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Suppose I have a free cash flow to the firm FCFF and the market cap is E. Obviously I don't believe the equity valuation and that's why I would even attempt valuating it. So one way to do it is find WACC

$$ R_{WACC} = \frac{E}{E+D} r_E + \frac{D}{D+E}(1-t) r_D $$

and value the equity as

$$ \tilde E = \frac{FCFF}{R_{WACC}} - D \tag{WACC method} $$

However, a more direct way is to evaluate the free cash flow to equity FCFE

$$ FCFE=FCFF - r_D D (1-t) $$

and use this directly

$$ \hat E=\frac{FCFE}{r_E} \tag{Direct Method} $$

Now the point is that in general the two valuations will NOT agree unless we have $E=\hat E$. That is unless the market cap is priced according to the Direct Method, the two methods will disagree.

One can try to 'fix' the WACC method by Newton's method to recursively put $\tilde E$ values in the $R_{WACC}$ expression till one reaches a fixed point but this will just be $\hat E$.

So my question is why do people use the WACC method in the first place to find equity value when the incorrect equity value is an input in it in the first place.

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  • $\begingroup$ ...and now you are on the path to enlighenment $\endgroup$ – David Addison Feb 14 '18 at 11:33
  • $\begingroup$ How familiar are you with the Modigliani Miller theorem? Often called the capital structure irrelevance principal, it lays out the theory for the paradox you've encountered. en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem $\endgroup$ – David Addison Feb 14 '18 at 19:14
  • $\begingroup$ @DavidAddison I know it quite well. I have a derivation of it somewhere here. I don't think it lays out the theory for the paradox as its a mathematical work and the paradox is behavioral. The point is MM values a equity as present value of equity flows (what I call Direct Method above) and debt at present value of debt flows. One can then define WACC so that the free cash flow discounted by WACC gives the same answer. However, to use WACC to value equity is crazy as you need the value of equity in there to evaluate value of equity. That is the problem I am describing above. $\endgroup$ – Borun Chowdhury Feb 15 '18 at 10:19
  • $\begingroup$ @DavidAddison Here is the derivation I refered to above quant.stackexchange.com/questions/33384/… $\endgroup$ – Borun Chowdhury Feb 15 '18 at 10:20
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If it helps, this class of problems has a non-existence proof tied to them. Mean-variance finance models have an assumption that all parameters are known built into the proofs. There is an existing proof that shows these models, if true, can never form an estimator for the parameters.

Consider the intertemporal budget constraint from the CAPM. It is commonly written in static models as $\tilde{w}=R\bar{w}+\epsilon.$ Let us assume that $R$ is unknown as I am sure you don't know it and neither do I. If you knew the valuation and so forth, then you wouldn't have needed to ask the question.

So future wealth equals current wealth ttimes a reward for investing plus a shock. You invested to make money so $R>1$. This is a static model of a general case where $w_{t+1}=R{w}_t+\varepsilon_{t+1},$ where $\varepsilon$ is drawn from any density centered on zero with finite variance greater than zero.

Mann and Wald showed that the maximum likelihood estimator, and the MVUE, for this AR(1) process is ordinary least squares subject to the restrictions on the mean and variance in the shock term. This meets all requirements for a Frequentist estimator for $R\in\mathbb{R}$. Mann and Wald showed the sampling distribution for $\hat{R}-R$ where $|R|<1$ is the normal distribution. White in 1958 showed that the sampling distribution for $|R|>1$ is the Cauchy distribution. Since least squares provides a version of the sample mean, you need to find the population mean of the Cauchy distribution for convergence, yet it has no population mean. Consequently, any such estimation has zero power with an infinite sample size.

With all of that said, there is a Bayesian solution to this class of problems, but it doesn't use a mean or a variance. I presented a distribution-free and parameter-free Bayesian solution at the Southwestern Finance Association Conference last week to price options. I also provided a parametric form. It takes advantage of the fact that while the distributions lack a sufficient statistic, predictions don't have that problem.

Save yourself time, ignore WACC. Do use the marginal cost of capital as that is a very real thing. Ignore WACC. Even if the math were valid, it has been shown that you can alway stochastically dominate the solution implying it cannot actually be a solution, even if the math is correct.

If you want to check the intuition to White's result above, consider $w_0=0$ and $\varepsilon_1=1$, where $R>1$. The shock would go to infinity as time went to infinity. Shocks go to zero when the normal distribution is the sampling distribution, implying learning.

Sorry, I am not at a place to provide a citation, but I believe Mann and Wald are either in 1941 or 1943 and White is 1958. Rao generalized White's result to all AR(n) processes, but I do not remember the date.

Proof of models like the CAPM, Black-Scholes, those built using Ito calculus are vacuous even if the assumptions are true in the strictest sense.

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Modigliani Miller (MM) tells us that leverage should not affect the value of a firm (under idealized conditions). I think the problem that your are encountering is due to the ambiguous "required return on equity", which as you point out, is subject to estimation error and bias. This is especially true when its parameters are estimated under the CAPM.

What follows is an attempt to show under what conditions required returns on equity does not result in any violation of MM.

First we begin with basic two basic equations.

(1) $\hat{E}=\frac{\text{FCFF}}{r_{\text{WACC}}}-D$

(2) $\hat{E}=\frac{\text{FCFE}}{r_e}$

Without resorting to Newton's method (or recursion), we can solve for $r_E$ which results in the condition that $(1) \equiv (2)$ if we are also given that:

$\text{FCFE}=\text{FCFF}-I_D \left(1-r_T\right)$

Therefore:

(3) $$r_E = \frac{r_{\text{WACC}} \left(\text{FCFF}+I_D \left(r_T-1\right)\right)}{\text{FCFF}-D* r_{\text{WACC}}}$$

Interestingly, equation (3) suggests that the required on equity also depends on operating leverage (i.e., the "beta" of free cash flows), in addition to financial leverage. In the derivation, low cash flows result in a lower required return on equity, which is probably what we should expected to see in the real world.

Given some arbitrary numbers for a firm, we can now show there equation (3) upholds the equality, $(1) \equiv (2)$.

$\left\{\text{FCFF}\to 20,r_{\text{WACC}}\to 0.03,r_D\to 0.02,r_T\to 0.2,I_D\to 10,D\to 200\right\}$

So, $r_E = 0.0257143$

$\hat{E}=\frac{\text{FCFF}}{r_{\text{WACC}}}-D = 466.667$

$\hat{E}=\frac{\text{FCFE}}{r_e} = 466.667$

Note: I just want to reiterate these are just models. The basic intuition that $NPV = \frac{C}{r}$ assumes that a) equity is a perpetuity with constant cash flows; and b) rates of return are homogeneous. I don't believe either of these conditions are true in the real world.

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  • $\begingroup$ I don't think you understand my question. Instead of explaining it again, let me ask, how do you calculate the $r_{WACC}$ of a firm? Remember you cannot use $\hat E$ and $r_E$ since you are trying to calculate both of these. $\endgroup$ – Borun Chowdhury Feb 16 '18 at 9:14
  • $\begingroup$ I gave up on CAPM. I just aggregate costs of equity, debt, non-controlling (minority) interests, and (effective/nominal???) tax rates at the peer group level. The I use your formula for $r_{WACC}$. In this method, cost of equity are computed in a way which is directly comparable to the numerator. So, $r_{\hat E, Group} = \frac{\sum_i^N FCFE_i}{\sum_i^N \hat E_i}$. I am not sure if its better to use historical average cost of equity through book value, or implied cost of equity through market cap. $\endgroup$ – David Addison Feb 16 '18 at 19:18
  • $\begingroup$ I meant what value of $E$ did you use to get the WACC. To get the correct WACC you need the correct $E$ in the first place. So its a chicken-n-egg problem. In your example you take it as a given. This is fine for an exam question but very unrealistic in practice. $\endgroup$ – Borun Chowdhury Feb 19 '18 at 9:44

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