The Adjusted Present Value approach (APV)APV approach begins with the value of the firm without debt. While debt is added to the firm, the net effect on the value of the firm is examined by considering both the benefits and the costs of borrowing. In these case, the assumption is that the primary benefit of borrowing funds is the tax benefit, and the most significant cost of borrowing is the additional risk of bankruptcy. Thus, in the adjusted present value approach, the value of the firm is written as the sum of the value of the firm without debt (the unlevered firm) and the effect of debt on firm value.Firm Value= unlevered firm value + (Tax benefit of debt Expected bankruptcy cost from the debt)Mechanics of APV valuationTo estimate the value of the firm, the steps are as follows:(i) Estimate the value of the firm with no leverage(ii) Consider the present value of the interest tax savings generated by borrowing (iii) Evaluate the effect of borrowing on the probability that the firm will go bankrupt and its expected cost of bankruptcyValue of unlevered firmValuation of a firm as if it had no debt can be done by discounting the expected free cash flow to the firm at the unlevered cost of equity. Value of unlevered firm = FCFF0(1 + g)/(Pu - g)Where FCFF0 is the current after-tax operating cash flow to the firm, ru is the unlevered cost of equity and g is the expected growth rate. The inputs needed for this valuation are the expected cash flows, growth rates and the unlevered cost of equity. Where unlevered cost of equity can be calculated by using unlevered beta of that firm, which is unlevered = current/[1 + (1 t)D/E]Where unlevered = Unlevered beta of the firmcurrent = Current equity beta of the firmt = Tax rate for the firmD/E = Current debt/equity ratio

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

Expected tax benefit from debtTax benefits can be estimated by the tax rate and the interest payment of the firm. This is discounted at cost of debt to show the riskiness of the cash flow. If the tax benefits are seen as a perpetuity then, Value of tax benefits = (Tax rate X Cost of debt X Debt)/Cost of Debt = Tax rate X Debt = tcDIt is assumed that the tax rate here is the marginal tax rate for the firm, and it is constant over time.

Estimating expected bankruptcy costs and effectsThe next step is to evaluate the level of debt on the risk of the firm, and on the expected cost of bankruptcy. This requires an estimation of probability of default with additional debt and the cost (direct and indirect) of bankruptcy. PV of expected bankruptcy cost = Probability of bankruptcy X PV of bankruptcy cost = aBCWhere a is the probability of default after additional debt and BC is the present value of the bankruptcy cost. As both probability of bankruptcy and bankruptcy cost cannot be estimated directly, so this method has major problems of estimation. To estimate the probability of bankruptcy, there are two methods. One method is to estimate the bond ratings and use empirical estimate of default probabilities for the corresponding ratings. For example, the figure below shows the probability of getting default with respect to bond ratings.

Bond RatingDefault Rate

D100.00%

C80.00%

CC65.00%

CCC46.61%

B-32.50%

B26.36%

B+19.28%

BB12.20%

BBB2.30%

A-1.41%

A0.53%

A+0.40%

AA0.28%

AAA0.01%

The other method is to use a statistical approach, such as a probit to estimate the probability of default, based on the firms observable characteristics, at each level of debt.The bankruptcy cost can be estimated 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 small, relative to firm value. The indirect costs of bankruptcy can be substantial, but the costs vary widely across different 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.

Finally, Firm Value= unlevered firms value + (Tax benefit of debt Expected bankruptcy cost from the debt)Or, Firm Value = FCFF0(1 + g)/(Pu - g) + tcD - aBC