## D

where bunlevered = Unlevered beta of the firm bcurrent = Current equity beta of the firm t = Tax rate for the firm D/E = Current debt/equity ratio

This unlevered beta can then be used to arrive at the unlevered cost of equity.

### Expected Tax Benefit from Borrowing

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,

= (Tax Rate )(Cost of Debt )(Debt) Cost of Debt Value of Tax Benefits = (Tax Rate)(Debt)

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.

### Estimating Expected Bankruptcy Costs and Net Effect

The third step is to evaluate the effect of the given level of debt on the default risk of the firm and on expected bankruptcy costs. In theory, at least, this requires the estimation of the probability of default with the additional debt and the direct and indirect cost of bankruptcy. If pa 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.

= (Probability of Bankruptcy )(PV of Bankruptcy Cost)

PV of Expected Bankruptcy cost

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 each rating.

For instance, Table 15.8, extracted from a study by Altman and Kishore, summarizes the probability of default over ten years by bond rating class in 1998.8

Table 15.8: 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. Research that has looked at the direct cost of bankruptcy concludes that they are small9, relative to firm value. The indirect costs of bankruptcy can be substantial, but the costs vary widely

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

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

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.

Illustration 15.5: Valuing a firm with the APV approach: Tube Investments

In Illustration 15.1, we valued Tube Investments, using a cost of capital approach. Here, we re-estimate the value of the firm using an adjusted present value approach in three steps.

Step 1: Unlevered firm value

To estimate the unlevered firm value, we first compute the unlevered beta. Tube Investment's beta is 1.17, its current market debt to equity ratio is 79% and the firm's tax rate is 30%.

1.17

Using the rupee riskfree rate of 10.5% and the risk premium of 9.23% for India, we estimate an unlevered cost of equity.

Unlevered cost of equity = 10.5% + 0.75(9.23%) = 17.45%

Using the free cash flow to the firm that we estimated in Illustration 15.1 of Rs 212.2 million and the stable growth rate of 5%, we estimate the unlevered firm value:

212.2

Unlevered firm value=-^-= \$1704.6 million

0.1745-0.05

Step 2: Tax benefits from debt

The tax benefits from debt are computed based upon Tube Investment's existing dollar debt of Rs. 1807.3 million and the tax rate of 30%:

Expected tax benefits in perpetuity = Tax rate (Debt) = 0.30 (1807.3) = Rs 542.2 million Step 3: Expected bankruptcy costs

To estimate this, we made two assumptions. First, based upon its existing rating, the probability of default at the existing debt level is 10%. Second,the cost of bankruptcy is 40% of unlevered firm value.

Expected bankruptcy cost =Probability of bankruptcy * Cost of bankruptcy * Unlevered firm value = 0.10*0.40*1704.6 = Rs 68.2 million

The value of the operating assets of the firm can now be estimated.

Value of the operating assets

= Unlevered firm value + PV of tax benefits - Expected Bankruptcy Costs = 1704.6 + 542.2 - 68.2 = Rs 2178.6 million

Adding to this the value of cash and marketable securities of Rs. 1365.3 million, we obtain a value for the firm of Rs 3543.9 million. In contrast, we valued the firm at Rs. 3367.3 million with the cost of capital approach.

Cost of Capital versus APV Valuation

In an APV valuation, the value of a levered firm is obtained by adding the net effect of debt to the unlevered firm value.

FCFF (1 + e) Value of Levered Firm =-^—^ + tcD -jtaBC

Pu-g

In the cost of capital approach, the effects of leverage show up in the cost of capital, with the tax benefit incorporated in the after-tax cost of debt and the bankruptcy costs in both the levered beta and the pre-tax cost of debt. Will the two approaches yield the same value? Not necessarily. The first reason for the differences is that the models consider bankruptcy costs very differently, with the adjusted present value approach providing more flexibility in allowing you to consider indirect bankruptcy costs. To the extent that these costs do not show up or show up inadequately in the pre-tax cost of debt, the APV approach will yield a more conservative estimate of value. The second reason is that the APV approach considers the tax benefit from a dollar debt value, usually based upon existing debt. The cost of capital approach estimates the tax benefit from a debt ratio that may require the firm to borrow increasing amounts in the future. For instance, assuming a market debt to capital ratio of 30% in perpetuity for a growing firm will require it to borrow more in the future and the tax benefit from expected future borrowings is incorporated into value today.

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