Suppose we have a zero coupon bond that pays $1 after one year if the issuing firm is not in default. If the firm is in default, the recovery rate is 42%. The risk free interest rate for one year is 5%. If the credit spread on the bond is 2.5%, what is the risk-neutral probability of default of the bond? Assume all yields are stated in simple terms with annual compounding.
A) 0.01
B) 0.02
C) 0.03
D) 0.04

Question

Quizplus · Accepted Answer

The price of the bond can be calculated using the formula: $ \text{Price} = \frac{\text{Expected Payoff}}{1 + \text{Risk-Free Rate} + \text{Credit Spread}} $.The expected payoff is $ (1 - \text{Probability of Default}) \times 1 + \text{Probability of Default} \times 0.42 $.Given that the risk-free rate is 5% (or 0.05) and the credit spread is 2.5% (or 0.025), the denominator becomes $ 1 + 0.05 + 0.025 = 1.075 $.Setting the price equal to the expected payoff divided by 1.075 and solving for the probability of default gives us the risk-neutral probability of default. Rearranging the formula and solving for the probability of default, we find it to be approximately 4%, or 0.04, under the given conditions.

Suppose We Have a Zero Coupon Bond That Pays $1