Question 14.4: A firm has a total asset value of 500. The expected rate of ...

Merton’s model gives the probability of default at time T as:

Pr(X_T\leq B)=\Phi \left(\frac{\ln (B/X_0)-(r_X-\sigma _X^2/2)T}{\sigma _X\sqrt{T} } \right) .

For this firm, B = 300, X_0 = 500, r_X = 0.10, \sigma _X = 0.30 and T = 1  Substituting these values into the above equation gives:

Pr(X_1\leq 300)=\Phi \left(\frac{\ln (300/500)-[0.10-(0.30^2/2)]}{0.30}\right)=0.0296.

The probability of insolvency is therefore 2.96%.