## D

Exhibit 13.5 WACC, Cost of Equity, and Cost of Debt vs. — with Corporate

Taxes

Exhibit 13.5 WACC, Cost of Equity, and Cost of Debt vs. — with Corporate

Taxes

debt capital as a function of the leverage ratio, D/E, when the corporate tax rate is 34 percent, based on the figures in Example 13.12. Note, as suggested earlier, that as D/E increases, more weight is placed on the bottom horizontal line, rD(1 - Tc), in the computation of the WACC. However, in contrast to the no-tax case, the increase in rE as D/E increases is insufficient to offset the additional weight on the lower debt cost rD(1 - Tc). Hence, unlike the pattern shown in Exhibit 13.3, the WACC in Exhibit 13.4 declines from the starting point of 14.16 percent (= rUA) as D/E increases. The WACC continues to decline as D/E increases, up to the point where the firm eliminates all corporate taxes. Thereafter the WACC is flat.

Dynamic Perpetual Risk-Free Debt. Up to this point we have assumed that the firm has a fixed amount of debt outstanding. Alternatively, if the firm wants to maintain a given ratio of debt to equity, it will need to issue new debt and repurchase equity as the firm's (or project's) value rises and issue equity to retire debt as the value of the firm (or project) falls.

In the Miles and Ezzell (1980, 1985) model, D is perfectly correlated with the value of the unlevered assets of the firm and thus the tax savings from debt issuance are perfectly correlated with the prior period's value of the unlevered assets. This implies that, at least approximately,

implying that the expected returns of assets, unlevered assets, and the debt tax shield are about the same. As periods become arbitrarily short, this equality between the expected returns of the assets and the unlevered assets holds exactly, rather than approximately, in which case equation (13.11) reduces to

Part III Valuing Real Assets

Dynamic updating of debt to maintain a constant leverage ratio has implications for both the WACC and the APV methods because it affects asset betas and, thus, the formulas for leveraging and unleveraging equity betas. The Miles and Ezzell model, for example, which assumes risk-free debt, implies that only the tax shield associated with the first interest payment, which has a present value of TcDrf¡(1 + rf) is certain, and the remainder of the tax shield has the same beta risk as the unlevered assets. Thus, the beta of all the assets is a portfolio weighted average of 0 and ¡3UA with the portfolio weight on 0 being TLDrf¡(1 + rf) and the portfolio weight on ¡3UA being (1 - Tc)Drf/1 + rf. This implies that the ¡3 of the assets can be expressed as

When the updating interval is short, the term in brackets is very close to one, and thus this formula says that with dynamic updating of debt, the beta of the assets and the beta of the unlevered assets are approximately the same. As periods become arbitrarily short, the value of the first debt interest payment becomes infinitesimal and the two betas become exactly the same. As suggested earlier, this implies that the no tax versions of equations (13.6) and (13.7) can be used to adjust equity betas for leverage when taxes exist, provided that the firm dynamically maintains a constant ratio of D to E.

One-Period Projects. If a project lasts only one period, there is only a single interest payment. In this case, the present value of the tax shield, TcDrf/(1 + rf), is the same as the present value of the riskless component of the tax shield in the Miles and Ezzell model. Thus, the beta of the assets when there is debt that lasts only a single period has the same formula as that in the Miles and Ezzell model. In general, the Miles and Ezzell formulas for the WACC and the beta of equity apply to one-period projects. This is because the fraction of the assets that is risk-free is identical in the two cases.

Which Set of Formulas Should Be Used? If the unlevered assets of the firm have a significant risk premium, the models of Hamada and Miles/Ezzell generate different adjusted cost of capital formulas, as well as different formulas for leverage adjustments of equity betas. Dynamic updating of debt and equity to maintain a constant leverage ratio, as in Miles and Ezzell, tends to generate larger WACC changes for a given leverage change than the Hamada model, which assumes that the amount of debt does not change.

Many firms, particularly those with low to moderate amounts of leverage, try to maintain a target debt to equity ratio, but update rather slowly, perhaps because the cost of frequently issuing and repurchasing debt and equity is prohibitive. For such firms, truth lies somewhere between the models of Hamada and Miles/Ezzell. The weighting of the two models depends on which behavioral assumption the firm conforms to better. Large firms, firms with existing shelf registrations, and those with a history of repurchasing equity are likely to be more active in dynamically updating. Also, for projects with short lives, truth is probably closer to that given by the Miles and Ezzell formulas (with the caveat that comparison firms are perpetual).

However, we noted earlier that for many highly leveraged firms, it is possible that the debt tax shield has a negative beta. For firms and projects with this property, it is important to use even a lower WACC and lower equity beta adjustments for leverage than those suggested by the Hamada model.

Chapter 13 Corporate Taxes and the Impact of Financing on Real Asset Valuation

Finally, it is important to recognize that the field of corporate finance has yet to develop formulas for how leverage changes affect the WACC, equity betas, and the expected returns of assets, unlevered assets, debt tax shields, and equity in many realistic situations. Foremost among these is the case of the growing firm with debt tied to its growth and reinvestment. This kind of problem, however, can often be analyzed by a skilled practitioner using the PV method. For this reason, we are still puzzled by the overwhelming popularity of the WACC method as a tool for valuation.

### Evaluating Individual Projects with the WACC Method

The appropriate discount rate for a particular project must reflect the risk and debt capacity of the project rather than the risk and debt capacity of the firm as a whole. While the tracking portfolio analysis in Chapter 11 emphasized this in great detail, some financial managers believe that they are losing money whenever a project returns less than the firm's WACC. Examples 13.13 and 13.14 provide an additional perspective on why this line of thinking is fallacious by examining an extreme case where a risky firm is evaluating a project that has no risk.

Riskless Project, Riskless Financing. In Example 13.13, the project has riskless cash flows. The project's financing also is riskless, consisting of debt with tax-deductible interest payments. In this case, analysts can evaluate the project directly by comparing the project's proceeds with its financing costs. In contrast to the analysis of riskless projects in Chapter 10, however, the analyst now has to account for the debt tax shield.

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