WACC and Direct Equity Valuation Under Uncertain Required Returns

# Valuing equity of a firm using WACC gives incorrect results


# Valuing equity of a firm using WACC gives incorrect results












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.

## Answer by David Addison (score 1)

https://quant.stackexchange.com/a/38264

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.

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/38839

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.