Problems and Solutions

Transcription

1 1 CHAPTER 1 Problems 1.1 Problems on Bonds Exercise 1.1 On 12/04/01, consider a fixed-coupon bond whose features are the following: face value: $1,000 coupon rate: 8% coupon frequency: semiannual maturity: 05/06/04 What are the future cash flows delivered by this bond? Solution The coupon cash flow is equal to $40 8% $1,000 Coupon = = $40 2 It is delivered on the following future dates: 05/06/02, 11/06/02, 05/06/03, 11/06/03 and 05/06/04. The redemption value is equal to the face value $1,000 and is delivered on maturity date 05/06/04. Exercise 1.3 Solution 1.3 Exercise 1.4 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 %. 1. What is the number of bonds he will buy? 2. Same question if the nominal value and the dirty price of the bond are respectively $100 and %. 1. The number of bonds he will buy is given by the following formula Cash Number of bonds bought = Nominal Value of the bond dirty price Here, the number of bonds is equal to 9, n is equal to 101,562 n = 10,000,000 1, % = 9, n = 10,000, % = 101, On 10/25/99, consider a fixed-coupon bond whose features are the following: face value: Eur 100

2 2 coupon rate: 10% coupon frequency: annual maturity: 04/15/08 Compute the accrued interest taking into account the four different day-count bases: Actual/Actual, Actual/365, Actual/360 and 30/360. Solution 1.4 The last coupon has been delivered on 04/15/99. There are 193 days between 04/15/99 and 10/25/99, and 366 days between the last coupon date (04/15/99) and the next coupon date (04/15/00). The accrued interest with the Actual/Actual day-count basis is equal to Eur % Eur 100 = Eur The accrued interest with the Actual/365 day-count basis is equal to Eur % Eur 100 = Eur The accrued interest with the Actual/360 day-count basis is equal to Eur % Eur 100 = Eur There are 15 days between 04/15/99 and 04/30/99, 5 months between May and September, and 25 days between 09/30/99 and 10/25/99, so that there are 190 days between 04/15/99 and 10/25/99 on the 30/360 day-count basis 15 + (5 30) + 25 = 190 Finally, the accrued interest with the 30/360 day-count basis is equal to Eur % Eur 100 = Eur Exercise 1.8 Solution 1.8 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? The answer is no. First, the coupon and yield frequency of the US corporate bond is semiannual, while it is annual for the French corporate bond. To compare the yields on the two instruments, you have to convert either the semiannual yield of the US bond into an equivalently annual yield or the annual yield of the French bond into an equivalently semiannual yield. Second, the two bonds do not necessarily have the same rating, that is, the same credit risk. Third, they do not necessarily have the same liquidity.

3 3 Exercise 1.15 What is the price P of the certificate of deposit issued by bank X on 06/06/00, with maturity 08/25/00, face value $10,000,000, an interest rate at issuance of 5% falling at maturity and a yield of 4.5% as of 07/31/00? Solution 1.15 Recall that the price P of such a product is given by ( 1 + c n c ) B P = F ( 1 + ym n ) m B where F is the face value, c the interest rate at issuance, n c is the number of days between issue and maturity, B is the year-basis (360 or 365), y m is the yield on a money-market basis and n m is the number of days between settlement and maturity. Then, the price P of the certificate of deposit issued by bank X is equal to P = $10,000,000 ( ) 1 + 5% ( ) = $10,079, % Indeed, there are 80 calendar days between 06/06/00 and 08/25/00, and 25 calendar days between 07/31/00 and 08/25/00. 2 CHAPTER 2 Problems Exercise 2.1 Suppose the 1-year continuously compounded interest rate is 12%. What is the effective annual interest rate? Solution 2.1 The effective annual interest rate is R = e = = 12.75%. Exercise 2.2 Solution 2.2 If you deposit $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? The account balance in 7.14 years will be $2,500.e 8% 7.14 = $4, Exercise 2.3 Solution 2.3 If an investment has a cumulative 63.45% rate of return over 3.78 years, what is the annual continuously compounded rate of return? The annual continuously compounded rate of return R is such that We find R c = ln(1.6345)/3.78 = 13% = e 3.78Rc

4 4 Exercise 2.7 Solution 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? 2. Same question for a yield to maturity of 8%, 9% and 10%. Conclude. 1. The price P of a bond is given by the formula n N c P = (1 + y) i + N (1 + y) n which simplifies into P = N c y i=1 [ 1 ] 1 (1 + y) n + N (1 + y) n where N, c, y and n are respectively the nominal value, the coupon rate, the yield to maturity and the number of years to maturity of the bond. Here, we obtain for P P = 10 7% [ 1 ] 1 (1 + 7%) (1 + 7%) 5 P is then equal to % of the nominal value or $ Note that we can also use the Excel function Price to obtain P. 2. Prices of the bond for different yields to maturity (YTM) are given in the following table YTM (%) Price ($) Bond prices decrease as rates increase. Exercise 2.14 We consider the following zero-coupon curve: Maturity (years) Zero-Coupon Rate (%) What is the price of a 5-year bond with a $100 face value, which delivers a 5% annual coupon rate? 2. What is the yield to maturity of this bond? 3. We suppose that the zero-coupon curve increases instantaneously and uniformly by 0.5%. What is the new price and the new yield to maturity of the bond? What is the impact of this rate increase for the bondholder?

5 5 4. We 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? Solution The price P of the bond is equal to the sum of its discounted cash flows and given by the following formula P = % + 5 ( %) ( %) ( %) (1 + 5%) 5 = $ The yield to maturity R of this bond verifies the following equation = (1 + R) i (1 + R) 5 Using the Excel function Yield, we obtain % for R. 3. The new price P of the bond is given by the following formula: P = i= % + 5 (1 + 5%) ( %) ( %) ( %) 5 = $ The new yield to maturity R of this bond verifies the following equation = (1 + R) i (1 + R) 5 i=1 Using the Excel function yield, we obtain % for R. The impact of this rate increase is an absolute capital loss of $2.137 for the bondholder. Absolute Loss = = $2.137 and a relative capital loss of 2.134% Relative Loss = = 2.134% 4. Before maturity, the bondholder receives intermediate coupons that he reinvests in the market: after one year, he receives $5 that he reinvests for 4 years at the 4-year zerocoupon rate to obtain on the maturity date of the bond 5 ( %) 4 = $ after two years, he receives $5 that he reinvests for 3 years at the 3-year zerocoupon rate to obtain on the maturity date of the bond 5 ( %) 3 = $5.7469

6 6 after three years, he receives $5 that he reinvests for 2 years at the 2-year zero-coupon rate to obtain on the maturity date of the bond 5 ( %) 2 = $ after four years, he receives $5 that he reinvests for 1 year at the 1-year zerocoupon rate to obtain on the maturity date of the bond 5 (1 + 4%) = $5.2 after five years, he receives the final cash flow equal to $105. The bondholder finally obtains $ five years later = $ which corresponds to a 4.944% annual return rate ( ) /5 1 = 4.944% This return rate is different from the yield to maturity of this bond (4.9686%) because the curve is not flat at a % level. With a flat curve at a % level, we obtain $ five years later = $ which corresponds exactly to a % annual return rate. ( ) /5 1 = % Exercise 2.20 We consider two bonds with the following features Bond Maturity (years) Coupon Rate (%) Price YTM (%) Bond , Bond YTM stands for yield to maturity. These two bonds have a $1,000 face value, and an annual coupon frequency. 1. 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. 2. 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? Solution We consider that the investor reinvests its intermediate cash flows at a unique 5.4% rate.

7 7 For Bond 1, the investor obtains the following sum at the maturity of the bond ( %) i + 1,100 = 2, i=1 which corresponds exactly to a % annual return rate. ( ) 2, /10 1 = % 1,352.2 For Bond 2, the investor obtains the following sum at the maturity of the bond 9 50 ( %) i + 1,050 = 1, i=1 which corresponds exactly to a % annual return rate. ( ) 1, /10 1 = % We have to find the value R, such that i=1 (1 + R) i + 1,100 1,352.2 = 50 9 i=1 (1 + R) i + 1, Using the Excel solver, we finally obtain % for R. The annual return rate of the two bonds is equal to % ( ) 100 9i=1 1/10 ( ) i + 1,100 1 = % 1,352.2 Exercise 2.24 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%. 1. What is the 6-month forward rate in six months? 2. What is the 1-year forward rate in six months? 3. What is the price of a semiannual 10% coupon Treasury bond that matures in exactly 18 months? Solution ( 1 + R ) 2(0, 1) 2 ( = 1 + R 2(0, 0.5) = ( 1 + F 2(0, 0.5, 0.5) 2 F 2 (0, 0.5, 0.5) = % )( 1 + F 2(0, 0.5, 0.5) 2 ) )

8 8 2. ( 1 + R ) 2(0, 1.5) 3 ( = 1 + R 2(0, 0.5) = ( 1 + F 2(0,0.5,1) 2 )( 1 + F 2(0, 0.5, 1) 2 ) 2 F 2 (0, 0.5, 1) = % 3. The cash flows are coupons of 5% in six months and a year, and coupon plus principal payment of 105% in 18 months. We can discount using the spot rates that we are given: 3 CHAPTER 3 Problems Exercise P = ( ) ( ) 2 + ( ) 3 = We consider three zero-coupon bonds (strips) with the following features: Each strip delivers $100 at maturity. Bond Maturity (years) Price Bond Bond Bond Extract the zero-coupon yield curve from the bond prices. 2. 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, and What is the zero-coupon yield curve anticipated in one year? Solution The 1-year zero-coupon rate denoted by R(0, 1) is equal to 3.702% R(0, 1) = = 3.702% The 2-year zero-coupon rate denoted by R(0, 2) is equal to 3.992% ( ) 100 1/2 R(0, 2) = 1 = 3.992% The 3-year zero-coupon rate denoted by R(0, 3) is equal to 4.365% ( ) 100 1/3 R(0, 2) = 1 = 4.365% The 1-year, 2-year and 3-year zero-coupon rates become respectively 4.286%, 4.846% and 5.887%. ) 2

9 9 Exercise 3.3 Solution 3.3 We consider the following decreasing zero-coupon yield curve: Maturity (years) R(0,t) (%) Maturity (years) R(0,t) (%) where R(0,t) is the zero-coupon rate at date 0 with maturity t. 1. Compute the par yield curve. 2. Compute the forward yield curve in one year. 3. Draw the three curves on the same graph. What can you say about their relative position? 1. Recall that the par yield c(n) for maturity n is given by the formula c(n) = 1 1 (1+R(0,n)) n ni=1 1 (1+R(0,i)) i Using this equation, we obtain the following par yields: Maturity (years) c(n) (%) Maturity (years) c(n) (%) Recall that F(0,x,y x), the forward rate as seen from date t = 0, starting at date t = x, and with residual maturity y x is defined as [ (1 + R(0,y)) y ] 1 y x F(0,x,y x) 1 (1 + R(0,x)) x Using the previous equation, we obtain the forward yield curve in one year Maturity (years) F(0, 1,n) (%) Maturity (years) F(0, 1,n) (%) The graph of the three curves shows that the forward yield curve is below the zero-coupon yield curve, which is below the par yield curve. This is always the case when the par yield curve is decreasing.

10 Par yield curve Zero-coupon yield curve Forward yield curve Yield (%) Maturity Exercise 3.13 Solution 3.13 Explain the basic difference that exists between the preferred habitat theory and the segmentation theory. In the segmentation theory, investors are supposed to be 100% risk-averse. So risk premia are infinite. It is as if their investment habitat were strictly constrained, exclusive. In the preferred habitat theory, investors are not supposed to be 100% risk averse. So, there exists a certain level of risk premia from which they are ready to change their habitual investment maturity. Their investment habitat is, in this case, not exclusive. 4 CHAPTER 4 Problems Exercise 4.1 At date t = 0, we consider five bonds with the following features: Annual Coupon Maturity Price Bond year P0 1 = 103 Bond years P0 2 = 102 Bond years P0 3 = 100 Bond years P0 4 = 104 Bond years P0 5 = 99 Derive the zero-coupon curve until the 5-year maturity.

11 11 Solution 4.1 Using the no-arbitrage relationship, we obtain the following equations for the five bond prices: 103 = 106B(0, 1) 102 = 5B(0, 1) + 105B(0, 2) 100 = 4B(0, 1) + 4B(0, 2) + 104B(0, 3) 104 = 6B(0, 1) + 6B(0, 2) + 6B(0, 3) + 106B(0, 4) 99 = 5B(0, 1) + 5B(0, 2) + 5B(0, 3) + 5B(0, 4) + 105B(0, 5) which can be expressed in a matrix form as = We get the following discount factors: B(0, 1) B(0, 2) B(0, 3) B(0, 4) = B(0, 5) and we find the zero-coupon rates R(0, 1) = 2.912% R(0, 2) = 3.966% R(0, 3) = 4.016% R(0, 4) = 4.976% R(0, 5) = 5.339% B(0, 1) B(0, 2) B(0, 3) B(0, 4) B(0, 5) Exercise 4.4 Solution The 10-year and 12-year zero-coupon rates are respectively equal to 4% and 4.5%. Compute the 11 1/4 and 11 3/4 -year zero-coupon rates using linear interpolation. 2. Same question when you know the 10-year and 15-year zero-coupon rates that are respectively equal to 8.6% and 9%. Assume that we know R(0,x) and R(0,z) respectively as the x-year and the z-year zero-coupon rates. We need to get R(0,y),they-year zero-coupon rate with y [x; z]. Using linear interpolation, R(0,y) is given by the following formula: (z y)r(0,x)+ (y x)r(0,z) R(0,y)= z x 1. The 11 1/4 and 11 3/4 -year zero-coupon rates are obtained as follows: R(0, 11 1/4 ) = % % 2 = %

12 12 R(0, 11 3/ % % ) = = % 2 2. The 11 1/4 and 11 3/4 -year zero-coupon rates are obtained as follows: R(0, 11 1/4 ) = % % 5 = 8.70% R(0, 11 3/4 ) = % % 5 = 8.74% Exercise 4.7 From the prices of zero-coupon bonds quoted in the market, we obtain the following zero-coupon curve: Maturity (years) Zero-coupon Rate R(0,t) (%) Discount Factor B(0,t) ?? ?? where R(0,t) is the zero-coupon rate at date 0 for maturity t, andb(0,t) is the discount factor at date 0 for maturity t. We need to know the value for the 5-year and the 8-year zero-coupon rates. We have to estimate them and test four different methods. 1. We use a linear interpolation with the zero-coupon rates. Find R(0, 5), R(0, 8) and the corresponding values for B(0, 5) and B(0, 8). 2. We use a linear interpolation with the discount factors. Find B(0, 5), B(0, 8) and the corresponding values for R(0, 5) and R(0, 8). 3. We postulate the following form for the zero-coupon rate function R(0,t): R(0,t)= a + bt + ct 2 + dt 3 Estimate the coefficients a, b, c and d, which best approximate the given zerocoupon rates using the following optimization program: (B(0,i) B(0,i)) 2 Min a,b,c,d i where B(0,i) are the zero-coupon rates given by the market. Find the value for R(0, 5) = R(0, 5), R(0, 8) = R(0, 8), and the corresponding values for B(0, 5) and B(0, 8).

13 13 4. We postulate the following form for the discount function B(0,t): B(0,t)= a + bt + ct 2 + dt 3 Estimate the coefficients a, b, c and d, which best approximate the given discount factors using the following optimization program: (B(0,i) B(0,i)) 2 Min a,b,c,d i where B(0,i) are the discount factors given by the market. Obtain the value for B(0, 5) = B(0, 5), B(0, 8) = B(0, 8), and the corresponding values for R(0, 5) and R(0, 8). 5. Conclude. Solution Consider that we know R(0,x) and R(0,z) respectively as the x-year and the z-year zero-coupon rates and that we need R(0,y),they-year zero-coupon rate with y [x; z]. Using linear interpolation, R(0,y) is given by the following formula: (z y)r(0,x)+ (y x)r(0,z) R(0,y)= z x From this equation, we find the value for R(0, 5) and R(0, 8) (6 5)R(0, 4) + (5 4)R(0, 6) R(0, 5) = 6 4 R(0, 4) + R(0, 6) = = 6.375% 2 (9 8)R(0, 7) + (8 7)R(0, 9) R(0, 8) = 9 7 R(0, 7) + R(0, 9) = = 6.740% 2 Using the following standard equation in which lies the zero-coupon rate R(0,t) and the discount factor B(0,t) 1 B(0,t)= (1 + R(0,t)) t we obtain for B(0, 5) and for B(0, 8). 2. Using the same formula as in question 1 but adapting to discount factors (z y)b(0,x)+ (y x)b(0,z) B(0,y)= z x we obtain for B(0, 5) and for B(0, 8). Using the following standard equation ( ) 1 1/t R(0,t)= 1 B(0,t) we obtain 6.358% for R(0, 5) and 6.717% for R(0, 8).

14 14 3. Using the Excel function Linest, we obtain the following values for the parameters: Parameters Value a b c d E-05 which provide us with the following values for the zero-coupon rates and associated discount factors: Maturity R(0,t) R(0,t) B(0,t) B(0,t) (%) (%) ? 6.403? ? 6.741? We first note that there is a constraint in the minimization because we must have B(0, 0) = 1 So, the value for a is necessarily equal to 1. Using the Excel function Linest, we obtain the following values for the parameters: Parameters Value a 1 b c d which provide us with the following values for the discount factors and associated zero-coupon rates:

15 15 Maturity B(0,t) B(0,t) R(0,t) (%) R(0,t) (%) ? ? ? ? The table below summarizes the results obtained using the four different methods of interpolation and minimization Rates Interpol. DF Interpol. Rates Min. DF Min. R(0, 5) 6.375% 6.358% 6.403% 6.328% R(0, 8) 6.740% 6.717% 6.741% 6.805% B(0, 5) B(0, 8) Rates Interpol. stands for interpolation on rates (question 1). DF Interpol. stands for interpolation on discount factors (question 2). Rates Min stands for minimization with rates (question 3). DF Min. stands for minimization with discount factors (question 4). The table shows that results are quite similar according to the two methods based on rates. Differences appear when we compare the four methods. In particular, we can obtain a spread of 7.5 bps for the estimation of R(0, 5) between Rates Min. and DF Min., and a spread of 8.8 bps for the estimation of R(0, 8) between the two methods based on discount factors. We conclude that the zerocoupon rate and discount factor estimations are sensitive to the method that is used: interpolation or minimization. Exercise 4.15 Consider the Nelson and Siegel Extended model ( ) R c (0,θ)= β 0 + β 1 1 exp ( ) τ θ 1 + β 2 1 exp τ θ 1 exp θ τ 1 + β 3 1 exp ( θ τ 2 ) θ τ 2 ( exp θ ) τ 2 θ τ 1 ) ( θτ1 with the following base-case parameter values: β 0 = 8%, β 1 = 3%, β 2 = 1%, β 3 = 1%, 1/τ 1 = 0.3 and1/τ 2 = 3.

16 16 We give successively five different values to the parameter β 3 : β 3 = 3%,β 3 = 2%, β 3 = 1%, β 3 = 0% and β 3 = 1%. The other parameters are fixed. Draw the five different yield curves to estimate the effect of the curvature factor β 3. Solution 4.15 The following graph shows clearly the effect of the curvature factor β 3 for the five different scenarios: Zero-coupon rate b 3 = 3% b 3 = 2% base case b 3 = 0% b 3 = 1% Maturity 5 CHAPTER 5 Problems Exercise 5.1 Calculate the percentage price change for 4 bonds with different annual coupon rates (5% and 10%) and different maturities (3 years and 10 years), starting with a common 7.5% YTM (with annual compounding frequency), and assuming successively a new yield of 5%, 7%, 7.49%, 7.51%, 8% and 10%. Solution 5.1 Results are given in the following table: New Change 5%/3yr 10%/3yr 5%/10yr 10%/10yr Yield (%) (bps)

17 17 Exercise 5.7 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 (1) annual; (2) semiannual and (3) quarterly. Bond Maturity (years) Coupon Rate (%) YTM (%) Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Solution 5.7 We use the following Excel functions Price, Duration and MDuration to obtain respectively the dirty price, the duration and the modified duration of each bond. The $duration is simply given by the following formula: The BPV is simply $duration = price modified duration BPV = $duration 10, When coupon frequency and compounding frequency are assumed to be annual, we obtain the following results: Price Duration Modified $Duration BPV Duration Bond Bond Bond Bond Bond Bond Bond , Bond , Bond Bond , When coupon frequency and compounding frequency are assumed to be semiannual, we obtain the following results:

18 18 Price Duration Modified Duration $Duration BPV Bond Bond Bond Bond Bond Bond Bond , Bond , Bond Bond , When coupon frequency and compounding frequency are assumed to be quarterly, we obtain the following results: Price Duration Modified Duration $Duration BPV Bond Bond Bond Bond Bond Bond Bond , Bond , Bond Bond , Exercise 5.11 Solution 5.11 Zero-coupon Bonds 1. 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. 2. What is the bond s modified duration? 3. Use the modified duration to find the approximate change in price if the bond yield rises by 15 basis points. 1. The price P is given by $100 P = ( ) 2 7 = $ % 2 2. The modified duration MD is given by MD = P 1 ( 1 + 7% 2 ) i t i PV(CF i ) = 6.763

19 19 3. The approximate change in price is $0.627 P MD y P = $ = $0.627 Exercise 5.19 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% $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% $ 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. 1. What is the quantity φ of the hedging instrument that the investor has to sell? 2. We suppose that the YTM curve increases instantaneously by 0.1%. (a) What happens if the bond portfolio has not been hedged? (b) And if it has been hedged? 3. Same question as the previous one when the YTM curve increases instantaneously by 2%. 4. Conclude. Solution The quantity φ of the hedging instrument is obtained as follows: 11,418, φ = = 91, The investor has to sell 91,793 units of the hedging instrument. 2. Prices of bonds with maturity 18 years and 20 years become respectively $ and $ (a) If the bond portfolio has not been hedged, the investor loses money. The loss incurred is given by the following formula (exactly $103,657 if we take all the decimals into account): Loss = $100,000 ( ) = $103,600 (b) If the bond portfolio has been hedged, the investor is quasi-neutral to an increase (and a decrease) of the YTM curve. The P&L of the position is given by the following formula: P&L = $103,600 + $91,793 ( ) = $57 3. Prices of bonds with maturity 18 years and 20 years become respectively $ and $100.

20 20 (a) If the bond portfolio has not been hedged, the loss incurred is given by the following formula: Loss = $100,000 ( ) = $1,831,800 (b) If the bond portfolio has been hedged, the P&L of the position is given by the following formula: P&L = $1,831,800 + $91,793 ( ) = $15, For a small move of the YTM curve, the quality of the hedge is good. For a large move of the YTM curve, we see that the hedge is not perfect because of the convexity term that is no more negligible (see Chapter 6). 6 CHAPTER 6 Problems Exercise 6.1 We consider a 20-year zero-coupon bond with a 6% YTM and $100 face value. Compounding frequency is assumed to be annual. 1. Compute its price, modified duration, $duration, convexity and $convexity? 2. On the same graph, draw the price change of the bond when YTM goes from 1% to 11% (a) by using the exact pricing formula; (b) by using the one-order Taylor estimation; (c) by using the second-order Taylor estimation. Solution The price P of the zero-coupon bond is simply $100 P = = $31.18 (1 + 6%) 20 Its modified duration is equal to 20/(1 + 6%) = Its $duration, denoted by $Dur, is equal to $Dur = = Its convexity, denoted by RC, is equal to 100 RC = = (1 + 6%) 22 Its $convexity, denoted by $Conv, is equal to $Conv = = 11, Using the one-order Taylor expansion, the new price of the bond is given by the following formula: New Price = $Dur (New YTM 6%)

21 21 Using the two-order Taylor expansion, the new price of the bond is given by the following formula: New Price = $Dur (New YTM 6%) + $Conv (New YTM 6%) 2 2 We finally obtain the following graph. The straight line is the one-order Taylor estimation. Using the two-order Taylor estimation, we underestimate the actual price for YTM inferior to 6%, and we overestimate it for YTM superior to 6% Actual price One-order taylor estimation Two-order taylor estimation % 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% Exercise 6.6 Assume a 2-year Euro-note, with a $100,000 face value, a coupon rate of 10% and a convexity of If today s YTM is 11.5% and term structure is flat. Coupon frequency and compounding frequency are assumed to be annual. 1. What is the Macaulay duration of this bond? 2. What does convexity measure? Why does convexity differ among bonds? What happens to convexity when interest rates rise? Why? 3. What is the exact price change in dollars if interest rates increase by 10 basis points (a uniform shift)? 4. Use the duration model to calculate the approximate price change in dollars if interest rates increase by 10 basis points. 5. Incorporate convexity to calculate the approximate price change in dollars if interest rates increase by 10 basis points.

22 22 Solution Duration D = 1 10, , , , , , = 1 10, , , = , Convexity measures the change in modified duration or the change in the slope of the price-yield curve. Holding maturity constant, the higher the coupon, the smaller the duration. Hence, for low duration levels the change in slope (convexity) is small. Alternatively, holding coupon constant, the higher the maturity, the higher the duration, and hence, the higher the convexity. When interest rates rise, duration (sensitivity of prices to changes in interest rates) becomes smaller. Hence, we move toward the flatter region of the price-yield curve. Therefore, convexity will decrease parallel to duration. 3. Price for a 11.6% YTM is P(11.6%) = 10, , , = $97, Price has decreased by $ from P(11.5%) = $97, to $97, We use P MD y P(11.5%) = D 1 + y y P = , = $ We use P MD y P RC ( y)2 P = , (0.001) 2 97, = $ Hedging error is smaller when we account for convexity. Exercise 6.8 Modified Duration/Convexity Bond Portfolio Hedge At date t, the portfolio P to be hedged is a portfolio of Treasury bonds with various possible maturities. Its characteristics are as follows: Price YTM MD Convexity $28,296, % We consider Treasury bonds as hedging assets, with the following features: Bond Price ($) Coupon Rate (%) Maturity date Bond years Bond years Bond years

23 23 Coupon frequency and compounding frequency are assumed to be annual. At date t, we force the hedging portfolio to have the opposite value of the portfolio to be hedged. 1. What is the number of hedging instruments necessary to implement a modified duration/convexity hedge? 2. Compute the YTM, modified duration and convexity of the three hedging assets. 3. Which quantities φ 1,φ 2 and φ 3 of each of the hedging asset 1, 2, 3dowehave to consider to hedge the portfolio P? Solution We need three hedging instruments. 2. We obtain the following results: Bond YTM (%) MD Convexity Bond Bond Bond We then are looking for the quantities φ 1,φ 2 and φ 3 of each hedging instrument 1, 2, 3 as solutions to the following linear system: φ φ 2 = ,296, , ,143,615 = 290,043 φ , , ,912,260, ,432 7 CHAPTER 7 Problems Exercise 7.1 Would you say it is easier to track a bond index or a stock index. Why or why not? Solution 7.1 As is often the case, the answer is yes and no. On the one hand, it is harder to perform perfect replication of a bond index compared to a stock index. This is because bond indices typically include a huge number of bonds. Other difficulties include that many of the bonds in the indices are thinly traded and the fact that the composition of the index changes regularly, as they mature. On the other hand, statistical replication on bond indices is easier to perform than statistical replication of stock indices, in the sense that a significantly lower tracking error can usually be achieved for a given number of instruments in the replicating portfolio. This is because bonds with different maturities tend to exhibit a fair amount of cross-sectional correlation so that a very limited number of factors account for a very large fraction of changes in bond returns. Typically, 2 or 3 factors (level, slope, curvature) account for more than 80% of these variations. Stocks typically exhibit much more idiosyncratic risk, so that one typically needs to use a large number of factors to account for not much more than 50% of the changes in stock prices.

24 24 8 CHAPTER 8 Problems Exercise 8.2 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 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? Solution 8.2 We compute the modified duration and the $duration of these five bonds. In the scenario anticipated by the portfolio manager, we then calculate the absolute gain and the relative gain that the portfolio manager will earn with each of these five bonds Maturity MD $Dur Absolute Relative (years) Gain ($) Gain (%) , , , MD stands for modified duration and $Dur for $duration. If he wants to optimize his absolute gain, the portfolio manager will choose the 22-year maturity bond. On the contrary, if he prefers to optimize his relative gain, he will invest in the 20-year maturity bond. Exercise 8.4 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.

25 25 1. Calculate the annualized total return rate of these two strategies assuming that the investor s anticipation is correct. 2. Same question when interest rates decrease by 1% after six months. Solution The annualized total return rate of the first strategy is, of course, 3.2% ( %) = = 3.2% ( %) By doing a rollover, the investor will invest at date t = 0, $ to obtain $100 six months later. Note that $ is obtained as follows: $100 = $ (1 + 3%) 12 6 Six months later, he will then( buy a quantity α = ) 100/ of a 6-month $100 T-bill, whose price is $ = $ and which pays $100 at (1+4%) 12 6 maturity so that the annualized total return rate of the second strategy is 3.5% 100 α = 3.5% The annualized total return rate of the first strategy is still 3.2% as it is only 2.5% for the rollover strategy 100 β = 2.5% where β = 100/ In this case, the rollover strategy is worse than the simple buy-and-hold strategy. Exercise 8.7 Butterfly We consider three bonds with short, medium and long maturities whose features are summarized in the following table: Maturity Coupon YTM Bond Price ($) $Duration Quantity (years) Rate (%) (%) q s , , q l 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, year bonds; we buy q s 2-year bonds and q l 30-year bonds.

26 26 1. Determine the quantities q s and q l so that the butterfly is cash-and-$durationneutral. 2. What is the P&L of the butterfly if the yield-to-maturity curve goes up to a 7% level? And down to a 5% level? 3. Draw the P&L of the butterfly depending on the value of the yield to maturity. Solution The quantities q s and q l, which are determined so that the butterfly is cashand-$duration-neutral, satisfy the following system: { (qs ) + (q l 1,376.48) = 10, (q s 100) + (q l 100) = 10, whose solution is ( ) qs = q l ( ) , ( ) 7,360,100 = 1,000,000 ( ) 5,368 4, If the yield-to-maturity curve goes up to a 7% level or goes down to a 5% level, bond prices become Maturity (years) Price if YTM = 7% Price if YTM = 5% P&Ls are respectively $3,051 and $3,969 when the yield-to-maturity curve goes up to a 7% level or goes down to a 5% level. 3. We draw below the P&L profile of the butterfly depending on the value of the yield to maturity Total return in $ Yield to maturity

27 27 9 CHAPTER 9 Problems Exercise 9.2 The butterfly has a positive convexity. Whatever the value of the yield to maturity, the strategy always generates a gain. This gain is all the more substantial as the yield to maturity reaches a level further away from 6%. We have registered for each month the end-of-month value of a bond index, which is as follows: Month Index Value Month Index Value January 98 July 112 February 101 August 110 March 104 September 111 April 107 October 112 May 111 November 110 June 110 December 113 At the beginning of January, the index value is equal to Compute the arithmetic average rate of return denoted by AARR, and calculate the final index value by using the following formula: Final Index Value = 100 (1 + AARR) Compute the time-weighted rate of return denoted by TWRR, and calculate the final index value by using the following formula: Final Index Value = 100 (1 + TWRR) Same question with the following index values: 4. Conclude. Month Index Value Month Index Value January 85 July 115 February 110 August 130 March 120 September 145 April 135 October 160 May 115 November 145 June 100 December 160 Solution We first compute the monthly rate of return Month Monthly Rate Month Monthly Rate of Return (%) of Return (%) January 2.00 July 1.82 February 3.06 August 1.79 March 2.97 September 0.91 April 2.88 October 0.90 May 3.74 November 1.79 June 0.90 December 2.73

28 28 The arithmetic average rate of return is the average of the monthly rate of return, and is equal to The final index value is equal to AARR = 1.045% Final Index Value = 100 (1 + AARR) 12 = The time-weighted rate of return is given by the following formula: TWRR = [(1 + R J ) (1 + R F ) (1 + R D ] where R J, R F and R D are the monthly rate of return registered in January, February and December respectively. We obtain The final index value is equal to TWRR = 1.024% Final Index Value = 100 (1 + TWRR) 12 = 113 which, of course, corresponds to the actual final index value. 3. We first compute the monthly rate of return Month Monthly Rate Month Monthly Rate of Return (%) of Return (%) January July February August March 9.09 September April October May November 9.38 June December The arithmetic average rate of return is the average of the monthly rate of return, and is equal to The final index value is equal to AARR = 4.920% Final Index Value = 100 (1 + AARR) 12 = The time-weighted rate of return is equal to TWRR = 1.024% and the final index value is, of course, equal to It is incorrect to take the arithmetic average rate of return as a measure of the average return over an evaluation period. In fact, the arithmetic average rate of return is always superior or equal to the time-weighted rate of return. The difference between these two indicators of performance is all the more substantial than the variation in the subperiods returns are large over the valuation period.

29 29 10 CHAPTER 10 Problems Exercise 10.4 We 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 % for a $10 million loan and a 5-year maturity. 1. We suppose that firm B prefers a floating-rate debt as firm A prefers a fixedrate debt. What is the swap they will structure to optimize their financial conditions? 2. 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. Solution Firm B has 2% better conditions at a fixed rate and 1.75% better conditions at a floating rate than firm A. The spread between the conditions obtained by firm A and firm B at a fixed rate and the spread between the conditions obtained by firm A and firm B at a floating rate is different from 0.25%. To optimize their financial conditions: Firm B borrows money at a 9% fixed rate, firm A contracts a loan at a Libor + 2% floating rate and they structure the following swap. Firm B pays Libor % and receives the fixed 9% as firm A receives Libor % and pays the fixed 9%. The financing operation is summarized in the following table: Firm A Firm B Initial Financing (Libor + 2%) (9%) Swap A to B (9%) 9% Swap B to A Libor % (Libor %) Financing Cost (10.875%) (Libor %) Financing Cost without Swap (11%) (Libor %) Gain 0.125% 0.125% By structuring a swap, firm A and firm B have optimized their financial conditions and each firm has gained 0.125%. 2. There is no swap to structure between the two firms so that they can improve their financial conditions. Exercise 10.7 We consider at date T 0, a 1-year Libor swap contract with maturity 10 years and with the following cash-flow schedule:

30 30 F F F F F F F F F F T 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 V 0 V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 Note that T i+1 T i = 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 T 0, zero-coupon rates with maturities T 1,T 2,...T 10 are given in the following table: Maturity ZC rates (%) Maturity ZC rates (%) T T T T T T T T T T Give the price of this swap. 2. What is the swap rate such that the price of this swap is zero? 3. An investor holds a bond portfolio whose price, yield to maturity and modified duration are respectively $9,991,565,452, 9.2% and He wants to be protected against an increase in rates. How many swaps must he sell to protect his bond portfolio? Solution The price of this swap is given by the following formula: SWAP T0 = $10,000,000 ( 10 i=1 9.55%.B(T 0,T i ) 1 + B(T 0,T 10 ) We first have to compute the discount factors B(T 0,T i ) for i = 1, 2,...,10 using the following formula: 1 B(T 0,T i ) = [1 + R(T 0,T i T 0 )] T i T 0 They are given below Maturity B(T 0,T i ) Maturity B(T 0,T i ) T T T T T T T T T T ) Finally, the price of the swap is $28, The swap rate is equal to 9.595%. 3. Assuming that the yield-to-maturity move of the bond portfolio to be hedged is equal to the yield to maturity of the bond contained in the swap, we find the number φ s of swaps that the investor has to sell according to the following

31 31 equation: P B MD B = φ s N S P S MD S where P B is the price in $ of the bond portfolio held by the investor, MD B,the modified duration of the bond portfolio, N S, the nominal amount of the swap. P S is the price in $ of the bond contained in the swap. MD S is the modified duration of the bond contained in the swap. The price P S is equal to %. Using the Excel functions Yield and MDuration, we obtain the yield to maturity and the modified duration of the bond contained in the swap. They are respectively 9.596% and We finally obtain 9,991,565, φ s = 10,000, % = so that the investor has to sell 948 swaps. 11 CHAPTER 11 Problems Exercise 11.1 The Cheapest-to-Deliver on the Repartition Date We are on the repartition date. Consider a futures contract with size Eur 100,000 whose price is 95% and three bonds denoted by A,B and C with the following features. What is the bond that the seller of the futures contract will choose to deliver? Clean Price (%) Conversion Factor (%) Bond A Bond B Bond C Solution 11.1 The seller of the futures contract will choose to deliver the bond that maximizes the difference between the invoice price IP and the cost of purchasing the bond CP, which is called the cheapest-to-deliver. The quantity IP CP is given by the following formula: IP-CP = Size [futures price CF-clean price] where CF is the conversion factor. Here, we obtain the following results: IP-CP Bond A 1,292 Bond B 1,187 Bond C 2,321 The seller of the futures contract will choose to deliver Bond C. Exercise 11.4 An investor holds a bond portfolio with principal value $10,000,000 whose price and modified duration are respectively 112 and He wishes to be hedged against a rise in interest rates by selling futures contracts written on a bond.

32 32 Suppose the futures price of the contract is The contract size is $100,000. The conversion factor for the cheapest-to-deliver is equal to 98.1%. The cheapestto-deliver has a modified duration equal to Give a proof of the hedge ratio φ f as obtained in equation (11.4). 2. Determine the number of contracts φ f he has to sell. Solution To be immunized, the investor has to sell φ f contracts so that (N P dp)+ (φ f N F df) = 0 where N P and N F are respectively the principal amount of the bond portfolio to be hedged and of the futures contract, whose prices (in % of the principal amount) are denoted by P and F.dP and df represent price changes of P and F. We now suppose that dp CTD = CF df where P CTD is the price (in % of the size) of the cheapest-to-deliver, CF its conversion factor. We then write ( (N P dp)+ φ f N F dp ) CTD = 0 CF We now denote by y and y 1, the yield to maturity of the bond portfolio to be hedged and of the cheapest-to-deliver, with the assumption that dy = dy 1 (the yield-to-maturity curve is only affected by parallel shifts). Using the one-dimensional version of Taylor expansion, we obtain [ [N P P (y)] + φ f N ] F CF P CTD (y 1) = 0 and [ [N P P MD P ] + φ f N ] F CF P CTD MD CTD = 0 where MD P and MD CTD are respectively the modified duration of the bond portfolio P and of the cheapest-to-deliver bond. Finally, φ f is given by φ f = N P P MD P CF N F P CTD MD CTD = $Dur P CF $Dur CTD 2. The investor has to sell 125 contracts as given in the following formula: 10,000, % 9.21 φ f = 1,000, % 105.2% 8 = Exercise 11.6 Hedging a Bond Position Using Eurodollar Futures

33 33 Suppose that you are a bond trader. You have just sold $10,000,000 par value of a 10% coupon bond that matures in exactly three-fourth of a year. The yield is 6.5%. Compounding frequency is semiannual. You do not have the bond in inventory, but think you can purchase it in the market tomorrow. You want to hedge your interest-rate risk overnight with Eurodollar futures. Do you buy or sell? How many contracts do you need? Solution 11.6 Exercise To answer those questions, we first need to find the price P and duration D of the bond. ) 1 10,000,000 ( % P = ( 10% 10,000,000 ( ) % 2 = 492, ,008,159 = 10,500, % 2 2 θ i P V (F (θ i )) = , ,008,159 i=1 ) = 123, ,506,119 = 7,629,136 D = 7,629,136 10,500,227 = If interest rates rise by one basis point, the value of the bond loses approximately MD y = D ( 1 ) P y % 2 1 = ( ) 10,500, = $ % 2 } {{ } =MD= Note that since we are short the bond, we will gain when rates go up. We also know that the Eurodollar futures contract loses $25 when Libor goes up by one basis point. That means we want to purchase futures. We gain on our short position in the bond and lose on the futures when rates go up, and vice versa when rates go down. We need to get enough contracts to cover our position. The proper number of contracts is = Since we can only buy a whole number of contracts, we should buy 30 contracts. Speculation with Futures Today, the gross price of a 10-year bond with $1,000 principal amount is At the same moment, the price of the 10-year futures contract which expires in two months is Its nominal amount is $100,000, and the deposit margin is $1,000. One month later, the price of the bond is as the futures price is