Fin 473 Excercise

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b. Repeat part a. for Macauley duration.

c. Give a brief explanation of your findings.

[pic 3]

4. True-False: Suppose the spread between the yields at two different maturities narrows. Then it is profitable to have a short position in the bond with lower maturity and a long position in the bond with higher maturity. Explain your answer.

ANSWER:

It depends. In the case of upward sloping yield curve, increasing short yields and decreasing long yields, then yes. In general this does not have to be the case.

5. Consider three zero-coupon bonds with 2, 10, and 30 years to maturity and with required yields 4%, 7%, and 9%, respectively.

a. Calculate the price and modified duration of the three bonds using annual compounding.

b. How can a trader use convexity to set up a profitable trade in case she expects the yield curve to move in a parallel way? (Hint: assume a short position of 100 10 year bonds and then solve the system of two linear equations in two unknowns)

c. Using Excel, produce a print-out of a table with the trade positions values and the trade profit generated by parallel yield curve shifts of 250 bp in steps of 50 bp.

d. Now assume that the yield curve does not shift in a parallel way but instead flattens for the shorter maturities. More precisely assume y2 increases 1%, y10 decreases 1% and y30 remains unchanged. What are the trade positions values and the trade profit in this case?

ANSWER:

a. P2 = $92.46, P10 = $50.83, P30 = $7.54. Modified duration: MD2 = 1.92, MD10 = 9.35, MD30 = 27.53.

b. Short 100 10Y bonds. For a duration neutral, self-financing portfolio we must have

100P10 − x2P2 + x30P30 = 0

100MD10P10 − x2MD2P2 − x30MD30P30 = 0.

Note that for the second equation we use that the modified (percentage) duration of a portfolio is the weighted sum of the individual modified durations. Solving the above system gives x2 = 39.04 and

x30 = 195.56.

c. [pic 4]

d. The trade loses some $568.95.

6. True-False. At each date an investor can choose between a one year and a two year zero-coupon bond. Assume that she wants to make a two year horizon investment and that she expects the one year rate, one year from now, to be higher than the current one year rate. Then she should invest

in one year bonds twice (now and in one year) rather than directly in the two-year bond. Explain your answer.

ANSWER: It depends. If her expectations of the one year one year forward rate is higher that the market expectations, then yes, otherwise no.

- Assume the following yield curve for zero-coupon bonds:

Maturity

YTM

1 year

2 years

3 years

4 years

5 years

5%

6%

7%

8%

9%

- What is the Macaulay duration of each of the bonds?

- Assume your HD is three years and you want to buy bonds with one- and four-year maturities. What percentage investment should be made in each to assure a fully immunized portfolio?

a. For zero-coupon bond, Macualey’s duration is equal to the bond’s maturity.

b. Bond immunization requires constructing a bond portfolio with a duration equal to your horizon date of 3 years: HD = Dp:

HD = Dp

3 = w1 D1 + w2 D2

3 = w1 D1 + (1-w1) D2

3 = w1 (1) + (1-w1)