PART
ONE
Exercises
Chapter 1Bond Prices, Discount Factors, and Arbitrage
1.1Write down the cash flow dates and the cash flows of $1,000 face value of the U.S. Treasury 4s of April 30, 2003, issued on April 30, 2001.
1.2Here is a list of bond transactions on May 15, 2001. For each transaction list the transaction price.
BondPriceFace Amount
10.75s of 5/15/2003112-25/8$10,000
4.25s of 11/15/200399-14+$1,000
7.25s of 5/15/2004107-4$1,000,000
1.3Use this list of Treasury bond prices as of May 15, 2001, to derive the discount factors for cash flows to be received in 6 months, 1 year, and 1.5 years.
BondPrice
7.5’s of 11/15/2001101-253/4
7.5’s of 5/15/2001103-1215/16
11.625’s of 11/15/2002110-211/4
1.4Suppose there existed a Treasury issue with a 7.5% coupon maturing on November 15, 2002. Using the discount factors derived in question 1.3, what would be the price of the 7.5s of November 15, 2002?
1.5Say that the 7.5s of November 15, 2002, existed and traded at a price of 105 instead of the price derived in question 1.4. How could one earn an arbitrage profit by trading the 7.5s of November 15, 2002, and the three bonds listed in question 1.3? Using the prices listed in question 1.3, how much arbitrage profit is available in this trade?
1.6Consider the following three bonds and bond prices:
BondPrice
0s of 5/15/200296-12
7.5s of 5/15/2002103-1215/16
15s of 5/15/2002106-2
Do these prices make sense relative to one another? Why or why not?
Chapter 2Bond Prices, Spot Rates, and Forward Rates
2.1You invest $100 for two years at 5%, compounded semiannually. How much do you have at the end of the two years?
2.2You invested $100 for three years and, at the end of those three years, your investment was worth $120. What was your semiannually compounded rate of return?
2.3Using your answers to question 1.3, derive the spot rates for 6 months, 1 year, and 1.5 years.
2.4Derive the relationship between discount factors and forward rates.
2.5Using your answers to either question 1.3 or 2.3, derive the six-month rates for 0 years, .5 years, and 1 year forward.
2.6Are the forward rates from question 2.5 above or below the spot rates of question 2.3? Why is this the case?
2.7Question 1.3 gives the price of the 7.5s of November 15, 2001, and the 7.5s of May 15, 2002. The answer to question 1.4 gives the price of the 7.5s of November 15, 2002. Are these prices rising, falling, or both rising and falling with maturity? Why?
Chapter 3Yield-to-Maturity
3.1On May 15, 2001, the price of the 11.625s of November 15, 2002, was 110-214/4. Verify that the yield-to-maturity was 4.2139%. Explain this yield relative to the spot rates from question 2.3.
3.2On May 15, 2001, the price of the 6.75s of May 15, 2005, was 106-211/8. Use a calculator or spreadsheet to find the yield of the bond.
3.3Consider a 10-year par bond yielding 5%. How much of the bond’s value comes from principal and how much from coupon payments? How does your answer change for a 30-year par bond yielding 5%?
3.4Why would anyone buy a bond selling at a premium when after holding that bond to maturity it will be worth only par?
3.5On May 15, 2001, the price and yield of the 11.625s of November 15, 2002, were 110-211/4 and 4.2139%, respectively. Say that on November 15, 2001, the yield of the bond is still 4.2139%. Calculate the annualized return on the bond over that six-month period.
3.6Consider the following bond yields on May 15, 2001:
BondYield
5.25s of 8/15/20034.3806
5.75s of 8/15/20034.3838
11.125s of 8/15/20034.4717
Do these yields make sense relative to one another? Assume that the yield curve on May 15, 2001, was upward-sloping.
3.7A 60-year-old retired woman is considering purchasing an annuity that pays $25,000 every six months for the rest of her life. Assume that the term structure of semiannually compounded rates is flat at 6%.
a.If the annuity cost $575,000 and the woman expects to live another 25 years, will she purchase the annuity? What if she expects to live another 15 years?
b.If law prohibits insurance companies from charging a different annuity price to men and to women and if everyone expects women to live longer than men, what would happen in the annuity market?
3.8A state lottery advertises a jackpot of $1,000,000. In the fine print it is written that the winner receives 40 annual payments of $25,000. If the term structure is flat at 6%, what is the true value of the jackpot?
Chapter 4Generalizations and Curve Fitting
4.1The Treasury 5s of February 15, 2011, which were issued on February 15, 2001, are purchased on May 15, 2001, for a quoted price of 96-231/2. What is the invoice price on $100,000 face amount?
4.2Bank 1 offers 4.85% compounded monthly for a one-year investment. Bank 2 offers 5% compounded semiannually. Which bank offers the better investment?
4.3Using simple interest and the actual/360 convention, how much interest is owed on a $1,000,000 loan from April 24, 2001, to May 2, 2001?
4.4Is the discount function in Figure 1.2 concave or convex?
4.5The following table gives spot rates for four terms:
TermSpot Rate
24.32%
55.10%
105.74%
30 6.07%
Fit a cubic using equation (4.21) through these points. Graph the resulting spot rate function. Does this function seem reasonable? Why or why not?
4.6A trader thinks that the 10.75s of August 15, 2005, are cheap relative to other bonds in that maturity sector. What risk does the trader face by buying that bond in the hope that its price will rise relative to other bonds in the sector? What if the trader buys that bond and sells the 6.5 of August 15, 2005?
4.7What is the .75-year discount factor if the .75-year rate, continuously compounded, is 6%?
Chapter 5One-Factor Measures of Price Sensitivity
The exercises for this chapter are built around a spreadsheet exercise. Set up a column of interest rates from 1.75% to 8.25% in 25 basis point increments. In the next column compute the price of a perpetuity with a face of 100 and a coupon of 5%: 100.05/y where y is the rate in the first column. In the next column compute the price of a one-year bond with a face of 100 and an annual coupon of 5%: 105/(1+y).
5.1Graph the prices of the perpetuity and the one-year bond as a function of the interest rate. Use the graph to determine which security is more sensitive to changes in rates. Use the graph to determine which security is more convex.
5.2On the spreadsheet compute the DV01 of the perpetuity and of the one-year bond numerically for all of the rates in the first column. To compute the DV01 at a rate y, use the prices at the rates y plus 25 basis points and y minus 25 basis points. Do the results match your answer to question 5.1? How can you tell from these results which security has the higher convexity?
5.3A trader buys 100 face of the perpetuity and hedges with the one-year bond. At a yield level of 5% what is the DV01 hedge? Why is the hedge so large? What is the hedge at a yield level of 2.50%? Explain why the hedge changes.
5.4Calculate the duration of the perpetuity and of the one-year bond in the spreadsheet. At a yield level of 5% interpret the duration numbers in the context of a 10 basis point interest rate move for a fixed income portfolio manager.
5.5Compute the convexity of the perpetuity and the one-year bond at all yield levels in the first column of your spreadsheet. To compute the convexity at y, compute the derivative using prices at y plus 25 basis points and at y. Then compute the derivative using prices at y and y minus 25 basis points.
5.6Is the hedged position at a yield level of 5% computed for question 5.3 long or short convexity? First answer intuitively and then calculate the exact answer.
5.7Estimate the price change of the perpetuity from a yield level of 5% to a level of 6% using its duration and convexity at 5%. How does this compare to the actual price change?
Chapter 6Measures of Price Sensitivity Based on Parallel Yield Shifts
6.1Order the following bonds by duration without doing any calculations:
CouponMaturityYield
4.25%11/15/20034.4820%
11.875%11/15/20034.5534%
4.625%5/15/20064.9315%
6.875%5/15/20065.0379%
6.2Try to order the bonds listed in question 6.1 by DV01 without doing any calculations. This is not so straightforward as question 6.1.
6.3Calculate the DV01 and modified duration for each of the following bonds as of May 15, 2001:
CouponMaturityYieldPrice
8.755/15/20205.9653%131-127/8
8.1255/15/20215.9857%124-241/8
Comment on the results.
6.4In a particular trading session, two-year Treasury notes declined by $19 per $1,000 face amount while 30-year bonds fell $11 per $1,000 face amount. What lesson does this session have to teach with respect to the use of yield-based duration to hedge bond positions?
6.5Calculate the Macaulay duration of 30-year and 100-year par bonds at a yield of 6%. Use the results to explain why Treasury STRIPS maturing in 20 to 30 years are in particularly high demand.
6.6Bond underwriters often agree to purchase a corporate client’s new bonds at a set price and then attempt to reoffer the bonds to investors. There can be a few days between the time the underwriter sets the price it will pay and the time it manages to sell all of its client’s bonds. Underwriting fees often increase with the maturity of the bonds being sold. Why might this be so?
Chapter 7Key Rate and Bucket Exposures
The following questions will lead to the design of a spreadsheet to calculate the two- and five-year key rate duration profile of four-year bonds.
7.1Column A should contain the coupon payment dates from .5 to 5 years in increments of .5 years. Let column B hold a spot rate curve flat at 4.50%. Put the discount factors corresponding to the spot rate curve in column C. Price a 12% and a 6.50% four-year bond under this initial spot rate curve.
7.2Create a new spot rate curve, by adding a two-year key rate shift of 10 basis points, in column D. Compute the new discount factors in column E. What are the new bond prices?
7.3Create a new spot rate curve, by adding a five-year key rate shift of 10 basis points, in column F. Compute the new discount factors in column G. What are the new bond prices?
7.4Use the results from questions 7.1 to 7.3 to calculate the key rate durations of each of the bonds.
7.5Sum the key rate durations to obtain the total duration of each bond. Calculate the percentage of the total duration accounted for by each key rate for each bond. Comment on the results.
7.6What would the key rate duration profile of a four-year zero coupon bond look like relative to those computed for question 7.4? How would your answer change for a five-year zero coupon bond?
Chapter 8Regression-Based Hedging
You consider hedging FNMA 6.5s of August 15, 2004, with FNMA 6s of May 15, 2011. Taking changes in the yield of the 6s of May 15, 2011, as the independent variable and changes in the yield of the 6.5s of August 15, 2004, as the independent variable from July 2001 to January 2002 gives the following regression results:
Number of observations 131
R-squared 77.93%
Standard error 4.0861
Regression CoefficientsValuet-Stat
Constant–.7549–2.1126
Change in yield of 6s of 5/15/2011.961921.3399
8.1What is surprising about the regression coefficients?
8.2The DV01 of the 6.50s of August 15, 2004, is 2.796, and the DV01 of the 6s of May 15, 2011, is 7.499. Using the regression results given, how much face value of the 6s of May 15, 2011, would you sell to hedge a $10,000,000 face value position in the 6.50s of August 15, 2004?
8.3How do the regression results given here compare with the regression results in Table 8.1? Explain the differences. How do the regression results given here make you feel about hedging FNMA 6.50s of August 15, 2004, with FNMA 6.5s of May 15, 2011?
Chapter 9The Science of Term Structure Models
9.1A fixed income analyst needs to estimate the price of an interest rate cap that pays $1,000,000 next year if the one-year Treasury rate exceeds 6% and pays nothing otherwise. Using a macroeconomic model developed in another area of the firm the analyst estimates that the one-year Treasury rate will exceed 6% with a probability of 25%. Since the current one-year rate is 5%, the analyst prices the cap as follows:
Comment on this pricing procedure.
9.2The following tree gives the true six-month rate process:
The prices of six-month, one-year, and 1.5-year zeros are 97.5610, 95.0908, and 92.5069. Find the risk-neutral probabilities for the six-month rate process. Assume, as in the text, that the risk-neutral probability of an up move from date 1 to date 2 is the same from both date 1 states. As a check to your work, write down the price trees for the six-month, one-year, and 1.5-year zeros.
9.3Using the risk-neutral tree derived for question 9.2, price $100 face amount of the following 1.5-year collared floater. Payments are made every six months according to this rule: If the short rate on date i is ri, then the interest payment of the collared floater on date i+1 is
In addition, at maturity, the collared floater returns the $100 principal amount.
9.4Using your answers to questions 9.2 and 9.3, find the portfolio of the originally one-year and 1.5-year zeros that replicates the collared floater from date 1, state 1, to date 2. Verify that the price of this replicating portfolio gives the same price for the collared floater at that node as derived for question 9.3.
9.5Using the risk-neutral tree from question 9.2, price $100 notional amount of a 1.5-year participating cap with a strike of 5% and a participation rate of 40%. Payments are made every six months according to this rule: If the short rate on date i is ri, then the cash flow from the participating cap on date i+1 is, as a percent of par,
There is no principal payment at maturity.
Chapter 10The Short-Rate Process and the Shape of the Term Structure
10.1On February 15, 2001, the yields on a 5-year and a 10-year interest STRIPS were 5.043% and 5.385%, respectively. Assuming that the expected yield change of each is zero and that the yield volatility is 95 basis points for both, use equation (10.27) to infer the risk premium in the marketplace. Hint: You will also need equations (6.24) and (6.36).
On May 15, 2001, the yields on a 5-year and 10-year interest STRIPS were 5.099% and 5.735%, respectively. Repeat the preceding exercise.
10.2Describe as fully as possible the qualitative effect of each of these changes on 10- and 30-year par yields.
a.The market risk premium increases.
b.Volatility across the curve increases.
c.Volatility of the 10-year rate decreases while the volatility of the 30-year par rate stays the same.
d.The expected values of future short-term rates fall. Hint: Make assumptions about which future rates change in expected value.
e.The market risk premium falls and the volatility across the curve falls in such a way as to keep the 10-year yield unchanged.
Chapter 11The Art of Term Structure Models: Drift
11.1Assume an initial interest rate of 5%. Using a binomial model to approximate normally distributed rates with weekly time steps, no drift, and an annualized volatility of 100 basis points, what are the two possible rates on date 1?
11.2Add a drift of 20 basis points per year to the model described in question 11.1. What are the two rates now?
11.3Consider the following segment of a binomial tree with six-month time steps. All transition probabilities equal .5.
Does this tree display mean reversion?
11.4What mean reversion parameter is required to achieve a half-life of 15 years?
Chapter 12The Art of Term Structure Models: Volatility and Distribution
12.1The yield volatility of a short-term interest rate is 20% at a level of 5%. Quote the basis point volatility and the Cox-Ingersoll-Ross (CIR) volatility parameter.
12.2You are told that the following tree was built with a constant volatility. All probabilities equal .5. Which volatility measure is, in fact, constant?
12.3Use the closed-form solution in Appendix 12A to compute spot rates of various maturities in the Vasicek model with the parameters =10%, k=.035, =.02, and r0=4%. Comment on the shape of the term structure.
Chapter 13Multi-Factor Term Structure Models
13.1The following trees give the processes for the two factors of a term structure model:
The correlation of the changes in the factors is –.5. Finally, the short-term rate equals the sum of the factors. Derive the two-dimensional tree for the short-term rate.
Chapter 14Trading with Term Structure Models
14.1Question 9.3 required the calculation of the price tree for a collared floater. Repeat this exercise, under the same assumptions, but assuming that the option-adjusted spread (OAS) of the collared floater is 10 basis points.
14.2Using the price trees from questions 9.3 and 14.1, calculate the return to a hedged and financed position in the collared floater from dates 0 to 1 assuming no convergence (i.e., the OAS on date 1 is also 10 basis points). Hint 1: Use all of the proceeds from selling the replicating portfolio to buy collared floaters. Hint 2: You do not need to know the composition of the replicating portfolio to answer this question.
Is your answer as you expected? Explain.
14.3What is the return if the collared floater converges on date 1 so its OAS equals 0 on that date?
Chapter 15Repo
The following data as of May 15, 2001, relates to the old 10-year Treasury bond and the on-the-run 10-year Treasury bond.
Overnight
CouponMaturityYieldPriceRepo RateDV01
5.75%8/15/20105.4709%101-317/83.80%.07273
5%2/15/20115.4346%96-23+0.10%.07343
15.1Calculate the carryover one day for $100 face of each of these bonds and comment on the difference. Note that there are 89 days between May 15, 2001, and August 15, 2001, and 181 days between February 15, 2001, and August 15, 2001.
15.2Calculate the return to an investment in each bond if their respective yields fall by one basis point immediately after purchase.
15.3By how many basis points does the yield spread between the two bonds have to change for the returns to be the same?
15.4Explain whether or not it is likely for the yield spread to move in the direction indicated by your answer to question 15.3. Also explain the conditions under which a one-day investment in the on-the-run 10-year will be superior to an investment in the old 10-year and vice versa.
Chapter 16Forward Contracts
16.1For delivery dates in the near future, the forward prices of zero coupon bonds are above spot prices while the forward prices of coupon bonds are usually below spot prices. Explain.