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 1 1 96.43 Bond 2 2 92.47 Bond 3 3 87.97 1. 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, 90.97 and 84.23. What is the zero-coupon yield curve anticipated in one year? Solution 3.1 1. The 1-year zero-coupon rate denoted by R(0, 1) is equal to 3.702% R(0, 1) = 100 96.43 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% 92.47 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% 87.97 2. 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) (%) 1 7.000 6 6.250 2 6.800 7 6.200 3 6.620 8 6.160 4 6.460 9 6.125 5 6.330 10 6.100 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 Solution 3.3 1. Recall that the par yield c(n) for maturity n is given by the formula c(n) = 1 1 (1+R(0,n)) n ni=1 1 (1+R(0,i)) i Using this equation, we obtain the following par yields: Maturity (years) c(n) (%) Maturity (years) c(n) (%) 1 7.000 6 6.293 2 6.807 7 6.246 3 6.636 8 6.209 4 6.487 9 6.177 5 6.367 10 6.154 2. Recall that F(0,x,yx), 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 yx 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) (%) 1 6.600 6 6.067 2 6.431 7 6.041 3 6.281 8 6.016 4 6.163 9 6.000 5 6.101 3. 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. 7.25 7.00 6.75 Par yield curve Zero-coupon yield curve Forward yield curve Yield (%) 6.50 6.25 6.00 5.75 1 2 3 4 5 6 7 8 9 10 Maturity
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) + 1 + c(2) (1 + c(2)) 2 = c(2) 1 + R(0, 1) + 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) n1 k.(c(n) c(k)) + c(k) 1 + c(n 1) n1 c(n1) as 1+c(n1) k=1 c(k) can be considered negligible since c(n1) 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(n1) 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 Maturity Zero-Coupon Maturity Zero-Coupon (years) Rate (%) (years) Rate (%) 1 5.00 4 6.80 2 6.00 5 7.00 3 6.50 1. 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 3.11 1. 1-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) = (1 + 6.8%) 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 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) 4.00 5.00 5.00 5.00 5.00 F a (1, 2, 1) 4.00 5.00 6.00 4.00 4.00 F a (1, 2, 2) 4.00 5.00 6.00 4.00 3.00 F a (1, 2, 3) 4.00 5.00 6.00 4.00 2.00 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 we find the spot zero-coupon yield curve in each scenario. Scenario a Scenario b Scenario c Scenario d Scenario e % % % % % R(1, 1) 4.00 5.00 5.00 5.00 5.00 R(1, 2) 4.00 5.00 5.50 4.50 4.50 R(1, 3) 4.00 5.00 5.67 4.33 4.00 R(1, 4) 4.00 5.00 5.75 4.25 3.49 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 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 1 6 1 year P0 1 = 103 Bond 2 5 2 years P0 2 = 102 Bond 3 4 3 years P0 3 = 100 Bond 4 6 4 years P0 4 = 104 Bond 5 5 5 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) 102 100 = B(0, 2) B(0, 3) 104 99 106 5 105 4 4 104 6 6 6 106 5 5 5 5 105 We get the following discount factors: B(0, 1) 0.97170 B(0, 2) B(0, 3) B(0, 4) = 0.92516 0.88858 0.82347 B(0, 5) 0.77100 B(0, 4) B(0, 5)
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 3.20 1 Month 3.30 2 Months 3.40 3 Months 3.50 6 Months 3.60 9 Months 3.80 1 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 4 102.8 1 Year and 6 Months 4.5 102.5 2 Years 3.5 98.3 3 Years 4 98.7 4 Years 5 101.6 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 4.3 1. 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: 4 102.8 = (1 + 3.5%) 1/4 + 104 (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 we have to extract the 3-year maturity zero-coupon rate, solving the following equation: 98.7 = 4 (1 + 4%) + 4 (1 + 4.41%) 2 + 104 (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: 101.6 = 5 (1 + 4%) + 5 (1 + 4.41%) 2 + 5 (1 + 4.48%) 3 + 105 (1 + z%) 4 z is equal to 4.57%. 2. Using the linear graph option in Excel, we draw the zero-coupon yield curve 4.80 4.60 4.40 Zero-coupon rate (%) 4.20 4.00 3.80 3.60 3.40 3.20 3.00 0 1 2 3 4 Maturity Exercise 4.4 Solution 4.4 1. 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 ) = 0.75 4% + 1.25 4.5% 2 = 4.3125%
24 R(0, 11 3/4 0.25 4% + 1.75 4.5% ) = = 4.4375% 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 ) = 3.75 8.6% + 1.25 9% 5 3.25 8.6% + 1.75 9% 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) 1 5.000 0.95238 2 5.500 0.89845 3 5.900 0.84200 4 6.200 0.78614 5?? 6 6.550 0.68341 7 6.650 0.63720 8?? 9 6.830 0.55177 10 6.900 0.51312 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 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 4.7 1. 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 0.73418 for B(0, 5) and 0.59345 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 0.73478 for B(0, 5) and 0.59449 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 Parameters Value a 0.04351367 b 0.00720757 c 0.000776521 d 3.11234E-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) 1 5.000 4.998 0.95238 0.95240 2 5.500 5.507 0.89845 0.89833 3 5.900 5.899 0.84200 0.84203 4 6.200 6.191 0.78614 0.78641 5? 6.403? 0.73322 6 6.550 6.553 0.68341 0.68330 7 6.650 6.659 0.63720 0.63681 8? 6.741? 0.59339 9 6.830 6.817 0.55177 0.55237 10 6.900 6.906 0.51312 0.51283 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 Linest, we obtain the following values for the parameters: Parameters Value a 1 b 0.04945479 c 0.001445358 d 0.000153698 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) (%) 1 0.95238 0.94925 5.000 5.346 2 0.89845 0.89654 5.500 5.613 3 0.84200 0.84278 5.900 5.867 4 0.78614 0.78889 6.200 6.107 5? 0.73580? 6.328 6 0.68341 0.68444 6.550 6.523 7 0.63720 0.63571 6.650 6.686 8? 0.59055? 6.805 9 0.55177 0.54988 6.830 6.871 10 0.51312 0.51461 6.900 6.869
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) 0.73418 0.73478 0.73322 0.73580 B(0, 8) 0.59345 0.59449 0.59339 0.59055 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) (%) 0.5 7.500 5 8.516 1 7.130 6 8.724 1.25 7.200 7 8.846 2 7.652 8 8.915 3 8.023 10 8.967 4 8.289 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 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 4.8 1. (a) Using the Excel function Linest, we obtain the following values for the parameters Parameters Value a 0.070774834 b 0.00254927 c 0.000175503 d 2.44996E-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) (%) 0.5 7.500 7.209 1 7.130 7.348 1.25 7.200 7.419 2 7.652 7.638 3 8.023 7.934 4 8.289 8.221 5 8.516 8.485 6 8.724 8.710 7 8.846 8.882 8 8.915 8.986 10 8.967 8.932
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) (%) 0.8 7.291 1.5 7.491 3.4 8.051 5.25 8.545 8.3 9.002 9 9.007 (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. 9.00 Zero-coupon rates (%) 8.50 8.00 7.50 7.00 0 1 2 3 4 5 6 7 8 9 10 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 0.06865628 c 0.000397845 d 0.000151057 which provide us with the following values for the discount factors and associated zero-coupon rates:
30 Maturity (years) B(0,t) B(0,t) R(0,t) (%) R(0,t) (%) 0.5 0.96449 0.96559 7.500 7.254 1 0.93345 0.93110 7.130 7.400 1.25 0.91676 0.91385 7.200 7.473 2 0.86289 0.86230 7.652 7.689 3 0.79333 0.79453 8.023 7.968 4 0.72721 0.72868 8.289 8.235 5 0.66456 0.66565 8.516 8.480 6 0.60541 0.60637 8.724 8.695 7 0.55248 0.55172 8.846 8.867 8 0.50501 0.50263 8.915 8.979 10 0.42369 0.42471 8.967 8.941 (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) (%) 0.8 0.94490 7.342 1.5 0.89663 7.545 3.4 0.76791 8.077 5.25 0.65045 8.537 8.3 0.48912 8.998 9 0.45999 9.012 (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. 9.00 Zero-coupon rates (%) 8.50 8.00 7.50 7.00 0 1 2 3 4 5 6 7 8 9 10 Maturity
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). 0.08 0.075 Zero-coupon rate 0.07 0.065 0.06 0.055 1/t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = 0.5 0.05 0 5 10 15 20 25 30 Maturity
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. 0.11 Zero-coupon rate 0.105 0.1 0.095 0.09 1/t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = 0.5 0.085 0.08 0 5 10 15 20 25 30 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 0.078 0.077 0.076 0.075 0.074 0.073 0.072 0.071 1/t = 0.1 1/t = 0.2 1/t = 0.3 1/t = 0.4 1/t = 0.5 0.07 0.069 0.068 0 5 10 15 20 25 30 Maturity
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: 0.08 0.075 Zero-coupon rate 0.07 0.065 0.06 0.055 0.05 0.045 b 3 = 3% b 3 = 2% base case b 3 = 0% b 3 = 1% 0.04 0 5 10 15 20 25 30 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)
34 Maturity Euribor Contract Futures Maturity Swap Rate Maturity Price Maturity Rate (%) 03/22/02 4.07% 06/15/02 95.2 03/15/04 5.20 03/29/02 4.11% 09/15/02 95.13 03/15/05 5.36 04/15/02 4.15% 12/15/02 94.93 03/15/06 5.49 05/15/02 4.26% 03/15/03 94.79 03/15/07 5.61 06/15/02 4.34% 06/15/03 94.69 03/15/08 5.71 07/15/02 4.44% 09/15/03 94.54 03/15/09 5.75 08/15/02 4.53% 03/15/10 5.79 03/15/11 5.82 03/15/12 5.84 Note that the underlying asset of the futures contract is a three-month Euribor rate. For example, the first contract matures on 06/15/02, and the underlying asset matures three months later on 09/15/02. 1. Extract the implied zero-coupon rates from market data. 2. Draw the zero-coupon yield curve by building a linear interpolation between the implied zero-coupon rates. Solution 4.16 1. We first extract the implied zero-coupon rates from the Euribor rates using the following formula: ( x ) ( R 0, = 1 + x ) 365 365 360.Euribor x x 1 where R ( 0, 365) x and Euriborx are respectively the zero-coupon rate and the Euribor rate with residual maturity x (as a number of days). We obtain the following results: x Euribor x (%) R ( 0, 365 x ) (%) 7 4.07 4.211 14 4.11 4.252 31 4.15 4.290 61 4.26 4.398 92 4.34 4.473 122 4.44 4.569 153 4.53 4.654 We now extract the implied zero-coupon rates from the futures price using the following formula: ( B f x 0, 365, y ) = B ( 0, y ) 365 365 B ( 0, 365 x )
35 which transforms into ( ( )) y x 1 + (100 Futures Price)%. = B ( 0, x ) 365 360 B ( 0, y ) 365 and finally enables to obtain [ ( y ) R 0, = 365 1 B ( 0, x 365 ( ). 1 + (100 Futures Price)%. ( )) ] 365 y x y 1 360 where B(0,t)is the discount factor with maturity t and B f (0,t,T)the forward discount factor determined at date 0, beginning at date t and finishing at date T. Using the last equation, we obtain the following results (FP stands for Futures Price) x y B ( 0, 365 x ) FP R ( 0, 365) y (%) 92 184 0.98903 95.2 4.714 184 275 0.97705 95.13 4.819 275 365 0.96516 94.93 4.923 365 457 0.95308 94.79 5.016 457 549 0.94056 94.69 5.096 549 640 0.92797 94.54 5.175 Detailing the calculations for the first line of the previous table, we obtain x = number of days between the 03/15/02 and the 06/15/02 y = number of days between the 03/15/02 and the 09/15/02 ( x ) 1 B 0, = = 0.98903 365 (1 + 4.473%) 92/365 FP = 95.20 ( y ) R 0, = 4.714% 365 We now extract the implied zero-coupon rates from the swap rates using the following formula: SR(n) 1 + R(0, 1) + SR(n) 1 + SR(n) + + (1 + R(0, 2)) 2 (1 + R(0,n)) n = 1 which enables us to obtain 1 n 1 R(0,n)= 1 SR(n) 1+R(0,1) SR(n) (1+R(0,n1)) n1 where SR(n) is the swap rate with maturity n. (1 + SR(n)) 1
36 We then obtain the following results: n SR(n) (%) R(0,n) (%) 2 5.20 5.207 3 5.36 5.374 4 5.49 5.512 5 5.61 5.642 6 5.71 5.753 7 5.75 5.795 8 5.79 5.839 9 5.82 5.872 10 5.84 5.893 2. We obtain the following interbank zero-coupon yield curve: 6.00 5.80 5.60 Zero-coupon rate (%) 5.40 5.20 5.00 4.80 4.60 4.40 4.20 4.00 0 1 2 3 4 5 6 7 8 9 10 Maturity 5 CHAPTER 5 Problems Exercise 5.1 Calculate the percentage price change for 4 bonds with different annual coupon rates (5% and 10%) and different maturities (3 years and 10 years), starting with a common 7.5% YTM (with annual compounding frequency), and assuming successively a new yield of 5%, 7%, 7.49%, 7.51%, 8% and 10%. Solution 5.1 Results are given in the following table: