Final answer:
To calculate the risk-free interest rate for a defaulting zero-coupon bond with a face value of $4,000 maturing in one year and paying $3,000, we set the future value equal to the present value compounded at the risk-free rate. Solving the formula, we find that the risk-free rate is 33.33%.
Step-by-step explanation:
The question centers on the concept of determining the risk-free interest rate given a zero-coupon bond that is going to default. The bond, with a face value of $4,000, is expected to pay only $3,000 at maturity, and its current yield to maturity (YTM) is 38.4%. To find the risk-free rate, we compare the situation with a risk-free bond.
Assuming a risk-free bond with the same face value ($4,000) and maturity of one year from today was issued, it would sell for a price that, when compounded at the risk-free rate, would equal the face value ($4,000) at maturity. Given that the risky bond sells for $3,000 (given its default risk), we can infer that the market requires a return of 38.4% to compensate for this risk.
To find the risk-free interest rate, we set the future value ($4,000) equal to the present value ($3,000) compounded at the risk-free rate for one year. This can be calculated using the formula: Future Value = Present Value * (1 + risk-free rate)^n, where n is the number of periods (1 year in this case).
Plugging in our known values, $4,000 = $3,000 * (1 + risk-free rate)^1. Solving for the risk-free rate, we get: risk-free rate = ($4,000/$3,000) - 1 = 1.3333 - 1 = 0.3333, or 33.33%.
Therefore, the zero-coupon bond’s risk-free interest rate is 33.33%.