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capital by industry group in the United States for the most recent quarter.


Adjusted Present Value Approach

In the adjusted present value (APV) approach, we begin with the value of the firm
without debt. As we add debt to the firm, we consider the net effect on value by
considering both the benefits and the costs of borrowing. The value of the levered firm
can then be estimated at different levels of the debt, and the debt level that maximizes
firm value is the optimal debt ratio.

Steps in the Adjusted Present Value approach

In the Adjusted Present Value approach, we assume that the primary benefit of
borrowing is a tax benefit, and that the most significant cost of borrowing is the added



22 The normal range for long term interest rates in the United States for the last 40 years has been between
4 and 8%. There was a short period between 1978 and 1982 when long term interest rates were much
higher.

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risk of bankruptcy. To estimate the value of the firm, with this assumption, we proceed
in three steps. We begin by estimating the value of the firm with no leverage. We then
consider the present value of the interest tax savings generated by borrowing a given
amount of money. Finally, we evaluate the effect of borrowing the amount on the
probability that the firm will go bankrupt, and the expected cost of bankruptcy.
Step 1: Estimate the value of the firm with no debt: The first step in this approach is the
estimation of the value of the unlevered firm. This can be accomplished by valuing the
firm as if it had no debt, i.e., by discounting the expected after-tax operating cash flows at
the unlevered cost of equity. In the special case where cash flows grow at a constant rate
in perpetuity,
Value of Unlevered Firm = FCFFo (1+g)/(ρu - g)
where FCFF0 is the current after-tax operating cash flow to the firm, ρu is the unlevered
cost of equity, and g is the expected growth rate. The inputs needed for this valuation are
the expected cashflows, growth rates and the unlevered cost of equity. To estimate the
latter, we can draw on our earlier analysis and compute the unlevered beta of the firm “
βunlevered = βcurrent/[1+(1-t)D/E]
where
βunlevered = Unlevered beta of the firm,
βcurrent = Current equity beta of the firm,
t = Tax rate for the firm and
D/E = Current debt/equity ratio.
This unlevered beta can then be used to arrive at the unlevered cost of equity.
Alternatively, we can take the current market value of the firm as a given and back out
the value of the unlevered firm by subtracting out the tax benefits and adding back the
expected bankruptcy cost from the existing debt.
Current Firm Value = Value of Unlevered firm + PV of tax benefits “ Expected
Bankruptcy cost
Value of Unlevered firm = Current Firm Value “ PV of tax benefits + Expected
Bankruptcy costs




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Step 2: Estimate the present value of tax benefits from debt: The second step in this
approach is the calculation of the expected tax benefit from a given level of debt. This tax
benefit is a function of the tax rate of the firm and is discounted at the cost of debt to
reflect the riskiness of this cash flow. If the tax savings are viewed as a perpetuity,
Value of Tax Benefits =[ Tax Rate * Cost of Debt * Debt] / Cost of Debt
= Tax Rate * Debt
= tc D
The tax rate used here is the firm™s marginal tax rate, and it is assumed to stay constant
over time. If we anticipate the tax rate changing over time, we can still compute the
present value of tax benefits over time, but we cannot use the perpetual growth equation
cited above.
Step 3: Estimate the expected bankruptcy costs Bankruptcy Cost: This is the cost
associated with going bankruptcy. It
as a result of the debt: The third step is to
includes both direct costs (from going
evaluate the effect of the given level of debt on
bankrupt) and indirect costs (arising from
the default risk of the firm and on expected
the perception that a firm may go
bankruptcy costs. In theory, at least, this requires bankrupt).
the estimation of the probability of default with
the additional debt and the direct and indirect cost of bankruptcy. If πa is the probability
of default after the additional debt and BC is the present value of the bankruptcy cost, the
present value of expected bankruptcy cost can be estimated“
PV of Expected Bankruptcy cost = Probability of Bankruptcy * PV of Bankruptcy Cost
= πa BC
This step of the adjusted present value approach poses the most significant estimation
problem, since neither the probability of bankruptcy nor the bankruptcy cost can be
estimated directly. There are two basic ways in which the probability of bankruptcy can
be estimated indirectly. One is to estimate a bond rating, as we did in the cost of capital
approach, at each level of debt and use the empirical estimates of default probabilities for




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each rating. For instance, table 8.18, extracted from a study by Altman and Kishore,
summarizes the probability of default over ten years by bond rating class in 1998.23
Table 8.18: Default Rates by Bond Rating Classes
Bond Rating Default Rate
D 100.00%
C 80.00%
CC 65.00%
CCC 46.61%
B- 32.50%
B 26.36%
B+ 19.28%
BB 12.20%
BBB 2.30%
A- 1.41%
A 0.53%
A+ 0.40%
AA 0.28%
AAA 0.01%

Source: Altman and Kishore (1998)
The other is to use a statistical approach, such as a probit to estimate the probability of
default, based upon the firm™s observable characteristics, at each level of debt.
The bankruptcy cost can be estimated, albeit with considerable error, from studies
that have looked at the magnitude of this cost in actual bankruptcies. Studies that have
looked at the direct cost of bankruptcy conclude that they are small24, relative to firm
value. The indirect costs of bankruptcy can be substantial, but the costs vary widely
across firms. Shapiro and Titman speculate that the indirect costs could be as large as 25
to 30% of firm value but provide no direct evidence of the costs.
The net effect of adding debt can be calculated by aggregating the costs and the
benefits at each level of debt.
Value of Levered Firm = FCFFo (1+g)/(ρu - g) + tc D - πa BC
We compute the value of the levered firm at different levels of debt. The debt level that
maximizes the value of the levered firm is the optimal debt ratio.


23 This study estimated default rates over ten years only for some of the ratings classes. We extrapolated
the rest of the ratings.


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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.


24 In Warner™s study of railroad bankruptcies, the direct cost of bankruptcy seems to be about 5%.

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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.25 Allowing for a range of 10-40% 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




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= $ 69,789 + $ 5,479 - $ 984
= $ 65,294 million
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.
Table 8.19: Tax Savings From Debt (tcD): Disney
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
Table 8.20: Expected Bankruptcy Cost, Disney
Expected
Probability of Bankruptcy
Debt Ratio Bond Rating Default Cost
0% AAA 0.01% $2
10% AAA 0.01% $2
20% A- 1.41% $246


25 This estimate is based upon the Warner study, which estimates bankruptcy costs for large companies to
be 10% of the value, and upon the qualitative analysis of indirect bankruptcy costs in Shapiro and Cornell.

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30% BB 7.00% $1,266
40% CCC 50.00% $9,158
50% C 80.00% $14,218
60% C 80.00% $14,218
70% C 80.00% $14,218
80% C 80.00% $14,218
90% C 80.00% $14,218

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