Derivatives Markets

Robert L. McDonald

Chapter 7 Interest Rate Forwards and Futures - all with Video Answers

Educators

Chapter Questions

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Problem 1

Suppose you observe the following zero-coupon bond prices per $\$ 1$ of maturity payment: $0.96154(1-\text { year ) }, 0.91573 \text { (2-year), } 0.87630 \text { (3-year), } 0.82270(4-\text { year ) }$ 0.77611 (5-year). For cach maturity year compute the zero-coupon bond yields (effective annual and continuously compounded), the par coupon rate, and the I-ycar implied forward rate.

Rashmi Sinha

Numerade Educator

01:18

Problem 2

Using the information in the previous problem, find the price of a 5 -year coupon bond that has a par payment of $\$ 1,000.00$ and annual coupon payments of $\$ 60.00$.

Narayan Hari

Numerade Educator

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Problem 3

Suppose you observe the following effective annual zero-coupon bond yields:
0.030 (1-year), 0.035 (2-year), 0.040 (3-year), 0.045 (4-year), 0.050 ( 5 -year). For cach maturity year compute the zero-coupon bond prices, continuously compounded zero-coupon bond yiclds, the par coupon rate, and the 1-year implied forward rate.

Rashmi Sinha

Numerade Educator

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Problem 4

Suppose you observe the following 1-year implied forward rates: 0.050000 (1year), 0.034061 (2-year), 0.036012 (3-year), 0.024092 (4-year), 0.001470 year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yiclds, and the par coupon rate.

Rashmi Sinha

Numerade Educator

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Problem 5

Suppose you observe the following continuously compounded zero-coupon bond yields: 0.06766 (1-year), 0.05827 ( 2 -year), 0.04879 (3-year), 0.04402 (4-year). 0.03922 ( ( 5 -year). For each maturity year compute the zero-coupon bond prices, effective annual zero-coupon bond yields, the par coupon rate, and the 1 -year implied forward rate.

Rashmi Sinha

Numerade Educator

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Problem 6

Suppose you observe the following par coupon bond yields: 0.03000 (1-year), 0.03491 (2-year), 0.03974 (3-year), 0.04629 (4-year), 0.05174 ( 5-year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the 1-year implied forward rate.

Rashmi Sinha

Numerade Educator

01:32

Problem 7

Using the information in Table 7.1
a. Compute the implied forward rate from time 1 to time 3 .
b. Compute the implied forward price of a par 2 -year coupon bond that will be issued at time 1.

Adriano Chikande

Numerade Educator

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Problem 8

Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that will guarantee a $6 \%$ effective annual interest rate for 1 year on $\$ 500,000.00 .$ On the date you borrow the $\$ 500,000.00,$ the actual interest rate is
$5 \% .$ Determine the dollar settlement of the FRA assuming
a. Settlement occurs on the date the loan is initiated.
b. Settlement occurs on the date the loan is repaid.

Taylor Jordan

Numerade Educator

04:00

Problem 9

Using the same information as the previous problem, suppose the interest rate on the borrowing date is $7.5 \%$. Determine the dollar settlement of the FRA assuming
a. Settlement occurs on the date the loan is initiated.
b. Settlement occurs on the date the loan is repaid.

Ankit Gupta

Numerade Educator

03:17

Problem 10

Use the following zero-coupon bond prices to answer the next three questions:
$$\begin{array}{cl}\begin{array}{c}\text { Days to } \\\text { Maturity }\end{array} & \begin{array}{l}\text { Zero-Coupon } \\\text { Bond Price }\end{array} \\\hline 90 & 0.99009 \\180 & 0.97943 \\270 & 0.96525 \\360 & 0.95238\end{array}$$
What is the rate on a synthetic FRA for a 90 -day loan commencing on day $90 ?$ A 180-day loan commencing on day 90? A 270-day loan commencing on day 90?

Narayan Hari

Numerade Educator

01:35

Problem 11

Use the following zero-coupon bond prices to answer the next three questions:
$$\begin{array}{cl}\begin{array}{c}\text { Days to } \\\text { Maturity }\end{array} & \begin{array}{l}\text { Zero-Coupon } \\\text { Bond Price }\end{array} \\\hline 90 & 0.99009 \\180 & 0.97943 \\270 & 0.96525 \\360 & 0.95238\end{array}$$
What is the rate on a synthetic FRA for a 180 -day loan commencing on day $180 ?$ Suppose you are the counterparty for a borrower who uscs the FRA to hedge the interest rate on a $\$ 10 \mathrm{m}$ loan. What positions in zero-coupon bonds would you use to hedge the risk on the FRA?

Neel Faucher

Numerade Educator

01:35

Problem 12

Use the following zero-coupon bond prices to answer the next three questions:
$$\begin{array}{cl}\begin{array}{c}\text { Days to } \\\text { Maturity }\end{array} & \begin{array}{l}\text { Zero-Coupon } \\\text { Bond Price }\end{array} \\\hline 90 & 0.99009 \\180 & 0.97943 \\270 & 0.96525 \\360 & 0.95238\end{array}$$
Suppose you are the counterparty for a lender who enters into an FRA to hedge the lending rate on $\$ 10 \mathrm{m}$ for a 90 -day loan commencing on day $270 .$ What positions in zero-coupon bonds would you use to hedge the risk on the FRA?

Neel Faucher

Numerade Educator

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Problem 13

Using the information in Table 7.1 , suppose you buy a 3 -year par coupon bond and hold it for 2 years, after which time you sell it. Assume that interest rates are certain not to change and that you reinvest the coupon received in year 1 at the 1-year rate prevailing at the time you receive the coupon. Verify that the 2 -year retura on this investment is $6.5 \%$.

Rashmi Sinha

Numerade Educator

07:43

Problem 14

As in the previous problem, consider holding a 3 -year bond for 2 years. Now suppose that intercst rates can change, but that at time 0 , the rates in Table 7.1 prevail. What transactions could you undertake using forward rate agreements to guarantee that your 2 -year return is $6.5 \% ?$

John Lee

Numerade Educator

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Problem 15

Consider the implied forward rate between year 1 and year $2,$ based on Table 7.1
a. Suppose that $r_{0}(1,2)=6.8 \%$. Show how buying the 2 -year zero-coupon bond and borrowing at the 1 -year rate and implied forward rate of $6.8 \%$ would earn you an arbitrage profit.
b. Suppose that $r_{0}(1,2)=7.2 \%$. Show how borrowing the 2 -year zerocoupon bond and lending at the 1 -year rate and implied forward rate of $7.2 \%$ would earn you an arbitrage profit.

Rashmi Sinha

Numerade Educator

05:25

Problem 16

Suppose the September Eurodollar futures contract has a price of $96.4 .$ You plan to borrow $\$ 50 \mathrm{m}$ for 3 months in September at LIBOR, and you intend to use the Eurodollar contract to hedge your borrowing rate.
a. What rate can you secure?
b. Will you be long or short the Eurodollar contract?
c. How many contracts will you enter into?
d. Assuming the true 3 -month LIBOR is $1 \%$ in September, what is the settlement in dollars at expiration of the futures contract? (For purposes of this question, ignore daily marking-to-market on the futures contract.)

Manasvee Singh

Numerade Educator

01:54

Problem 17

A lender plans to invest $\$ 100 \mathrm{m}$ for 150 days, 60 days from today. (That is, if today is day $0,$ the loan will be initiated on day 60 and will mature on day $210 .$ ) The implied forward rate over 150 days, and hence the rate on a 150 -day $\mathrm{FRA}$, is $2.5 \% .$ The actual interest rate over that period could be either $2.2 \%$ or $2.8 \%$
a. If the interest rate on day 60 is $2.8 \%$, how much will the lender have to pay if the FRA is settled on day $60 ?$ How much if it is settled on day $210 ?$
b. If the interest rate on day 60 is $2.2 \%$, how much will the lender have to pay if the FRA is settled on day $60 ?$ How much if it is settled on day $210 ?$

Catt Huth

Numerade Educator

05:25

Problem 18

Consider the same facts as the previous problem, only now consider hedging with the 3 -month Eurodollar futures. Suppose the Eurodollar futures contract that matures 60 days from today has a price on day 0 of 94
a. What issues arise in using the 3 -month Eurodollar contract to hedge a 150 -day loan?
b. If you wish to hedge a lending position, should you go long or short the contract?
c. What 3 -month LIBOR is implied by the Eurodollar futures price? Approximately what lending rate should you be able to lock in?
d. What position in Eurodollar futures would you use to lock in a lending rate? In doing this, what assumptions are you making about the relationship between 90 -day LIBOR and the 150 -day lending rate?

Manasvee Singh

Numerade Educator

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Problem 19

Consider the bonds in Example $7.8 .$ What hedge ratio would have exactly hedged the portfolio if interest rates had decreased by 25 basis points? Increased by 25 basis points? Repeat assuming a 50 -basis-point change.

Majid Borumand

Numerade Educator

04:33

Problem 20

Compute Macaulay and modified durations for the following bonds:
a. A 5-year bond paying annual coupons of 4.432\% and sclling at par.
b. An 8 -year bond paying semiannual coupons with a coupon rate of $8 \%$ and a yield of $7 \%$
c. A 10 -year bond paying annual coupons of $6 \%$ with a price of $\$ 92$ and maturity value of $\$ 100$.

Breanna Ollech

Numerade Educator

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Problem 21

Consider the following two bonds which make semiannual coupon payments: a 20 -year bond with a $6 \%$ coupon and $20 \%$ yield, and a 30 -year bond with a $6 \%$ coupon and a $20 \%$ yield.
a. For each bond, compute the price value of a basis point.
b. For each bond, compute Macaulay duration.
c. "For otherwise identical bonds, Macaulay duration is increasing in time to maturity." Is this statement always true? Discuss.

Rashmi Sinha

Numerade Educator

00:33

Problem 22

An 8 -year bond with $6 \%$ annual coupons and a $5.004 \%$ yield sells for $\$ 106.44$ with a Macaulay duration of $6.631864 .$ A 9 -year bond has $7 \%$ "annual coupons with a $5.252 \%$ yield and sells for $\$ 112.29$ with a Macaulay duration of 7.098302 You wish to duration-hedge the 8 -year bond using a 9 -year bond. How many 9-year bonds must we short for every 8 -year bond?

Trinity Steen

Numerade Educator

01:23

Problem 23

A 6 -year bond with a $4 \%$ coupon sells for $\$ 102.46$ with a $3.5384 \%$ yield. The conversion factor for the bond is $0.90046 .$ An 8 -year bond with $5.5 \%$ coupons sells for $\$ 113.564$ with a conversion factor of $0.9686 .$ (All coupon payments are semiannual.) Which bond is cheaper to deliver given a T-note futures price of $113.81 ?$

Jennifer Stoner

Numerade Educator

04:33

Problem 24

a. Compute the convexity of a 3 -year bond paying annual coupons of $4.5 \%$ and selling at par.
b. Compute the convexity of a 3 -year $4.5 \%$ coupon bond that makes semiannual coupon payments and that currently sells at par.
c. Is the convexity different in the two cases? Why?

Breanna Ollech

Numerade Educator

04:33

Problem 25

Suppose a 10-year zero coupon bond with a face value of $\$ 100$ trades at $\$ 69.20205$
a. What is the yield to maturity and modified duration of the zero-coupon bond?
b. Calculate the approximate bond price change for a 50 basis point increase in the yield, based on the modified duration you calculated in parta). Also calculate the exact new bond price based on the new yield to maturity.
c. Calculate the convexity of the 10 -year zero-coupon bond.
d. Now use the formula (equation 7.15 ) that takes into account both duration and convexity to approximate the new bond price. Compare your result to that in part b).

Breanna Ollech

Numerade Educator