Interest Rate Markets Spring 2011
Problem Set No. 4 Due: Friday, June 24, at 5:00 PM. Please note: Email your solutions to your new TA, Tal Moshe, at:
[email protected]
Emails received after 5:00 PM on Friday will not be considered.
1. Suppose that you are reviewing a price sheet for bonds and see the following f ollowing reported prices
(per $100 par value). You observe what seem to be several errors. Without calculating the price of each bond, indicate which bonds seem to be reported r eported incorrectly, and explain why.
Bond
Price
Coupon Rate (%)
Required Yield (%)
A
90
6
9
B
96
9
8
C
110
8
6
D
105
0
5
E
107
7
9
F
100
6
6
2. Suppose that you purchased a debt obligation three years ago at its par value of $100,000
and nine years remaining to maturity. The market price of this debt obligation today is $90,000. What are some reasons why the price of this debt obligation could have declined from the time you purchased it three years ago?
3. Suppose that an investor with a five-year investment horizon is considering purchasing a
seven-year 9% coupon bond selling at par. The investor expects that he can reinvest the coupon payments at an annual interest rate of 9.4% and that at the end of the investment horizon twoyear bonds will be selling to offer a yield to maturity of 11.2%. 1 1.2%. What is the total percentage return for this bond?
4. Calculate the yield to maturity of a portfolio that includes the following four bonds:
Bond
Price
Coupon Rate (%)
Years to Maturity
A
884.20
7
5
B
948.90
8
7
C
967.70
9
4
D
456.39
0
10
5. Consider the following bond:
Coupon rate = 11% Maturity = 18 years Par value = $1,000 First par call in 13 years Only put date is in five years and putable at par value Suppose that the market price for this bond $1,169. Calculate the following three different measures of yields: yield to maturity, yield to first par call, and yield to put.
6. Following are three government bonds maturing at the end of Year 3. Their prices today and
cash flows at the end of each year are as follows:
Bond
Price
CF Year 1
CF Year 2
CF Year 3
A
94
4
4
104
B
100
6
6
106
C
?
12
12
112
Given that you can buy or sell short any amount of these bonds (including fractions), what is the no-arbitrage price of bond C? (Hint: How would you replicate bond C using different amounts and/or combinations of bonds A and B? Use this combination to price bond C.)
7. Consider the following four government bonds maturing at or before the end of Year 4. Their
prices today and cash flows at the end of each year are as follows:
Bond
Price
CF Year 1
CF Year 2
CF Year 3
CF Year 4
A
127.20
10
10
110
110
B
18.80
0
0
0
0
C
55.80
0
30
10
10
D
27.80
10
10
10
10
In a market with no-arbitrage opportunities, what would be the price of a new bond E that matures at the end of year 4 with the following cash flow: 50, 20, 20, and 150 on years 1 through 4, respectively. 8. Consider the following information about bonds A, B, C, and D:
Bond
Price
A
Csah Flow at the end of: Year 1
Year 2
Year 3
96
100
0
0
B
91
0
100
0
C
80
0
0
100
D
?
10
10
110
Assume that bond prices admit no arbitrage strategies. What is the modified duration of bond D?
9. You observe the following 4 default-free non-callable Treasury securities.
Bond
Price
Coupon Rate (%)
Yield
Years to Maturity
A
97.2006
0.000
5.76%
0.50
B
94.4337
0.000
5.81%
1.00
C
100.1265
6.250
6.18%
2.00
D
100.3579
6.375
6.18%
2.00
A. Solve for the 0.5, 1.0, 1.5, and 2.0 year spot or zero-coupon discount rates. (Notice: there is no bond maturing in 1.5 years). (Hint: Think how to artificially create a cash flow that has payments only in 0.5,1.0, and 1.5 years. Recall you can buy or sell short any amount of bonds, including fractions of bonds). B. Price a 1.5-year bond with a coupon of 6%. C. Compute the implied one-period ahead forward rates f (0,1), f (1,2), f (2,3) and f (3,4).
10. You observe the following default-free no callable bonds:
Maturity
Modified
Macaulay
Duration
Duration
94.434
0.97
1.00
1.417
6.180%
100.130
1.85
1.91
4.426
6.375%
6.340%
100.094
2.69
2.78
8.895
5
6.625%
6.560%
100.273
4.20
4.34
21.278
10
7.000%
6.790%
101.507
7.14
7.38
64.694
30
6.000%
6.970%
87.865
12.84
13.29
264.667
Coupon
Yield
Price
1
0.000%
5.810%
2
6.250%
3
(yrs)
Convexity
You are a manager of a $100m pension fund and can only invest in the bonds listed above. You want a portfolio with Macaulay duration of 5 years. There are many ways using these securities to create portfolios with Macaulay durations of 5 years. Create two different portfolios both having Macaulay durations of 5 years. Explain your steps. Which portfolio would you prefer? Why? 11. Consider the data from the previous question again. You are a bond trader and want to do
the following “yield curve” trade: Go long the 30-year bond and go short the 2-year bond. A. Suppose you want to buy $10 million face value of the 30-year bond. How much of the 2year bond do you need to short in order to have a modified duration of zero? What is the convexity of this portfolio? B. What is this trade betting on? C. Using modified duration only, approximate the amount of money you can make or lose with respect to an immediate 50 basis point (i.e., 0.5%) movement of the 2- year and 30-year rates? Note that there are four cases to consider (i.e., each bond can experience a different direction of change of 50 basis points to its rate). Interpret your results in light of your answer to part B.
