Assignment
Problem 1: Reinvestment Risk
Fisher is a bond portfolio manager at GS Capital Partners. His fund recently purchased $500 million principal value of 2-year 10% coupon bonds at par value. Assume that the term structure of interest rates is flat.
Under each of the following 3 scenarios, calculate the actual rate of return earned on this bond investment, assuming GS Capital Partners holds the bonds until maturity.
A. Assume the term structure instantaneously shifts (right after his purchase) from 10% to 15%. Compute the new price of the bonds. Assume GS Capital Partners holds the bonds until maturity. Compute the holding period return for the bonds assuming the term structure remains at 15% over the remainder of the holding period.
B. Repeat part A assuming an instantaneous shift from 10% to 5%.
C. Explain the roles of interest rate risk and reinvestment rate risk in this example.
Problem 4: Bond Price Volatility
Consider the following four bonds:
i. 5 years to maturity, 0% coupon
ii. 20 years to maturity, 0% coupon
iii. 5 years to maturity, 5% coupon
iv. 20 years to maturity, 5% coupon
A. Assuming the term structure is flat at 5%, compute the price of each bond i. to iv.
B. What is the current yield for each bond from Part A?
C. Compute the new price and the percentage price change for each bond if the term structure instantaneously shifts from 5% to 5.5%.
D. Compute the new price and the percentage price change for each bond if the term structure instantaneously shifts from 5% to 4.5%.
E. How does the bond coupon rate relate to the magnitude of the percentage price change in C. and D.?
F. How does the term-to-maturity relate to the magnitude of the percentage price change in C. and D.?
G. Are the magnitudes of the percentage price changes in C. and D. different for each bond? If so, what contributed to that difference?
Problem 5: Interest Rate Risk Consider a 10-year zero coupon bond.
A. Using the bond pricing equation, P(y), illustrate the derivation (using calculus) of this bond's modified duration. Note: the correct answer is an equation, not a number.
B. Assuming the yield-to-maturity is 5%, compute the bond's modified duration.
C. Use your answer to part B to estimate the price of the bond and the percentage price change if interest rates instantaneously rise to 6%.
D. Use your answer to part B to estimate the price of the bond and the percentage price change if interest rates instantaneously decline to 4%.
E. How do the estimates from C. and D. compare to the true prices using the bond pricing equation P(y) if yields change to 6% and 4%? Is the estimate more accurate for part C. or D., and why?
Problem 6: Bond Values Through Time
Consider a 17-year zero coupon bond.
A. Plot the value of the bond P(y) for yields ranging from 0% to 30%.
B. Add to the graph the price of the same bond after 12 years have passed (i.e. it will be a 5-year zero coupon bond).
C. Explain what happens through time (at any given yield) to:
a. The bond's price
b. The bond's interest rate risk
c. The bond's convexity.
Clarification: "Explain" means "using your graph, explain to me how I can understand your answer by looking at the graph."
Problem 7: Bootstrapping The Yield Curve
Based on the U.S. Treasury bond information below, answer the following series of questions.
|
Bond
|
Coupon Rate
|
Maturity (years)
|
Price
|
|
1
|
3%
|
0.5
|
100.000
|
|
2
|
4%
|
1.0
|
100.500
|
|
3
|
3%
|
1.5
|
98.750
|
|
4
|
0%
|
2.0
|
92.000
|
A. From the given information, compute the 6-month, 1-year, 1.5-year, and 2-year spot rates. Do not round excessively: use at least 5 decimal places (for example 12.345%).
B. Is the term structure inverted, normal, or flat?
C. Use the liquidity preference theory to explain the shape of the term structure (your answer to B).
Problem 8: Forward Rates
According to the pure expectations theory of the term structure, what are the market's expectations of the short rate for the next three semi-annual periods (meaning the 6-month periods beginning in 6 months, 12 months, and 18 months), i.e. what are the implied forward rates? As in problem 7, do not round excessively.
|
Maturity (years)
|
Spot Rate
|
|
0.5
|
5.000%
|
|
1.0
|
4.750%
|
|
1.5
|
4.600%
|
|
2.0
|
4.400%
|