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# sol10 - Solutions for Unit 10 Questions 1 A rms asset value...

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Solutions for Unit 10 Questions 1. A firm’s asset value is modeled using Merton’s model. Assume V 0 = 1 , 000 , 000, and that the return rate on the assets is μ 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 σ V of the firm’s return on its asset value if its probability of default is 0.06. Solution: We need to write the probability of default under Merton’s model and solve for σ V : 0 . 06 = P ( V T < B ) = P ( N (0 , 1) < (ln B - ln( V 0 ) - ( μ V - σ 2 V / 2) T ) / ( σ V T )) because under Merton’s model, we have that ln V T N (ln V 0 + ( μ V - σ 2 V / 2) T, σ 2 V T ), where here V 0 = 10 6 , μ V = 0 . 1, B = 500 , 000, T = 3. Hence we need to have z 0 . 06 = (ln B - ln( V 0 ) - ( μ V - σ 2 V / 2) T ) / ( σ V T ) , where z 0 . 06 1 . 5551. Solving for σ V we find σ 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) φ ((ln( c ) - a - b ) / b ) , where, as usual, φ ( 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 0 + ( μ V - σ 2 V / 2) T and b = σ 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) φ ((ln( c ) - a - b ) / b )) / 0 . 06 = 0 . 80635 , where c = B . Note that we take the expectation under the physical measure here, because we are not pricing. 1

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2. A firm’s asset value is modeled using Merton’s model. Assume V 0 = 750 , 000, that the return rate μ V = 0 . 08 and σ V = 0 . 25. The firm
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