Helpful hints for Problem Set 9

A student asked me whether it’s okay to solve Problem Set 9 by using Excel. While I generally encourage students to use Excel for the purpose of validating their work (especially for computationally challenging problem sets such as the present one), I also expect students to demonstrate understanding and knowledge of the logical framework upon which any given problem is based. In other words, I expect you to show and explain your work on this problem set just as you would have to show and explain your work if this was an exam question.

By all means, create your own spreadsheet model of Problem Set 9 to validate your answers for this problem set. But start out by devising you own computation strategy using a piece of paper, pen or pencil, and calculator. Since you know that the value of risky debt is equal to the value of safe debt minus the value of the limited liability put option, one approach to solving this problem set would be to start out by calculating the value of a riskless bond, and the value of the limited liability put option. The value of a riskless bond is V(B) = B{e^{ - rT}}, where B corresponds to the promised payment to creditors. The value of the option to default (V(put)) can be calculated by applying the BSM put equation (see the second bullet point on page 8 of http://fin4335.garven.com/spring2018/lecture16.pdf); this requires 1) calculating {d_1} and {d_2}, 2) using the Standard Normal Distribution Function (“z”) Table to find 1-N({d_1}) and 1- N({d_2}) , and inputting these probabilities into the BSM put equation, where the exercise price corresponds to the promised payment to creditors and the value of the underlying asset corresponds to the value of the firm’s assets (keep in mind that the (risk neutral) probability of default corresponds to  1- N({d_2}) for reasons explained during yesterday’s class meeting). Once you obtain the value of the safe bond (V(B)) and the value of the option to default (V(put)) for each firm, then the fair value for each firm’s debt is simply the difference between these two values; i.e., V(D) = V(B) – V(put). Upon finding V(D) for firm 1 and firm 2, then you can obtain these bonds’ yields to maturity (YTM ) by solving for YTM in the following equation: V(D) = B{e^{ - YTM(T)}}; the credit risk premium is equal to the difference between the yield to maturity and the riskless rate of interest.

Since the value of equity corresponds to a call option written on the firm’s assets with exercise price equal to the promised payment to creditors, you could also solve this problem by first calculating the value of each firm’s equity (V(E)) using BSM call equation (see the second bullet point on page 7 of http://fin4335.garven.com/spring2018/lecture16.pdf and substitute the value of assets (V(F)) in place of S and the promised payment of $B in place of K in that equation). Once you know V(E) for each firm, then the value of risky debt (V(D)) is equal to the difference between the value of assets V(F) and the value of equity V(E) Then YTM and credit risk premium follow in the manner described in the previous paragraph.

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