Answer: 22.7%

The correct answer to the question is D, which indicates a risk-neutral 1-year probability of default of 22.7% for Company PQR. This conclusion is reached using a risk-neutral binomial tree methodology and the risk-free rate of 3% per year. 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 the bond's market price is USD 1,500,000. Given that the recovery rate in the event of default is 0%, the risk-neutral valuation framework is used to equate the present value of the bond's payoffs to the risk-free investment's payoff. The equation to find the risk-neutral probability of default (PD) is set up as follows: \[ 1.5 \times e^{0.03 \times 1} = 0 \times PD + 2 \times (1 - PD) \] This equation represents the expected payoff from the bond in a risk-neutral world. The left side of the equation is the present value of the bond's payoff if it matures without default, which is calculated by multiplying the bond's market price (USD 1,500,000) by the exponential of the continuously compounded risk-free rate (3% per year) over 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%) multiplied by the risk-neutral probability of default (PD), plus the face value of the bond (USD 2,000,000) multiplied by the probability of no default (1 - PD). Solving for PD: \[ PD = 1 - \frac{1.5 \times e^{0.03}}{2} \] \[ PD = 1 - \frac{1.5 \times 1.03}{2} \] \[ PD = 1 - 0.765 \] \[ PD = 0.235 \] This results in a risk-neutral probability of default of approximately 23.5%, which corresponds to option D, 22.7%, when rounded to one decimal place. This calculation assumes that the market has already priced in the credit risk, and the bond's current trading price reflects the market's expectation of the default probability under a risk-neutral measure.

Author: LeetQuiz Editorial Team