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Ch 13: End-of-Chapter Problems – Bond Pricing and Yields

by | Mar 23, 2020 | Uncategorized | 0 comments

1.  Problem 13-01

eBookeBookProblem 13-01

A $1,000 bond has a coupon of 8 percent and matures after ten years. Assume that the bond pays interest annually.

  1. What would be the bond’s price if comparable debt yields 9 percent? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $   

  2. What would be the price if comparable debt yields 9 percent and the bond matures after five years? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $   

  3. Why are the prices different in a and b?
    The price of the bond in a is   than the price of the bond in b as the principal payment of the bond in a is   than the principal payment of the bond in b (in time).
  4. What are the current yields and the yields to maturity in a and b? Round your answers to two decimal places.

    The bond matures after ten years:

    CY:   %
    YTM:   %

    The bond matures after five years:

    CY:   %
    YTM:   %

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Solution
  1. Price of the bond (annual payments):
    PB =
    = $80(6.418)+$1,000(0.422)
    = $513+$422
    = $935

    (PV = ?; N = 10; I = 9; PMT = 80, and FV = 1,000. PV = -936.)

  2. Price of the bond (annual payments):
    PB =
    = $80(3.890)+$1,000(0.650)
    = $311+$650
    = $961

    (PV = ?; N = 5; I = 9; PMT = 80, and FV = 1,000. PV = -961.)

  3. In both cases the bond’s price is less than the face amount (principal) because the current rate of interest is greater than the rate paid on these bonds (i.e., 9% versus 8%). However, the amount of price decline is affected by the term of the bond, and the bond with the longer term experiences the larger price decline because the investors will collect the smaller interest payments for a longer period of time.
  4. The current yields are

    ;

    .

    The yields to maturity are 9% in both cases.
Solution
Correct Response
eBookProblem 13-01

A $1,000 bond has a coupon of 8 percent and matures after ten years. Assume that the bond pays interest annually.

  1. What would be the bond’s price if comparable debt yields 9 percent? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $  

  2. What would be the price if comparable debt yields 9 percent and the bond matures after five years? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $  

  3. Why are the prices different in a and b?
    The price of the bond in a is  than the price of the bond in b as the principal payment of the bond in a is  than the principal payment of the bond in b (in time).
  4. What are the current yields and the yields to maturity in a and b? Round your answers to two decimal places.

    The bond matures after ten years:

    CY:  %
    YTM:  %

    The bond matures after five years:

    CY:  %
    YTM:  %

2.  Problem 13-02

eBookeBookProblem 13-02

  1. A $1,000 bond has a 5.5 percent coupon and matures after ten years. If current interest rates are 9 percent, what should be the price of the bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $   

  2. If after four years interest rates are still 9 percent, what should be the price of the bond? Use Appendix B and Appendix D to answer the question. Assume that the bond pays interest annually. Round your answer to the nearest dollar.

    $   

  3. Even though interest rates did not change in a and b, why did the price of the bond change?

    The price of the bond with the longer term is   than the price of the bond with the shorter term as the investors will collect the   interest payments and receive the principal within a longer period of time.

  4. Change the interest rate in a and b to 3 percent and rework your answers. Assume that the bond pays interest annually. Round your answers to the nearest dollar.

    Price of the bond (ten years to maturity): $   
    Price of the bond (six years to maturity): $   

    Even though the interest rate is 3 percent in both calculations, why are the bond prices different?

    The price of the bond with the longer term is   than the price of the bond with the shorter term as the investors will collect the   interest payments for a longer period of time.

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Solution
  1. Price of the bond:
    PB =
    = $55(6.418)+$1,000(0.422)
    = $775

    (PV = ?; N = 10; I = 9; PMT = 55, and FV = 1,000. PV = -775.)

  2. Price of the bond:
    PB =
    = $55(4.486)+$1,000(0.596)
    = $843

    (PV = ?; N = 6; I = 9; PMT = 55, and FV = 1,000. PV = -843.)

  3. The term to maturity has diminished which increases the value of the bond (i.e., the investor gets the principal back in only six instead of ten years). Moreover, the bond with the longer term experiences the larger price decline because the investors will collect the smaller interest payments for a longer period of time.
  4. Price of the bond (ten years to maturity):
    PB =
    = $55(8.530)+$1,000(0.744)
    = $1,213

    (PV = ?; N = 10; I = 3; PMT = 55, and FV = 1,000. PV = -1,213.)Price of the bond (six years to maturity):

    PB =
    = $55(5.417)+$1,000(0.837)
    = $1,135

    (PV = ?; N = 6; I = 3; PMT = 55, and FV = 1,000. PV = -1,135.)The term to maturity is less, but in this illustration the value of the bond declines. In a and b, the bond sold for a discount. Now it sells for a premium, which declines as the bond approaches maturity. The investor earns the higher coupon interest for a shorter time period, which decreases the attractiveness of the bond. Considering the bond with a longer term, the investors will collect the higher coupon interest for a longer time period so the price of this bond is higher.
Solution
Correct Response
eBookProblem 13-02

  1. A $1,000 bond has a 5.5 percent coupon and matures after ten years. If current interest rates are 9 percent, what should be the price of the bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $  

  2. If after four years interest rates are still 9 percent, what should be the price of the bond? Use Appendix B and Appendix D to answer the question. Assume that the bond pays interest annually. Round your answer to the nearest dollar.

    $  

  3. Even though interest rates did not change in a and b, why did the price of the bond change?

    The price of the bond with the longer term is  than the price of the bond with the shorter term as the investors will collect the  interest payments and receive the principal within a longer period of time.

  4. Change the interest rate in a and b to 3 percent and rework your answers. Assume that the bond pays interest annually. Round your answers to the nearest dollar.

    Price of the bond (ten years to maturity): $  
    Price of the bond (six years to maturity): $  

    Even though the interest rate is 3 percent in both calculations, why are the bond prices different?

    The price of the bond with the longer term is  than the price of the bond with the shorter term as the investors will collect the  interest payments for a longer period of time.

3.  Problem 13-03

eBookeBookProblem 13-03

A bond with 16 years to maturity has an annual interest payment of $30. If the bond sells for its par value, what are the bond’s current yield and yield to maturity? Round your answers to two decimal places.

CY:   %

YTM:   %

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Solution
The current yield: $30/$1,000 = 3.00%

The yield to maturity: 3.00%

The current yield and the yield to maturity are equal when the bond sells for face value (principal amount).
Solution
Correct Response
eBookProblem 13-03

A bond with 16 years to maturity has an annual interest payment of $30. If the bond sells for its par value, what are the bond’s current yield and yield to maturity? Round your answers to two decimal places.

CY:  %

YTM:  %

4.  Problem 13-04

eBookeBookProblem 13-04

Carrie’s Clothes, Inc. has a seven-year bond outstanding that pays $50 annually. The face value of each bond is $1,000, and the bond sells for $890.

  1. What is the bond’s coupon rate? Round your answer to two decimal places.

      %

  2. What is the current yield? Round your answer to two decimal places.

      %

  3. What is the yield to maturity? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest whole number.

      %

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Solution
  1. The coupon rate: $50/$1,000 = 5.00%.
  2. The current yield: $50/$890 = 5.62%.
  3. Determination of the yield to maturity (r):
    $890 =

    Select an interest rate (e.g., 7%) and substitute into the equation:

    $50(5.389)+$1,000(0.623) = $892

    Repeat the process (e.g., 8%):

    $50(5.206)+$1,000(0.583) = $843

    The yield to maturity is between 7% and 8%.

    Using a financial calculator:

    PV = -890; PMT = 50 FV = 1,000; N = 7, and I = ?. I = 7.04%.
Solution
Correct Response
eBookProblem 13-04

Carrie’s Clothes, Inc. has a seven-year bond outstanding that pays $50 annually. The face value of each bond is $1,000, and the bond sells for $890.

  1. What is the bond’s coupon rate? Round your answer to two decimal places.

     %

  2. What is the current yield? Round your answer to two decimal places.

     %

  3. What is the yield to maturity? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest whole number.

     %

5.  Problem 13-05

eBookeBookProblem 13-05

Charlotte’s Clothing issued a 5 percent bond with a maturity date of 15 years. Five years have passed and the bond is selling for $690. Assume that the bond pays interest annually.

  1. What is the current yield? Round your answer to two decimal places.

      %

  2. What is the yield to maturity? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest whole number.

      %

  3. If four years later the yield to maturity is 10 percent, what will be the price of the bond? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $   

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Solution
Notice that five years have passed since the bond was issued, so the bond has ten years to maturity. The yield to maturity in part b is based on the remaining ten years and not the initial fifteen years.

  1. The current yield = $50/$690 = 7.25%.
  2. Determination of the yield to maturity (r):
    $690 =

    Select an interest rate (e.g., 10%) and substitute into the equation:

    $50(6.145)+$1,000(0.386) = $693

    The yield to maturity is approximately 10 percent.

    (PV = -690; PMT = 50 FV = 1,000; N = 10, and I = ?. I = 10.06%.)

  3. Since the bond is selling for a discount, that discount will diminish over time even if interest rates remain stable. If yields are 10 percent four years later, the price of the bond is
    PB =
    = $50(4.355)+$1,000(0.564)
    = $782
    (PV = ?; N = 6; I = 10; PMT = 50, and FV = 1,000. PV = -782.)

    Interest rates have not changed but the bond matures in six years; its value rises from $693 to $782.
Solution
Correct Response
eBookProblem 13-05

Charlotte’s Clothing issued a 5 percent bond with a maturity date of 15 years. Five years have passed and the bond is selling for $690. Assume that the bond pays interest annually.

  1. What is the current yield? Round your answer to two decimal places.

     %

  2. What is the yield to maturity? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest whole number.

     %

  3. If four years later the yield to maturity is 10 percent, what will be the price of the bond? Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $  

6.  Problem 13-06

eBookeBookProblem 13-06

Your broker offers to sell for $1,220 a AAA-rated bond with a coupon rate of 9 percent and a maturity of nine years. Given that the interest rate on comparable debt is 6 percent, calculate the bond’s price. Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

$   

Is your broker fairly pricing the bond?

 , so the bond   be purchased.

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Solution
Price of the bond (annual payments):

PB =
= $90(6.802)+$1,000(0.592)
= $1,204

(PV = ?; N = 9; I = 6; PMT = 90, and FV = 1,000. PV = -1,204.)The bond is overpriced ($1,220 versus $1,204) and should not be purchased.
Solution
Correct Response
eBookProblem 13-06

Your broker offers to sell for $1,220 a AAA-rated bond with a coupon rate of 9 percent and a maturity of nine years. Given that the interest rate on comparable debt is 6 percent, calculate the bond’s price. Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

$  

Is your broker fairly pricing the bond?

, so the bond  be purchased.

7.  Problem 13-07

eBookeBookProblem 13-07

Fifteen years ago your grandfather purchased for you a 30-year $1,000 bond with a coupon rate of 10 percent. You now wish to sell the bond and read that yields are 8 percent. What price should you receive for the bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

$   

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Solution
PB =
= $100(8.559)+$1,000(0.315)
= $1,171

(PV = ?; N = 15; I = 8; PMT = 100, and FV = 1,000. PV = -1,171.)You should receive $1,171 for the bond.
Solution
Correct Response
eBookProblem 13-07

Fifteen years ago your grandfather purchased for you a 30-year $1,000 bond with a coupon rate of 10 percent. You now wish to sell the bond and read that yields are 8 percent. What price should you receive for the bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

8.  Problem 13-08

eBookeBookProblem 13-08

Bond A has the following terms:

  • Coupon rate of interest (paid annually): 12 percent
  • Principal: $1,000
  • Term to maturity: Ten years

Bond B has the following terms:

  • Coupon rate of interest (paid annually): 6 percent
  • Principal: $1,000
  • Term to maturity: Ten years
  1. What should be the price of each bond if interest rate is 12 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $   
    Price of bond B: $   
  2. What will be the price of each bond if, after three years have elapsed, interest rate is 12 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $   
    Price of bond B: $   
  3. What will be the price of each bond if, after ten years have elapsed, interest rate is 9 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $   
    Price of bond B: $   

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Solution
  1. Price of the bond A:
    PB =
    = $120(5.650)+$1,000(0.322)
    = $1,000

    (PV = ?; N = 10; I = 12; PMT = 120, and FV = 1,000. PV = -1,000.)Price of the bond B:

    PB =
    = $60(5.650)+$1,000(0.322)
    = $661

    (PV = ?; N = 10; I = 12; PMT = 60, and FV = 1,000. PV = -661.)

  2. In this question three years have passed but interest rates have not changed.

    Price of the bond A:

    PB =
    = $120(4.564)+$1,000(0.452)
    = $1,000

    (PV = ?; N = 7; I = 12; PMT = 120, and FV = 1,000. PV = -1,000.)Price of the bond B:

    PB =
    = $60(4.564)+$1,000(0.452)
    = $726

    (PV = ?; N = 7; I = 12; PMT = 60, and FV = 1,000. PV = -726.)

  3. After ten years have elapsed, both bonds mature, and the investors receive $1,000.
Solution
Correct Response
eBookProblem 13-08

Bond A has the following terms:

  • Coupon rate of interest (paid annually): 12 percent
  • Principal: $1,000
  • Term to maturity: Ten years

Bond B has the following terms:

  • Coupon rate of interest (paid annually): 6 percent
  • Principal: $1,000
  • Term to maturity: Ten years
  1. What should be the price of each bond if interest rate is 12 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $  
    Price of bond B: $  
  2. What will be the price of each bond if, after three years have elapsed, interest rate is 12 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $  
    Price of bond B: $  
  3. What will be the price of each bond if, after ten years have elapsed, interest rate is 9 percent? Use Appendix B and Appendix D to answer the question. Round your answers to the nearest dollar.
    Price of bond A: $  
    Price of bond B: $  

9.  Problem 13-09

eBookeBookProblem 13-09

A bond has the following features:

  • Coupon rate of interest (paid annually): 5 percent
  • Principal: $1,000
  • Term to maturity: 8 years
  1. What will the holder receive when the bond matures?

     

  2. If the current rate of interest on comparable debt is 7 percent, what should be the price of this bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $   

    Would you expect the firm to call this bond? Why?

     , since the bond is selling for a  .

  3. If the bond has a sinking fund that requires the firm to set aside annually with a trustee sufficient funds to retire the entire issue at maturity, how much must the firm remit each year for eight years if the funds earn 7 percent annually and there is $90 million outstanding? Use Appendix C to answer the question. Round your answer to the nearest dollar.

    $   

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Solution
  1. The investor receives the principal at maturity (i.e., $1,000).
  2. Price of the bond with 8 years to maturity:
    PB =
    = $50(5.971)+$1,000(0.582)
    = $881

    (PV = ?; N = 8; I = 7; PMT = 50, and FV = 1,000. PV = -881.)Since the bond is selling for a discount, there is no reason to expect the firm to call the bond.

  3. This is an example of the future value of an annuity.

    The amount the firm must set aside annually (X) is

    X(10.260) = $90,000,000

    X = $90,000,000/10.260 = $8,771,930

    10.260 is the interest factor for the future value of an annuity of $1 at 7% for eight years.

    (PV = 0; FV = 90,000,000; N = 8; I = 7, and PMT = ?. PMT = -8,772,099.)
Solution
Correct Response
eBookProblem 13-09

A bond has the following features:

  • Coupon rate of interest (paid annually): 5 percent
  • Principal: $1,000
  • Term to maturity: 8 years
  1. What will the holder receive when the bond matures?

  2. If the current rate of interest on comparable debt is 7 percent, what should be the price of this bond? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Round your answer to the nearest dollar.

    $  

    Would you expect the firm to call this bond? Why?

    , since the bond is selling for a .

  3. If the bond has a sinking fund that requires the firm to set aside annually with a trustee sufficient funds to retire the entire issue at maturity, how much must the firm remit each year for eight years if the funds earn 7 percent annually and there is $90 million outstanding? Use Appendix C to answer the question. Round your answer to the nearest dollar.

    $  

10.  Problem 13-10

eBookeBookProblem 13-10

You are given the following information concerning a noncallable, sinking fund debenture:

  • Principal: $1,000
  • Coupon rate of interest: 8 percent
  • Term to maturity: 16 years
  • Sinking fund: 4 percent of outstanding bonds retired annually; the balance at maturity
  1. If you buy the bond today at its face amount and interest rates rise to 14 percent after three years have passed, what is your capital gain or loss? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Use a minus sign to enter the loss amount, if any, as a negative value. Round your answer to the nearest dollar.

    $   

  2. If you hold the bond 16 years, what do you receive at maturity?

     

  3. What is the bond’s current yield as of right now? Round your answer to the nearest whole number.

      %

  4. Given your price in a, what is the yield at maturity? Round your answer to the nearest whole number.

      %

  5. What proportion of the total debt issue is retired by the sinking fund? Round your answer to the nearest whole number.

      %

  6. If the final payment to retire this bond is $1,200,000, how much must the firm invest annually to accumulate this sum if the firm is able to earn 8 percent on the invested funds? Use Appendix C to answer the question. Round your answer to the nearest dollar.

    $   

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Solution
a. PB =
= $80(5.842)+$1,000(0.182)
= $649
(PV = ?; N = 13; I = 14; PMT = 80, and FV = 1,000. PV = -649.)The loss is $649 – $1,000 = -$351.
  1. The principal amount: $1,000.
  2. Current yield right now: $80/$1,000 = 8%

    Current yield after three years: $80/$649 = 12.3%

  3. The yield to maturity is the rate used to determine the bond’s price in (a): 14%.
  4. 4% annually × 16 years = 64%.
  5. This is an example of the future value of an annuity.

    The amount the firm must set aside annually (X) is

    X(30.324) = $1,200,000

    X = $1,200,000/30.324 = $39,573

    30.324 is the interest factor for the future value of an annuity of $1 at 8% for sixteen years.

    (PV = 0; FV = 1,200,000; N = 16; I = 8, and PMT = ?. PMT = -39,572.)
Solution
Correct Response
eBookProblem 13-10

You are given the following information concerning a noncallable, sinking fund debenture:

  • Principal: $1,000
  • Coupon rate of interest: 8 percent
  • Term to maturity: 16 years
  • Sinking fund: 4 percent of outstanding bonds retired annually; the balance at maturity
  1. If you buy the bond today at its face amount and interest rates rise to 14 percent after three years have passed, what is your capital gain or loss? Assume that the bond pays interest annually. Use Appendix B and Appendix D to answer the question. Use a minus sign to enter the loss amount, if any, as a negative value. Round your answer to the nearest dollar.

    $  

  2. If you hold the bond 16 years, what do you receive at maturity?

  3. What is the bond’s current yield as of right now? Round your answer to the nearest whole number.

     %

  4. Given your price in a, what is the yield at maturity? Round your answer to the nearest whole number.

     %

  5. What proportion of the total debt issue is retired by the sinking fund? Round your answer to the nearest whole number.

     %

  6. If the final payment to retire this bond is $1,200,000, how much must the firm invest annually to accumulate this sum if the firm is able to earn 8 percent on the invested funds? Use Appendix C to answer the question. Round your answer to the nearest dollar.

    $