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To evaluate an enterprise we can discount free cash flow by either the unlevered required rate of return or the WACC.

With Tax we have:

$WACC=R_e \frac{E}{E+D}+R_f\frac{D}{E+D}(1-t)$

where $R_e$ is return on equity, $R_f$ is return on debt or risk free rate, $E$ is market value of equity, $D$ is market value of debt and $t$ is tax.

If the equity beta of the firm is $\beta_e$ we have the unlevered beta or asset beta as: $\beta_A=\beta_e/(1+\frac{D}{E+D}(1-t))$. Also by CAPM:

$R_e=Rf+\beta_e(R_m-R_f)$

$R_A=Rf+\beta_A(R_m-R_f)$

by plugging in the expression for $\beta_A$ and simplifying I get:

$R_A=R_e\frac{E}{E+D(1-t)}+R_f\frac{D(1-t)}{E+D(1-t)}$

This is different from the WACC expression while I think it should be the same. What am I doing wrong?

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There may be two points you are missing:

  1. You are allowed to apply the CAPM to calculate the cost of equity $R_e$. However, one of the CAPM assumptions is, that taxes are not taken into account into the model.
  2. The unlevered WACC gives a theoretical solution under the assumption that there is no debt at all. In conventional WACC, the tax part $t$ only impacts the cost of debt, but if the weight of debt is zero, tax is not relevant any more.

In summary, the difference from your formulas arises, because the WACC-approach explicitly includes the tax-shield $t$ and the CAPM is an economic model without taxes.


Furthermore, i would like you to point to this wonderful answer, as it may be useful for further details on WACC.

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In fact, the WACC is misinforming us; it is neither a cost nor a required return but a weighted average of a cost and a required return. The corporate finance community promotes the WACC as a comprehensive cost of capital to guide investment decisions. This practice is wrong. Suppose that the risk-free rate is 5% (simple rate), and the required rate of return is 10%. If the current D/S=1/4, then the WACC is 9%. Given a project with a constant rate of return at 6%, it is far below the WACC, but the project is profitable because it is an arbitrage opportunity to finance the project by borrowing money. We see that we cannot take the WACC as a hurdle rate

Using the WACC to guide investment decisions, and minimizing the WACC for the optimal capital structure, both are following faulty ideologies. As brainchildren of deterministic and isolated thinking, just as the scientific community finally abandoned the “phlogiston” and “aether” doctrines, we must discard the hypothetical cost of equity capital and renounce the postulated WACC as soon as possible.

A common mistake with WACC is that increasing debt financing can reduce WACC and thus increase the value of a firm. In fact, as long as the total input is invariant, thus future payoff (output) remain the same, the value of a company's securities must remain unchanged in the perfect market. Let WACC be $\rho$, since $\rho=\operatorname{E}(X)/V$, WACC will not be affected by its capital structure. When the debt financing is changed, the cost of equity can not be fixed. As an example, if $\mu>r$ and we replace all the debt $D$ by equity, then the value of stock is increased by $D$, but its payoff is increased by $rD$. Since $$ \mu_{\mathrm{new}}-\mu=\frac{\operatorname{E}(X)}{V}-\frac {\operatorname{E}(X-rD)}{V-D}=-\frac{D}{V}(\mu-r)<0 $$ the cost of equity is decreased.

For more, see The Circular Justification in the MM Proposition: A Rethink or Perfect Market, Arbitrage, and Value Creation in the MM Proposition

It is extremely wrong to use beta to estimate the so-called cost of capital: In CAPM (security market line), the returns are endogenous. It is the result of the equilibrium of the entire market according to the mean-variance criterion, and is not determined by the so-called risk (variance, beta, or covariance). The beta value is calculated from the equilibrium return, using beta value to explain the expected return is a circular argument.

For more on CAPM, see CAPM: Absolute Pricing, or Relative Pricing? or Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium

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