### 1. The P/E Ratio

In the introductory chapter to the course Firm Valuation we have argued that the market value of a financial assets corresponds to the the present value of all future expected cash flows that the asset generates for its owners. We have also argued that the logic behind relative valuation is to "outsource" the work associated with such intrinsic valuations: Instead of deriving financial values based on a detailed model of the financial future of the firm, relative valuation uses existing market valuations of other firms and applies them to the firm in questions.

In what follows, we show that relative valuation is indeed consistent with a valuation framework that is based on the valuation of the future cash flows of a firm. We start by showing this for the P/E ratio, as this is the typical work horse of financial analysts.

The P/E Ratio from a Discounted Cash Flows Model

The stock price of a firm corresponds to the present value of all future dividend payments that shareholders expect to receive. If we assume for simplicity that these future dividend payments take the form of a growing perpetuity, we can estimate a firm's theoretical stock price (P) as follows:

P = $\frac{Div_1}{k_E-g}$

with: P = current stock price; Div1 = expected dividend in 1 year; kE = cost of equity; g = future annual dividend growth rate.

This is the so-called Gordon-Shapiro model, named after the two economist Myron Gordon and Eli Shapiro, who published it back in the 1950ies.

Example: A firm is expected to pay a dividend of 10 per share one year from now. It's cost of equity is 12% and the future expected dividend growth rate is 2%. Based on this information, and assuming a growing perpetuity, the theoretical stock price of the firm is 100:

P = $\frac{Div_1}{k_E-g} = \frac{10}{0.12-0.02}$ = 100.

But what exactly does this have to do with the P/E ratio? It turns out that it is rather straightforward to derive the P/E ratio from the above valuation model. To do so, two additional steps are needed:

First, remember that the dividend corresponds to the fraction of the firm's net income that is paid out to shareholders. On a per-share basis, we can therefore link dividend payments and earnings per share (EPS) as follows:

Div = EPS × payout ratio.

If we plug this into the above equation, we can find the firm's theoretical stock price as follows:

P = $\frac{EPS_1 \times \text{payout ratio}}{k_E-g}$

In the above example, the dividend payment of 10 could be the result of EPS of 20 and a payout ratio of 50%.

The second step is to divide both sides of the above equation by EPS to find the (forward) P/E ratio:

P/E ratio = $\frac{P}{EPS_1}= \frac{\text{payout ratio}}{k_E-g}$

In our example, we have assumed a payout ratio of 50%, a cost of equity of 12% and a dividend growth rate of 2%. Consequently, the firm's P/E ratio should be 5:

P/E ratio =$\frac{P}{EPS_1}= \frac{\text{payout ratio}}{k_E-g} = \frac{0.5}{0.12 - 0.02}$ = 5.

We can easily verify that this information is correct: SInce we have assumed a stock price of 100 and EPS of 20, the stock price is indeed five times the firm's EPS.

Relevance and Application

The above expressions allow us to understand how the P/E-ratio comes about and what (implicit) assumptions we make when working with the P/E ratio. This information is important for example also with respect to the choice of comparable firms. More sepcifically, we have seen that the P/E ratio is a function of the firm's payout ratio, its riskiness (as reflected by the cost of equity), as well as its expected growth rate. When working with the P/E ratio, we should therefore make sure that the peer group consists of firms that are comparable with respect to:

• Payout policy (fraction of net income the firm distributes to shareholders rather than reinvesting it in the company)
• Risk (usually mainly driven by the firm's industry as well as its financing policy)
• Expected growth rate

The relations also show us what drives the P/E ratio. More specifically, and all else the same, we can conclude that:

• Firms with higher payout ratios should have higher P/E ratios
• Risky firms should have lower P/E ratios
• Firms with better growth prospects should have higher P/E ratios