(1) Adjust the bond price quotes in Table 1 for accrued interest. (2) Determine the term structure of interest rates by constructing a zero coupon bond for each maturity date. This can be done by a suitable linear combination of two coupon bonds for each maturity date. Use continuous compounding to calculate the spot rates from the prices of these zero coupon bonds: d(t) = e8 where d(t) is the discount factor at time t, sy is the spot rate, and t is the time in units of years. For simplicity assume 365 days in a year. Show the equations used and plot the obtained term structure. (3) Estimate the term structure as a smooth (polynomial) function: s(t) = 00 +01+ + agt? + azt + aft" by determining the best values of the parameters ao, 01, 02, 03, 04 by solving a nonlinear least squares problem using the coupon bond data directly. Let P, be the quoted market price (with accrued interest) of bond ; and P, the price of bond j calculated by using s(t) = 40 + ant + a2t2 + 343 + axt4 as the spot rate (in the bond price formula). Then the best fit is obtained by min, (P; - ;) Assume continuous compounding. Present the optimization problem, give the values of the obtained coefficients, and plot the obtained term structure. Assume all bonds have a face value of $100 and all cash flows happen exactly on 2/15 and 8/15 of each year. Maturity Coupon Ask Price 02/15/22 0.02 100:23 1/8 02/15/22 0.025 100:28 7/8 08/15/22 0.015 101:07 3/8 08/15/22 0.01625 101:10 7/8 02/15/23 0.02 102:18 02/15/23 0.01375 101:22 08/15/23 0.025 104:08 08/15/230.00125 99:26 3/8 02/15/24 0.0275 105:23 1/4 02/15/24 0.00125 99:28.17 08/15/24 0.00375 99:24 1/4 08/15/24 0.02375 105:16 1/4 02/15/25 0.02 104:24 02/15/25 0.07625 123:23 08/15/25 0.02 104:30 1/4 08/15/25 0.06875 123:23 3/4 02/15/26 0.01625 103:14 1/4 02/15/26 0.06 122:15 Table 1: Bond quotations as of 7/17/2019 (1) Adjust the bond price quotes in Table 1 for accrued interest. (2) Determine the term structure of interest rates by constructing a zero coupon bond for each maturity date. This can be done by a suitable linear combination of two coupon bonds for each maturity date. Use continuous compounding to calculate the spot rates from the prices of these zero coupon bonds: d(t) = e8 where d(t) is the discount factor at time t, sy is the spot rate, and t is the time in units of years. For simplicity assume 365 days in a year. Show the equations used and plot the obtained term structure. (3) Estimate the term structure as a smooth (polynomial) function: s(t) = 00 +01+ + agt? + azt + aft" by determining the best values of the parameters ao, 01, 02, 03, 04 by solving a nonlinear least squares problem using the coupon bond data directly. Let P, be the quoted market price (with accrued interest) of bond ; and P, the price of bond j calculated by using s(t) = 40 + ant + a2t2 + 343 + axt4 as the spot rate (in the bond price formula). Then the best fit is obtained by min, (P; - ;) Assume continuous compounding. Present the optimization problem, give the values of the obtained coefficients, and plot the obtained term structure. Assume all bonds have a face value of $100 and all cash flows happen exactly on 2/15 and 8/15 of each year. Maturity Coupon Ask Price 02/15/22 0.02 100:23 1/8 02/15/22 0.025 100:28 7/8 08/15/22 0.015 101:07 3/8 08/15/22 0.01625 101:10 7/8 02/15/23 0.02 102:18 02/15/23 0.01375 101:22 08/15/23 0.025 104:08 08/15/230.00125 99:26 3/8 02/15/24 0.0275 105:23 1/4 02/15/24 0.00125 99:28.17 08/15/24 0.00375 99:24 1/4 08/15/24 0.02375 105:16 1/4 02/15/25 0.02 104:24 02/15/25 0.07625 123:23 08/15/25 0.02 104:30 1/4 08/15/25 0.06875 123:23 3/4 02/15/26 0.01625 103:14 1/4 02/15/26 0.06 122:15 Table 1: Bond quotations as of 7/17/2019