Price of a Stock: What is it?

Question

My limited understanding of stock prices is that according to theoretical arguments, the price of an asset is generally given as:$$P_{A}=E_{0}\,\sum_{t=0}^{\infty}\frac{C_{t}}{(1+r)^{t}}$$ whereby $C_{t}$ is the coupon promised by the asset and r is our discount rate. The way I have written this above expression corresponds to a consol/perpetuity, but it can (I think) be conceptually extended to stocks and bonds as well.

I guess what I am trying to get at is the fact that asset pricing theory suggests that the price of an asset is the Net Present Value of the future income stream from that asset.To some extent, this is applicable to stocks. The Gordon Growth model is based on this idea.

My question is how does this relate to the market cap? I mean, the market cap is the stock prices times the number of shares. However, given that the the market cap (under perfect information and only tangible assets) would coincide with the book value, how is the net present value of the income stream related to owner's equity ?

Answer 1

I'll attempt an answer here --- but really, this is a relatively straightforward corporate finance question, and indeed, I'd argue this is effectively an accounting question.

We have the accounting identity that,

$$ Assets = Equity + Debt $$

At this point, I don't want to get into tax shields, financial distress costs and things of that sort (it's effectively Modigliani and Miller theorems). So to keep things simple, suppose there are no corporate taxes and hence there are no tax shields.

And to furthermore keep things simple, suppose the number of shares outstanding is $N$.

Using the simple discounted cashflow idea, the value of the assets / value of the unlevered firm $U_A$ (note my change in notations to make things clearer) is what you wrote,

$$ U_A = E_0 \sum_{t = 0}^{\infty} \frac{C_t}{(1 + r)^t} $$

where the discount rate $r$ is some appropriate cost of capital for discounting the assets. If we invoke the Modigliani and Miller theorems, we can even link this to the weighted average cost of capital (WACC).

Suppose you have outstanding market value of debt as $D$. Then the value of equity for the firm --- again, from the accounting identity --- is,

$$ E = U_A - D $$

The equity price per share (again, another accounting identity) is simply,

$$ P = \frac{E}{N} $$

Note the special case when there is no debt, so $D = 0$, the value of the assets would equal to the value of equity.

Thus, the key point of my response: Your question is nothing more than playing around with accounting.

Some remarks:

1. You mention in your comments that you allow for stochastic cash flows and stochastic discount rate. The equation that you wrote is simply impossible to have any meaningful stochastic discounting. The discount rate $r$, as you write it now, is clearly known at $t = 0$. Thus, you can pull the entire discount factor $1 / (1 + r)^t$ outside of the expectation, and indeed, move the expectation operator under the summation to just consider $E_0 C_t$. I'm not going to be so worried here about integrability issues of doing this.
2. There honestly isn't any "asset pricing theory" in this question (at least to me). Your equation is simply a discounted cash flow equation --- this is just a basic concept of "time value of money", which everybody will understand and accept as long as you agree that "\$1 today is not the same \$1 tomorrow". The key asset pricing problems are really the determination of the expectation $E_0$ and also the discount rate $r$ (and it's vastly more interesting if you allow the discount rate to be time-varying and stochastic --- which again, you do don't have as you'd written it). The most interesting part is really what drives the potential time-varying and stochastic behavior of the discount rates --- this is the true study of "asset pricing". Another highly important study in asset pricing is also how investors form expectation $E_t$ based on their time $t$ information sets of a firm's cash flows.

Answer 2

To elaborate a bit and to try to link the theoretical models to today’s news: In a 1960’s world of stable market shares and oligopolistic pricing power, the dividend growth model worked well as an approximation of the value of an equity as a cash flow asset. Notice that the cash flows from debt are much more certain than equity cash flows due to covenants and well established precedence in case of financial distress. So why buy a given company’s equity if you can buy its debt, or equivalent rated debt? This question was especially relevant before the recent tax laws change. The reason to buy equities is not the dividends, it is the appreciation. To get a sense of the relative magnitudes, take the S&P or NYSE, define a period of 5 years and a set of companies, look at the total change in market cap, subtract out the dividends and arguably what is left (assuming it is positive) is price appreciation. Note that dividends are paid out of operating earnings or the proceeds of debt. But price appreciation is provided by OPM. That is why the cash flow growth models do not capture the (potential) value of an equity.