Stockholm School of Economics in Riga Financial Economics, Spring 2010 Tālis Putniņš

Problem Set VII: Bond pricing, duration, convexity and immunization Exercise 1: Bond pricing Suppose five-year government bonds are selling on a yield of 4% p.a. and have a coupon of 6% p.a. a) Calculate the price of the bonds assuming they are issued by a European government and the coupon payments are made annually. b) Calculate the price of the bonds assuming they are issued by the US Treasury so the coupon payments are made semi-annually and the given yield refers to a semiannually compounded rate.

Exercise 2: Bond pricing Two Treasury bonds have face values of $100,000 and pay coupons at the rate of 10%, semi-annually. Bond P has four years to maturity and bond Q has eight years to maturity. a) If the yield on the bonds is 7.5% p.a., what are the prices of the two bonds? b) If the yield rises to 12% p.a., what are the prices of the two bonds? c) What do the prices illustrate about the relations between price, yield, coupon rate and maturity?

Exercise 3: Bond pricing Which security has a higher effective annual interest rate? a) A 3-month Treasury note selling at $97,645 with par value $100,000. b) A coupon bond selling at par and paying a 10% coupon semi-annually.

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Exercise 4: Bond pricing Consider a bond that pays a coupon rate of 10% p.a. semi-annually when the market interest rate is only 4% per half year. The bond has 3 years to maturity. a) Find the bond’s price today and in 6 months from now after the next coupon is paid (assuming interest rates do not change). b) What is the total (6 month) rate of return on the bond if you buy it today and sell it in 6 months? c) Repeat (b), but instead of interest rates remaining unchanged, in 6 months time they have fallen to 3% per half year.

Exercise 5: Bond pricing A $1000 face value bond with a coupon rate of 7% makes semi-annual coupon payments on January 15 and July 15 of each year. The Wall Street Journal reports the asked price for the bond on 30 January at 100:02. What is the invoice price of the bond? The coupon period has 182 days.

Exercise 6: Bond pricing Brengulis Corp. issues two bonds with 20-year maturities, face values of $1000 and annual coupons. Both bonds are callable at $1050. The first bond is issued at a deep discount with a coupon rate of 4% and a price of $580 to yield 8.4%. The second bond is issued at par value with a coupon rate of 8.75%. a) What is the yield to maturity on the par bond? Why is it higher than the yield of the discount bond? b) If you expect rates to fall substantially in the next 2 years, which bond would you prefer to hold? c) In what sense does the discount bond offer “implicit call protection”?

Exercise 7: Bond pricing A 10-year bond of a firm in severe financial distress pays annual coupons with a coupon rate of 14% and sells for $900. The firm is currently renegotiating the debt, and it appears that the lenders will allow the firm to reduce the coupon payments on the bond to one-half of the originally contracted amount. The firm can handle these lower payments. What is the stated and expected yield to maturity on the bonds?

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Exercise 8: Duration, convexity and immunization Consider the following three bonds: Bond Bond 1 Bond 2 Bond 3

Time to maturity 1 year 3 years 4 years

Coupon 10 % 0% 20 %

The yield curve is flat and the yield to maturity is 9%. The face value is 100 for all three bonds. a) Compute the duration for all three bonds. b) Suppose that you purchase one Bond 1 and one Bond 3. Compute the duration of this portfolio. c) How can an investor adjust the portfolio in b) to get a duration of 2 years?

Exercise 9: Duration, convexity and immunization Rank the effective durations of the following pairs of bonds: a) Bond A is an 8% coupon bond with 20 years to maturity selling at par value. Bond B is an 8% coupon bond with 20 years to maturity selling below par value. b) Bond A is a 20-year non-callable 8% coupon bond selling at par value. Bond B is a 20-year callable coupon bond with a coupon of 9%, also selling at par. c) Bond A is a 3-year 6% coupon bond making annual coupon payments priced at a yield of 4%. Bond B is a 3-year 6% coupon bond making semiannual coupon payments, also priced at a yield of 4%. d) Bond A is a 3-year 6% coupon bond making annual coupon payments priced at a yield of 4%. Bond B is a 3-year 8% coupon bond making annual coupon payments priced at a yield of 4%. e) Bond A is Baa-rated with an 8% coupon and 20 years to maturity. Bond B is Aaarated with an 8% coupon and 20 years to maturity.

Exercise 10: Duration, convexity and immunization An insurance company must make payments to a customer of $10 million in 1 year and $4 million in 5 years. The yield curve is flat at 10%. a) If it wants of fully fund and immunize the obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?

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b) What must be the face value and market value of that zero-coupon bond?

Exercise 11: Duration, convexity and immunization Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely, the pension fund obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is 16%. a) If the duration of 5-year maturity bonds with annual coupons of 12% is 4 years and the duration of 20-year maturity bonds with 6% annual coupons is 11 years, how much of each of these coupon bonds (in market value) should you hold to fully fund and immunize your obligation? b) What will be the par value of your holdings in the 20-year bond?

Exercise 12: Duration, convexity and immunization A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% annual coupon bond also selling at a yield to maturity of 8% has nearly identical duration (11.79 years) but considerably higher convexity: 231.2. a) Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each of the bonds? What percentage capital loss would be predicted by the duration-with-convexity rule? b) Repeat (a), but this time assume the yield to maturity decreases to 7%. c) Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other involving a decrease. Based on the comparative investment performance, explain the attraction of convexity. d) In view of your answer to (c), do you think it is possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? Would anyone be willing to buy the bond with lower convexity under these circumstances?

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