Answer: 22.7%

The correct answer to the question is D, which indicates a risk-neutral probability of default of 22.7% for Company PQR. This conclusion is derived using the risk-neutral binomial tree methodology and the given information about the bond and the market conditions. The bond in question is a zero-coupon bond with a face value of USD 2,000,000 and is currently trading at 75% of its face value, which means it is priced at USD 1,500,000. The risk-free rate is given as 3% per year, and the recovery rate in the event of default is 0%. The risk-neutral valuation approach is used to calculate the expected payoff of the bond, which is then equated to the risk-free investment payoff. The equation used to find the risk-neutral probability of default (PD) is: \[ 1.5 \times e^{0.03} = 0 \times PD + 2 \times (1 - PD) \] This equation represents the present value of the bond's payoff in a risk-neutral world. The left side of the equation is the present value of the bond's price, which is USD 1,500,000 compounded at the risk-free rate for one year. The right side of the equation is the expected payoff from the bond, which is zero in the case of default (since the recovery rate is 0%) and the full face value of USD 2,000,000 with a probability of (1 - PD), which is the probability of no default. Solving for PD: \[ PD = 1 - \left(\frac{1.5 \times e^{0.03}}{2}\right) \] \[ PD = 1 - \left(\frac{1.5 \times 1.03}{2}\right) \] \[ PD = 1 - 0.76575 \] \[ PD = 0.227 \] This calculation shows that the risk-neutral probability of default is approximately 22.7%, which corresponds to option D. This probability reflects the market's expectation of default under the assumption of a risk-neutral valuation, where investors are indifferent to risk and only the expected return is considered.

Author: LeetQuiz Editorial Team