Problem Set 9 (due at the beginning of class on Thursday, April 25) is essentially a reparameterized version of the class problem that we will work on during tomorrow’s class meeting (also described in pp. 6-8 of the Credit Risk lecture note).
In order to fully comprehend the pricing of credit risk in the Black-Scholes-Merton framework, it is advised that students begin by solving the problem manually, followed by creating a spreadsheet model to validate their work. The computation strategy for completing this problem set is best described as follows:
- Calculate and , where and . Since and represent critical values for the standard normal distribution, we calculate and accordingly. Since corresponds to the risk neutral probability that at date T, it follows that 1 – corresponds to the risk neutral probability that at date T; i.e., this is the risk neutral probability that the firm defaults on its promised debt payment. Also, because of the symmetry of the standard normal distribution, 1 – = .
- Note that the value of risky debt, corresponds to the value of safe debt () minus the value of the limited liability put option , where F is the terminal value of risky assets, B is the terminal (date T) value of a riskless zero coupon (also known as a “pure discount”) bond and . Thus, the “fair market value for the bond” is determined by calculating . The dollar value of the limited liability put option is given by , which also corresponds to the “fair premium” for credit insurance (cf. part 3 of Problem Set 9).
- The class problem and Problem Set 9 also ask for the yield to maturity and credit risk premium. The yield to maturity (YTM) for a T period pure discount bond corresponds to the rate of interest which must be earned from date 0 to date T in order for the future value of to be equal to B; i.e., . Solving for YTM in this equation, we find that . The credit risk premium corresponds to the difference between the yield to maturity (YTM) and the riskless rate of interest r. This risk premium compensates investors for bearing default risk costs. Intuitively, it makes a lot of sense that there is a positive relationship between the risk of default and the credit risk premium.