The standard Discounted Cash Flow (DCF) method is the typical tool used to value firms, but one drawback its inability to handle changing debt to equity levels. As we know, higher debt offers a higher tax shield, which in turn increases both firm value and equity value. To explicitly account for the debt tax shield, the Adjusted Present Value (APV) method is used. APV values the firm without leverage, and then values the debt tax shields to determine the value of the whole firm.

### Using the APV Method

#### Step 1: Determine the Discount Rate

Determining the discount rate is done with a similar process to the cost of equity calculations in a normal DCF. The equation used is almost identical to the CAPM model, except for the fact that we use unlevered Beta.

*r _{U }= r_{f }+ β_{U }× E(r_{m }– r_{f})*

Where unlevered Beta can be calculated with this equation:

*β _{U} = E / (D + E) × β_{E}*

As per the equation, unlevered Beta is always less or equal to the levered Beta. This should make sense, as the variation of cash flows to equityholders is higher for levered firms. This in turn increases its systematic risk and Beta.

#### Step 2: Discount Free Cash Flows

As with the DCF method, we need to discount the free cash flows of the firm. Rather than using the weighted average cost of capital, we treat the firm as one with no debt and use the unlevered discount rate determined above. The terminal value is also determined with the same unlevered discount rate in a perpetuity formula.

*V _{U }= ∑ FCF_{t} / (1 + r_{U})^{t} + TV / (1 + r_{U})^{T}*

*TV = FCF _{T} / (r_{U }– g)*

As discussed above, the value we get here is the firm value, if the firm is not holding any debt. The unlevered firm value is typical lower than the levered firm value, unless costs of financial distress are greater than the value of debt tax shields.

#### Step 3: Discount Debt Tax Shields

To account for debt tax shields explicitly, we discount the value of the tax shields from each period. We typically use the cost of debt as the discount rate, as we can assume the tax shields to have similar risks as debt. In some cases, we may use a different discount rate. Similar to the discount cash flow, we assume a terminal value at the end, where we would use the perpetuity formula. The terminal period is the period in which debt level stops changing. The equation for PV of tax shields is:

_{PV(tax shields) }= ∑ (D_{t-1} × r_{D} × t) / (1 + r_{D})^{t} + TV(tax shields) / (1 + r_{D})^{T}

*TV(tax shields) = D*_{T-}_{1} × t

#### Step 4: Determine Enterprise Value and Equity Value

To determine the enterprise value of the firm, we take the sum of the present value of the unlevered firm and the present value of the debt shield. If the firm has equity investments (20% – 50% ownership in another firm), those investments should be valued separately from the unlevered firm. If there is risk of financial distress, we must incorporate the present value of the costs involved. If the debt levels do not change, we should get the same firm value with the normal DCF method.

To find the equity value, we take out the net debt from the enterprise value. Usually, this means we subtract debt and add cash. This is exactly the same as the DCF method.

### When to Use APV Method

APV method explicitly accounts for the debt tax shields, and should be used for firms with changing debt levels. Theoretically, DCF can also handle changing debt levels, as the debt tax shield is implicitly contained in the WACC. However, having a different WACC every period makes it difficult to implement, and prone to mistakes.

One specific situation where APV method really shines is in a debt funded acquisition. Some acquisitions are highly levered, at unsustainable levels of debt to equity. After the acquisition, debt is then brought back to a reasonable level as quickly as possible with the free cash flows from the business. The combined firm then transitions to a sustainable and static capital structure, where it can be valued as a perpetuity.

Another situation could be a shift in the corporate capital structure. Strategically, a company may want to change its debt to equity ratio, and the APV method will explicitly map out the benefits and costs of that strategy.

### Financial Distress and Other Costs

To determine the costs of financial distress, we can break it down into probability of financial distress and loss given financial distress. Probability can be calculated with data from credit rating agencies, who keeps statistics on defaults for each credit rating. Loss given financial distress is much harder to estimate, and will differ from firm to firm. For example, a law firm may lose a significant number of their star lawyers in event of a default, which reduces their assets and equity value, but a telecom firm would be relatively resilient to a loss of talent.

Statistics 101 | · · · |
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