FIXED INCOME I EXERCISES

FIXED INCOME I EXERCISES This version: 25.09.2011 Interplay between macro and financial variables 1. Read the paper: The Bond Yield Conundrum from a Macro-Finance Perspective, Glenn D. Rudebusch, Eric T. Swanson, and TaoWu, Monetary and Economic Studies (Special Edition), December 2006. a) Establish a clear link between changes of the yield curve and the behavior of macroeconomic variables; b) What s the Conundrum? Compounding, Clean/dirty prices, Accrued interest, Bid/ask prices 1. Which is cheaper to issue: a 1-year Treasury bill issued at a discount of 5% or a 1- year zero coupon bond issued at a yield of 5%? What zero rate would make you indifferent? 2. On 25/10/99, consider a fixed-coupon bond whose features are the following: Face value: 100 Coupon rate: 10% Coupon frequency: annual Maturity: 15/04/08 Compute the accrued interest taking into account the four different day-count basis: Actual/Actual, Actual/365, Actual/360 and 30/360. Paulo Leiria Page 1 Fixed Income I

3. What is the price (P) of the certificate of deposit issued by bank X on 06/06/00, with maturity 25/08/00, face value 10,000,000, an interest rate at issuance of 5% falling at maturity, and a yield of 4.5% as of 31/07/00? 4. Suppose the 1-year continuously compounded interest rate is 12%. What is the effective annual interest rate? 5. If you make a deposit of 2,500 in a bank account that earns 8% annually on a continuously compounded basis, what will be the account balance in 7.14 years? 6. If an investment has a cumulative 63.45% rate of return over 3.78 years, what is the annual continuously compounded rate of return? 7. What is the price of a 5-year bond with a nominal value of 100, a yield-to-maturity of 7% (with annual compounding frequency), a 10% coupon rate and an annual coupon frequency? Same question for a yield-to-maturity of 8%, 9% and 10%. Conclude. 8. Your bank has won a mandate to issue an 8-year, annual coupon, 200 million bullet bond for a client. Suppose the risk-free (spot) interest rate curve is the following: Risk free Maturity Curve 1 3,55% 2 4,06% 3 4,50% 4 4,89% 5 5,21% 6 5,48% 7 5,68% 8 5,83% After running the credit risk numbers and consulting with the trading desk, it was determined that the term structure of credit spreads applicable to the client is: Paulo Leiria Page 2 Fixed Income I

Credit Maturity spread 1 0,02% 2 0,02% 3 0,03% 4 0,03% 5 0,04% 6 0,04% 7 0,05% 8 0,05% What will be the coupon rate and the issuance price? Is the bond above or below PAR? Hint: To solve this problem, you should do the following: a) Set P equal to PAR, that is, P = 100, and solve the bond price equation for c (the coupon rate); b) Round c (the coupon rate) to 10 basis points [in excel, use the function round(c,3)], and set the coupon rate to this (rounded) figure; c) Compute the present value of the bond s cash-flows using the coupon rate set in b). That s the issuance price. 9. Bank 1 offers 4.85% compounded monthly for a one-year investment. Bank 2 offers 5% compounded semiannually. Which bank offers the better investment? 10. Using simple interest of 6% and the actual/360 convention, how much interest is owed on a 1,000,000 loan from April 24, 2001, to May 2, 2001? 11. A 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? 12. Consider the following three bonds and bond prices: 0% coupon maturing 5/15/2005 96.3750 Paulo Leiria Page 3 Fixed Income I

7.5% coupon maturing 5/15/2005 103.4043 15% coupon maturing 5/15/2005 106.0625 Do these prices make sense relative to each other? Demonstrate why or why not. 13. An investor has a cash of 10,000,000 at disposal. He wants to invest in a bond with 1,000 nominal value and whose dirty price is equal to 107.457%. a) What is the number of bonds he will buy? b) Same question if the nominal value and the dirty price of the bond are respectively 100 and 98.453%. 14. An investor wants to buy a bullet bond of the automotive sector. He has two choices: either invest in a US corporate bond denominated in euros or in a French corporate bond with same maturity and coupon. Are the two bonds comparable? Spot/forward rates, Term structure, Strips, Bootstrapping 1. Assume that the following bond yields, compounded semiannually: 6-month Treasury Strip: 5.00%; 1-year Treasury Strip: 5.25%; 18-month Treasury Strip: 5.75%. a) What is the 6-month forward rate in six months? b) What is the 1-year forward rate in six months? c) What is the price of a semiannual 10% coupon Treasury bond that matures in exactly 18 months? 2. Evidence from trading in Treasury strips indicates the following spot rates: Term Spot rate 1 year 1.86% 2 years 2.67% 3 years 3.27% 4 years 3.70% 5 years 4.01% a) What is the 1-year forward rate for each period? Paulo Leiria Page 4 Fixed Income I

b) What is the three-year forward rate from year 2 through year 5? c) If the expectations theory is correct, what do investors think will be the 1-year spot rate four years from now? 3. Consider three zero-coupon bonds (strips) with the following features: Bond Maturity (years) Price Bond 1 1 96.43 Bond 2 2 92.47 Bond 3 3 87.97 Each strip delivers 100 at maturity. a) Extract the zero-coupon yield curve from the bond prices. b) We anticipate a rate increase in one year so the prices of strips with residual maturity 1 year, 2 years and 3 years are respectively 95.89, 90.97 and 84.23. What is the zero-coupon yield curve anticipated in one year? 4. Consider the following decreasing zero-coupon yield curve: Maturity (years) R(0,t) (%) Maturity (years) R(0,t) (%) 1 7.000 6 6.250 2 6.800 7 6.200 3 6.620 8 6.160 4 6.460 9 6.125 5 6.330 10 6.100 where R(0, t) is the zero-coupon rate at date 0 with maturity t. a) Compute the par yield curve. b) Compute the forward yield curve in one year. c) Draw the three curves on the same graph. What can you say about their relative position? Paulo Leiria Page 5 Fixed Income I

5. At date t = 0, we consider five bonds with the following features: Annual Coupon Maturity (years) Price Bond 1 6 1 P 1 = 103 Bond 2 5 2 P 2 = 102 Bond 3 4 3 P 3 = 100 Bond 4 6 4 P 4 = 104 Bond 5 5 5 P 5 = 99 Derive the zero-coupon curve until the 5-year maturity. 6. a) The 10-year and 12-year zero-coupon rates are respectively equal to 4% and 4.5%. Compute the 111/4 and 113/4-year zero-coupon rates using linear interpolation. b) Same question when you know the 10-year and 15-year zero-coupon rates that are respectively equal to 8.6% and 9%. Yields, Rates of return, Par rate, Linear interpolation 1. Consider the following zero-coupon curve: Maturity (years) Zero-Coupon Rate (%) 1 4.00 2 4.50 3 4.75 4 4.90 5 5.00 a) What is the price of a 5-year bond with a 100 face value, which delivers a 5% annual coupon rate? What is the yield-to-maturity of this bond? b) Suppose that the zero-coupon curve increases instantaneously and uniformly by 0.5%. What are the new price and the new yield-to-maturity of the bond? What is the impact of this rate increase for the bondholder? Paulo Leiria Page 6 Fixed Income I

c) Suppose now that the zero-coupon curve remains stable over time. You hold the bond until maturity. What is the annual return rate of your investment? Why is this rate different from the yield-to-maturity? 2. Consider two bonds with the following features: Bond Maturity (years) Coupon Rate (%) Price YTM (%) Bond 1 10 10 1,352.2 5.359 Bond 2 10 5 964.3 5.473 YTM stands for yield-to-maturity. These two bonds have a 1,000 face value, and an annual coupon frequency. a) An investor buys these two bonds and holds them until maturity. Compute the annual return rate over the period, supposing that the yield curve becomes instantaneously flat at a 5.4% level and remains stable at this level during 10 years. b) What is the rate level such that these two bonds provide the same annual return rate? In this case, what is the annual return rate of the two bonds? 3. Suppose you are an investor with a 5-year investment horizon and are considering purchasing a seven-year 9% coupon bond selling at par. You expect that you can reinvest the coupon payments at a semi-annually compounded APR of 9.4% and that at the end of your investment horizon two-year bonds will have a yield to maturity of 11.2%. What is your expected return for this bond? 4. Suppose today, 10.05.2008, you are given the following information from the cash market: Euribor 1-month = 5.60% (maturing in 10.06.2008) Euribor 2-months = 5.71% (maturing in 10.07.2008) You also know, from the futures market, the 3-months interest rate (from 18.06.2008 to 16.09.2008) is 6%. Paulo Leiria Page 7 Fixed Income I

Days from......to 10-05-2008 10-06-2008 18-06-2008 10-06-2008 31 18-06-2008 39 8 10-07-2008 61 30 22 16-09-2008 129 98 90 Compute the spot rate from 10.05.2008 to 16.09.2008. Hint: To solve this problem, you should do the following: a) Compute the spot rate from 10.05 to 18.06 by linear interpolation, from the 2 money market rates. b) Compute the spot discount factor from 10.05 to 18.06, using the spot rate found in a). c) Compute the forward discount factor from 16.09 to 18.06, using the futures rate of 6.0%. d) Compute the spot discount factor from 16.09 to 10.05, using the two previously computed discount factors, in b) and c). e) Compute the spot rate from 10.05.2008 to 16.09.2008, using the discount factor found in d). The spot rate is 5.9135%. Duration, PVBP, Duration of a portfolio, Duration over time, Convexity 1. Each of the following bonds has 100 face value, and coupons are paid annually. What can you say about the relative durations of the bonds? Can you rank the bonds according to duration, from lowest to highest, without any numerical calculations? If so, explain why. If not, calculate the durations. Bond Coupon Time to Maturity (years) Yield to Maturity A 6% 5 6% B 6% 10 6% C 8% 5 9% D 8% 10 6% E 0% 5 6% 2. State whether the following claims are true, false, or uncertain. Paulo Leiria Page 8 Fixed Income I

a) Since the Macaulay duration of a zero-coupon bond is equal to its time to maturity, the percent change in the bond price for a one unit change in the yield-to-maturity on a given day is the same regardless of the yield level. b) A 4.250% coupon bond maturing on 11/15/2008 with a yield-to-maturity of 4.482% has a lower duration than a 11.875% coupon bond maturing also on 11/15/2008 with a yield to maturity of 4.553%. c) Using the price of a perpetuity to approximate the price of a very long maturity coupon bond is more accurate the higher the coupon rate and the lower the yield-to-maturity. d) If two portfolios have the same duration, the change in their value when discount rates change will be the same. 3. You think the price of a 10.75% bond maturing in August 2005 is cheap relative to other bonds in its maturity range. a) What risks do you face by buying that bond in the hope that its price will rise relative to other bonds in its maturity range? b) What risks do you face by buying that bond and selling an appropriate amount of a 6.5% bond maturing also in August 2005 to be duration neutral? 4. Explain what a 10-year key rate modified duration of 0.35 means. Describe step-bystep how you would compute this key-rate duration. 5. A major weakness of using one single duration figure for the entire portfolio is that we are assuming that movements in the entire term structure can be described by only one interest rate factor. So, for example, a (naive) duration analysis would allow for the hedging of a position in 2- or 10-year bonds with, for example, a 5-year security. Suppose you have 2 bonds in your portfolio: 8Y bond, 12% annual coupon 8Y bond, 6.5% annual coupon The initial yield curve is flat at 4.5%. Paulo Leiria Page 9 Fixed Income I

a) Price these two bonds. Admit now that we define 3 key rates (1, 2, and 3) for the following buckets of maturities, respectively, defined in years: Bucket 1: [0, 5[ Bucket 2: [5, 10[ Bucket 3: [10] Each key rate affects yields from the term of the previous key rate bucket (or zero) to the term of the next key rate bucket (or the last term). Key rate 1 affects all yields of term zero to 5, the key rate 2 affects yields of term 5 to 10, and the key rate 3 affects the yield of term 10. The impact of each key rate is x basis points at its own maturity bucket and declines linearly to zero at the term of the adjacent key rate (see the hint). b) Suppose key rate 1 increases by 10bp. Price again the two bonds. c) Beginning with the initial yield curve, assume key rate 3 increases by 10bp. Price the bonds. d) Compute the percentual price change of each bond (which means, the modified duration) in the following 2 cases: i) a change by 10bp in key rate 1; ii) a change by 10bp in key rate 3. e) Which bond is more exposed to key rate 1? Explain why. Hint: The yield curves should look like this: 4,7% Yield Curve 4,6% 4,5% 4,4% 1 2 3 4 5 6 7 8 9 10 Initial YC +10bp key rate 1 +10bp key rate 3 Paulo Leiria Page 10 Fixed Income I

6. Consider the following portfolio consisting of 4 bonds: Bond Market Value Modified Duration Convexity A 13 2 5 B 27 7 13 C 60 8 26 D 40 14 184 a) What is the modified duration of the portfolio? b) If the yields to maturity for all maturities increase by 50 basis points, what is the approximate percent change in the value of the portfolio? c) What is the approximate duration of the portfolio after the 50 basis point yield to maturity increase described in part b)? 7. Compute the dirty price, the duration, the modified duration, the $duration and the BPV (basis point value) of the following bonds with 100 face value assuming that coupon frequency and compounding frequency are annual: Bond Maturity (years) Coupon Rate (%) YTM (%) 1 1 5 5 2 1 10 6 3 5 5 5 4 5 10 6 5 5 5 7 6 5 10 8 7 20 5 5 8 20 10 6 9 20 5 7 10 20 10 8 8. Zero-coupon bonds. a) What is the price of a zero-coupon bond with 100 face value that matures in seven years and has a yield of 7%? We assume that the compounding frequency is semiannual. b) What is the bond s modified duration? Paulo Leiria Page 11 Fixed Income I

c) Use the modified duration to find the approximate change in price if the bond yield rises by 15 basis points. 9. An investor holds 100,000 units of a bond whose features are summarized in the following table. He wishes to be hedged against a rise in interest rates. Maturity Coupon Rate YTM Duration Price 18 Years 9.5% 8% 9.5055 114,181 Characteristics of the hedging instrument, which is here a bond, are as follows: Maturity Coupon Rate YTM Duration Price 20 Years 10% 8% 9.8703 119,792 Coupon frequency and compounding frequency are assumed to be semiannual. YTM stands for yield to maturity. The YTM curve is flat at an 8% level. a) What is the quantity φ of the hedging instrument that the investor has to sell? b) We suppose that the YTM curve increases instantaneously by 0.1%. i. What happens if the bond portfolio has not been hedged? ii. And if it has been hedged? c) Same question as the previous one when the YTM curve increases instantaneously by 2%. d) Conclude. 10. Consider a 20-year zero-coupon bond with a 6% YTM and 100 face value. Compounding frequency is assumed to be annual. a) Compute its price, modified duration, $duration, convexity and $convexity? b) On the same graph, draw the price change of the bond when YTM goes from 1% to 11%: (i) by using the exact pricing formula; (ii) by using the one-order Taylor estimation; (iii) by using the second-order Taylor estimation. Paulo Leiria Page 12 Fixed Income I

11. Assume a 2-year Euro-note, with a 100,000 face value, a coupon rate of 10% and a convexity of 4.53. Today s YTM is 11.5% and term structure is flat. Coupon frequency and compounding frequency are assumed to be annual. a) What is the Macaulay duration of this bond? b) What does convexity measure? Why does convexity differ among bonds? What happens to convexity when interest rates rise? Why? c) What is the exact price change in euros if interest rates increase by 10 basis points (a uniform shift)? d) Use the duration model to calculate the approximate price change in euros if interest rates increase by 10 basis points. e) Incorporate convexity to calculate the approximate price change in euros if interest rates increase by 10 basis points. Horizon date immunization, Cash-flow matching, Net-worth immunization Pricing a swap, Swap curve 1. Consider two firms A and B that have the same financial needs in terms of maturity and principal. The two firms can borrow money in the market at the following conditions: Firm A: 11% at a fixed rate or Libor + 2% for a $10 million loan and a 5-year maturity. Firm B: 9% at a fixed rate or Libor + 0.25% for a $10 million loan and a 5-year maturity. a) Suppose that firm B prefers a floating-rate debt as firm A prefers a fixed rate debt. What is the swap they will structure to optimize their financial conditions? b) If firm B prefers a fixed-rate debt as firm A prefers a floating-rate debt, is there a swap to structure so that the two firms optimize their financial conditions? Conclude. Paulo Leiria Page 13 Fixed Income I

2. We consider at date T0, a 1-year Libor swap contract with maturity 10 years and with the following cash-flow schedule: F F F F F F F F F F T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 -V0 -V1 -V2 -V3 -V4 -V5 -V6 -V7 -V8 -V9 Note that Ti+1 Ti = 1 year, i {0, 1, 2,..., 9}. We suppose that the swap nominal amount is 10 million, and that the rate F of the fixed leg is 9.55%. At date T0, zerocoupon rates with maturities T1, T2,...,T10 are given in the following table: Maturity ZC rates (%) Maturity ZC rates (%) T1 8.005 T6 9.235 T2 7.856 T7 9.478 T3 8.235 T8 9.656 T4 8.669 T9 9.789 T5 8.963 T10 9.883 a) Give the price of this swap. b) What is the swap rate such that the price of this swap is zero? c) An investor holds a bond portfolio whose price, yield to maturity and modified duration are respectively 9,991,565,452, 9.2% and 5.92. He wants to be protected against an increase in rates. How many swaps must he sell to protect his bond portfolio? Duration of a swap Paulo Leiria Page 14 Fixed Income I

Bond portfolio management 1. Choosing a portfolio with the Maximum $Duration or Modified Duration Possible. Consider at date t, five bonds delivering annual coupon rates with the following features: Maturity (years) CR (%) YTM (%) Price 5 7 4.00 113.355 7 6 4.50 108.839 15 8 5.00 131.139 20 5 5.25 96.949 22 7 5.35 121.042 CR stands for coupon rate and YTM for yield to maturity. A portfolio manager believes that the YTM curve will very rapidly decrease by 0.3% in level. Which of these bonds provides the maximum absolute gain? Which of these bonds provides the maximum relative gain? 2. Rollover Strategy. An investor has funds to invest over one year. He anticipates a 1% increase in the curve in six months. The 6-month and 1-year zero-coupon rates are respectively 3% and 3.2%. He has two different opportunities: he can buy the 1-year zero-coupon T-bond and hold it until maturity, or he can choose a rollover strategy by buying the 6-month T-bill, holding it until maturity, and buying a new 6-month T-bill in six months, and holding it until maturity. a) Calculate the annualized total return rate of these two strategies assuming that the investor s anticipation is correct. b) Same question when interest rates decrease by 1% after six months. 3. Butterfly. Consider three bonds with short, medium and long maturities whose features are summarized in the following table: Paulo Leiria Page 15 Fixed Income I

Maturity Coupon Rate YTM Bond $Duration Quantity (years) (%) (%) Price 2 6 6 100-183.34 qs 10 6 6 100-736.01-10,000 30 6 6 100-1,376.48 ql YTM stands for yield to maturity, bond prices are dirty prices, and we assume a flat yield-to-maturity curve in the exercise. We structure a butterfly in the following way: we sell 10,000 10-year bonds; we buy qs 2-year bonds and ql 30-year bonds. a) Determine the quantities qs and ql so that the butterfly is cash-and $duration neutral. b) What is the P&L of the butterfly if the yield-to-maturity curve goes up to a 7% level? And down to a 5% level? c) Draw the P&L of the butterfly depending on the value of the yield to maturity. Paulo Leiria Page 16 Fixed Income I