Clarification of expectations for and hints concerning Problem Set 9

A student asked me whether it’s okay to use an Excel spreadsheet to solve problem set 9.  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.

As I am sure you are already well aware, you can obtain most of the “correct” answers for problem set 9 by simply downloading and opening up the Credit Risk Spreadsheet in Excel and performing the following steps:

  1. For part A, open the Firm 1 worksheet, replace the “.4” in cell B3 with “.3”.  Then the answers for the three questions are in cells F4, F3, and F5 respectively.
  2. For part B, open the Firm 2 worksheet, replace the “.4” in cell B3 with “.5”.  Then the answers for the three questions are in cells F4, F3, and F5 respectively.
  3. For part C, assuming that  you are able to follow the logic presented in my On the economics of financial guarantees blog post from yesterday afternoon, the fair insurance premiums appear on both of the worksheets, and presumably you also understand from our study of financial derivatives that the expected return on a default-free bond is the riskless rate of interest.

The problem with simply plugging and chugging the spreadsheet template is that one can mechanically follow the steps outlined above without necessarily understanding the credit risk problem.  The key takeaway from our study of credit risk is that limited liability causes prices of bonds issued by risky (poor credit quality) firms to be lower than prices of bonds issued by safe (good credit quality) firms.  In the case of this problem set, firms 1 and 2 are identical in all respects expect for asset risk, and because of limited liability, this implies that in the absence of a financial guarantee, firm 2’s bonds are riskier than firm 1’s bonds. Thus, firm 2’s bonds have a lower market value (and a correspondingly higher yield, or expected return) than firm 1’s bonds, and firm 2 can expect to have to pay more than firm 1 for a financial guarantee which transfers the default risk from investors over to a financial guarantor.  In a competitive market, the fair premium for such a guarantee is given by the value of the limited liability put option.

By all means, make use of the Credit Risk Spreadsheet to validate your answers for the problem set.  But start out by devising you own coherent 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/fall2017/lecture16.pdf); this requires 1) calculating {d_1} and {d_2}, 2) using the Standard Normal Distribution Function (“z”) Table to find N({d_1}) and N({d_2}), and 3) inputting your N({d_1}) and N({d_2}) values into the BSM put equation, where the exercise price corresponds to the promised payment to creditors of $500,000, and the value of the underlying asset corresponds to the value of the firm, which is $1,000,000.  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)}}.

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/fall2017/lecture16.pdf and substitute the value of assets ($1,000,000) in place of S and the promised payment of $500,000 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) = $1,000,000) and V(E).  Upon calculating V(D) in this manner, then obtain these these bonds’ yields to maturity (YTM ) by solving for YTM in the following equation: V(D) = B{e^{ - YTM(T)}}.

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