问一道题：NO.PZ2016082406000037

问题如下：

The capital structure of HighGear Corporation consists of two parts: one five- year zero-coupon bond with a face value of $100 million and the rest is equity. The current market value of the firm’s assets is $130 million and the expected rate of change of the firm’s value is 25%. The firm’s assets have an annual volatility of 30%. Assume that firm value is lognormally distributed, with constant volatility. The firm’s risk management division estimates the distance to default using the Merton model, or $\frac{\ln(\frac{\text{K}}{\text{V}})-\delta\tau+0.5\sigma^2\tau}{\sigma\sqrt\tau}$Given the distance to default, the estimated default probability is

选项：

2.74%

12.78%

12.79%

30.56%

解释：

ANSWER: A

We compute $z=\frac{\ln(\frac{\text{K}}{\text{V}})-\delta\tau+0.5\sigma^2\tau}{\sigma\sqrt\tau}=\frac{\ln(\frac{\text{100}}{\text{130}})-25\%\times5+0.5\times30\%^2\times5}{30\%\sqrt5}=-1.919$. The PD is then $N{(z)}=N{(-1.919)}=2.749\%$.