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FINANCE 1 DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES Philippe PRIAULET 1 Exercise 1 On 12/04/01 consider a fixed coupon bond whose features are

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FINANCE 1 DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES Philippe PRIAULET 1 Exercise 1 On 12/04/01 consider a fixed coupon bond whose features are the following: face value: $1,000 coupon rate: 8% coupon frequency: semi-annual maturity: 05/06/04 What are the future cash-flows delivered by this bond? Solution 2 The coupon cash-flow is equal to $40 Coupon = 8% $1, =$40 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 3 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 4 1- 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 = $ Exercise 5 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 %. Solution 6 1- The number of bonds he will buy is given by the following formula Number of bonds bought = Here the number of bonds is equal to n is equal to 101,562 n = n = Cash Nomin al Value of the bond dirty price 10, 000, 000 1, % = , 000, % = Exercise 7 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. Solution 8 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, five 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 9 Treasury bills are quoted using the yield on a discount basis or on a money market basis. 1- The yield on a discount basis denoted y d is computed as y d = F P F B 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 following equation µ P = F 1 n y d B 4 2- The yield on a money market basis denoted 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 following equation P = ³ F 1+ n ym B Solution From the equation y d = F P F B n we deduce n y d B 1= P F and finally we obtain µ P = F 1 n y d B 2- From the equation y m = B y d B n y d we deduce y m = B F F P B n F F P B n B n = ³ B n F F P 1 F P F Then we have Finally we obtain n y m B = F P F P F P = ³ = F P P F 1+ n ym B = F P 1 Exercise What is the yield on a discount basis of a bill whose face value F is 1,000, price P is 975 and n the number of calendar days remaining to maturity is 126? We assume that the year-basis is What is the yield on a money market basis of the same bill? 5 Solution The yield y d on a discount basis which satisfies the following equation is equal to 7.143% y d = 1, , =7.143% 2- The yield y m on a money market basis which satisfies the following equation is equal to 7.326% y m = % % =7.326% Exercise 13 Suppose the interest rate is 12% per year compounded continuously. What is the effective annual interest rate? Solution 14 The effective annual interest rate is obtained as R = e = = 12.74%. Exercise 15 If you deposit $2,500 in a bank account which earns 8% annually on a continuously compounded basis, what will be the account balance in 7.14 years. Solution 16 The account balance in 7.14 years will be $2500.e 8%.7.14 = $ Exercise 17 If an investment has a cumulative 63.45% rate of return over 3.78 years, what is the annual continuously compounded rate of return? Solution 18 The annual continuously compounded rate of return R C is such that = e 3.78RC We deduce R C =ln(1.6345)/3.78 = 13%. Exercise 19 How long does it take to double a $100 initial investment when investing at a 5% continuously compounded interest rate? 6 Solution 20 In general, the solution is given by xe RCT =2x or T = ln 2 R C Note that this does not depend on the principle x. In this example, we obtain T = ln 2 = years 0.05 Exercise 21 A invests $1000 at 5% per annum continuously compounded. B invests $200 at 20% per annum continuously compounded. Does B ever catch up? How long does it take? Solution 22 B catches up if the difference between them becomes zero, Alternatively if the ratio of their amounts becomes 1. The equality condition can be stated as 1000e 0.05t =200e 0.2t equivalent to ln(5) t =0.2t or t = ln(5) 0.15 =10.73 Exercise What is the price of a 5-year bond with a nominal value of $100, a yield to maturity of 7%, a 10% coupon rate, and an annual coupon frequency? 2- Same question for a yield to maturity of 8%, 9% and 10%. Conclude Solution The price P of a bond is given by the following formula P = nx i=1 N c N (1 + y) n + (1 + y) n 7 which simplifies into P = N c y 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 1 7% # 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 8% $ % $ % $100 Bond prices decrease with an increase in rates. Exercise What is the price of a 5-year bond with a nominal value of $100, a yield to maturity of 7%, a 10% coupon rate, and semi-annual coupon payments? 2- Same question for a yield to maturity of 8%, 9% and 10%. Solution The price P of this bond is given by the following formula which simplifies into P = P = N c y 2nX i=1 N c/2 (1 + y/2) i + N (1 + y/2) 2n 1 # 1 N (1 + y/2) 2n + (1 + y/2) 2n 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. 8 Here we obtain for P P = 10 7% 1 1 ³ 1+ 7% ³ 1+ 7% 2 10 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 8% $ % $ % $100 Exercise 27 We consider the following zero-coupon curve Maturity Zero-Coupon Rate 1year 4.00% 2years 4.50% 3years 4.75% 4years 4.90% 5years 5.00% 1- 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 rates increase for the bondholder? 4- We suppose now that the zero-coupon curve will remain 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? 9 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+4% + 5 ( %) ( %) ( %) (1 + 5%) 5 = $ The yield to maturity R of this bond verifies the following equation = 4X i=1 5 (1 + R) i (1 + R) 5 Using the Excel function yield, we obtain % for R. 3-ThenewpricePofthebondisgivenbythefollowingformula P = % + 5 (1 + 5%) ( %) ( %) ( %) 5 =$ The new yield to maturity R of this bond verifies the following equation = 4X i=1 5 (1 + R) i (1 + R) 5 Using the Excel function yield, we obtain % for R. The impact of this rates increase for the bondholder is an absolute capital loss of $2.137 Absolute Loss = = $2.137 and a relative capital loss of 2.134% Re lative Loss = = 2.134% 4-Before maturity, the bondholder receives intermediate coupons that he reinvests on the market: - after one year, he receives $5 that he reinvests for 4 years at the 4-year zero-coupon rate 10 to obtain at the maturity date of the bond 5 ( %) 4 =$ after two years, he receives $5 that he reinvests for 3 years at the 3-year zero-coupon rate to obtain at the maturity date of the bond 5 ( %) 3 =$ after three years, he receives $5 that he reinvests for 2 years at the 2-year zero-coupon rate to obtain at the maturity date of the bond 5 ( %) 2 =$ after four years, he receives $5 that he reinvests for 1 year at the 1-year zero-coupon rate to obtain at 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 µ /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 %, we obtain $ five years later = $ which corresponds exactly to a % annual return rate. µ /5 1=4.9686% Exercise 29 We consider the three zero-coupon bonds (strips) with the following features Bond Maturity Price Bond 1 1year Bond 2 2years Bond 3 3years Each strip delivers $100 at maturity. 1- Deduce the zero-coupon yield curve from the bond prices 2- We anticipate a rates increase in one year so the prices of strips with residual maturity 1 year, 2 years and three 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 R(0,1) is equal to 3.702% R(0, 1) = =3.702% The 2-year zero-coupon rate denoted R(0,2) is equal to 3.702% R(0, 2) = µ 100 1/2 1=3.992% The 3-year zero-coupon rate denoted R(0,3) is equal to 3.702% R(0, 2) = µ 100 1/3 1=4.365% The 1-year, 2-year and 3-year zero-coupon rates become respectively 4.286%, 4.846% and 5.887%. 12 Exercise 31 We consider the following increasing zero-coupon yield curve Maturity R(0,t) Maturity R(0,t) 1year 5.000% 6years 6.550% 2years 5.500% 7years 6.650% 3years 5.900% 8years 6.741% 4years 6.200% 9years 6.830% 5years 6.382% 10 years 6.900% where R(0,t) is the zero-coupon rate at date 0 with maturity t. 1- Compute the par curve. 2- Compute the forward rate curve in one year. 3- Draw the three curves in the same graph. Wwhat can you say about their relative position? Solution Recall that the par yield c(n) for maturity n is given by the following formula c(n) = 1 1 (1+R(0,n)) n np i=1 1 (1+R(0,i)) i Using this equation we obtain the following par yields Maturity c(n) Maturity c(n) % % % % % % % % % % 2- Recall that F (0,x,y x), the forward rate as seen from date t =0,startingatdatet = x, andwithresidualmaturityy x is defined as (1 + R(0,y)) y F (0,x,y x) (1 + R(0,x)) x 1 y x 1 13 Using the previous equation,we obtain the forward rate curve in one year Maturity F(0,1,n) Maturity F(0,1,n) % % % % % % % % % 3- The graph of the three curves shows that the forward yield curve is above the zero-coupon yield curve, which is above the par yield curve. This is always the case when the par yield curve is increasing. 7,5% 7,0% 6,5% Yield 6,0% Par Yield Curve 5,5% Zero-Coupon Yield Curve Forward Yield Curve 5,0% Maturity Exercise 33 At date t=0, we get in the market three bonds with the following features Coupon Maturity Price Bond years P0 1 = Bond years P0 2 = Bond years P0 3 = Derive the zero-coupon curve until the five-year maturity. Solution 34 Using the no-arbitrage relation, we obtain the following equations for the five bond prices 108 = 10B(0, 1) + 110B(0, 2) = 7.5B(0, 1) + 7.5B(0, 2) B(0, 3) =9B(0, 1) + 9B(0, 2) + 109B(0, 3) which can be expressed in a matrix form = B(0, 1) B(0, 2) B(0, 3) Then we get the following discount factors B(0, 1) B(0, 2) B(0, 3) = we deduce the zero-coupon rates R(0, 1) = 5.901% R(0, 2) = 5.646% R(0, 3) = 7.288% Exercise 35 Suppose we know from market prices the following zero-coupon rates with matu- 15 rities inferior or equal to one year: Maturity Zero-Coupon Rate 1Day 3.20% 1Month 3.30% 2Months 3.40% 3Months 3.50% 6Months 3.60% 9Months 3.80% 1Year 4.00% Now we consider bonds priced by the market until the 4-year maturity: Maturity Coupon Gross Price 1Yearand3Months 4% Yearand6Months 4.5% Years 3.5% Years 4% Years 5% Using the bootstrapping method, compute the zero-coupon rates for the following maturities1yearand3months,1yearand6months,2years,3yearsand4years. 2- Draw the zero-coupon yield curve using a linear interpolation Solution We first extract the one-year-and-three-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: = 4 ( %) 1/ (1 + x) 1+1/4 where x is the one-year-and-three-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 one-year-and-six-month maturity and the two-year maturity bonds we obtain respectively 4.32% and 4.41% for x. Next we have to extract the 3-year maturity zero-coupon 16 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 z by solving the following equation z is equal to 4.57% = 5 (1 + 4%) + 5 ( %) ( %) (1 + z%) 4 2- Using the linear graph option in Excel we draw the zero-coupon yield curve 4,80% 4,60% 4,40% 4,20% Zero-Coupon Rate 4,00% 3,80% 3,60% 3,40% 3,20% 3,00% Maturity Exercise 37 From the prices of zero-coupon bonds quoted in the market, we obtain the following 17 zero-coupon curve Maturity Zero-Coupon Rate R(0,t) Discount Factor B(0,t) 1year 5.000% years 5.500% years 5.900% years 6.200% years?? 6years 6.550% years 6.650% years?? 9years 6.830% years 6.900% where R(0,t) is the zero-coupon rate at date 0 for maturity t, and B(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. Deduce 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. Deduce 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 zero-coupon rates using the following optimization program Min a,b,c,d X i ³ R(0,i) R _ 2 (0,i) 18 where R(0,i) are the zero-coupon rates given by the market. Deduce 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). 4- We postulate the following form for the discount factor 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 Min a,b,c,d X i ³ B(0,i) B _ 2 (0,i) where B(0,i) are the discount factors given by the market. Deduce 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, the x-year and the z-year maturity zero-coupon rates and that we need R(0,y) the y-years maturity zero-coupon rate with y [x; z]. Using the linear interpolation, R(0,y) is given by the following formula R(0,y)= (z y)r(0,x)+(y x)r(0,z) z x From this equation, we deduce 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 standard following equation 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 19 we obtain for B(0,5) and for B(0,8). 2- We use the same formula as in question 1 but adapted to discount factors B(0,y)= (z y)b(0,x)+(y x)b(0,z) z x we obtain for B(0,5) and for B(0,8). Using the standard following equation which lies the zero-coupon rate R(0,t) and the discount factor B(0,t) R(0,t)= µ 1 1/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 DroiteReg, we obtain the following values for the parameters Parameters Value a b c d E-05 which provides 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) % 4.998% % 5.507% % 5.899% % 6.191% ? 6.403%? % 6.553% % 6.659% ? 6.741%? % 6.817% % 6.906% 4- 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 DroiteReg, we obtain the following values for the parameters Parameters Value a 1 b c d which provides 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) % 5.346% % 5.613% % 5.867% % 6.107% 5? ? 6.328% % 6.523% % 6.686% 8? ? 6.805% % 6.871% % 6.869% 5- The table below provides the results obtained using the four different methods of interpo- 21 lation 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. is for interpolation on rates (question 1). DF Interpol. is for interpolation on discount factors (question 2). Rates Min is for minimization with rates (question 3). DF Min. is 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 zero-coupon rates and discount factors estimations are sensitive to the method of interpolation or minimization used. Exercise 39 We want to derive the current zero-coupon yield curve for maturities inferior to 10 years. For that goal, we use a basket of bonds quoted by the market and a discount function modelled as a three-order polynomial spline. The features of the bonds used to derive this curve 22 are summarized in the following table Bond Coupon Rate Maturity Market Price Bond 1 0% 7/365 year Bond 2 0% 1/12 year Bond 3 0% 0.25 year Bond 4 0% 0.5 year Bond 5 5% 1year Bond 6 6% 2years Bond 7 5% 2.5 years Bond 8 7% 3.25 years Bond 9 8% 4years Bond 10 5% 4.5 years Bond 11 7% 5.5 years Bond 12 7% 7years Bond 13 6% 8.75 years Bond 14 7% 10 years The coupon frequency of these bonds is annual, and the face value is Eur 100. We model the discount function B(0,s) as a standard

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