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Unformatted text preview: Solutions for Unit 10 Questions 1. A firms asset value is modeled using Mertons model. Assume V = 1 , 000 , 000, and that the return rate on the assets is V = 0 . 1 The firm has issued a 3-year zero coupon bond with a face value of B = 500 , 000. (a) Find the volatility V of the firms return on its asset value if its probability of default is 0.06. Solution: We need to write the probability of default under Mertons model and solve for V : . 06 = P ( V T < B ) = P ( N (0 , 1) < (ln B- ln( V )- ( V- 2 V / 2) T ) / ( V T )) because under Mertons model, we have that ln V T N (ln V + ( V- 2 V / 2) T, 2 V T ), where here V = 10 6 , V = 0 . 1, B = 500 , 000, T = 3. Hence we need to have z . 06 = (ln B- ln( V )- ( V- 2 V / 2) T ) / ( V T ) , where z . 06 1 . 5551. Solving for V we find V = 0 . 31386. (b) Determine the expected fraction of money recovered by the lender when the firm defaults (given that there has been default). Use the fact that if X N ( a,b ) then E( e X 1 e X <c ) = exp( a + b/ 2) ((ln( c )- a- b ) / b ) , where, as usual, ( x ) = P ( N (0 , 1) x ). Solution: We need to compute E V T B | V T < B where ln V T N ( a,b ) with a = ln V + ( V- 2 V / 2) T and b = 2 V T . Hence the above expectation is equal to E V T B 1 V T <B /P ( V T < B ) = (1 /B )(exp( a + b/ 2) ((ln( c )- a- b ) / b )) / . 06 = 0 . 80635 , where c = B . Note that we take the expectation under the physical measure here, because we are not pricing....
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This note was uploaded on 12/12/2010 for the course ACTSC 445 taught by Professor Christianelemieux during the Fall '09 term at Waterloo.
- Fall '09