##### Anyone knows what is the amount of default? In details

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Present value of the amount of default

The adjusted present value (APV) approach begins with the value of the firm without debt. As debt is added to the firm, the net effect on value is examined by considering both the benefits and the costs of borrowing. To do this, it is assumed that the primary benefit of borrowing is a tax benefit, and that the most significant cost of borrowing is the added risk of bankruptcy.

## Mechanics of APV Valuation

We estimate the value of the firm in three steps:

- Estimate the value of the firm with no leverage.
- Consider the present value of the interest tax savings generated by borrowing a given amount of money.
- Evaluate the effect of borrowing the amount on the probability that the firm will go bankrupt, and the expected cost of bankruptcy.

### Value of Unlevered Firm

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 free cash flow to the firm at the unlevered cost of equity). In the special case where cash flows grow at a constant rate in perpetuity,

_{1})/(ρ

_{u}− g)

where FCFF_{1} is the expected after-tax operating cash flow to the firm, ρ_{u} is the unlevered cost of equity, and g is the expected growth rate. In the more general case, you can value the firm using any set of growth assumptions you believe are reasonable for the firm.

The inputs needed for this valuation are the expected cash flows, growth rates, and the unlevered cost of equity. To estimate the unlevered cost of equity, we can draw on our earlier analysis and compute the unlevered beta of the firm:

_{unlevered}= β

_{current}/[1 + (1 − t)D/E]

where

β

_{unlevered}=Unlevered beta of the firm

β

_{current}=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 and interest payments 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 = t

_{c}D

The tax rate used here is the firm’s marginal tax rate, and it is assumed to stay constant over time. If you anticipate the tax rate changing over time, you can still compute the present value of tax benefits over time, but you cannot use the perpetual growth equation cited earlier. In addition, you would have to modify this equation if the current interest expenses do not reflect the current cost of debt.

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