% This is lecture no. 5 of BU8201/: Business Finance.

\chapter{Risk \& Rates of Return}%
\label{cha:risk_&_rates_of_return}

\section{The Risk-Return Trade-Off}%
\label{sec:the_risk_return_trade_off}

Investors expect to be compensated when they take on risk.
When they invest in stocks, they compute the required rate of return of the stock based off how risky the stock is.
The required rate of return on all stocks that have risk will rise when investors are more risk averse, and vice versa.

\section{Stand-Alone Risk}%
\label{sec:stand_alone_risk}

An asset's risk can be analysed in two ways:
\begin{enumerate}
	\item On a standalone basis, where the asset is considered by itself.
	\item On a portfolio basis, where the asset is held as one of a number of assets in a portfolio.
\end{enumerate}

\begin{definition}
	Stand-alone Risk.
	An asset's standalone risk is the risk an investor would face if he or she held only this one asset.
\end{definition}

\subsection{Statistical Measures of Standalone Risk}%
\label{sub:statistical_measures_of_standalone_risk}

The expected rate of return is calculated as such:
\begin{align*}
	\hat{r} & = P_{1}r_{1} +P_{2}r_{2} + \ldots + P_{n}r_{n}\\
		& = \sum_{i=1}^n P_ir_i
\end{align*}
Where $P_{i}$ is the probability of the different states of the stock market occuring (e.g. recession, average, boom), and where $r_i$ is the rate of return for the state of economy.
There is a probability distribution curve for the different rates of return for all stocks that resembles the normal distribution curve.

The standard deviation of this normal distribution curve is given by the formula:
\[
	\sigma = \sqrt{\sum_{i=1}^n (r_i -\hat{r})^2 P_i}
.\]
We know that since we cannot know all the data values, we cannot actually calculate the standard deviation from this formula as the standerd error can be calculated from the following formula:
\[
	\sigma_{est.} = \sqrt{\frac{\displaystyle{\sum_{t=1}^n (\overline{r}_t - \overline{r}_{avg})^2}}{n-1}}
.\]
Here $\overline{r}_t$ denotes the past realised rate of return in period $t$, and $\overline{r}_{avg}$ is the average annual return earned over the last $n$ years.

\subsection{The Coefficient of Variation \& the Sharpe Ratio}%
\label{sub:the_coefficient_of_variation_&_the_sharpe_ratio}

A stock that has a high standard deviation means that it deviates more than average and we can conclude that it is risky.
But since investors only choose riskier stocks if their expected return of the stock is high, we can use the coefficient of variation to compare stocks based on their risk and their expected return rate:
\[
	\text{Coefficient of Variation} = CV = \frac{\sigma}{\hat{r}}
.\]
The coefficient of variation shows the risk per unit of return, and it provides a more meaningful risk measure when the expected returns of two alternatives are not the same.

Another alternative risk measure is the \textbf{Sharpe ratio}. The Sharpe ratio compares the asset's realised excess return to its standard deviation over a specific period:
\[
	\text{Sharpe ratio} = \frac{\text{Return}-\text{Risk free rate}}{\sigma}
.\]
Excess returns measure the amount that investment returns are above the risk free rate, so investments with returns equal to the risk free rate will have a zero Sharpe ratio. Stocks with a higher Sharpe ratio perform better.

\subsection{Risk Aversion \& Required Returns}%
\label{sub:risk_aversion_&_required_returns}

The expected rate of return is given as follows:
\[
	\text{Expected rate of return} = \frac{\text{Expected ending value}-\text{Cost}}{\text{Cost}}
.\]

\section{Risk in a Portfolio Context: The CAPM}%
\label{sec:risk_in_a_portfolio_context_the_capm}

The CAPM model was developed in the 60's to explain how risk should be considered in a world where stocks and other assets are held in portfolios.
The risk of a stock held in a portfolio is typically lower than the stock's risk when it is held alone.
The risk and return of an individual stock should be analysed in terms of how the security affects the risk and return of the portfolio in which it is held.

Diversification can get rid of risk without affecting the return on the investment by reducing the effects of the random ups and downs in a particular stock to the return of a portfolio.

\subsection{Expected Portfolio Returns: $\hat{r}_p$}%
\label{sub:expected_portfolio_returns_r_p_}

The expected return on a portfolio $\hat{r}_p$ is the weighted average of the expected returns of the individual assets in the portfolio, with the weights being the percentage of the total portfolio invested in each asset:
\begin{align*}
	\hat{r}_p & = w_1\hat{r}_1 + w_2\hat{r}_2 + \ldots + w_n\hat{r}_n\\
		  & = \sum_{i=1}^n w_i\hat{r}_i
\end{align*}
Where $\hat{r}_i$ is the expected return on the $i$-th stock; the $w_i$ are the stock's weights, or the percentage of the total value of the portfolio invested in each stock; and $n$ is the number of stocks in the portfolio.

If you added another stock to the portfolio with a higher expected return, the portfolio’s expected return would increase, and vice versa if you added a stock with a lower expected return.

\begin{mdframed}
	The key point to remember is that the expected return on a portfolio is a weighted average of expected returns on the stocks in the portfolio.
\end{mdframed}

\subsection{Portfolio Risk}%
\label{sub:portfolio_risk}

Although the expected return on a portfolio is simply the weighted average of the expected returns on its individual stocks, the portfolio’s risk, $\sigma_p$, is not the weighted average of the individual stocks’ standard deviations.

\bigbreak
The portfolio’s risk is generally smaller than the average of the stocks’ $\sigma$s because diversification lowers the portfolio’s risk.

\bigbreak
To illustrate this imagine the extreme case of a portfolio containing two stocks: M and W.
Imagine that these stocks prices always move in opposite directions, like a luxury watchmaker and a debt collection agency.

\bigbreak
Stocks W and M can be combined to form a riskless portfolio because their returns move countercyclically to each other; when W’s falls, M’s rises, and vice versa.
The tendency of two variables to move together is called correlation, and the correlation coefficient, $\rho$, measures this tendency.

\bigbreak
In statistical terms, we say that the returns on Stocks W and M are perfectly negatively correlated, with $\rho = –1.0$.
The opposite of perfect negative correlation is perfect positive correlation, with $\rho = +1.0$.
If returns are not related to one another at all, they are said to be independent and $\rho = 0$.

\begin{enumerate}
	\item A portfolio’s risk declines as stocks are added, but at a decreasing rate; once 40 to 50 stocks are in the portfolio, additional stocks do little to reduce risk.
	\item A portfolio’s total risk can be divided into two parts, diversifiable risk and market risk. Diversifiable risk is the risk that is eliminated by adding stocks. Market risk is the risk that remains even if the portfolio holds every stock in the market.
	\item Diversifiable risk is caused by such random, unsystematic events as lawsuits, strikes, and other events that are unique to the particular firm.
\end{enumerate}

\subsection{The Beta Coefficient}%
\label{sub:the_beta_coefficient}

The risk that remains once a stock is in a diversified portfolio is its contribution to the portfolio’s market risk, and that risk can be measured by the extent to which the stock moves up or down with the market.

The tendency of a stock to move with the market is measured by its beta coefficient, $b$, which is estimated using the stock's historical data to describe how the stock will move relative to the market in the future.

Thus, because a stock’s beta reflects its contribution to the riskiness of a portfolio, beta is the theoretically correct measure of the stock’s riskiness.

A portfolio consisting of low-beta stocks will also have a low beta because the beta of a portfolio is a weighted average of its individual securities’ betas, found using this equation:
\begin{align*}
	b_p & = w_1b_1 + w_2b_2 + \dotsb + w_nb_n\\
	    & = \sum_{i=1}^n w_ib_i
\end{align*}

\subsection{Relationship Between Risk \& Rates of Return}%
\label{sub:relationship_between_risk_&_rates_of_return}

The market risk premium, $\textit{RP}_m$, shows the premium that investors require for bearing the risk of an average stock. The size of this premium depends on how risky investors think the stock market is and on their degree of risk aversion:

\[
	\textit{RP}_m = r_M - r_{RF}
.\]

The required return for any stock can be found as follows:
\[
	\text{Required return on a stock} = \text{Risk-free return} + \text{Premium for the stock's risk}
.\]
The risk-free return includes a premium for expected inflation.
The required return on a stock can be found using the security market line equation:

\[
	r_n = r_{RF} + (r_m - r_{RF})b_n
.\]

In a SML graph, required rates of return are shown on the horizontal axis, and risk as measured by beta is shown on the vertical axis.
The slope is the risk premium.

