A zero coupon bond with a maturity of one-year pays $1,000 if the issuing firm is not in default. If the firm is in default, the recovery rate is 35%. The risk-free interest rate for one year is 5% and the risk-neutral probability that the firm defaults is 20%. What is the credit spread (over the risk-free rate) on the bond? All yields are in simple terms with annual compounding.

Question

Quizplus · Accepted Answer

The credit spread is the difference between the yield on the risky bond and the yield on the risk-free bond. The yield on the risky bond is the expected return on the bond, which is equal to the risk-free rate plus the credit spread. The expected return on the bond is equal to the face value of the bond times the probability of default times the recovery rate plus the face value of the bond times the probability of no default. The probability of default is 20%, the recovery rate is 35%, and the face value of the bond is $1,000. Therefore, the expected return on the bond is $1,000 * 0.20 * 0.35 + $1,000 * 0.80 = $870. The yield on the risky bond is equal to the expected return on the bond divided by the price of the bond. The price of the bond is equal to the face value of the bond times the present value of the expected return on the bond. The present value of the expected return on the bond is equal to the expected return on the bond divided by the risk-free rate. Therefore, the price of the bond is equal to $1,000 * $870 / 1.05 = $828.57. The yield on the risky bond is equal to $870 / $828.57 = 10.50%. The credit spread is equal to the yield on the risky bond minus the yield on the risk-free bond. The yield on the risk-free bond is 5%. Therefore, the credit spread is 10.50% - 5% = 5.50%.

A Zero Coupon Bond with a Maturity of One-Year Pays