FINANCE 1. DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES. Philippe PRIAULET


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1 FINANCE 1 DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES Philippe PRIAULET 1
2 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: semiannual maturity: 05/06/04 What are the future cashflows delivered by this bond? Solution 2 The coupon cashflow 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 daycount 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 = $
3 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 daycount 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 daycount basis is equal to Eur % Eur 100 = Eur
4 the accrued interest with the Actual/365 daycount basis is equal to Eur % Eur 100 = Eur the accrued interest with the Actual/360 daycount 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 daycount basis 15 + (5 30) + 25 = 190 finally the accrued interest with the 30/360 daycount 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 yearbasis (365 or 360) and n is the number of calendar days remaining to maturity. Prove in this case that the price of the Tbill is obtained using the following equation µ P = F 1 n y d B 4
5 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 Tbill 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 yearbasis is What is the yield on a money market basis of the same bill? 5
6 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
7 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 5year 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
8 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 5year bond with a nominal value of $100, a yield to maturity of 7%, a 10% coupon rate, and semiannual 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
9 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 zerocoupon curve Maturity ZeroCoupon Rate 1year 4.00% 2years 4.50% 3years 4.75% 4years 4.90% 5years 5.00% 1 What is the price of a 5year 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 zerocoupon 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 zerocoupon 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
10 Solution The price P of the bond is equal to the sum of its discounted cashflows 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. 3ThenewpricePofthebondisgivenbythefollowingformula 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% 4Before maturity, the bondholder receives intermediate coupons that he reinvests on the market:  after one year, he receives $5 that he reinvests for 4 years at the 4year zerocoupon rate 10
11 to obtain at the maturity date of the bond 5 ( %) 4 =$ after two years, he receives $5 that he reinvests for 3 years at the 3year zerocoupon rate to obtain at the maturity date of the bond 5 ( %) 3 =$ after three years, he receives $5 that he reinvests for 2 years at the 2year zerocoupon rate to obtain at the maturity date of the bond 5 ( %) 2 =$ after four years, he receives $5 that he reinvests for 1 year at the 1year zerocoupon rate to obtain at the maturity date of the bond 5 (1 + 4%) = $5.2  after five years, he receives the final cashflow 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 = $
12 which corresponds exactly to a % annual return rate. µ /5 1=4.9686% Exercise 29 We consider the three zerocoupon 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 zerocoupon 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 zerocoupon yield curve anticipated in one year? Solution The 1year zerocoupon rate denoted R(0,1) is equal to 3.702% R(0, 1) = =3.702% The 2year zerocoupon rate denoted R(0,2) is equal to 3.702% R(0, 2) = µ 100 1/2 1=3.992% The 3year zerocoupon rate denoted R(0,3) is equal to 3.702% R(0, 2) = µ 100 1/3 1=4.365% The 1year, 2year and 3year zerocoupon rates become respectively 4.286%, 4.846% and 5.887%. 12
13 Exercise 31 We consider the following increasing zerocoupon 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 zerocoupon 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
14 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 zerocoupon 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% ZeroCoupon 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 =
15 Derive the zerocoupon curve until the fiveyear maturity. Solution 34 Using the noarbitrage 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 zerocoupon 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 zerocoupon rates with matu 15
16 rities inferior or equal to one year: Maturity ZeroCoupon 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 4year maturity: Maturity Coupon Gross Price 1Yearand3Months 4% Yearand6Months 4.5% Years 3.5% Years 4% Years 5% Using the bootstrapping method, compute the zerocoupon rates for the following maturities1yearand3months,1yearand6months,2years,3yearsand4years. 2 Draw the zerocoupon yield curve using a linear interpolation Solution We first extract the oneyearandthreemonth maturity zerocoupon rate. In the absence of arbitrage opportunities, the price of this bond is the sum of its future discounted cashflows: = 4 ( %) 1/ (1 + x) 1+1/4 where x is the oneyearandthreemonth maturity zerocoupon 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 oneyearandsixmonth maturity and the twoyear maturity bonds we obtain respectively 4.32% and 4.41% for x. Next we have to extract the 3year maturity zerocoupon 16
17 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 4year maturity zerocoupon 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 zerocoupon yield curve 4,80% 4,60% 4,40% 4,20% ZeroCoupon Rate 4,00% 3,80% 3,60% 3,40% 3,20% 3,00% Maturity Exercise 37 From the prices of zerocoupon bonds quoted in the market, we obtain the following 17
18 zerocoupon curve Maturity ZeroCoupon 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 zerocoupon 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 5year and the 8year zerocoupon rates. We have to estimate them, and test four different methods. 1 We use a linear interpolation with the zerocoupon 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 zerocoupon 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 Min a,b,c,d X i ³ R(0,i) R _ 2 (0,i) 18
19 where R(0,i) are the zerocoupon 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 xyear and the zyear maturity zerocoupon rates and that we need R(0,y) the yyears maturity zerocoupon 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 zerocoupon rate R(0,t) and the discount factor B(0,t) B(0,t)= 1 (1 + R(0,t)) t 19
20 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 zerocoupon 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 E05 which provides us with the following values for the zerocoupon 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%
21 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 zerocoupon 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
22 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 zerocoupon rates and discount factors estimations are sensitive to the method of interpolation or minimization used. Exercise 39 We want to derive the current zerocoupon 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 threeorder polynomial spline. The features of the bonds used to derive this curve 22
23 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 polynomial spline with two splines B 0 (s) =1+c 0 s + b 0 s 2 + a 0 s 3 for s [0, 3] B (0,s)= B 10 (s) =1+c 0 s + b 0 s 2 + a 0 hs 3 (s 3) 3i + a 1 (s 3) 3 for s [3, 10] 1 Write the theoretical price of bond Calculate the coefficients behind each parameter and the constant number for bond 1 3 Do the same job for bond 2 to bond Estimate the coefficients c 0,b 0,a 0 and a 1 which best approximate the market prices of given bonds using the following optimization program Min c 0,b 0,a 0,a 1 14X i=1 ³ P i P _ 2 i 23
24 where P i are the market prices and _ P ithe theoretical prices. We suppose that residual are homoscedastic so that each bond has the same weight in the minimization program. 5 Calculate P 14 i=1 ³P i P _ 2 i 6 For each bond, calculate the spread between the market price and the theoretical price. 7 Draw the graph of the zerocoupon yield curve. 8 Draw the graph of the forward yield curve in one, two and three months. Solution The theoretical price _ P 1ofbond1is µ " _ 7 P i= 100.B 0, = c b 0. µ 7 2 µ # a From question 1, we deduce the coefficients behind each parameter and the constant number for bond 1 constant number c 0 b 0 a ³ ³ For each bond, we obtain the coefficients behind each parameter and the constant number 24
25 in the following table Bond constant number c 0 b 0 a 0 a 1 Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Note that Z, what we call the coefficients matrix; is the matrix with dimension 14 4 which is represented by the four last columns of the table. We give below the details for Bond 8: constant number n n = = 128 coefficient behind c 0 7 [(3.25 3) + (3.25 2) + (3.25 1)] = 374 coefficient behind b 0 h 7 (3.25 3) 2 +(3.25 2) 2 +(3.25 1) 2i =
26 coefficient behind a 0 7 h (3.25 3) 3 +(3.25 2) 3 +(3.25 1) 3i +107 h (3.25 3) 3i = coefficient behind a (3.25 3) 3 = β =(c 0,b 0,a 0,a 1 ) T, the vector of parameters is the solution of the standard OLS (Ordinary Least Squared) procedure. It is the result of the following matricial calculation β = ³ Z T Z 1 Z T P where Z T is the transposed matrix of Z, X 1 is the inverse matrix of X and P is the following vector resulting for each bond of the difference between the constant number and the market price of the bond. P = =
27 We finally obtain for β β T = E 05 β. Note that we can also use the Excel function Droitereg to obtain the vector of parameters 5 The sum of squared spreads P 14 i=1 ³P i P _ i 2is equal to In the following table we examine for each bond the spread between the market price and the theoretical price Bond Market Price Theoretical Price Spread Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond
28 7 We draw below the graph of the zerocoupon yield curve 6,25% 5,75% ZeroCoupon Rate 5,25% 4,75% 4,25% Maturity We draw below the graph of the three forward yields curves beginining in one, two and three months. Recall first that F (0,x,y x), the forward rate as seen from date t =0,startingat date t = x, and with residual maturity y x is defined as (1 + R(0,y)) y F (0,x,y x) (1 + R(0,x)) x 1 y x 1 28
29 We give successively the values 1/12, 2/12 and 3/12 to x. 6,10% 5,90% 5,70% Forward Yield 5,50% 5,30% 5,10% 4,90% Forward Yield Curve in 3 Months Forward Yield Curve in 2 Months Forward Yield Curve in 1 Month 4,70% 4,50% Maturity in Months Because the zerocoupon curve is increasing, the forward yield curve in three months is above the forward yield curve in two months, which is above the forward yield curve in one month. Exercise 41 We consider three bonds with the following features: Bond Maturity Coupon Rate YTM Bond 1 2years 5% 5% Bond 2 10 years 6% 5.5% Bond 3 30 years 7% 6% YTM is for yield to maturity. Coupon frequency is annual. 1 Compute the dirty price and the modified duration of each bond 2 a) The YTM of each of these bonds decreases instantaneously by 0.2%. For each bond, compute the new exact price given by discounting its future cashflows, the price approximation given using the first order Taylor expansion, and the difference between these two prices. b) Same question if the YTM of each of these bonds decreases instantaneously by 1%. Conclude 29
30 c) For bond 3, draw the difference between the two prices (the new exact price given by discounting its future cashflowsandthepricegivenusingthefirst order Taylor expansion) depending on the YTM change. 3 Compute the convexity of each bond 4 We suppose that the YTM of each of these bonds decreases instantaneously by 1%. Compute the price approximation given using the second order Taylor expansion. Compare it to the exact price given by discounting its future cashflows. Solution 42 1Thedirtypriceandthemodified duration of each of these bonds is given in the following table: Bond Price Modified Duration Bond Bond Bond a) When the YTM of each of these bonds decreases by 0.2%, we obtain the following results Bond New Exact Price FOTE Price Spread Bond Bond Bond Recall that the FOTE price is given by the following formula FOTE Pr ice = P + MD y P where P is the original price, MD the modified duration and y, the YTM change. b) When the YTM of each of these bonds decreases by 1%, we obtain the following results Bond New Exact Price FOTE Price Spread Bond Bond Bond
31 When the YTM change is high, the spread between the two prices is not negligible. We have to use the second order approximation. c) For bond 3, we draw below the difference between the new exact price given by discounting its future cashflows and the price given using the first order Taylor expansion depending on the YTM change. 20 Difference between the Two Prices % 3.3% 3.6% 3.9% 4.2% 4.5% 4.8% 5.1% 5.4% 5.7% 6.0% YTM Level 3 The convexity of each of these bonds is given in the following table 6.3% 6.6% 6.9% 7.2% 7.5% 7.8% 8.1% 8.4% 8.7% 9.0% Bond Convexity Bond Bond Bond Recall that the SOTE (second order Taylor expansion) price is given by the following formula SOTE Pr ice = P + MD P y + RC P ( y) 2 /2 where RC is the (relative) convexity. 31
32 When the YTM of each of these bonds decreases by 1%, we obtain the following results Bond Exact Price SOTE Price Spread Bond Bond Bond Exercise 43 Today is 1/1/98. On 6/30/99 we will have to make a payment of $100. We can only invest in a riskfree pure discount bond (nominal $100) that matures on 12/31/98 and in a riskfree 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? Solution 44 We first have to compute the present value PV of the debt, which is the amount we will have to deposit PV = =90.35 (1.07) We also compute the price P 1 of the oneyear pure discount bond P 1 = =93.46 Similarly, the price P 3 of the three year coupon bond is P 3 = (1.07) (1.07) 3 =102.6 The duration of the oneyear pure discount bond is obviously one. threeyear coupon bond is The duration D 3 of the D 3 = (1.07) (1.07) =2.786 We now compute the number of units of the oneyear and the treeyear bonds (q 1 and q 3 respectively), so as to achieve a dollar duration equal to that of debt, and also a present value of the portfolio equal to that of debt. We know that the duration of the debt we are trying to 32
Problems and Solutions
1 CHAPTER 1 Problems 1.1 Problems on Bonds Exercise 1.1 On 12/04/01, consider a fixedcoupon bond whose features are the following: face value: $1,000 coupon rate: 8% coupon frequency: semiannual maturity:
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