At the end of 2019, Treefern Ltd. had BBB-rated, 5-year bonds outstanding with a yield to maturity of 15%. At the time, government bonds with similar maturity had a yield of 2%. Suppose the expected return of the market portfolio is 7% and you believe Treefern Ltd.’s bonds have a beta of 0.45. If the expected loss rate of these bonds in the event of default is 55%. What annual probability of default would be consistent with the yield to maturity of Treefern Ltd.’s bonds at the end of the year 2019?A.19.6%B.22.7%C.23.4%D.20.3%
Question
At the end of 2019, Treefern Ltd. had BBB-rated, 5-year bonds outstanding with a yield to maturity of 15%. At the time, government bonds with similar maturity had a yield of 2%. Suppose the expected return of the market portfolio is 7% and you believe Treefern Ltd.’s bonds have a beta of 0.45. If the expected loss rate of these bonds in the event of default is 55%. What annual probability of default would be consistent with the yield to maturity of Treefern Ltd.’s bonds at the end of the year 2019?A.19.6%B.22.7%C.23.4%D.20.3%
Solution
The yield on a corporate bond is typically composed of the risk-free rate (in this case, the yield on government bonds), a premium for default risk, and a premium for systematic risk.
The default risk premium is the product of the probability of default and the loss given default. We are asked to find the probability of default, so we can rearrange the formula for the default risk premium to solve for this:
Probability of Default = Default Risk Premium / Loss Given Default
The systematic risk premium can be estimated using the Capital Asset Pricing Model (CAPM), which is the product of the bond's beta and the market risk premium (the difference between the expected return of the market portfolio and the risk-free rate). In this case, the beta is 0.45 and the market risk premium is 7% - 2% = 5%, so the systematic risk premium is 0.45 * 5% = 2.25%.
The yield on Treefern Ltd.'s bonds is 15%, so the default risk premium is this yield minus the risk-free rate and the systematic risk premium:
Default Risk Premium = Yield - Risk-free rate - Systematic risk premium = 15% - 2% - 2.25% = 10.75%
Substituting this and the loss given default of 55% into the formula for the probability of default gives:
Probability of Default = 10.75% / 55% = 19.55%
Therefore, the annual probability of default that would be consistent with the yield to maturity of Treefern Ltd.'s bonds at the end of 2019 is closest to 19.55%. The closest option to this value is A. 19.6%.
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