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Investments

Zvi Bodie, Alex Kane, Alan J. Marcus

Chapter 16

Managing Bond Portfolios - all with Video Answers

Educators

Chapter Questions

Problem 1

Prices of long-term bonds are more volatile than prices of short-term bonds. However, yields to maturity of short-term bonds fluctuate more than yields of long-term bonds. How do you reconcile these two empirical observations?

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Problem 2

How can a perpetuity, which has an infinite maturity, have a duration as short as 10 or 20 years?

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Problem 3

A 9 -year bond paying coupons annually has a yield of $10 \%$ and a duration of 7.194 years. If the market yield changes by 50 basis points, what is the percentage change in the bond's price?

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Problem 4

a. Find the duration of a $6 \%$ coupon bond making annual coupon payments if if has three years until maturity and has a yield to maturity of $6 \%$.
$b$. What is the duration if the yield to maturity is $10 \%$ ?

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Problem 5

Repeat Problem 4, but now assume the coupons are paid semiannually.

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Problem 6

a. The historical yield spread between AAA bonds and Treasury bonds widened dramatically during the financial crisis in 2008 . If you believed that the spread would soon return to more typical historical levels, what should you have done?
b. This would be an example of what sort of bond swap?

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 7

You predict that interest rates are about to fall. Which bond will give you the highest capital gain?
a. Low coupon, long maturity.
b. High coupon, short maturity.
c. High coupon, long maturity.
d. Zero coupon, long maturity.

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Problem 8

Rank the durations or effective durations of the following pairs of bonds:
a. Bond $A$ is a $6 \%$ coupon bond, with a 20 -year time to maturity selling at par value. Bond $B$ is a $6 \%$ coupon bond, with a 20 -year time to maturity selling below par value.
b. Bond $A$ is a 20 -year noncallable coupon bond with a coupon rate of $6 \%$, selling at par. Bond $B$ is a 20 -year callable bond with a coupon rate of $7 \%$, also selling at par.

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01:01

Problem 9

An insurance company must make payments to a customer of $$\$ 10$$ million in one year and $$\$ 4$$ million in five years. The yield curve is flat at $10 \%$.
a. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
b. What must be the face value and market value of that zero-coupon bond?

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 10

Long-term Treasury bonds currently are selling at yields to maturity of nearly $6 \%$. You expect interest rates to fall. The rest of the market thinks that they will remain unchanged over the coming year. In each question, choose the bond that will provide the higher holding-period return over the next year if you are correct. Briefly explain your answer.
a. i. A Baa-rated bond with coupon rate $6 \%$ and time to maturity 20 years.
ii. An Aas-rated bond with coupon rate of $6 \%$ and time to maturity 20 years.
b. 1. An A-rated bond with coupon rate $3 \%$ and maturity 20 years, callable at 105 .
ii. An A-rated bond with coupon rate $6 \%$ and maturity 20 years, callable at 105 .
c. i. A $4 \%$ coupon noncallable T-bond with maturity 20 years and $\mathrm{YTM}=6 \%$.
ii. A.7\% coupoa noncallable T-bond with maturity 20 years and $\mathrm{YTM}=6 \%$.

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Problem 11

Currently, the term structure is as follows: 1 -year zero-coupon bonds yield $7 \% ; 2$-year zerocoupon bonds yield $8 \% ; 3$-year and longer-maturity zero-coupon bonds all yield $9 \%$. You are choosing between 1-, 2-, and 3-year maturity bonds all paying annual coupons of $8 \%$.
a. What is the price of each bond today?
b. What will be the price of each bond in one year if the yield curve is flat at $9 \%$ at that time?
c. What will be the rate of return on each bond?

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Problem 12

You will be paying $$\$ 10,000$$ a year in tuition expenses at the end of the next two years. Bonds currently yield $8 \%$.
a. What are the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to $9 \%$. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation?
d. What if rates fall immediately to $7 \%$ ?

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Problem 13

Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perperual payments of $$\$ 2$$ million per year to beneficiaries. The yield to maturity on all bonds is $16 \%$.
a. If the duration of 5 -year-maturity bonds with coupon rates of $12 \%$ (paid annually) is four years and the duration of 20 -year-maturity bonds with coupon rates of $6 \%$ (paid annually) is 11 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation?
b. What will be the par value of your holdings in the 20 -year coupon bond?

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Problem 14

You are managing a portfolio of $$\$ 1$$ million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a perpetuity, each currently yielding $5 \%$.
a. How much of (i) the zero-coupon bond and (ii) the perpetuity will you hold in your portfolio?
b. How will these fractions change next year if target duration is now nine years?

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Problem 15

My pension plan will pay me $$\$ 10,000$$ once a year for a 10 -year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.
a. What is the duration of its obligation to me? The current interest rate is $10 \%$ per year.
b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond?
c. What will be the face value of the holdings in each zero?

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05:50

Problem 16

A 30 -year-maturity bond making annual coupon payments with a coupon rate of $12 \%$ has duration of 11.54 years and convexity of 192.4 . The bond currently sells at a yield to maturity of $8 \%$.
a. Use a financial calculator or spreadsheet to find the price of the bond if its yield to maturity falls to $7 \%$.
b. What price would be predicted by the duration rule?
c. What price would be predicted by the duration-with-convexity rule?
d. What is the percent error for each rule? What do you conclude about the accuracy of the two rules?
e. Repeat your analysis if the bond's yield to maturity increases to $9 \%$. Are your conclusions about the accuracy of the two rules consistent with parts $(a)-(d)$ ?

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 17

Frank Meyers, CFA, is a fixed-income portfolio manager for a large pension fund. A member of the Investment Committee, Fred Spice, is very interested in learning about the management of fixed-income portfolios. Spice has approached Meyers with several questions.
Meyers decides to illustrate fixed-income trading strategies to Spice using a fixed-rate bond and note. Both the bond and note have semiannual coupon periods. Unless otherwise stated, all interest rate changes are parallel. The characteristics of these securities are shown in the following table. He also considers a 9 -year floating-rate bond (floater) that pays a floating rate semiannually and is currently yielding $5 \%$.
Characteristics of Fixed-Rate Bond and Fixed-Rate Note
$$
\begin{array}{lcc}
\hline & \text { Fixed-Rate Bond } & \text { Fixed-Rate Note } \\
\hline \text { Price } & 107.18 & 100.00 \\
\text { Yeld to maturly } & 5.00 \% & 5.00 \% \\
\text { Time to maturity (years) } & 9 & 4 \\
\text { Modifed duration (years) } & 6.9848 & 3.5851
\end{array}
$$
Spice asks Meyers about how a fixed-income manager would position his portfolio to capitalize on expectations of increasing interest rates. Which of the following would be the most appropriate strategy?
a. Shorten his portfolio duration.
b. Buy fixed-rate bonds.
c. Lengthen his portfolio duration.

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01:12

Problem 18

Spice asks Meyers (see Problem 17) to quantify price changes from changes in interest rates. To illustrate, Meyers computes the value change for the fixed-rate note in the table. Specifically, he assumes an increase in the level of interest rate of 100 basis points. Using the information in the table, what is the predicted change in the price of the fixed-rate note?

Breanna Ollech
Breanna Ollech
Numerade Educator

Problem 19

Find the duration of a bond with a settlement date of May 27, 2025, and maturity date November 15,2036 . The coupon rate of the bond is $7 \%$, and the bond pays coupons semiannually. The bond is selling at a bond-equivalent yield to maturity of $8 \%$. You can use Spreadsheet 16.3. available in Connect or through your course instructor.

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Problem 20

A 30 -year-maturity bond has a $7 \%$ coupon rate, paid annually. It sells today for $$\$ 867.42$$. A 20 -year-maturity bond has a $6.5 \%$ coupon rate, also paid annually. It sells today for $$\$ 879.50$$. A bond market analyst forecasts that in five years, 25 -year-maturity bonds will sell at yields to maturity of $8 \%$ and 15 -year-maturity bonds will sell at yields of $7.5 \%$. Because the yield curve is upward-sloping, the analyst believes that coupons will be invested in short-term securities at a rate of $6 \%$.
a. Calculate the (annualized) expected rate of return of the 30 -year bond over the 5 -year period.
b. What is the (annualized) expected return of the 20 -year bond?

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Problem 21

a. Use a spreadsheet to calculate the durations of the two bonds in Spreadsheet 16.1 if the market interest rate increases to $12 \%$. Why does the duration of the coupon bond fall while that of the zero remains unchanged?
b. Use the same spreadsheet to calculate the duration of the coupon bond if the coupon is $12 \%$ instead of $8 \%$ and the semiannual interest rate is again $5 \%$. Explain why duration is lower than in Spreadsheet 16.1. (Again, start by looking at column F.)

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Problem 22

a. Footnote 7 presents the formula for the convexity of a bond. Build a spreadsheet to calculate the convexity of a 5 -year, $8 \%$ coupon bond making annual payments at the initial yield to maturity of $10 \%$.
b. What is the convexity of a 5 -year zero-coupon bond?

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05:50

Problem 23

A 12.75 -year-maturity zero-coupon bond selling at a yield to maturity of $8 \%$ (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30 -year-maturity $6 \%$ coupon bond making annual coupon payments also selling at a yield to maturity of $8 \%$ has nearly identical duration- 11.79 years-but considerably higher convexity of 231.2 .
a. Suppose the yield to maturity on both bonds increases to $9 \%$. (i) What will be the actual percentage capital loss on each bond? (ii) What percentage capital loss would be predicted by the duration-with-convexity rule?
b. Repeat part (a), but this time assume the yield to maturity decreases to $7 \%$.
c. Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on the comparative investment performance, explain the attraction of convexity.
d. In view of your answer to part (c), do you think it would be possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:50

Problem 24

A newly issued bond has a maturity of 10 years and pays a $7 \%$ coupon rate (with coupon payments coming once annually). The bond sells at par value.
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7 .
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from $7 \%$ to $8 \%$ (with maturity still 10 years).
c. What price would be predicted by the modified duration rule (Equation 16.3)? What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule (Equation 16.5)? What is the percentage error of that rule?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:50

Problem 25

a. Use a spreadsheet to answer this question and assume the yield curve is flat at a level of $4 \%$. Calculate the convexity of a "bullet" fixed-income portfolio, that is, a portfolio with a single cash flow. Suppose a single $$\$ 1,000$$ cash flow is paid in year 5.
b. Now calculate the convexity of a "ladder" fixed-income portfolio, that is, a portfolio with equal cash flows over time. Suppose the security makes $$\$ 100$$ cash flows in each of years 1-9, so that its duration is close to the bullet in part (a).
c. Do ladders or bullets have greater convexity?

Rashmi Sinha
Rashmi Sinha
Numerade Educator