Individual HW 1 — Bond Pricing & Risk Attribution

Due: April 17 (Week 3) at the start of class

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1. Plot the US yield curve including the Fed Funds Target, 3-month T-bill, 1-, 2-, 5-, 10-, and 30-year bonds. Choose 3 of the following dates: (a) Today, (b) When your mom or dad was born, (c) When you were born, (d) This day in 2004, (e) This day in 2001, (f) This day in 1990.

2. Suppose a 3-year bond has a face value of 1000 and annual coupon payments of 80.

(a) If the bond sells at $1000, what is YTM?

(b) If the bond sells at $900, what is YTM?

(d) If the YTM is higher than the coupon rate, then what must be true about the price?

3. (a) Suppose that a 3-year zero-coupon bond has a YTM of 5%. What is the bond’s current price?

(b) Next year, the YTM changes to 7%. What is the price in that year? What is the realized (holding period) return over the one year period?

(c) What if the YTM in year 1 had remained 5%? What would be the price that year? What would be the realized return over the one year period?

4. You are holding a 3-year bond with coupon rate 10%. Coupon payments are annual and par values are 100. Spot rates are: r(1)=5%, r(2)=6%, r(3)=6.5%.

(a) Determine the bond’s price and YTM.

(b) Determine as many forward rates as you can, based on the spot rates above.

5. Relative value analysis based on relative richness or cheapness, usually expressed in spreads.

(a) Spot rates of 1-, 2-, and 3-year US Treasury are all 4%. What is the price of a 3-year risk-free bond with coupon rate 10%? Coupon payments are annual and par values are 100.

(b) Compute the yield to maturity on the risk-free bond in (a). Because the term structure is flat, the yield equals the spot rates so just show the formula.

(c) Suppose Apple’s 3-year bond with coupon rate 10% is traded at 110.69. Is this bond more or less expensive than a risk-free bond? Why do you think this makes sense (or not)?

(d) Compute the spread s of Apple’s bond such that 110.69 = 10/(1.04+s) + 10/(1.04+s)² + 110/(1.04+s)³. What is s? What does the sign of s represent?

6. Compute the Macaulay duration and modified duration of a bond with the following assumptions: coupon rate is 8%, maturity is 5 years, and the yield to maturity is 6%. What is the expected effect of a 1% increase in the bond yield on the bond price?

Synthesis Question

7. Client Question — Positioning a Bond Portfolio for the Current Rate Environment

You are a fixed income analyst at a wealth management firm. A client comes to you and says: “I keep hearing that the Fed is done cutting rates and may even need to raise them again if inflation stays sticky. I have $1 million in a bond portfolio and I’m worried. What should I do?”

The current spot rate curve is: r(1) = 4.3%, r(2) = 4.2%, r(3) = 4.0%, r(5) = 3.9%, r(10) = 4.1%.

Consider two bonds, both with face value $1,000 and annual coupon payments:

Bond A: 2-year US Treasury, 4% coupon rate
Bond B: 10-year investment-grade corporate bond (BBB-rated), 5.5% coupon rate, trading at a credit spread of 150bp over Treasuries

(a) Notice that the yield curve above is slightly inverted at the short end and then steepens. Compute the 1-year forward rate f(1,2). What does this tell you about where the market expects the 1-year rate to be next year? Is the market pricing in rate hikes or cuts?

(b) Price Bond A using the spot rates above. Compute its approximate modified duration. If rates rise by 75bp across the curve, estimate the percentage price decline.

(c) For Bond B, compute its yield to maturity (YTM = Treasury spot rate + 150bp spread at each maturity) and approximate modified duration. If rates rise by 75bp and credit spreads widen by 50bp (as often happens when the economy slows), estimate the total percentage price decline.

(d) Your client’s current portfolio has an average duration of 6 years. Based on your analysis, would you recommend shortening or lengthening duration given the risk of rising rates? Should the client be concerned about credit spread widening in addition to rate moves?

(e) Write a 1-paragraph client memo (4–6 sentences) explaining your portfolio recommendation. Reference the shape of the yield curve, what the forward rates imply about market expectations, the difference in interest rate risk (duration) between short and long bonds, and the additional risk from credit spreads on corporate bonds. Use plain language — your client is a successful entrepreneur, not a bond trader.