The price-earnings, or P/E, ratio is one of the first things investors consider when looking at a stock. They tend to have rules of thumb like: having a P/E ratio below 15 makes a stock "cheap". But, where does that notion come from? More importantly, how do these numbers relate to what we know about intrinsic valuation? We have the Benjamin Graham formula that helps relate the P/E ratio to expected growth, but let's see what we can derive from the basics of valuation.

Let’s start with the textbook definition of an asset’s value (here $P$ is the fair price, $CF_n$ is the cash flow for year $n$ and $r$ is the discount rate): $$P_{asset}=\frac{CF_1}{(1+r)}+\frac{CF_2}{(1+r)^2}+...+\frac{CF_N}{(1+r)^N}$$

As mentioned in my first post, we value an asset that bears income by adding up the cash flows we expect to receive by holding it, discounting them by an appropriate rate based on how long we have to wait and the risk of not getting each cash flow. For stocks the dividends are technically the expected cash flows, but not all companies pay a dividend. Here, we can use an arbitrage argument to say: we can value the company based on what could be paid out as dividends, because if we don’t then somebody could buy up the shares cheaply and pay themselves those potential dividend payments. We refer to the amount that could be paid out to shareholders as a dividend (after all expenses, interest payments, taxes, capex, etc) as the free cash flow to equity. So, our updated formula should be (here $FCF_n$ is the free cash flow to equity in year $n$): $$P_{stock}=\frac{FCF_1}{1+r}+\frac{FCF_2}{(1+r)^2}+...+\frac{FCF_N}{(1+r)^N}$$

Now, I promised to discuss the P/E ratio, so I’m going to pitch substituting the free cash flow to equity per share for earnings per share (EPS). The problem with EPS is it’s based on accrual accounting and has non-cash charges, like amortization of goodwill, included. To get free cash flow we typically add back depreciation and amortization, subtract the amount required to maintain working capital and account for actual capex. But here we will assume that EPS does roughly approximate free cash flow to equity per share. If you strongly disagree with this then feel free to imagine that I’m doing the rest of the analysis with FCF instead of EPS (or at least verify that they aren't drastically different for the company you have in mind): $$\tilde{P}_{stock}=\frac{EPS_1}{1+r}+\frac{EPS_2}{(1+r)^2}+...+\frac{EPS_N}{(1+r)^N}$$

We have a lot of EPS values here. Instead, we want to try to get to an expression in terms of this year’s recorded EPS ($EPS$) only, so let’s rewrite it in terms of the growth rates for each year in the future ($g_n$) instead: $$\tilde{P}_{stock}=\frac{EPS(1+g_1)}{1+r}+\frac{EPS(1+g_1)(1+g_2)}{(1+r)^2}+...+\frac{EPS\prod_{n=1}^{N}(1+g_n)}{(1+r)^N}$$

At last we can get to our P/E ratio by factoring out the EPS and dividing both sides by it: $$PE=\frac{1+g_1}{1+r}+\frac{(1+g_1)(1+g_2)}{(1+r)^2}+...+\frac{\prod_{n=1}^{N}(1+g_n)}{(1+r)^N}$$

Here we see that the P/E ratio tells us what the future growth in EPS is expected to be, given your discount rate. But, the problem is there are a lot of growth rates to consider and only a single value for P/E. You can’t exactly solve for all these unknowns given only one equation.

So, we have to make some simplifying assumptions. Let’s start with the easiest: let’s assume the growth rate will be constant over all future years. Immediately, we can rule this out for companies with high growth. You can’t grow at a higher rate than the economy forever or else you’ll become larger than the economy and that doesn’t make sense. However, let’s assume the company has reached a realistic steady state and proceed with our constant growth rate assumption: $$PE=\frac{1+g}{1+r}+\frac{(1+g)^2}{(1+r)^2}+...+\frac{(1+g)^N}{(1+r)^N}$$ $$PE=\sum_{n=1}^{N}\frac{(1+g)^n}{(1+r)^n}$$

Since the growth rate is constant, our company will last forever, so if we take the limit as N goes to infinity the series converges to: $$PE=\frac{1+g}{r-g}$$

I think the most interesting case though is to see what the P/E ratio should be for a company with zero EPS growth. Here we’re assuming that the company won’t growth or shrink it’s earnings, just maintain it, as is, forever: $$PE=\frac{1}{r}$$

We see that P/E ratio for a company that doesn’t grow or shrink should be the inverse of the discount rate. So, more risky companies should have a lower P/E and vice versa. The way you might use this is to find companies that have a P/E multiple that is lower than the inverse of the discount rate and those will have an implicit assumption that they will shrink. Then you can ask yourself the question: “will this company actually grow?” and if the answer is yes, you might be looking at a cheap stock. Of course, we can’t forget our constant growth assumption. There are ways to justify a P/E that is the inverse of the discount rate, by having some years where the growth rate is negative and some where it’s positive, so you have to keep that in mind.

One recent example is Apple (AAPL) back in May 2016. It’s P/E ratio dipped below 10, which implied, using the constant growth rate assumption, negative growth rates based on my discount rate for the stock. Knowing that Apple has such strong economic moats, I made a bet that it would, in fact, grow and as of writing it's returned 40% vs the market’s return of 15%.

What if we want to use a more realistic model for a company growing faster than the economy. Let’s assume that the company will grow at higher near-term rate until we reach a terminal year (when the market for the product is saturated) and then it will grow at a low constant rate (under the rate of economic growth). Here's how we could formulate that (where $g_{near}$ is the near-term growth rate, $g_{long}$ is the long-term growth rate, and T is the number of years until long-term growth is reached): $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + \sum_{n=T+1}^{\infty}\frac{(1+g_{near})^{T}(1+g_{long})^{n-T}}{(1+r)^n}$$

The first sum takes care of the value of the near-term growth and the second sum takes care of the terminal growth. Now, let's factor out the parts of the second sum that don't depend on $n$: $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + (1+g_{near})^{T}\bigg(\sum_{n=T+1}^{\infty}\frac{(1+g_{long})^{n-T}}{(1+r)^n}\bigg)$$ $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + (1+g_{near})^{T}\bigg(\sum_{n=T+1}^{\infty}\frac{(1+g_{long})^{n}}{(1+g_{long})^{T}(1+r)^n}\bigg)$$ $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + \frac{(1+g_{near})^{T}}{(1+g_{long})^{T}}\bigg(\sum_{n=T+1}^{\infty}\frac{(1+g_{long})^{n}}{(1+r)^n}\bigg)$$

We can then split up the second sum into an infinite sum starting at one and subtract out the missing first T years: $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + \frac{(1+g_{near})^{T}}{(1+g_{long})^{T}}\bigg(\sum_{n=1}^{\infty}\frac{(1+g_{long})^{n}}{(1+r)^n}-\sum_{n=1}^{T}\frac{(1+g_{long})^{n}}{(1+r)^n}\bigg)$$

We've seen the infinite sum before. Let's substitute the solution in: $$PE=\sum_{n=1}^{T}\frac{(1+g_{near})^n}{(1+r)^n} + \frac{(1+g_{near})^{T}}{(1+g_{long})^{T}}\bigg(\frac{1+g_{long}}{r-g_{long}}-\sum_{n=1}^{T}\frac{(1+g_{long})^{n}}{(1+r)^n}\bigg)$$

Finally, let's rearrange to express this as a single finite sum and a constant: $$PE=\sum_{n=1}^{T}\bigg[\frac{(1+g_{near})^n}{(1+r)^n} - \frac{(1+g_{near})^{T}}{(1+g_{long})^{T}}\frac{(1+g_{long})^{n}}{(1+r)^n}\bigg] + \frac{(1+g_{near})^{T}}{(1+g_{long})^{T}}\frac{(1+g_{long})}{(r-g_{long})}$$ $$PE=\sum_{n=1}^{T}\bigg[\frac{(1+g_{near})^n-(1+g_{near})^{T}(1+g_{long})^{n-T}}{(1+r)^n}\bigg] + \frac{(1+g_{near})^{T}}{(1+g_{long})^{T-1}(r-g_{long})}$$

This still doesn't look particularly pretty, but the key thing is we can plug in our values to compute this. With this formulation we can make an estimate for the terminal year and then solve for the near-term growth. I made a Constant Near-term Growth Model tool to help you with the calculations. Again, this can be compared to historical growth, and more importantly, whether people are optimistic or pessimistic given the current P/E ratio.

You can keep thinking of ways to improve the model. For example, you might have the near-term growth starting high and then linearly slowing down until the terminal year. But, ultimately, the more complicated you make it, the more variables you add and the closer you get to just doing a straight-up DCF.

As you can see the P/E ratio is a rather blunt indicator for what investors expect EPS growth to be in the future. When you think of it this way, it has some implications. People tend to assume a low P/E is synonymous with “cheap”. But really it’s all relative to what growth a company can actually achieve. If you have good reason to believe a company will achieve a very high growth rate, then maybe a higher P/E does not mean the stock is expensive. But you should proceed with caution, because there are more ways for things to go wrong if your expectations are too high, than if you have relatively low expectations for a stock.

I should also bring up the fact that lower P/E portfolios tend to outperform higher P/E portfolios. The key word here is “portfolio”. On average, a lower P/E basket does better than a higher P/E basket, because of human psychology. People, on average, tend to be overly pessimistic when a company is doing badly and overly optimistic when a stock is doing well. But, it’s a statistical effect, which means you only capture it by having a large enough basket of the stocks in the category to overcome idiosyncratic effects of each individual stock.

When looking at an individual stock, you can’t count on the statistical value effect. You should be thinking about the stock’s very individual situation and comparing your forecast to the market’s forecast implied by the P/E ratio and other valuation tools.

The key is to use these kinds of models to get a grasp on what expectations are for growth. If they differ greatly from past growth, they why is that the case? Can you form a contrarian viewpoint and be right? In the end, you can only beat the market if you do both.