Question 14.7: The firm X has a total asset value of £500m, whilst the firm...
The firm X has a total asset value of £500m, whilst the firm Y has a total asset value of £800m. The expected rate of growth of X’s asset value is 10% per annum, whilst its volatility is 30% per annum. For Y, the expected rate of growth is 5% per annum with a volatility of 10% per annum. The returns of the two firms are linked by a Frank copula with a parameter, α, of 2.5. If the total borrowing for firm X consists of a fixed repayment of £300m, whilst for Y it is £750m, in each case repayable in exactly one year’s time, what is the probability that the both firms will be insolvent at this point?
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) .
The default probability for firm X was established in Example 14.5.3 as 2.96%. For firm Y, B = 750, Y_0 = 800, r_Y = 0.05, \sigma _Y = 0.10 and T = 1 Substituting these values into the above equation gives:
Pr(Y_1\leq 800)=\Phi \left(\frac{\ln (750/800)-[0.50-(0.10^2/2)]}{0.10}\right)=0.1367.The joint probability under a Frank copula in terms of firm values at time T is given by:
Pr(X_T\leq x\;and\;Y_T\leq y)=-\frac{1}{a}\ln \left[1+\frac{(e^{-\alpha F(x)}-1)(e^{-\alpha F(y) }-1)}{e^{-\alpha }-1} \right] .
Here, for T = 1, the values needed are F(x) = 0.0296, F(y) = 0.1367
and α =2.5. This gives:
Pr(X_1\leq 300\;and\;Y_1\leq 750)=-\frac{1}{2.5}\ln \left[1+\frac{-0.0713\times -0.2895}{-0.9179} \right]=0.0091.
The probability that both firms will be insolvent is, therefore, 0.91%.