Math 618 Answer to Selected Problems in Section 6.1 Autumn 2005 Problem 6.1.5 We want to find the spot rates i(2).5 , i(2)1 , i(2)1.5, and i(2)2 (where i(2)k = i0,k is the nominal rate of zero-coupon bond with maturity of k-year convertible semiannually). Let jk = i(2)k /2 be the corresponding effective half-year rate. Suppose the bonds face value and redemption amount are 100. • 12-year bond: (r = .02, j = .025) P Fr + F Using the actual yield, the price of the bond (at time 0) is P = (100 + 100r)vj = (100 + 2) 11 + .025 = 1021.025. Using the (effective half-year) spot rate j.5, the price is P = (100 + 100r)vj.5 = 1021 + j .5 . The (blue) prices are equal. So, 1021 + j .5 = 1021.025. Solving, we get j.5 = .025. (j.5 = .025) • 1-year bond: (r = .03, j = .05) P Fr = 3a27 j.5 Fr + F = 103a27 j1 Using the actual yield, the price of the bond (at time 0) is P = ···[F + F(r − j)an|j, or K + rj(F − K), or ···,]··· = 96.281179.... (N = 2, FV = 100, PMT = 3, and I/Y = 5% on a BA II PLUS gives PV = −96.28117914.) Using the (effective half-year) spot rate j.5 and j1, the price is P = Frvj.5 + (Fr + F)v2j1 = 31.025 + 103(1 + j 1)2 The (blue) prices are equal. So, 31.025 + 103(1 + j 1)2 = 96.281179. Solving, we get j1 = .050391825. (j.5 = .025, j1 = .050392) • 1 1/2-year bond: (r = .02, j = .075) P Fr = 2a27 j.5 Fr = 2a27 j1 Fr + F = 102a27 j1.5 Using the actual yield, the price of the bond (at time 0) is P = ···[F + F(r − j)an|j, or K + rj(F − K), or ···,]··· = 85.69710843 (N = 3, FV = 100, PMT = 2, and I/Y = 7.5% on a BA II PLUS gives PV = −85.69710843.) Using the (effective half-year) spot rate j.5, j1, and j1.5 the price is P = (Fr)vj.5 + (Fr)v2j1 + (Fr + F)v3j1.5 = 21.025 + 21.050392 + 102(1 + j 1.5)3 The (blue) prices are equal. So, 21.025 + 21.050392 + 102(1 + j 1.5)3 = 85.69710843. Solving, we get j1.5 = 0.07575521278. (j.5 = .025, j1 = .050392, j1.5 = .0757552) • 2-year bond: (r = .04, j = .075) P Fr = 4a27 j.5 Fr = 4a27 j1 Fr = 4a27 j1.5 Fr + F = 104a27 j2 Using the actual yield, the price of the bond (at time 0) is P = ···[F + F(r − j)an|j, or K + rj(F − K), or ···,]··· = 88.27735806 (N = 4, FV = 100, PMT = 4, and I/Y = 7.5% on a BA II PLUS gives PV = −88.27735806.) Using the (effective half-year) spot rate j.5, j1, j1.5, and j2 the price is P = (Fr)vj.5 +(Fr)v2j1 +(Fr)v3j1.5 +(Fr+F)v4j2 = 41.025 + 41.050392 + 41.07575523 + 104(1 + j 2)4 The (blue) prices are equal. So, 41.025 + 41.050392 + 41.07575523 + 104(1 + j 2)4 = 88.27735806. Solving, we get j2 = 0.07617249008. (j.5 = .025, j1 = .050392, j1.5 = .0757552, j2 = .0761725) Using i(2)k = 2jk, we have i(2).5 = 2j.5 = 5%, i(2)1 = 2j1 = 10.784%, i(2)1.5 = 2j1.5 = 15.151%, i(2).5 = 2j.5 = 15.2345%.