# sol8 - ACTSC 445 Asset-Liability Management Department of...

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ACTSC 445: Asset-Liability Management Department of Statistics and Actuarial Science, University of Waterloo Solutions for Unit 8 Exercises 1. Figure 1 shows the value of the bond obtained at the different nodes of the lattice. The price at time 0 is 70.84. The corresponding spot rate is (100 / 70 . 84) 0 . 25 - 1 = 0 . 09. 84.13 87.88 90.83 93.12 78.56 70.84 71.63 75.54 81.37 86.00 Figure 1: Finding P (0 , 4) 2. Using spot rates, we get 10 1 1 . 06 + 1 1 . 07 2 + 1 1 . 08 3 + 110 1 1 . 09 4 = 104 . 033 . 3. We get V (1 , 1) = 102 . 4387, V (1 , 0) = 103 . 4039 and V (0 , 0) = 107 . 1703. 4. (i) At time 1, in node (1,1) we sell 124 . 69 - 84 . 155 units of the zero-coupon bond with maturity at t = 3, and each unit has a value of 0.816608. We thus get \$33.10. We add 105 . 32 - 69 . 583 × 1 . 06 = 31 . 562 in the bank account, so the net cash-flow is 1.54, as desired. At node (1,0), we buy 84 . 155 - 74 . 486 units of the zero-coupon bond, each unit being worth 0.866354, for a total amount of \$8.377, and withdraw 69 . 583 × 1 . 06 - 65 . 38 = 8 . 378 from the bank account, for a net cash-flow of 0 . 001 0. (ii) The value of the portfolio at time 0 is 69 . 583 - 84 . 155092 × 0 . 793847 = 2 . 78 .

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