In Practice Using a Probit to Estimate the Probability of Bankruptcy
It is possible to estimate the probability of default using statistical techniques, when there is sufficient data avaialble. For instance, if we have a database that lists all firms that went bankrupt during a period of time, as well as firms that did not go bankrupt during the same period, together with descriptive characteristics on these firms, a probit analysis can be used to estimate the likelihood of bankruptcy as a function of these characteristics. The steps involved in a probit analysis are as follows:
1. Identify the event of interest: Probits work best when the event either occurs or it does not. For bankruptcy, the event might be the filing for bankruptcy protection under the law.
2. Over a specified time period, collect information on all the firms that were exposed to the event. In the bankruptcy case, this would imply collecting information on which firms that filed for bankruptcy over a certain period (say, 5 years).
3. Based upon your knowledge of the event, and other research on it, specify measurable and observable variables that are likely to be good predictors of that event. In the case of bankruptcy, these might include excessive debt ratios, declining income, poor project returns and small market capitalization.
4. Collect information on these variables for the firms that filed for bankruptcy, at the time of the filing. Collect the same information for all other firms that were in existence at the same time, and which have data available on them on these variables. (If this is too data intensive, a random sampling of the firms that were not exposed to the event can be used.) In the bankruptcy analysis, this would imply collecting information on debt ratios, income trends, project returns and market capitalization on the firms that filed for bankruptcy at the time of the filing, and all other firms across the period.
5. In a probit, the dependent variable is the occurrence of the specified event (1 if it occurs, 0 if it does not) and the independent variables are the variables specified in step 3. The output from the probit looks very much like the output from a multiple regression, with statistical significance attached to each of the independent variables.
25 In Warner's study of railroad bankruptcies, the direct cost of bankruptcy seems to be about 5%.
Once the probit has been done, the probability of a firm defaulting can be estimated by plugging in that firm's values for the independent variables into the probit. The predicted value that emerges from the probit is the probability of default.
Illustration 8.8: Using the Adjusted Present Value Approach to calculate Optimal Debt Ratio for Disney in 2004
This approach can be applied to estimating the optimal capital structure for Disney. The first step is to estimate the value of the unlevered firm. To do so, we start with the firm value of Disney in 2004 and net out the effect of the tax savings and bankruptcy costs arising from the existing debt.
Current Market Value of Disney = Value of Equity + Value of Debt = $55,101+$14,668 = $ 69,789
We first compute the present value of the tax savings from the existing debt, assuming that the interest payment on the debt constitutes a perpetuity, using a marginal tax rate for Disney of 37.30%.
PV of Tax Savings from Existing Debt = Existing Debt * Tax Rate
= $14,668* 0.373 = $ 5,479 million Based upon Disney's current rating of BBB+, we estimate a probability of bankruptcy of 1.41% from Table 8.18. The bankruptcy cost is assumed to be 25% of the firm value, prior to the tax savings.26 Allowing for a range of 1040% for bankruptcy costs, we have put Disney's exposure to expected bankruptcy costs in the middle of the range. There are some businesses that Disney is in where the perception of distress can be damaging theme parks, for instance  but the movie and broadcasting businesses are less likely to be affected since projects tend be shorter term and on a smaller scale. PV of Expected Bankruptcy Cost = Probability of Default * Bankruptcy cost
= 1.41% * (0.25* 69,789) = $ 984 million We then compute the value of Disney as an unlevered firm. Value of Disney as an Unlevered Firm
= Current Market Value  PV of Tax Savings + Expected Bankruptcy Costs
The next step in the process is to estimate the tax savings in table 8.19 at different levels of debt. While we use the standard approach of assuming that the present value is calculated over a perpetuity, we reduce the tax rate used in the calculation, if interest expenses exceed the earnings before interest and taxes. The adjustment to the tax rate was described more fully earlier in the cost of capital approach.
Debt Ratio 
$ Debt 
Tax Rate 
Tax Benefits 
0% 
$0 
37.30% 
$0 
10% 
$6,979 
37.30% 
$2,603 
20% 
$13,958 
37.30% 
$5,206 
30% 
$20,937 
37.30% 
$7,809 
40% 
$27,916 
31.20% 
$8,708 
50% 
$34,894 
18.72% 
$6,531 
60% 
$41,873 
15.60% 
$6,531 
70% 
$48,852 
13.37% 
$6,531 
80% 
$55,831 
11.70% 
$6,531 
90% 
$62,810 
10.40% 
$6,531 
The final step in the process is to estimate the expected bankruptcy cost, based upon the bond ratings, the probabilities of default, and the assumption that the bankruptcy cost is 25% of firm value. Table 8.20 summarizes these probabilities and the expected bankruptcy cost, computed based on the levered firm value Expected Bankruptcy Cost at x% debt
= (Unlevered firm value + Tax benefits from debt at x% debt) * (Bankruptcy cost as % of firm value) * Probability of bankruptcy
Debt Ratio 
Bond Rating 
Probability of Default 
Expected Bankruptcy Cost 
0% 
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