% This is lecture no. 6 of BU8201/: Business Finance.

\chapter[Stock Valuation \& Market Efficiency]{Stock Valuation \& Stock Market Efficiency}%
\label{cha:stock_valuation_&_stock_market_efficiency}

\section{Stock Price vs. Intrinsic Value}%
\label{sec:stock_price_vs_intrinsic_value}

The stock price of a stock is the last selling and buying price of the stock in question.
The intrinsic value of a stock is something that only the CFO of the company in question knows for sure.

When the stock price is above the intrinsic value, it is said to be \textbf{overpriced}, and vice versa.
If stocks in a market tend to vary little from their intrinsic value, the market is said to be \textbf{efficient}.

\section{The Discounted Dividend Model}%
\label{sec:the_discounted_dividend_model}

The value of a share of common stock depends on the cash flows it is expected to provide, and those flows consist of two elements:
\begin{enumerate}
	\item The dividends the investor receives each year while he or she holds the stock.
	\item The price received when the stock is sold. The final price includes the original price paid plus an expected capital gain. 
\end{enumerate}

The analysis as performed by the marginal investor, whose actions actually determine the equilibrium stock price, is critical, but every investor, marginal or not, implicitly goes through the same type of analysis.

\begin{definition}
	Marginal Investor.
	A representative investor whose actions reflect the beliefs of those people who are currently trading a stock. It is the marginal investor who determines a stock's price.
\end{definition}

\subsection{Expected Dividends as the Basis For Stock Values}%
\label{sub:expected_dividends_as_the_basis_for_stock_values}

In pricing bonds, the equation is the present value of interest payments over the bond’s life plus the present value of its maturity (or par) value.
Stock prices are likewise determined as the present value of a stream of cash flows.
If you intend to hold the stock forever, you will receive a stream of dividends, and the value of the stock today can be calculated as the present value of an infinite stream of dividends:
\begin{align*}
	\hat{P}_0 & = \frac{D_1}{(1+r_s)^1} + \frac{D_2}{(1+r_s)^2} + \dotsb + \frac{D_{\infty}}{(1+r_s)^{\infty}}\\
		  & = \sum_{t=1}^{\infty} \frac{D_t}{(1+r_s)^t}
\end{align*}

Where $\hat{P}_0$ represents the current value of the stock, $D_n$ represents the dividend a stockholder expects to receive at the end of year $n$, and $r_s$ represents the required rate of return on the stock.

This same equation is used to determine the price for the more typical case where you expect to hold the stock for a finite period and then sell it.

\subsection{Constant Growth Stocks}%
\label{sub:constant_growth_stocks}

In subsection \ref{sub:expected_dividends_as_the_basis_for_stock_values}, the price model of stocks are generalised, in that the time pattern of $D_n$ can be anything.
If however, we predict that dividends will grow at a constant rate, then the price model equation can be rewritten as follows:
\begin{align*}
	\hat{P}_0 & = \frac{D_0(1+g)^1}{(1+r_s)^1} + \frac{D_0(1+g)^2}{(1+r_s)^2} + \dotsb + \frac{D_0(1+g)^{\infty}}{(1+r_s)^{\infty}}\\
		  & = \frac{D_0(1+g)}{r_s-g}\\
		  & = \frac{D_1}{r_s-g}
\end{align*}
Where $g$ is the expected growth rate in dividends as predicted by an investor.
This equation is known as the constant growth model.
It assumes that $r_s > g$, and that $g$ remains constant.

If a stock is in equilibrium, the required rate of return, $r_s$, must equal the expected rate of return, which is the expected dividend yield plus an expected capital gains yield.
So we can formulate the following equation:
\[
	\hat{r}_s = \frac{D_1}{P_0} + g
.\]

\subsection{Dividends vs. Growth}%
\label{sub:dividends_vs_growth}

The discounted dividend model as expressed in the equation in section \ref{sub:constant_growth_stocks} shows that, other things held constant, a higher value for $D_1$ increases a stock’s price. However, the equation in section \ref{sub:constant_growth_stocks} shows that a higher growth rate also increases the stock’s price. But now recognize the following:
 
\begin{itemize}
	\item Dividends are paid out of earnings.
	\item Therefore, growth in dividends requires growth in earnings.
\end{itemize}

\noindent
This results in the following equation:
\[
	g = (1-\text{Payout Ratio})\textit{ROE}
.\]
Where the payout ratio is the ratio of profit payed out as dividends, and $\textit{ROE}$ is the expected return on equity, or how much the expected rate of return is on the company.

\subsection{Valuing Non constant Growth Stocks}%
\label{sub:valuing_nonconstant_growth_stocks}

It is not appropriate to assume that dividends will grow at a constant rate.
Firms generally go through \textit{life cycles} where they experience different growth rates during different parts of the cycle.
When firms are young, they tend to grow much faster than the economy as a whole; then they match the economy's growth; and finally they grow at a slower rate than the economy.

\begin{definition}
	Supernormal (Non-constant) growth.
	The part of the firm’s life cycle in which it grows much faster than the economy as a whole.
\end{definition}

In any firm experiencing supernormal growth we can say that sooner or later the growth of the company will no longer be supernormal, but rather, it will be constant at a lower rate.
The stock's intrinsic value today, $\hat{P}_0$, is the present value of the dividends during the non-constant growth period plus the present value of the stock during its constant growth period:
\begin{align*}
	\hat{P}_0 & = \underbrace{\frac{D_1}{(1+r_s)^1} + \frac{D_2}{(1+r_s)^2} + \dotsb + \frac{D_n}{(1+r_s)^n}}_{\substack{\text{PV of dividends during non-constant growth} \\ t=1,\dotsc,n}}
	+
	\underbrace{\frac{D_{n+1}}{(1+r_s)^{n+1}} + \dotsb + \frac{D_{\infty}}{(1+r_s)^{\infty}}}_{\substack{\text{PV of dividends during constant} \\ \text{growth; horizon value} \\ t=n+1,\dotsc,\infty}}\\
		  & = \underbrace{\frac{D_1}{(1+r_s)^1} + \frac{D_2}{(1+r_s)^2} + \dotsb + \frac{D_n}{(1+r_s)^n}}_{\substack{\text{PV of dividends during non-constant growth} \\ t=1,\dotsc,n}} + \underbrace{\frac{\hat{P}_n}{(1+r_s)^n}}_{\substack{\text{PV of} \\ \text{horizon} \\ \text{value:} \\ \hat{P}_n=\frac{D_{n+1}}{(r_s-g)}}}
\end{align*}

\section{Multiples of Comparable Firms Method}%
\label{sec:multiples_of_comparable_firms_method}

Analysts often use the following multiples to value stocks:

\bigbreak
\noindent
Stock price divided by earnings per share:
\[
	\frac{P}{E} = \frac{\text{Stock Price}}{\text{Earnings per Share}}
.\]
where:
\[
	\textit{EPS} = \frac{\text{Net Income}}{\text{Common Shares Outstanding}}
.\]

\bigbreak
\noindent
Stock price divided by dividends per share:
\[
	\frac{P}{D} = \frac{\text{Stock Price}}{\text{Dividends per Share}}
.\]
where:
\[
	\textit{DPS} = \frac{\text{Dividends paid to stockholders}}{\text{Common Shares Outstanding}}
.\]

\section{Preferred Stock}%
\label{sec:preferred_stock}

Preferred stock is a hybrid—it is similar to a bond in some respects and to common stock in others. 
Like bonds, preferred stock has a par value and a fixed dividend that must be paid before dividends can be paid on the common stock.

However, the directors can omit the preferred dividend without throwing the company into bankruptcy.
So although preferred stock calls for a fixed payment like bonds, skipping the payment will not lead to bankruptcy.

A preferred stock entitles its owners to regular, fixed dividend payments.
If the payments last forever, the issue is a perpetuity whose value, $V_p$, is found as follows:
\[
V_p = \frac{D_p}{r_p}
.\]
Where $V_p$ is the value of the preferred stock, $D_p$ is the preferred dividend, and $r_p$ is the required rate of return on the preferred stock.
