5 CHAPTER 5 Problems. We then obtain the following results:

Transcription

1 36 We then obtain the following results: n SR(n) (%) R(0,n) (%) We obtain the following interbank zero-coupon yield curve: Zero-coupon rate (%) 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:

2 37 New Yield (%) Change (bps) 5%/3yr 10%/3yr 5%/10yr 10%/10yr Exercise 5.4 Solution 5.4 Show that the duration of a perpetual bond delivering annually a coupon c with a YTM equal to y is 1+y y. The price P of the perpetual bond is given by the following formula: N c P = (1 + y) i = N c y i=1 where N is the face value of the perpetual bond. The duration D of the perpetual bond is D = (1 + y) P c (y) P(y) = (1 + y) y 2 (1 + y) = y c y Exercise 5.5 Show that the duration of a portfolio P invested in n bonds with weights w i, denominated in the same currency, is the weighted average of each bond s duration: n D p = w i D i i=1 Solution 5.5 Consider n bond prices denoted by P i for i = 1,...n, and a bond portfolio that is the sum of each of these n bonds. We denote by P, the price of this portfolio and suppose that all the bonds have the same YTM equal to y. Then n P(y) = P i (y) and P (y) = i=1 n i=1 P i (y) Dividing the previous equation by P(y) and multiplying it by (1 + y), we obtain (1 + y) P (y) n P(y) = (1 + y) P i (y) P(y) i=1 or n D P = w i D i i=1

3 38 where D P is the portfolio duration, D i the duration of bond, i, andw i = P i(y) P(y) is the weight of bond i in the portfolio P. 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 ,

4 39 2. When coupon frequency and compounding frequency are assumed to be semiannual, we obtain the following results: 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

5 40 3. The approximate change in price is $0.627 P MD y P = $ = $0.627 Exercise 5.13 Solution 5.13 Exercise 5.15 Solution 5.15 You own a 7% Treasury bond with $100 face value that has a modified duration of 6.3. The clean price is You have just received a coupon payment 12 days ago. Coupons are received semiannually. 1. If there are 182 days in this coupon period, what is the accrued interest? 2. Is the yield greater than the coupon rate or less than the coupon rate? How do you know? 3. Use the modified duration to find the approximate change in value if the yield were to suddenly rise by 8 basis points. 4. Will the actual value change more or less than this amount? Why? 1. The accrued interest AI is given by AI = = Since the price is so far below par, the yield must be higher than the coupon rate. 3. The approximate change in price is given by the following equation P MD y P = 6.3 $( ) = $ The actual loss will be less than this amount, due to convexity (see Chapter 6). Today is 01/01/98. On 06/30/99, we make a payment of $100. We can only invest in a risk-free pure discount bond (nominal $100) that matures on 12/31/98 and in a risk-free coupon bond, nominal $100 that pays an annual interest (on 12/31) of 8% and matures on 12/31/00. Assume a flat term structure of 7%. How many units of each of the bonds should we buy in order to be perfectly immunized? We first have to compute the present value PV of the debt, which is the amount we will have to deposit PV = 100 = (1.07) 1.5 We also compute the price P 1 of the 1-year pure discount bond P 1 = = Similarly, the price P 3 of the 3-year coupon bond is P 3 = (1.07) (1.07) 3 = 102.6

6 41 The duration of the 1-year pure discount bond is obviously 1. The duration D 3 of the 3-year coupon bond is 8 8 D 3 = (1.07) (1.07) = We now compute the number of units of the 1-year and the 3-year bonds (q 1 and q 3 respectively), so as to achieve a $duration equal to that of the debt, and also a present value of the portfolio equal to that of the debt. We know that the duration of the debt we are trying to immunize is 1.5. Therefore, q 1 and q 3 are given as the unique solution to the following system of equations: { q q 3 = q q 3 = { q1 = q 3 = 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.

7 42 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. (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). Exercise 5.22 A trader implements a duration-neutral strategy, which consists in buying a cheap bond and selling a rich bond. This is the rich and cheap bond strategy. Today, the rich and cheap bonds have the following characteristics: Bond Coupon (%) Maturity (years) YTM (%) Rich Cheap Coupon frequency and compounding frequency are assumed to be annual. Face value are $100 for the two bonds. Compute the BPV of the two bonds and find the hedge position. Solution 5.22 We first calculate the price, modified duration (MD) and BPV of each bond. Bond Price MD BPV Rich Cheap

8 43 We take a long position of $100,000,000 in the 5.5%/12-year bond. The hedge ratio HR is equal to HR = = Then we have to take a short position of x in the 5%/10-year bond, where x is given by x = HR $100,000,000 = $110,818,000 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 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%)

9 44 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 6.3 Solution Compute the modified duration and convexity of a 6%, 25-year bond selling at a yield of 9%. Coupon frequency and compounding frequency are assumed to be semiannual. 2. What is its estimated percentage price change for a yield change from 9% to 11% using the one-order Taylor expansion? Using the two-order Taylor expansion? Compare both of them with the actual change? 3. Same question when the yield decreases by 200 basis points. Conclude. 1. For a 6%, 25-year bond selling at a yield of 9%, modified duration amounts to 10.62, while convexity is equal to The estimated percentage price change, for a yield change from 9% to 11% is equal to (0.02) (182.92) (0.02) 2 = = 17.58%, while the actual change is 18.03%.

10 45 3. If the yield decreases by 200 basis points, instead, then the estimated price change is % due to duration, and +3.66% due to convexity, that is 24.90%; as a whole, the actual price change is 25.46%. The estimated price change is no longer symmetric around the current yield because the price function has curvature. Exercise 6.6 Solution 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. 1. 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 = , = $

11 46 5. 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 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 = 279, ,043 φ , , ,912,260, ,432 Exercise 6.10 Computing the Level, Slope and Curvature $Durations of a Bond Portfolio using the Nelson and Siegel Extended Model

12 47 On 09/02/02, the values of the Nelson and Siegel Extended parameters are as follows: β 0 β 1 β 2 τ 1 β 3 τ 2 5.9% 1.6% 0.5% 5 1% 0.5 Recall from Chapter 4 that the continuously compounded zero-coupon rate R c (0,θ) is given by the following formula: ( ) R c (0,θ)= β 0 + β 1 1 exp τ θ 1 θ τ 1 ( ) + β 2 1 exp τ θ 1 exp θ τ 1 ( ) + β 3 1 exp τ θ 2 exp θ τ 2 ) ( θτ1 ) ( θτ2 1. Draw the zero-coupon yield curve associated with this set of parameters. 2. We consider three bonds with the following features. Coupon frequency is annual. Maturity Coupon (years) (%) Bond Bond Bond Compute the price and the level, slope and curvature $durations of each bond. Compute also the same $durations for a portfolio with 100 units of Bond 1, 200 units of Bond 2 and 100 units of Bond The parameters of the Nelson and Siegel Extended model change instantaneously to become β 0 β 1 β 2 τ 1 β 3 τ 2 5.5% 1% 0.1% 5 2% 0.5 (a) Draw the new zero-coupon yield curve. (b) Compute the new price of the bond portfolio and compare it with the value given by the following equation: New Estimated Price = Former Price + β 0.D 0,P + β 1.D 1,P + β 2.D 2,P + β 3.D 3,P where β i is the change in value of parameter β i,andd i,p is the $duration of the bond portfolio associated with parameter β i. 4. Same questions when the coupon frequency is semiannual.

13 48 Solution We obtain the following zero-coupon yield curve Zero-coupon rate (%) Maturity 2. For each bond, the price and level, slope and curvature $durations denoted respectively by D 0,D 1,D 2 and D 3 are given in the following table: Price ($) D 0 D 1 D 2 D 3 Bond Bond Bond , Portfolio 40, , , , , The level, slope and curvature $durations of the bond portfolio denoted by D 0,P, D 1,P,D 2,P and D 3,P are simply obtained by using the following formulas: D 0,P = 100 ( ) ( ) ( 1,109.21) = 260, D 1,P = 100 ( ) ( ) ( ) = 130, D 2,P = 100 ( 56.49) ( ) ( ) = 69, D 3,P = 100 ( 47.08) ( 48.57) ( 51.82) = 19, (a) We draw the new curve on the following graph:

14 49 Zero-coupon rate (%) Curve at the origin New curve Maturity (b) The new price is equal to New Price = 39,977 whereas the new estimated price obtained by using the equation given in the exercise is New Estimated Price = 40,333 + ( 0.4%).( 260,075) + 0.6%.( 130,698) +0.6%.( 69,832) + 1%.( 19,604) = 39,975 We conclude that the price change of the bond portfolio is well explained by Nelson Siegel $durations multiplied by the change in value of the different parameters. 4. When the coupon frequency is semiannual, we obtain the following results: Price ($) D 0 D 1 D 2 D 3 Bond Bond Bond , Portfolio 40, , , , , Exercise 6.12 Bond Portfolio Hedge using the Nelson Siegel Extended Model We consider the Nelson Siegel Extended zero-coupon rate function ( ) R c (t, θ) = β 0 + β 1 1 exp ( ) τ θ 1 + β 2 1 exp τ θ ) 1 exp ( θτ1 θ τ 1 +β 3 1 exp ( θ τ 2 ) θ τ 2 ( exp θ ) τ 2 θ τ 1

15 50 where R c (t, θ) is the continuously compounded zero-coupon rate at date t with maturity θ. On 09/02/02, the model is calibrated, parameters being as follows: β 0 β 1 β 2 τ 1 β 3 τ 2 5.9% 1.6% 0.5% 5 1% 0.5 At the same date, a manager wants to hedge its bond portfolio P against interestrate risk. The portfolio contains the following Treasury bonds (delivering annual coupons, with a $100 face value): Bond Maturity Coupon Quantity Bond 1 01/12/ ,000 Bond 2 04/12/ ,000 Bond 3 07/12/ ,000 Bond 4 10/12/ ,000 Bond 5 03/12/ ,000 Bond 6 10/12/ ,000 Bond 7 01/12/ ,000 Bond 8 03/12/ ,000 Bond 9 07/12/ ,000 Bond 10 01/12/ ,000 Bond 11 07/12/ ,000 Bond 12 01/12/ ,000 Bond 13 07/12/ ,000 We consider Treasury bonds as hedging instruments with the following features: Hedging Asset Coupon Maturity Hedging Asset /15/06 Hedging Asset /28/12 Hedging Asset /05/15 Hedging Asset /10/20 Hedging Asset /10/31 Coupons are assumed to be paid annually, and the face value of each bond is $100. At date t, we force the hedging portfolio to have the opposite value of the portfolio to be hedged. 1. Compute the price and level, slope and curvature $durations of portfolio P. 2. Compute the price and level, slope and curvature $durations of the five hedging assets. 3. Which quantities φ 1,φ 2,φ 3, φ 4 and φ 5 of each hedging asset 1, 2, 3, 4and5do we have to consider to hedge the portfolio P? 4. The parameters of the Nelson and Siegel Extended model change instantaneously to become

16 51 β 0 β 1 β 2 τ 1 β 3 τ 2 6.5% 1% 0.1% 5 2% 0.5 (a) What is the price of the bond portfolio after this change? If the manager has not hedged its portfolio, how much money has he lost? (b) What is the variation in price of the global portfolio (where the global portfolio is the bond portfolio plus the hedging instruments)? (c) Conclusion. Solution The price and level, slope and curvature $durations of bond portfolio P are given in the following table: Price D 0 D 1 D 2 D 3 $12,723, ,075,273 42,504,124 25,924,204 6,021,090 where D i = β P i for i = 0, 1, 2 and 3 are the level, slope and curvature $durations of P in the Nelson Siegel Extended model. 2. The level, slope and curvature $durations of the five hedging instruments are given in the following table: Hedging Asset Price ($) D 0 D 1 D 2 D 3 Hedging Asset Hedging Asset Hedging Asset Hedging Asset Hedging Asset We are looking for the quantities φ 1,φ 2,φ 3, φ 4 and φ 5 of each hedging asset as the solutions to the following linear system: 1 φ φ 2 φ 3 φ 4 = φ ,004,516 42,733,844 26,093,954 6,044,277 12,769, (a) The price of the bond portfolio P, after the change in parameters, is equal to $11,663,433. With no hedge, the manager has lost $1,060,170 Loss = $11,715,756 $12,769,376 = $1,053,620

17 52 7 CHAPTER 7 Problems Exercise 7.1 (b) The prices of the five hedging assets, after the change in parameters, are given in the following table Hedging Asset Price ($) Hedging Asset Hedging Asset Hedging Asset Hedging Asset Hedging Asset With the hedging portfolio (which contains the five hedging assets in adequate quantity), the manager gains $1,054,624. Gain = 39,507.( ) 74,586.( ) + 52,726.( ) 37,764.( ) 23,649.( ) = $1,054,624 The variation in price of the global portfolio (bond portfolio + hedging instruments) is then equal to $5,546 ($1,054,624 $1,060,170). (c) The hedge is efficient. Would you say it is easier to track a bond index or a stock index. Why or why not? Solution 7.1 Exercise 7.2 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. What are the pros and cons of popular indexing methodologies in the fixed-income universe?