bond with a 9.08% coupon rate and semiannual coupons is trading with a yield to maturity of 7.30%. a.

Is this bond currently trading at a discount, at par, or at a premium? Explain.

b. If the yield to maturity of the bond rises to 8.25% (APR with semiannual compounding), what price will the bond trade for? a. b.

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

P=$ 45.40 ×

(

)

$ 1,000 1 1 + =$ 475.64 + $ 567.84 =$ 1,043.48 1− 14 14 0.04125 1.04125 1.04125

Solving using financial calculator

Given

N 14

I/Y 4.12

PV

PMT 45.40

FV 1,000

©2017 Pearson Education, Inc.

Chapter 6/Valuing Bonds 5

5 Solve for PV 6-11.

-1,043.48

Suppose that Ally Financial Inc. issued a bond with 10 years until maturity, a face value of $1000, and a coupon rate of 6% (annual payments). The yield to maturity on this bond when it was issued was 10%. a.

What was the price of this bond when it was issued?

b. Assuming the yield to maturity remains constant, what is the price of the bond immediately before it makes its first coupon payment? c.

a.

Assuming the yield to maturity remains constant, what is the price of the bond immediately after it makes its first coupon payment?

P=$ 60 ×

(

)

$ 1,000 1 1 =$ 368.68 +$ 385.54=$ 754.22 1− 10 + 10 0.1 1.1 1.1

Solving using financial calculator

Given Solve for PV

6-12.

N 10

I/Y 10

PV

PMT 60

FV 1,000

-754.22

b.

P=$ 60+ $ 60 ×

c.

P=$ 60 ×

(

)

$ 1,000 1 1 =$ 60 + $ 345.54 + $ 424.10=$ 829.64 1− 9 + 9 0.1 1.1 1.1

(

)

$ 1,000 1 1 =$ 345.54 + $ 424.10=$ 769.64 1− 9 + 9 0.1 1.1 1.1

Suppose you purchase a 10-year bond with 4% annual coupons. You hold the bond for four years, and sell it immediately after receiving the fourth coupon. If the bond’s yield to maturity was 3.75% when you purchased and sold the bond, a.

What cash flows will you pay and receive from your investment in the bond per $100 face value?

b. What is the internal rate of return of your investment? a.

First, we compute the initial price of the bond by discounting its 10 annual coupons of $4 and final face value of $100 at the 3.75% yield to maturity. Solving using financial calculator N I/Y PV PMT FV Given 10 3.75 4 100 Solve for PV -102.05 Thus, the initial price of the bond = $102.05. (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 bond’s cash flows when only 6 years remain until maturity. Solving using financial calculator

Given

N 6

I/Y 3.75

PV

PMT 4

FV 100

©2017 Pearson Education, Inc.

6

Berk/DeMarzo, Corporate Finance, Fourth Edition

Solve for PV

-101.32

Therefore, the bond was sold for a price of $101.32. The cash flows from the investment are therefore as shown in the following timeline.

b.

Year

0

Purchase Bond Receive Coupons Sell Bond Cash Flows

–$102.05

–$102.05

1

2

3

4

$4

$4

$4

$11.00

$11.00

$11.00

$4 $101.32 $101.32

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 = 3.75%. Because the YTM was the same at the time of purchase and sale, the IRR of the investment matches the YTM. Solving using financial calculator

Given

N 4

Solve for I/Y 6-13.

I/Y

PV -102.05

PMT 4

FV 101.3 2

3.75

Consider the following bonds:

a.

Bond

Coupon Rate (annual payments)

Maturity (years)

A

0%

16

B

0%

8

C

4%

16

D

7%

8

What is the percentage change in the price of each bond if its yield to maturity falls from 6% to 5%?

b. Which of the bonds A–D is most sensitive to a 1% drop in interest rates from 6% to 5% and why? Which bond is least sensitive? Provide an intuitive explanation for your answer. a.

We can compute the price of each bond at each YTM.

P (bond A , 6 % YTM )=

$ 100 =$ 39.36 1.0616

©2017 Pearson Education, Inc.

Chapter 6/Valuing Bonds 7

P (bond B ,6 % YTM ) =

$ 100 =$ 62.74 1.068

(

)

(

)

P (bond C , 6 % YTM )=$ 4 ×

1 1 $ 100 1− + =$ 79.79 16 16 0.06 1.06 1.06

P (bond D , 6 % YTM )=$ 7 ×

$ 100 1 1 + =$ 106.21 1− 8 8 0.06 1.06 1.06

We calculate the price of each bond in the same fashion when the YTM = 5%. 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.

Given Solve for I/Y b.

6-14.

N 20

I/Y 6

PV

PMT 7

FV 100

-111.47

Bond A is most sensitive, because it has the longest maturity and no coupons, so its future cash flows have the highest discount factors. Bond D is the least sensitive. Intuitively, higher coupon rates and a shorter maturity mean that relatively more of the bond’s cash flows happen early and thus cannot be as greatly affected by changes in interest rates as bonds with low coupon rates and longer maturities.

Suppose you purchase a 30-year, zero-coupon bond with a yield to maturity of 6%. You hold the bond for five years before selling it. a.

If the bond’s yield to maturity is 6% when you sell it, what is the internal rate of return of your investment?

b. If the bond’s yield to maturity is 7% when you sell it, what is the internal rate of return of your investment? c.

If the bond’s yield to maturity is 5% when you sell it, what is the internal rate of return of your investment?

d. Even if a bond has no chance of default, is your investment risk free if you plan to sell it before it matures? Explain.

6-15.

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.

Suppose you purchase a 30-year Treasury bond with a 7% annual coupon, initially trading at par. In 10 years’ time, the bond’s yield to maturity has risen to 6% (EAR). a.

If you sell the bond now, what internal rate of return will you have earned on your investment in the bond?

b. If instead you hold the bond to maturity, what internal rate of return will you earn on your investment in the bond?

©2017 Pearson Education, Inc.

8

Berk/DeMarzo, Corporate Finance, Fourth Edition

c.

Is comparing the IRRs in (a) versus (b) a useful way to evaluate the decision to sell the bond? Explain.

a.

You initially purchase the bond at $100 (par value). Use the financial calculator to calculate the price you will be able to sell the bond at in 10 years (20 years remaining) Solving using financial calculator

Bond A B C D

Coupon Rate 0% 0% 4% 7%

Maturity 16 8 16 8

Price at 6% $39.36 $62.74 $79.79 $106.21

Price at 5% $45.81 $67.68 $89.16 $112.93

Change 16.39% 7.87% 11.74% 6.33%

Then, use the financial calculator again to calculate the IRR of the investment: Solving using financial calculator

Given Solve for I/Y b. c.

6-16.

N 10

I/Y

PV -100

PMT 7

FV 111.47

7.80

7% (since it’s trading at par) We can’t simply compare IRRs. By not selling the bond for its current (at year 10) price of $111.47, we will earn the current market return of 6% on that amount going forward.

Suppose the current yield on a one-year, zero coupon bond is 3%, while the yield on a five-year, zero coupon bond is 4%. Neither bond has any risk of default. Suppose you plan to invest for one year. You will earn more over the year by investing in the five-year bond as long as its yield does not rise above what level?

The return from investing in the one-year is the yield. The return for investing in the five-year bond for initial price p 0 and selling after one year at price p1 is

p0=

1 5 ( 1.04 )

,

p1=

1 4 ( 1+ y )

.

p1 −1 . We have p0

So you break even when:

1 4 p1 (1+ y ) −1= −1=0.03 p0 1 5 (1.04 ) y=

1.045 / 4 −1=0.0425 =4.25 % 1.031 /4

©2017 Pearson Education, Inc.

Chapter 6/Valuing Bonds 9

6-17.

What is the price today of a two-year, default-free security with a face value of $1000 and an annual coupon rate of 6%? Does this bond trade at a discount, at par, or at a premium? P

60 60  1000 CPN CPN CPN  FV   ...    $1032.09 1 YTM1 (1  YTM 2 ) 2 (1  YTM N ) N (1  .04) (1  .043) 2

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 6-18.

What is the price of a five-year, zero-coupon, default-free security with a face value of $1000? The price of the zero-coupon bond is P

6-19.

FV 1000  $791.03 (1  YTM N ) N (1  0.048) 5

What is the price of a three-year, default-free security with a face value of $1000 and an annual coupon rate of 4%? What is the yield to maturity for this bond? The price of the bond is P

40 40 40  1000 CPN CPN CPN  FV   ...     $986.58. 1 YTM1 (1  YTM 2 ) 2 (1  YTM N ) N (1 .04) (1  .043) 2 (1  .045) 3

The yield to maturity is CPN CPN CPN  FV P   ...  1 YTM (1 YTM ) 2 (1 YTM ) N $986.58  6-20.

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

What is the maturity of a default-free security with annual coupon payments and a yield to maturity of 4%? Why? The maturity must be one year. If the maturity were longer than one year, there would be an arbitrage opportunity.

6-21.

Consider a four-year, default-free security with annual coupon payments and a face value of $1000 that is issued at par. What is the coupon rate of this bond? Solve the following equation:   1 1 1 1 1000     1000 CPN  2 3 4       (1 .04) (1 .043) (1 .045) (1 .047) (1 .047) 4   CPN $46.76. Therefore, the par coupon rate is 4.676%.

6-22.

Consider a five-year, default-free bond with annual coupons of 5% and a face value of $1000. a.

Without doing any calculations, determine whether this bond is trading at a premium or at a discount. Explain.

b. What is the yield to maturity on this bond? c.

If the yield to maturity on this bond increased to 5.2%, what would the new price be?

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.

©2017 Pearson Education, Inc.

10

Berk/DeMarzo, Corporate Finance, Fourth Edition

P 

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