Options, Futures, and Other Derivatives

John C. Hull

Chapter 4 Interest rates - all with Video Answers

Educators

Chapter Questions

01:35

Problem 1

A bank quotes an interest rate of $7 \%$ per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?

Manisha Sarker

Numerade Educator

Problem 2

Explain how LIBOR is determined.

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02:23

The 6-month and 1 -year zero rates are both $5 \%$ per annum. For a bond that has a life of 18 months and pays a coupon of $4 \%$ per annum (with semiannual payments and one having just been made), the yield is $5.2 \%$ per annum. What is the bond's price? What is the 18-month zero rate? All rates are quoted with semiannual compounding.

Penny Riley

Numerade Educator

03:11

Problem 4

An investor receives $$\$ 1,100$$ in one year in return for an investment of $$\$ 1,000$$ now. Calculate the percentage return per annum with:
(a) Annual compounding
(b) Semiannual compounding
(c) Monthly compounding
(d) Continuous compounding.

Narayan Hari

Numerade Educator

Problem 5

Suppose that risk-free zero interest rates with continuous compounding are as follows:
$$
\begin{array}{cc}
\hline \begin{array}{c}
\text { Maturity } \\
\text { (months) }
\end{array} & \begin{array}{c}
\text { Rate } \\
\text { (\% per annum) })
\end{array} \\
\hline 3 & 3.0 \\
6 & 3.2 \\
9 & 3.4 \\
12 & 3.5 \\
15 & 3.6 \\
18 & 3.7 \\
\hline
\end{array}
$$
Calculate forward interest rates for the second, third, fourth, fifth, and sixth quarters.

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01:14

Problem 6

Assuming that risk-free zero rates are as in Problem 4.5, what is the value of an FRA where the holder will pay LIBOR and receive $4.5 \%$ (quarterly compounded) for a threemonth period starting in one year on a principal of $$\$ 1,000,000$$ ? The forward LIBOR rate for the three-month period is $5 \%$ (quarterly compounded).

Narayan Hari

Numerade Educator

02:20

Problem 7

The term structure of interest rates is upward-sloping. Put the following in order of magnitude:
(a) The 5-year zero rate
(b) The yield on a 5 -year coupon-bearing bond
(c) The forward rate corresponding to the period between 4.75 and 5 years in the future. What is the answer when the term structure of interest rates is downward-sloping?

Jennifer Stoner

Numerade Educator

Problem 8

What does duration tell you about the sensitivity of a bond portfolio to interest rates. What are the limitations of the duration measure?

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Problem 9

What rate of interest with continuous compounding is equivalent to $8 \%$ per annum with monthly compounding?

Nick Johnson

Numerade Educator

01:13

Problem 10

A deposit account pays $4 \%$ per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $$\$ 10,000$$ deposit?

Monica Miller

Numerade Educator

02:40

Problem 11

Suppose that 6-month, 12 -month, 18-month, 24 -month, and 30 -month zero rates are, respectively, $4 \%, 4.2 \%, 4.4 \%, 4.6 \%$, and $4.8 \%$ per annum, with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pay a coupon of $4 \%$ per annum semiannually.

Anand Jangid

Numerade Educator

03:17

Problem 12

A 3-year bond provides a coupon of $8 \%$ semiannually and has a cash price of 104 . What is the bond's yield?

Narayan Hari

Numerade Educator

02:35

Problem 13

Suppose that the 6-month, 12 -month, 18 -month, and 24 -month zero rates are $5 \%, 6 \%$, $6.5 \%$, and $7 \%$, respectively. What is the 2 -year par yield?

Anand Jangid

Numerade Educator

Problem 14

Suppose that risk-free zero interest rates with continuous compounding are as follows:
$$
\begin{array}{cc}
\hline \begin{array}{c}
\text { Maturity } \\
\text { (years) }
\end{array} & \begin{array}{c}
\text { Rate } \\
(\% \text { per annum })
\end{array} \\
\hline 1 & 2.0 \\
2 & 3.0 \\
3 & 3.7 \\
4 & 4.2 \\
5 & 4.5 \\
\hline
\end{array}
$$
Calculate forward interest rates for the second, third, fourth, and fifth years

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Problem 15

Use the risk-free rates in Problem 4.14 to value an FRA where you will pay $5 \%$ (annually compounded) and receive LIBOR for the third year on $$
\text { \$1 million. }$$ The forward LIBOR rate (annually compounded) for the third year is $5.5 \%$.

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Problem 16

A 10 -year $8 \%$ coupon bond currently sells for $$\$ 90$$. A 10 -year $4 \%$ coupon bond currently sells for $$\$ 80$$. What is the 10-year zero rate?

Oluwadamilola Ameobi

Numerade Educator

01:37

Problem 17

Explain carefully why liquidity preference theory is consistent with the observation that the term structure of interest rates tends to be upward-sloping more often than it is downward-sloping.

Lottie Adams

Numerade Educator

01:07

Problem 18

"When the zero curve is upward-sloping, the zero rate for a particular maturity is greater than the par yield for that maturity. When the zero curve is downward-sloping the reverse is true." Explain why this is so.

Sandile Ndlovu

Numerade Educator

01:32

Problem 19

Why are U.S. Treasury rates significantly lower than other rates that are close to risk-free?

Sandile Ndlovu

Numerade Educator

00:47

Problem 20

Why does a loan in the repo market involve very little credit risk?

Amrita Bhasin

Numerade Educator

01:10

Problem 21

Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixed rate of interest.

Brooke Bussoletti

Numerade Educator

02:40

Problem 22

A 5 -year bond with a yield of $7 \%$ (continuously compounded) pays an $8 \%$ coupon at the end of each year.
(a) What is the bond's price?
(b) What is the bond's duration?
(c) Use the duration to calculate the effect on the bond's price of a $0.2 \%$ decrease in its yield.
(d) Recalculate the bond's price on the basis of a $6.8 \%$ per annum yield and verify that the result is in agreement with your answer to (c).

Anand Jangid

Numerade Educator

Problem 23

The cash prices of 6-month and 1-year Treasury bills are 94.0 and 89.0. A 1.5-year Treasury bond that will pay coupons of $$\$ 4$$ every 6 months currently sells for $$\$ 94.84$$. A 2-year Treasury bond that will pay coupons of $$\$ 5$$ every 6 months currently sells for $$\$ 97.12$$. Calculate the 6-month, 1-year, 1.5-year, and 2-year Treasury zero rates.

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Problem 24

"An interest rate swap where 6-month LIBOR is exchanged for a fixed rate of $5 \%$ on a principal of $$\$ 100$$ million for 5 years involves a known cash flow and a portfolio of nine FRAs." Explain this statement.

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04:45

Problem 25

When compounded annually an interest rate is $11 \%$. What is the rate when expressed with (a) semiannual compounding, (b) quarterly compounding, (c) monthly compounding, (d) weekly compounding, and (e) daily compounding.

Willis James

Numerade Educator

00:52

Problem 26

The table below gives Treasury zero rates and cash flows on a Treasury bond. Zero rates are continuously compounded.
(a) What is the bond's theoretical price?
(b) What is the bond's yield assuming it sells for its theoretical price?
$$
\begin{array}{cccc}
\hline \text { Maturity (years) } & \text { Zero rate } & \text { Coupon payment } & \text { Principal } \\
\hline 0.5 & 2.0 \% & \$ 20 & \\
1.0 & 2.3 \% & \$ 20 & \\
1.5 & 2.7 \% & \$ 20 & \\
2.0 & 3.2 \% & \$ 20 & \$ 1,000 \\
\hline
\end{array}
$$

Sheryl Ezze

Numerade Educator

01:21

Problem 27

A 5 -year bond provides a coupon of $5 \%$ per annum payable semiannually. Its price is 104 . What is the bond's yield? You may find Excel's Solver useful.

Gregory Higby

Numerade Educator

Problem 28

Suppose that 3-month, 6-month, 12 -month, 2 -year, and 3-year OIS rates are $2.0 \%, 2.5 \%$, $3.2 \%, 4.5 \%$, and $5 \%$, respectively. The 3-month, 6-month, and 12 -month OISs involve a single exchange at maturity; the 2 -year and 3 -year OISs involve quarterly exchanges. The compounding frequencies used for expressing the rates correspond to the frequency of exchanges. Calculate the OIS zero rates using continuous compounding. Interpolate between continuously compounded rates linearly to determine rates between 6 months and 12 months, between 12 months and 2 years, and between 2 years and 3 years. You may find Excel's Solver useful.

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04:03

Problem 29

An interest rate is quoted as $5 \%$ per annum with semiannual compounding. What is the equivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuous compounding.

Narayan Hari

Numerade Educator

02:35

Problem 30

The 6-month, 12 -month, 18 -month, and 24 -month zero rates are $4 \%, 4.5 \%, 4.75 \%$, and $5 \%$, with semiannual compounding.
(a) What are the rates with continuous compounding?
(b) What is the forward rate for the 6-month period beginning in 18 months?
(c) What is the two-year par yield?

Anand Jangid

Numerade Educator

Problem 31

Suppose the risk-free rates are as in Problem 4.30. What is the value of an FRA where the holder pays LIBOR and receives $7\%$ (semiannually compounded) for a six-month period beginning in 18 months? The current forward rate for this period is $6 \%$ (semiannually compounded) and the principal is $$\$ 10$$ million.

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Problem 32

The following table gives the prices of bonds:
$$
\begin{array}{cccc}
\hline \text { Bond principal }(\$) & \text { Time to maturity (years) } & \text { Annual coupon }^*(\$) & \text { Bond price }(\$) \\
\hline 100 & 0.50 & 0.0 & 98 \\
100 & 1.00 & 0.0 & 95 \\
100 & 1.50 & 6.2 & 101 \\
100 & 2.00 & 8.0 & 104 \\
\hline
\end{array}
$$
Half the stated coupon is assumed to be paid every six months.
(a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months.
(b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 months to 24 months?
(c) What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual coupon payments?
(d) Estimate the price and yield of a 2 -year bond providing a semiannual coupon of $7 \%$ per annum.

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Problem 33

Portfolio A consists of a l-year zero-coupon bond with a face value of $$\$ 2,000$$ and a 10-year zero-coupon bond with a face value of $$\$ 6,000$$. Portfolio B consists of a 5.95 -year zero-coupon bond with a face value of $$\$ 5,000$$. The current yield on all bonds is $10 \%$ per annum.
(a) Show that both portfolios have the same duration.
(b) Show that the percentage changes in the values of the two portfolios for a $0.1 \%$ per annum increase in yields are the same.
(c) What are the percentage changes in the values of the two portfolios for a $5 \%$ per annum increase in yields?

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04:33

Problem 34

Verify that DerivaGem agrees with the price of the bond in Section 4.6. Test how well DV01 predicts the effect of a 1-basis-point increase in all rates. Estimate the duration of the bond from DV01. Use DV01 and Gamma to predict the effect of a 200-basis-point increase in all rates. Use Gamma to estimate the bond's convexity.

Breanna Ollech

Numerade Educator