sol5 - 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 5 Exercises 1. (i) We have that A * = 200(1 . 06) - 2 + 500(1 . 07) - 4 + 300(1 . 075) - 5 = 768 . 41 , and the ytm is obtained by solving A * = 200(1 + y ) - 2 + 500(1 + y ) - 4 + 300(1 + y ) - 5 . Using GoalSeek, we find y = 7 . 0588%. (ii) We have that D m = 1 A * (2 × 200 × (1 + y ) - 3 + 4 × 500 × (1 + y ) - 5 + 5 × 300 × (1 + y ) - 6 ) = 3 . 5713 D = D m × (1 + y ) = 3 . 8234 C = 1 A * (2 × 3 × 200 × (1 + y ) - 4 + 4 × 5 × 500 × (1 + y ) - 6 + 5 × 6 × 300 × (1 + y ) - 7 ) = 17 . 0978 . (iii) (1) (a) The approximate % price change is - D m × 0 . 005 = - 1 . 786%; (b) The approximate % price change is - D m × 0 . 005 + C × 0 . 005 2 / 2 = - 1 . 764%; The true % price change is (754 . 8558 - 768 . 4145) / 768 . 4145 = - 1 . 765%. (2) The approximate % price change is - D m × - 0 . 0075 = 2 . 6785%; (b) The approximate % price change is - D m × - 0 . 0075 + C × ( - 0 . 0075) 2 / 2 = 2 . 7266%; The true % price change is (789 . 3715 - 768 . 4145) / 768 . 4145 = 2 . 7273%. (iv) The quasi-modified duration is given by D Q = 1 A * ( 2 × 200 × (1 . 06) - 3 + 4 × 500 × (1 . 07) - 5 + 5 × 300 × (1 . 075) - 6 ) = 3 . 5577 . The approximate % price change is (1) 3 . 5577 × 0 . 005 = - 1 . 779%; (2) 3 . 5577 × - 0 . 0075 = 2 . 6682%.

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