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DCF valuation models can be used to estimate the justified P/E for a stock and to gain insight into the sources of valuation differences when the method of comparables is used. Connecting P/Es to a DCF model helps us address what value the market should place on the EPS.

The forward P/E (justified forward P/E) of a share can be related to the value of a stock as calculated by the Gordon growth model as:

$$\frac{\text{P}_{0}}{\text{E}_{1}}=\frac{\frac{\text{D}_{1}}{\text{E}_{1}}}{\text{r}-\text{g}}=\frac{1-\text{b}}{\text{r}-\text{g}}$$

The trailing P/E (justified trailing P/E) can be calculated using the Gordon growth model as:

$$\frac{\text{P}_0}{\text{E}_0} = \frac{\text{D}_0 (1+\text{g}))⁄\text{E}_0}{\text{r}-\text{g}}=\frac{(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}$$

Where:

\(\text{P}\) = Price.

\(\text{E}\) = Earnings.

\(\text{D}\) = Dividends.

*\(\text{r}\) = *Required rate of return.

\(\text{g}\) = Dividend growth rate.

\(\text{b}\) = Retention rate.

The justified P/E is:

- Inversely related to the stock’s required rate of return, all else equal.
- Positively related to the growth rate of expected cash flows, all else equal.

A company’s fundamentals can be used to estimate a stock’s justified P/B.

From the Gordon growth model,

$$\text{P}_0=\frac{\text{E}_1×(1-\text{b})}{\text{r}-\text{g}}$$

Where:

\(\text{b}=\) Retention rate

ROE can be estimated as:

$$\text{ROE}= \frac{\text{E}_1}{\text{B}_0}$$

Substituting \(\text{E}_{1}=\text{B}_{0}\times\text{ROE}\) into the Gordon growth model,

$$\text{P}_0= \frac{(\text{B}_0×\text{ROE})×(1-\text{b})}{\text{r}-\text{g}}$$

The sustainable growth rate is:

$$\text{g}=\text{b}\times\text{ROE}$$

Substituting \(\text{b}=\frac{\text{g}}{\text{ROE}}\) into the Gordon growth model,

$$\frac{\text{P}_0}{\text{B}_0} = \frac{\text{ROE}×\bigg(1-\frac{\text{g}}{\text{ROE}}\bigg)}{\text{r}-\text{g}}$$

Therefore, \(\frac{\text{P}_{0}}{\text{B}_{0}}\)

$$\frac{\text{P}_0}{\text{B}_0} =\frac{\text{ROE}-\text{g}}{\text{r}-\text{g}}$$

Where:

\(\text{ROE}\) = Return on equity

\(\text{r}\) = Required rate of return

\(\text{g}\) = Sustainable growth rate

Comparing stocks with the same P/B, the one with the higher ROE is relatively undervalued.

The larger the ROE relative to \(r\), the higher the justified P/B multiple based on fundamentals. This should make sense because the more significant the difference between the return on equity (ROE) and the required return on equity (\(r\)), the greater the value created by the company for its shareholders.

P/S can be linked to DCF models. From the Gordon growth model:

$$\text{P}_{0}=\frac{\text{D}_{0}(1+\text{g})}{\text{r}-\text{g}}$$

Substituting \(\text{D}_{0}=\text{E}_{0}(1-\text{b})\) into the Gordon growth model:

$$\text{P}_0= \frac{\text{E}_0(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}$$

Dividing both sides by *S _{0}*, we get:

$$\frac{\text{P}_{0}}{\text{S}_{0}}=\frac{(\text{E}_{0}/\text{S}_{0})(1-\text{b})(1+\text{g})}{(\text{r}-\text{g})}$$

Where:

\(\text{E}_{0}/\text{S}_{0}\) = Net profit margin.

\(1-\text{b}\) = Payout ration.

Profit margin is an element of the justified P/S. An increase in the profit margin produces a higher sustainable growth rate as long as sales do not decrease proportionately.

The justified price to cash flow is inversely related to the stock’s required rate of return and positively related to the growth rate of expected future cash flows. Justified price to cash flow based on fundamentals can be calculated by estimating the stock’s value using the DCF model and dividing that number by the cash flow figure.

The constant growth FCFE model is:

$$\text{V}_{0}=\frac{\text{FCFE}_{0}(1+\text{g})}{\text{r}-\text{g}}$$

Dividing both sides of the equation by cash flow gives us the justified P/CF multiple.

From the Gordon growth model, the justified dividend yield based on fundamentals can be expressed as:

$$\begin{align*}\text{P}_{0}&=\frac{\text{D}_{0}(1+\text{g})}{\text{r}-\text{g}}\\\frac{\text{D}_{0}}{\text{P}_{0}}&=\frac{\text{r}-\text{g}}{1+\text{g}}\end{align*}$$

The dividend yield is negatively correlated to the expected rate of growth in dividends and positively correlated to the stock’s required rate of return. Investing in shares with relatively high dividend yields is a value investment strategy rather than a growth investment style.

This involves comparing a company’s dividend yield with peers to determine whether it is attractively priced. Things to consider:

- Can the differences in the dividend yield be explained by the differences in expected growth?
- What is the probability that the dividends will reduce or be eliminated? This is analyzed using the payout ratio. A high payout ratio is less secure than a low one.

## Question

Which of the following would

most likelyresult in a higher justified P/B ratio?

- A higher \(r\) relative to ROE.
- A higher ROE relative to \(r\).
- An ROE that is equal to \(r\).
## Solution

The correct answer is B.A higher ROE relative to \(r\) would result in a higher justified P/B ratio. The greater the difference between the ROE and the required return on equity, the more the value created for its shareholders.

A is incorrect.A higher \(r\) relative to ROE is an indication that the company is not creating any value for its shareholders and therefore justified is expected to reduce.

C is incorrect.An ROE equal to \(r\) indicates the firm is not creating or destroying value, and therefore justified P/B should remain constant.

Reading 25: Market-Based Valuation: Price and Enterprise Value Multiples

*LOS 25 (g) Describe fundamental factors that influence alternative price multiples and dividend yield.*