Question 3

In class, we typically infer the value of the conversion option on a convertible bond by subtracting the value of the straight bond component of the convertible bond from the price of the convertible bond itself. In this question, assume you are an investment banker who wants to determine the value of the conversion option on a convertible bond without relying on the price of the convertible bond. Consider a convertible bond with a face value of $100M, a maturity of 10 years, and an annual coupon rate of 6 percent. The conversion option is European and can only be exercised exactly four years before maturity ( = 6). The conversion ratio is 2M shares. Based on your firms credit rating, you determine the expected return on straight debt with a similar maturity is 5 percent per year. Assume that the firms stock price can take on only one of three values at exactly four years before maturity: $30 (State B), $60 (State M), or $90 (State G). The respective risk-neutral probabilities of these states are 40 percent, 40 percent, and 20 percent. Because we are given the risk-neutral probabilities, the proper discount rate for the expected payoff from the conversion option is the risk-free rate, which we will assume is 4 percent per year.

a) What is the value of the straight bond component of the convertible bond at = 0?

b) What is the payoff of the conversion option in each of the three states of the world at = 6? The option payoff is given by max{ 2 ( = 6), 0}.

c) What is the value of the conversion option today? This is calculated as the present value of the expected payoff from the conversion option. d) Based on your answers in (a) and (c), what is the value of the convertible bond?