# Why is market cap used to value equity instead of a self consistent solution?

My claim is that if we use the cost of equity of a levered firm via the DCF method then we make errors. Specifically if we find the firm is under-valued then in truth its more under-valued than we estimate and conversely if we find the firm is over-valued then its more over-valued then we estimate. Moreover, it is very easy to correct this "mistake" as I show below so my question is why don't we do it?

I give the argument and method for the correct valuation below. Suppose a firm has the following:

• Market Cap $E_0$
• Debt $D$
• Cost of (levered) equity $r_{E_0}$
• Cost of debt $r_D$
• Perpetual cash flow to equity $\mathcal E$

The usual way people value the equity is

$$E= \frac{\mathcal E}{r_{E_0}} \tag{Naive Equity}$$

which may or may not (usually not) be equal to $E_0$.

My question is why don't people do the following:

(a) Unlevered cost of equity is $$r_U = \frac{r_{E_0} E_0+ r_D D(1-t)}{E_0 + D(1-t)} \tag{unlevered COE}$$ (b) The true levered cost of equity if the true equity is $\tilde E$ is $$\tilde r_E = r_U + (r_U - r_D) \frac{D}{\tilde E} (1-t) \tag{true levered COE}$$ (c) The equity values is then

$$\tilde E = \frac{\mathcal E}{\tilde r_E} \tag{true Equity}$$

giving

$$\tilde E = \frac{\mathcal E - (r_U - r_D) D(1-t)}{r_U} \\$$

This expression gives the self-consistent equity value. We can write it in terms of the original data as

$$\tilde E = \frac{\mathcal E - \frac{r_{E_0}-r_D}{1+ \frac{D}{E_0} (1-t)} D(1-t)}{\frac{r_{E_0} + r_D \frac{D}{E_0} (1-t)}{1+ \frac{D}{E_0}(1-t)}}$$

Finally we can eliminate the cash flow to equity to obtain the relation between the naive and the true estimates of equity

$$\frac{\tilde E}{E_0} (1 + \frac{r_D}{r_{E_0}} \frac{D}{E_0}(1-t)) =\frac{E}{E_0} (1+ \frac{D}{E_0}(1-t)) - (1 -\frac{r_D}{r_{E_0}}) \frac{D}{E_0}(1-t)$$

Below I have plotted the ratio of the true valuation to naive valuation for $D/E_0=.6$,$r_D/r_{E_0}=.3$ and $t=.35$. The results are that when the naive method shows the stock is underpriced then the naive method itself is underpriced wrt the true price and vice-versa.

The differences vanish with $D/E_0 (1-t) \to 0$ which can either come from vanishing debt or high taxes. The differences vanish also when $r_D/r_{E_0} \to 1$.

Another plot to show this effect is $\tilde E/E_0$ vs $E/E_0$. We see that if the stock using the naive method is twice the market cap then its actually 2.2 times. Conversely if the naive method says its half the market cap its actually .4 times the same.

Now some may argue that the cost of levered equity is what it is and we shouldn't be correcting it. While this argument is true for short time scales but since we are considering a perpetuity, we assume the markets are fairly efficient and that soon the price will adjust to the correct value and when this starts happening the cost of levered equity will also correct. So it makes sense to use the correct formula. If one wants to be even more careful one can smoothen out the correction over a couple of years so let the market adjust.

As a further aid to intuition I give below the value of the slope of the line relating $\tilde E/E_0$ vs $E/E_0$ for various values of $r_D/r_{E_0}$ and $D/E_0$. It's useful to keep in mind that the lower right corner is inaccessible as a high $D/E_0$ ratio would push up $r_D/r_{E_0}$. Notice the slope fixes the line completely as it has to pass through $(1,1)$.

• While I provided a self consistent solution above in one-shot, it can also be arrived at by recursion. One first finds $r_U$ as above. Then $E_1 = \frac{\mathcal E}{r_{E_0}}$ gives the first estimate. Using this one finds $r_{E_1}$ to get $E_2 = \frac{\mathcal E}{r_{E_1}}$ and so on till one reaches a fixed point. This fixed point will presumably be $\tilde E$ as the system doesn't seem to be chaotic. – Borun Chowdhury Mar 1 '18 at 15:50
• First of all, why are assuming that the market is discounting a perpetual and deterministic cash flow? In my experience, that is almost never the case. – David Addison Mar 1 '18 at 16:14
• $E= \frac{\mathcal E}{r_{E_0}}$ is not "naive", it is the correct valuation equation for an all equity firm as long as the firm earns $\mathcal E$ in perpetuity and the cost of equity is $r_E$ (essentially by definition). It is a very simple case, as David Addison said, but OK. I don't understand why you "adjust" this based on the amount of debt and so forth to find the true equity. What does "true equity" mean ? You started with the assumption that the equity was $E$, then you say it is $E_1$ why? – Alex C Mar 1 '18 at 20:25
• @AlexC I started with market cap $E_0$ and cost of equity $r_{E_0}$. All I am saying is that if with this we find the value $E$ (which because I am anticipating iteration I call $E=E_1$ also) and I assume everyone has access to this information soon, then the cost of equity will change to reflect value $E_1$. Then I should price again with this new cost of equity $E_1$ and get value $E_2$ and so on. The value at which iterations stop changing the value (called a fixed point) is the "true" value (as per my definition). – Borun Chowdhury Mar 2 '18 at 8:31