Problem 3 (15): The value of a company's equity is $4 million and its volatility is 60%. The debt that will have to be repaid in two years is $15 million. The risk-free interest rate is 6% per annum. Use Merton's model to estimate the probability of default. (Problem 19.25, Hull). The Black-Scholes formula for the value of a European call of term T, struck at K on a non- dividend-paying stock of price S and volatility o isc=S,N(d) Ker (d,), where S. otd, = In + +(r+o? /2)T and ovId = In (r-o/2)T with risk-free rate r and K where N is the cumulative normal distribution -- () )+(v- a) Why is this result helpful? b) Rewrite the above option formula in a form explicitly representing the value of the company's assets, V, the value of its equity, E, and the amount of debt to be repaid, D, at time T. Carefully define any other variables and subscripts needed. c) We know that the value of a company's equity, E, will be a function of the value of its assets, V. Use Ito's Lemma to create a relationship that, when taken with your result from b), will provide a second equation to solve for the two unknowns the value of the company's assets today, V., and the volatility of the assets, ov. d) Use the results from b) and c) in the solver function of Excel to determine the probability of default. e) What is the market value of the debt? What is the present value of the debt? What is the expected loss on the debt as a percent of its no default value? f) Compare the expected loss with the probability of default to get the expected recovery in event of default