###### Effective Convexities

Effective convexity is the sensitivity of duration to changes in interest rates. $$... **Read More**

$$\text{Justified leading}\ \frac{\text{P}_{0}}{\text{E}_1} =\frac{\text{D}_{1}⁄\text{E}_1}{\text{r}-\text{g}}=\frac{1-\text{b}}{\text{r}-\text{g}}$$

Where:

\(1-\text{b}=\) Payout Ratio

$$\begin{align*}\text{Justified leading}\ \frac{\text{P}_{0}}{\text{E}_0} &=\frac{\frac{\text{D}_{0}(1+\text{g})}{\text{E}_0}}{\text{r}-\text{g}}=\frac{(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}=\bigg(\frac{1-\text{b}}{\text{r}-\text{g}}\bigg)(1+\text{g})\\ \text{Justified trailing}&=\text{Justified leading}\ \frac{\text{P}}{\text{E}}\times(1+\text{g})\end{align*}$$

- If earnings are expected to grow by \(g\), next year’s earnings will be greater than last year’s earnings by the growth rate, and the justified trailing P/E ratio will be greater than the justified, leading P/E ratio by (\(1 + g\)).
- A higher justified P/E ratio indicates relative overvaluation.

Given the following forecasted fundamentals:

- Retention ratio = 40%
- Required rate of return = 10%
- Dividend growth rate = 3%

Calculate the justified trailing and justified leading multiples based on the above-forecasted fundamentals.

$$\begin{align*}\text{Justified trailing P/E} &= \frac{(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}\\ &=\frac{(1-40\%)(1.03)}{0.10-0.03}\\&=8.83\\ \\ \text{Justified leading P/E}&=\frac{1-0.40}{0.10-0.03}\\&=8.57\end{align*}$$

$$\text{Justified}\ \frac{\text{P}_{0}}{\text{B}_{0}}=\frac{\text{ROE}-\text{g}}{\text{r}-\text{g}}$$

Where:

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

\(\text{r}=\) Required return on equity.

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

The following information relates to ABC Ltd:

- ROE = 16%
- Required rate of return = 12%
- Expected growth rate = 10%

The firm’s justified P/B based on the above fundamentals is *closest* to:

$$\begin{align*}\text{Justified}\ \frac{\text{P}_{0}}{\text{B}_{0}}&=\frac{\text{ROE}-\text{g}}{\text{r}-\text{g}}\\&=\frac{0.16-0.10}{0.12-0.10}\\&=3\end{align*}$$

All else equal, P/B is positively related to ROE.

- The bigger the spread between ROE and \(r\), the higher the P/B ratio, all else equal.
- A higher justified P/B ratio indicates relative overvaluation.

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

Where:

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

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

P/S ratio increases with an:

- Increase in profit margin
- Increase in earnings growth rate

Consider the following information:

$$\small{\begin{array}{l|r}\text{Dividend payout ratio} & 30\% \\ \hline\text{ROE} & 12\% \\ \hline\text{EPS} & \$6 \\ \hline\text{Sales per share} & \$328 \\ \hline{\text{Expected growth rate in}\\ \text{dividend and earnings}} & 7.50\% \\ \hline\text{Required rate of return} & 15\%\\ \end{array}}$$

Calculate justified P/S based on these fundamentals.

$$\begin{align*}\text{Justified P/S}&=\frac{(\text{E}_{0}/\text{S}_{0})(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}\\&=\frac{(\frac{6}{328})(0.30)(1.075)}{0.15-0.075}=0.0786\end{align*}$$

A low justified P/S ratio may indicate the stock is undervalued, while a significantly above-average ratio may suggest overvaluation.

## Question

Consider the following information:

- Earnings growth ratio = 12%
- Required rate of return = 13%
- Long term profit margin = 6.5%
- Dividend payout ratio = 30%
The justified P/S ratio is

closest to:

- 2.184.
- 2.451.
- 2.662.
## Solution

The correct answer is A.$$\begin{align*}\text{Justified P/S}&=\frac{(\text{E}_{0}/\text{S}_{0})(1-\text{b})(1+\text{g})}{\text{r}-\text{g}}\\&=\frac{(0.065)(0.30)(1.12)}{0.13-0.12}= 2.184\end{align*}$$

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

*LOS 25 (h) Calculate and interpret the justified price-to-earnings ratio (P/E), price-to-book ratio (P/B), and price-to-sales ratio (P/S) for a stock, based on forecasted fundamentals.*