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.