Problems and Solutions

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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 1.2 Consider the same bond as in the previous exercise. We are still on 12/04/ Compute the accrued interest taking into account the Actual/Actual day-count basis. 2. Same question if we are now on 09/06/02. Solution The last coupon has been delivered on 11/06/01. There are 28 days between 11/06/01 and 12/04/01, and 181 days between the last coupon date (11/06/01) and the next coupon date (05/06/02). Hence, the accrued interest is equal to $6.188 Accrued Interest = 28 $40 = $ The last coupon has been delivered on 05/06/02. There are 123 days between 05/06/02 and 09/06/02, and 184 days between the last coupon date (05/06/02) and the next coupon date (11/06/02). Hence, the accrued interest is equal to $ Accrued Interest = 123 $40 = $

2 2 Exercise 1.3 Solution 1.3 Exercise 1.4 Solution 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 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. 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

3 3 are 190 days between 04/15/99 and 10/25/99 on the 30/360 day-count basis 15 + (5 30) + 25 = 190 Exercise 1.5 Finally, the accrued interest with the 30/360 day-count basis is equal to Eur % Eur 100 = Eur Some bonds have irregular first coupons. A long first coupon is paid on the second anniversary date of the bond and starts accruing on the issue date. So, the first coupon value is greater than the normal coupon rate. A long first coupon with regular value is paid on the second anniversary date of the bond and starts accruing on the first anniversary date. So, the first coupon value is equal to the normal coupon rate. A short first coupon is paid on the first anniversary date of the bond and starts accruing on the issue date. The first coupon value is smaller than the normal coupon rate. A short first coupon with regular value is paid on the first anniversary date of the bond and has a value equal to the normal coupon rate. Consider the four following bonds with nominal value equal to 1 million euros and annual coupon frequency: Bond 1: issue date 05/21/96, coupon 5%, maturity date 05/21/02, long first coupon, redemption value 100%; Bond 2: issue date 02/21/96, coupon 5%, maturity date 02/21/02, long first coupon with regular value, redemption value 99%; Bond 3: issue date 11/21/95, coupon 3%, maturity date 3 years and 2 months, short first coupon, redemption value 100%; Bond 4: issue date 08/21/95, coupon 4.5%, maturity date 08/21/00, short first coupon with regular value, redemption value 100%. Compute the future cash flows of each of these bonds. Solution 1.5 Bond 1 pays 100,000 euros on 05/21/98, 50,000 euros on 05/21/99, 05/21/00, 05/21/01 and 1,050,000 euros on 05/21/02. Bond 2 pays 50,000 euros on 02/21/98, 02/21/99, 02/21/00, 02/21/01 and 1,040,000 euros on 05/21/02. Bond 3 pays 5,000 euros on 01/21/96, 30,000 euros on 01/21/97, 01/21/98 and 1,030,000 euros on 01/21/99. Bond 4 pays 45,000 euros on 08/21/96, 08/21/97, 08/21/98, 08/21/99 and 1,045,000 euros on 08/21/00. 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

4 4 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. Exercise 1.13 Treasury bills are quoted using the yield on a discount basis or on a moneymarket basis. 1. The yield on a discount basis denoted by y d is computed as y d = F P B F n where F is the face value, P the price, B the year-basis (365 or 360) and n is the number of calendar days remaining to maturity. Prove in this case that the price of the T-bill is obtained using the equation ( P = F 1 n y ) d B 2. The yield on a money-market basis denoted by y m is computed as y m = B y d B n y d Prove in this case that the price of the T-bill is obtained using the equation 3. Show that F P = ( 1 + n y m ) B y d = B y m B + n y m Solution From the equation we find and finally, we obtain 2. From the equation y d = F P F n y d B P = F B n 1 = P F ( 1 n y ) d B y m = B y d B n y d

5 5 we find Then, we have Finally, we obtain 3. From the equation y m = n y m B B F F P B n F P F = F P F P F B n B n = = F P P F P = ( 1 + n y m ) B B n F P F ( 1 F P F = F P 1 ) we find Then, we have y m = B y d B n y d y m (B n y d ) B y d = 0 y d ( n y m B) = B y m Finally, we obtain y d = B y m B + n y m 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 ( 1 + 5% P = $10,000,000 ( ) = $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 )

6 6 Exercise 1.16 Solution 1.16 On 01/03/2002, an investor buys $1 million US T-Bill with maturity date 06/27/2002 and discount yield 1.76% on the settlement date. 1. What is the price of the T-Bill? 2. What is the equivalent money-market yield? 1. The settlement date of the transaction is 01/04/2002 (trading date plus 1 working day). There are 174 calendar days between the settlement date and the maturity date. The price P of the T-Bill is equal to ( % 174 ) = The equivalent money-market yield is equal to 1.775% 1.76% % = 1.775% 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 Exercise 2.7 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 1. 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.

7 7 Solution 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 2.10 Solution What is the yield to maturity of a 5-year bond with a nominal value of $100, a 10% coupon rate, an annual coupon frequency and a price of ? 2. Same question for a price of 100 and The yield to maturity y of this bond is the solution to the following equation [ ] 1 + P = N c y 1 (1 + y) n N (1 + y) n where N, c, P and n are respectively the nominal value, the coupon rate, the price and the number of years to maturity of the bond. Here, y is solution to = 10 y [ 1 ] 1 (1 + y) (1 + y) 5 Using, for example, Newton s three points method (or the Solver function in Excel), we obtain %. Note that we can also use the Excel function Yield to obtain y. 2. Yields to maturity (YTM) of the bond for different prices are given in the following table

8 8 Price YTM (%) Exercise 2.13 Solution 2.13 Consider the following bond: annual coupon 5%, maturity 5 years, annual compounding frequency. 1. What is its relative price change if its required yield increases from 10% to 11%? 2. What is its relative price change if its required yield increases from 5% to 6%? 3. What conclusion can you draw from these examples? Explain why. 1. The initial price P is equal to P = 5 (1 + 10%) + 5 (1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%) 5 = After the yield change, the price becomes P = 5 (1 + 11%) + 5 (1 + 11%) (1 + 11%) (1 + 11%) (1 + 11%) 5 = Hence, the bond price has decreased by P P = 3.97% P 2. The initial price P is equal to P = 5 (1 + 5%) + 5 (1 + 5%) (1 + 5%) (1 + 5%) (1 + 5%) 5 = 100 After the yield change, the price becomes P = 5 (1 + 6%) + 5 (1 + 6%) (1 + 6%) (1 + 6%) (1 + 6%) 5 = Hence, the bond price has decreased by P P = 4.21% P 3. In low interest-rate environments, the relative price volatility of a bond is higher than in high interest-rate environments for the same yield change (here, in our example +1%). This is due to the convexity relationship between the price of a bond and its yield.

9 9 Exercise 2.14 Solution 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? 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? 1. 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

10 10 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 = $ 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 2.15 Let us consider the two following French Treasury bonds whose characteristics are the following:

11 11 Name Maturity Coupon Price (years) Rate (%) Bond Bond Your investment horizon is 6 years. Which of the two bonds will you select? Solution 2.15 Exercise 2.18 It depends on the level of the reinvestment rate, at which you can reinvest the coupons of Bond 1, as well as on the yield to maturity of Bond 2 at horizon. If you suppose, for example, that the reinvestment rate is equal to the yield to maturity of Bond 2 at horizon, then the total return of Bond 2 will decrease as the reinvestment rate increases, as opposed to Bond 1. Indeed, while the unique source of return for Bond 1 is its reinvested coupons, it lies for Bond 2 in its price appreciation. Bond 1 and Bond 2 will yield nearly the same annualized return (5.15%) for a reinvestment rate of 6.365%. We consider three bonds with the following features Bond Maturity (years) Annual Coupon Price Bond Bond Bond Find the 1-year, 2-year and 3-year zero-coupon rates from the table above. 2. We consider another bond with the following features Bond Maturity Annual Coupon Price Bond 4 3 years Use the zero-coupon curve to price this bond. 3. Find an arbitrage strategy. Solution The 1-year zero-coupon rate denoted by R(0, 1), verifies R(0, 1) = We find the expression R(0, 1) = = 3.228% The 2-year zero-coupon rate denoted by R(0, 2), verifies % (1 + R(0, 2)) 2 = We find the expression ( ) 1/2 108 R(0, 2) = 1 = 4.738% %

12 12 The 3-year zero-coupon rate denoted by R(0, 3), verifies % + 8 ( %) (1 + R(0, 3)) 3 = We find the expression ( ) 1/3 108 R(0, 3) = % 1 = 5.718% 8 ( %) 2 2. The price P of Bond 4 using the zero-coupon curve is given by the following formula: 9 P = % + 9 ( %) ( %) 3 = This bond is underpriced by the market compared to its theoretical value. There is an arbitrage if the market price of this bond reverts to the theoretical value. We have to simply buy the bond at a $ price and hope that it is mispriced by the market and will soon revert to around $ 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. 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

13 13 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 ( ) i 1/10 + 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 + R ) 2(0, 1.5) 3 ( = 1 + R 2(0, 0.5) = )( 1 + F 2(0, 0.5, 0.5) 2 ) ( 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 ) ) 2

14 14 that we are given: P = ( ) ( ) 2 + ( ) 3 = Exercise 2.26 Solution 2.26 Consider a coupon bond with n = 20 semesters (i.e., 10 years) to maturity, an annual coupon rate c = 6.5% (coupons are paid semiannually), and nominal value N = $1,000. Suppose that the semiannually compounded yield to maturity (YTM) of this bond is y 2 = 5.5%. 1. Compute the current price of the bond using the annuity formula. 2. Compute the annually compounded YTM and the current yield of the bond. Compare them with y If the yield to maturity on the bond does not change over the next semester, what is the Holding Period Return (HPR) obtained from buying the bond now and selling it one semester from now, just after coupon payment? At what price will the bond sell one semester from now just after coupon payment? 1. For the current price of the bond, we use the formula P 0 = N c ( ) 1 + y 2 so that P 0 = 1, % 5.5% ( 1 1 (1 + y 2 /2) n ) 1 ( ) 20 + N (1 + y 2 /2) n 1,000 = 1, ( ) The annually compounded yield to maturity (YTM) denoted by y and the current yield denoted by y c are obtained using the following formulas: ( y = 1 + y ) ( = ) 2 1 = y c = cn ,000 = = P 0 1, Therefore, they are both larger than y First, we compute P 1, the price of the bond one semester from now: P 1 = N c ( ) 1 N 1 + y 2 = 1, % 5.5% = 1,073.2 (1 + y 2 /2) n 1 ( 1 1 ( ) 19 (1 + y 2 /2) n 1 ) + 1,000 ( ) 19 The Holding Period Return from buying the bond now and selling it one semester from now is then: HPR = P 1 P 0 + cn 2 1, , = = 2.75% P 0 1,076.14

15 15 3 CHAPTER 3 Problems Exercise 3.1 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%. Exercise 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?

16 16 Solution 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 Par yield curve Zero-coupon yield curve Forward yield curve Yield (%) Maturity

17 17 Exercise 3.8 Solution 3.8 Exercise 3.11 When the current par yield curve is increasing (respectively, decreasing), the current zero-coupon rate curve is above (respectively, below) it, so as to offset the fact that the sum of the coupons discounted at the coupon rate is inferior (respectively, superior) to the sum of the coupons discounted at the zero-coupon rate. Give a proof of this assertion. Let us denote by c(i), the par yield with maturity i and by R(0,i), the zero-coupon rate with maturity i. Let us assume for k<nthat c(n) > c(k) At the first rank, we have R(0, 1) = c(1) At the second rank, c(2) 1 + c(2) c(2) (1 + c(2)) 2 = c(2) 1 + R(0, 1) c(2) [(1 + R(0, 2)] 2 Let us do a limited development at the first order of this last expression. Then c(2).(1 c(2)) + (1 + c(2)).(1 2c(2)) = c(2).(1 R(0, 1)) + (1 + c(2)).(1 2R(0, 2)) R(0, 2) c(2) = 1 2. c(2).(c(2) c(1)) 1 + c(2) As c(2) >c(1), R(0, 2) >c(2). The proposition is true at the first and second ranks. Let us assume that it is true at rank n 1 and let us prove it at rank n. n c(n) (1 + c(n)) k + 1 n (1 + c(n)) n = c(n) (1 + R(0,k)) k + 1 (1 + R(0,n)) n k=1 Let us do again a limited development at the first order. Then n n c(n).(1 kc(n)) + 1 nc(n) = c(n).(1 kr(0,k))+ 1 nr(0,n) k=1 k=1 k=1 k=1 After simplification [ n ] R(0,n) c(n) = 1 n. c(n) 1 + c(n). c(n 1) n 1 k.(c(n) c(k)) + c(k) 1 + c(n 1) n 1 c(n 1) as 1+c(n 1) k=1 c(k) can be considered negligible since c(n 1) zero, we can make the following approximation: R(0,n) c(n) = 1 n. c(n) n 1 + c(n). k.(c(n) c(k)) k=1 1+c(n 1) As c(n) > c(k), we obtain R(0,n)>c(n), which proves the assertion. At date t = 0, we observe the following zero-coupon rates in the market: k=1 is close to

18 18 Maturity Zero-Coupon Maturity Zero-Coupon (years) Rate (%) (years) Rate (%) What are the 1-year maturity forward rates implied by the current term structure? 2. Over a long period, we observe the mean spreads between 1-year maturity forward rates and 1-year maturity realized rates in the future. We find the following liquidity premiums: L 2 = 0.1% L 3 = 0.175% L 4 = 0.225% L 5 = 0.250% Taking into account these liquidity premiums, what are the 1-year maturity future rates expected by the market? Solution year maturity forward rates are given by the following formula: [1 + R(0,T)] T = [1 + R(0,T 1)] T 1. [1 + F(0,T 1, 1)] where R(0,T) is the zero-coupon rate at date t = 0 with T -year maturity and F(0,T 1, 1) is the 1-year maturity forward rate observed at date t = 0, starting at date t = T 1 and maturing one year later. F(0, 4, 1) is obtained by solving the following equation: (1 + 7%)5 F(0, 4, 1) = ( %) 4 1 = 7.804% Using the same equation, we obtain Forward Rates F(0, 1, 1) 7.009% F(0, 2, 1) 7.507% F(0, 3, 1) 7.705% 2. 1-year maturity future rates expected by the market are given by the following formula: [1 + R(0,T)] T = [1 + R(0,T 1)] T 1.[1 + F a (0,T 1, 1) + L T ] where F a (0,T 1, 1) is the 1-year maturity future rate expected by the market at date t = 0, starting at date t = T 1 and finishing one year later. Using the last equation, we find the relation between the forward rate and the future rate expected by the market F a (0,T 1, 1) = F(0,T 1, 1) L T

19 19 We finally obtain Expected Future Rates F a (0, 1, 1) 6.909% F a (0, 2, 1) 7.332% F a (0, 3, 1) 7.480% F a (0, 4, 1) 7.554% Exercise 3.12 Solution 3.12 Monetary policy and long-term interest rates Consider an investor with a 4-year investment horizon. The short-term (longterm respectively) yield is taken as the 1-year (4-year respectively) yield. The medium-term yields are taken as the 2-year and 3-year yields. We assume, furthermore, that the assumptions of the pure expectations theory are valid. For each of the following five scenarios, determine the spot-yield curve at date t = 1. The yield curve is supposed to be initially flat at the level of 4%, at date t = 0. (a) Investors do not expect any Central Bank rate increase over four years. (b) The Central Bank increases its prime rate, leading the short-term rate from 4 to 5%. Investors do not expect any other increase over four years. (c) The Central Bank increases its prime rate, leading the short-term rate from 4 to 5%. Investors expect another short-term rate increase by 1% at the beginning of the second year, then no other increase over the last two years. (d) The Central Bank increases its prime rate, leading the short-term rate from 4 to 5%. Nevertheless, investors expect a short-term rate decrease by 1% at the beginning of the second year, then no other change over the last two years. (e) The Central Bank increases its prime rate, leading the short-term rate from 4 to 5%. Nevertheless, investors expect a short-term rate decrease by 1% each year, over the following three years. What conclusions do you draw from that as regards the relationship existing between monetary policy and interest rates? Let us denote F a (1,n,m),them-year maturity future rate anticipated by the market at date t = 1 and starting at date t = n, and R(1,n) the n-year maturity zero-coupon rate at date t = 1. In each scenario, we have Scenario a % Scenario b % Scenario c % Scenario d % Scenario e % R(1, 1) F a (1, 2, 1) F a (1, 2, 2) F a (1, 2, 3) Using the following equation: 1 + R(t, n) = [(1 + R(t, 1))(1 + F a (t, t + 1, 1))(1 + F a (t, t + 2, 1))... (1 + F a (t, t + n 1, 1))] 1/n

20 20 we find the spot zero-coupon yield curve in each scenario. Scenario a Scenario b Scenario c Scenario d Scenario e % % % % % R(1, 1) R(1, 2) R(1, 3) R(1, 4) In the framework of the pure expectations theory, monetary policy affects long-term rates by directly impacting spot and forward short-term rates, which are supposed to be equal to market short-term rate expectations. But what about these expectations? The purpose of the exercise is to show that market shortterm rate expectations play a determining role in the response of the yield curve to monetary policy. More meaningful than the Central Bank action itself is the way market participants interpret this action. Is it a temporary action or rather the beginning of a series of similar actions...? We can draw three conclusions from the exercise. First, the direction taken by interest rates compared with that of the Central Bank prime rate depends on the likelihood, perceived by the market, that the Central Bank will question its action in the future through reversing its stance. Under the (b) and (c) scenarios, the Central bank action is perceived to either further increase its prime rate or leave things as they are. Consequently, long-term rates increase following the increase in the prime rate. Under the (d) scenario, the Central Bank is expected to exactly offset its increasing action in the future. Nevertheless, its action on short-term rates still remains positive over the period. As a result, long-term interest rates still increase. In contrast, under the (e) scenario, the Central Bank is expected to completely reverse its stance through a decreasing action in the future, that more than offsets its initial action. Consequently, long-term interest rates decrease. Second, the magnitude of the response of long-term rates to monetary policy depends on the degree of monetary policy persistence that is expected by the market. Under the (b) and (c) scenarios, the Central Bank action is viewed as relatively persistent. Consequently, the long-term interest-rate change either reflects the instantaneous change in the prime rate or exceeds it. Under the (d) scenario, as the Central Bank action is perceived as temporary, the change in long-term rates is smaller than the change in the prime rate. Third, the reaction of long-term rates to monetary policy is more volatile than that of short-term rates. That is, the significance of the impact of market expectations on interest rates increases with the maturity of interest rates. These expectations only play a very small role on short-term rates. As illustrated by the exercise, the variation margin of the 2-year interest rate following a 100-bps increase of the Central Bank prime rate is contained between 50 and 150 bps, while the variation margin of the 4-year interest rate is more volatile (between 50 bp and +175 bps).

21 21 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. 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 103 B(0, 1) = B(0, 2) B(0, 3) We get the following discount factors: B(0, 1) B(0, 2) B(0, 3) B(0, 4) = B(0, 5) B(0, 4) B(0, 5)

22 22 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% Exercise 4.3 Suppose we know from market prices, the following zero-coupon rates with maturities inferior or equal to one year: Maturity Zero-coupon Rate (%) 1Day Month Months Months Months Months Year 4.00 Now, we consider the following bonds priced by the market until the 4-year maturity: Maturity Annual Coupon (%) Gross Price 1 Year and 3 Months Year and 6 Months Years Years Years The compounding frequency is assumed to be annual. 1. Using the bootstrapping method, compute the zero-coupon rates for the following maturities: 1 year and 3 months, 1 year and 6 months, 2 years, 3 years and 4 years. 2. Draw the zero-coupon yield curve using a linear interpolation. Solution We first extract the 1-year-and-3-month maturity zero-coupon rate. In the absence of arbitrage opportunities, the price of this bond is the sum of its future discounted cash flows: = ( %) 1/ (1 + x) 1+1/4 where x is the 1-year-and-3-month maturity zero-coupon rate to be determined. Solving this equation (for example with the Excel solver), we obtain 4.16% for x. Applying the same procedure with the 1-year and 6-month maturity and the 2-year maturity bonds, we obtain respectively 4.32% and 4.41% for x. Next,

23 23 we have to extract the 3-year maturity zero-coupon rate, solving the following equation: 98.7 = 4 (1 + 4%) + 4 ( %) (1 + y%) 3 y is equal to 4.48% and finally, we extract the 4-year maturity zero-coupon rate denoted by z, solving the following equation: = 5 (1 + 4%) + 5 ( %) ( %) (1 + z%) 4 z is equal to 4.57%. 2. Using the linear graph option in Excel, we draw the zero-coupon yield curve Zero-coupon rate (%) Maturity 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 = %

24 24 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 ) = R(0, 11 3/4 ) = % % % % 5 = 8.70% = 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).

25 25 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) R(0, 5) = R(0, 8) = (6 5)R(0, 4) + (5 4)R(0, 6) 6 4 (9 8)R(0, 7) + (8 7)R(0, 9) 9 7 = = R(0, 4) + R(0, 6) 2 R(0, 7) + R(0, 9) 2 = 6.375% = 6.740% Using the following standard equation in which lies the zero-coupon rate R(0,t) and the discount factor B(0,t) B(0,t)= 1 (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). 3. Using the Excel function Linest, we obtain the following values for the parameters:

26 26 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: Maturity B(0,t) B(0,t) R(0,t) (%) R(0,t) (%) ? ? ? ?

27 27 5. 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 4.8 From the prices of zero-coupon bonds quoted in the market, we obtain the following zero-coupon 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, andb(0,t) is the discount factor at date 0 with maturity t. We need to know the value for R(0, 0.8), R(0, 1.5), R(0, 3.4), R(0, 5.25), R(0, 8.3) and R(0, 9),whereR(0,i)is the zero-coupon rate at date 0 with maturity i. We have to estimate them, and test two different methods. 1. We postulate the following form for the zero-coupon rate function R(0,t): R(0,t)= a + bt + ct 2 + dt 3 (a) Estimate the coefficients a, b, c and d, which best approximate the given zero-coupon rates using the following optimization program: (R(0,i) R(0,i)) 2 Min a,b,c,d i

28 28 where R(0,i)are the zero-coupon rates given by the market. Compare these rates R(0,i) to the rates R(0,i) given by the model. (b) Find the value for the six zero-coupon rates that we are looking for. (c) Draw the two following curves on the same graph: The market curve by plotting the market points. The theoretical curve as derived from the prespecified functional form. 2. Same question as the previous one. But we now 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 where B(0,i) are the discount factors given by the market. 3. Conclude. i Solution (a) 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 theoretical values for the zero-coupon rates R(0,t) given by the model and compared with the market values R(0,t) Maturity (years) R(0,t) (%) R(0,t) (%)

29 29 (b) We find the value for the six other zero-coupon rates we are looking for in the following table: Maturity (years) R(0,t) (%) (c) We now draw the graph of the market curve and the theoretical curve. We see that the three-order polynomial form used to model the zero-coupon rates is not well adapted to the market configuration, which is an inverted curve at the short-term segment Zero-coupon rates (%) Maturity 2. (a) 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:

30 30 Maturity (years) B(0,t) B(0,t) R(0,t) (%) R(0,t) (%) (b) By using the standard relationship between the discount factor and the zerocoupon rate ( ) 1/t 1 R(0,t)= 1 B(0,t) we find the value for the six other zero-coupon rates we are looking for in the following table: Maturity (years) B(0,t) R(0,t) (%) (c) We now draw the graph of the market curve and the theoretical curve. We can see that the three-order polynomial form used to model the discount function is not well adapted to the market configuration considered Zero-coupon rates (%) Maturity

31 31 3. Note first that this is a case of an inverted zero-coupon curve at the short-term end. We conclude that the two functional forms we have tested are unadapted to fit with accuracy the observed market zero-coupon rates. Exercise 4.10 Solution 4.10 Consider the Nelson and Siegel model [ ( )] [ ( ) 1 exp θ R c (0,θ)= β 0 + β τ 1 exp θ 1 + β τ 2 θ τ θ τ ( exp θ ) ] τ Our goal is to analyze the impact of the parameter 1/τ on the zero-coupon curve for three different configurations, an increasing curve, a decreasing curve and an inverted curve at the short-term end. 1. We consider the increasing curve corresponding to the following base-case parameter values: β 0 = 8%, β 1 = 3%, β 2 = 1% and 1/τ = 0.3. We give successively five different values to the parameter 1/τ: 1/τ = 0.1, 1/τ = 0.2, 1/τ = 0.3, 1/τ = 0.4 and1/τ = 0.5. The other parameters are fixed. Draw the five different yield curves to estimate the effect of the parameter 1/τ. 2. We consider the decreasing curve corresponding to the following base-case parameter values: β 0 = 8%, β 1 = 3%, β 2 = 1% and 1/τ = 0.3. We give successively five different values to the parameter 1/τ: 1/τ = 0.1, 1/τ = 0.2, 1/τ = 0.3, 1/τ = 0.4 and1/τ = 0.5. The other parameters are fixed. Draw the five different yield curves to estimate the effect of the parameter 1/τ. 3. We consider the inverted curve corresponding to the following base-case parameter values: β 0 = 8%, β 1 = 1%, β 2 = 2% and 1/τ = 0.3. We give successively five different values to the parameter 1/τ: 1/τ = 0.1, 1/τ = 0.2, 1/τ = 0.3, 1/τ = 0.4 and1/τ = 0.5. The other parameters are fixed. Draw the five different yield curves to estimate the effect of the parameter 1/τ. 1. The following graph shows clearly the effect of the parameter 1/τ in the five different scenarios for an increasing curve. The parameter 1/τ affects the slope of the curve. The higher the 1/τ, the more rapidly the curve goes to its long-term level (8% in the exercise) Zero-coupon rate /t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = Maturity

32 32 2. The following graph shows clearly the effect of the parameter 1/τ in the five different scenarios for a decreasing curve. The parameter 1/τ affects the slope of the curve. The higher 1/τ, the more rapidly the curve goes to its long-term level (8% in the exercise). The effect for a decreasing curve is exactly symmetrical to the effect for an increasing curve Zero-coupon rate /t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = Maturity 3. The following graph shows clearly the effect of the parameter 1/τ in the five different scenarios for an inverted curve. The parameter 1/τ affects the slope of the curve, and the maturity point where the curve becomes increasing. The higher 1/τ, the lower the maturity point where the curve becomes increasing. For example, this maturity point is around 1.5 years for 1/τ equal to 0.5, and around 8 years for 1/τ equal to 0.1. Zero-coupon rate /t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = Maturity

33 33 Exercise 4.15 Solution 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. 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. 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 Exercise 4.16 Deriving the Interbank Zero-coupon Yield Curve On 03/15/02, we get from the market the following Euribor rates, futures contract prices and swap rates (see Chapters 10 and 11 for more details about swaps and futures)