<p>Beta (β) measures the volatility of a stock in relation to a market such as S&amp;P 500 or any other index. It is an important measure to gauge the risk of a security.</p>
<p>The market itself is considered to have a Beta of 1. Using regression analysis, the beta of the stock is calculated. If the beta of the stock is greater than 1, this means the stock’s prices are more volatile than the market, and vice verse. For example, if a stock has a beta of 1.2, this means that a 1% change in the market index will bring about a 1.2% change in the stock’s price. Stocks with high beta are considered to be more risky compared to the ones with low beta.</p>
<h2>Formula:</h2>
<p>[latex]\beta = Covar \left ( r_{s}-r_{m} \right )/Var\left ( r_{m} \right )[/latex]</p>
<p>Where,</p>
<p>rs represents the returns of the stock, and rm represents the returns of the market or benchmark.</p>
<h2>Calculating Beta in Excel</h2>
<p>Beta of a stock can be calculated in excel using two methods.</p>
<ol start="1">
<li>Using the formula , i.e. [latex]\beta = Covar \left ( r_{s}-r_{m} \right )/Var\left ( r_{m} \right )[/latex]</li>
<li>Using the slope function in Excel</li>
</ol>
<h2>Steps</h2>
<h3>Step 1</h3>
<p>Download the historical prices for the stock and the benchmark index. For this example, let’s take IBM stock and S&amp;P500 benchmark index. You can do this using the <a title="Get Yahoo Finance Stock Quotes in Excel" href="https://financetrain.com/get-yahoo-finance-stock-quotes-in-excel/" target="_blank" rel="noopener noreferrer">download stock data spreadsheet</a>.</p>
<h3>Step 2</h3>
<p>The next step is to calculate the percentage returns for both the stock and the index. To do so, you can use the formula =((Cell2-Cell1)/Cell1)*100, where cell 1 refers to the previous period data point, and cell 2 refers to a current period data point.</p>
<h3>Step 3</h3>
<p>Calculate Stock’s Beta using one of the two methods.</p>
<p><strong>Method 1</strong> – Calculate Beta using the formula</p>
<p>[latex]\beta = Covar \left ( r_{s}-r_{m} \right )/Var\left ( r_{m} \right )[/latex]</p>
<p> </p>
<p><strong>Method 2</strong> – Calculate Beta using excel’s slope function</p>
<p>Beta = SLOPE(range of % change of equity, range of % change of index).</p>

How to Calculate Stock Beta in Excel

The Capital Asset Pricing Model is a popular asset pricing model in Finance. It is used to determine the expected rate of return of a risky asset. It says that the expected return on a risky asset is equal to the risk-free rate plus a risk premium.

The risk premium is a measure of non-diversifiable risk and is calculated using the asset’s <a title="How to Calculate Stock Beta in Excel" href="https://financetrain.com/how-to-calculate-stock-beta-in-excel/">Beta</a>. CAPM considers only systematic risk (non-diversifiable risk) as the security specific risk (unsystematic risk) can be diversified away.

According to the CAPM model, the expected return on a security is given as follows:
<div class="formula1">[latex]E(r)=R_{f}+\beta\left (E(R_{m}) - R_{f} \right )[/latex]

Where:
<ul>
	<li>E(Ri) is the expected return on the security</li>
	<li>Rf is the risk-free rate of return</li>
	<li>B is the beta of the stock</li>
	<li>Rm is the expected return from the market</li>
</ul>
</div>
The key idea behind the CAPM model is that the investors should be compensated for the time value of money for holding the asset and also for the additional risk taken by them.
<h3><strong>Assumptions of the CAPM Model</strong></h3>
The CAPM model is a single-period model and makes a few assumptions:
<ul>
	<li>Investors are risk-averse and want to maximize their wealth</li>
	<li>All investors have same expectations from the asset returns</li>
	<li>The asset returns are normally distributed</li>
	<li>The investors can borrow and lend unlimited amount of money at risk-free rate</li>
	<li>There are no transaction costs</li>
	<li>Investors cannot influence the price, that is, they are only price takers</li>
	<li>All information is available to everyone.</li>
</ul>
If the risk-free rate is 5.4%, the stock's Beta is 1.5, and the expected market return is 8.4, then the stock is expected to return 9.9% (=5.4 + 1.5 * (8.4 - 5.4)).

If the actual returns offered by the stock are less than 9.9%, then you might want to avoid the investment.

The Capital Asset Pricing Model

The securities Market Line (SML) is a graphical representation of the Capital Asset Pricing Model (CAPM). Essentially, it displays the expected rate of return of an individual security as a function of systematic, non-diversifiable risk (its beta).

Assume that the historical market return is 12%, and the risk-free rate is 5%. The expected return of a security can then be represented as:

<div class="formula1">[latex]R^{_{i}} = 5\% + \beta \times \left ( 12\% - 5\% \right )[/latex]</div>

The Securities Market Line will therefore represent the linear relationship between Ri and Beta.

<img src="https://financetrain.com/wp-content/uploads/2011/09/SML.gif" alt="" title="SML" width="567" height="375" class="aligncenter size-full wp-image-2893" />

If the CAPM model holds true, then all the securities should lie on the SML.

As we can see, the market has a beta of 1. If the beta of the stock is greater than 1, this means the stock’s prices are more volatile than the market, and vice versa. For example, if a stock has a beta of 1.2, this means that a 1% change in the market index will bring about a 1.2% change in the stock’s price. Stocks with high beta are considered to be more risky compared to the ones with low beta.

SML provides a cross-sectional analysis across different stocks. The average returns of two stocks will be proportional to their betas. 

One important use of SML is that it can help in identifying underprices and overpriced stocks.

<img src="https://financetrain.com/wp-content/uploads/2011/09/SH02.gif" alt="" title="SH02" width="565" height="373" class="aligncenter size-full wp-image-2894" />

Consider two stocks X and Y. Both have a Beta of 0.5. However, X gives less returns than suggested by the SML, and Y gives more market returns. Both these securities are mis-priced. An arbitrageur can see this opportunity. He will short-sell Security X and use the money to purchase Security Y. If you spotted this opportunity earlier, then you will make good profits in your trade.



Securities Market Line (SML)

While deciding on about what investments to make, one should weigh the rewards versus the risks of the investment opportunity. The Sharpe ratio is one popular measure of return on risk. It is named after Nobel Laureate professor William F. Sharpe. 

The Sharpe ratio measures the reward (or excess return) of an asset per unit of risk.

The Sharpe ratio is also commonly expressed as:

<div class="formula1">[latex]Sharpe \hspace{1 mm} Ratio = \frac{Return\hspace - \hspaceRisk\hspace{1 mm} free\hspace{1 mm} Rate}{Standard\hspace{1 mm} Deviation\hspace{1 mm} of\hspace{1 mm} Returns}[/latex]</div>

Both the return and the standard deviation are annualized. To annualize returns, you multiply linearly by time. For example, a monthly return of 1 % converts to an annualized return of 12%. Standard deviation of return is a measure of risk, or uncertainty, of returns. To annualize standard deviation, multiply by the square root of time.

For example, a monthly standard deviation of return of 1 % converts to annualized standard deviation of 1 %*sqrt(12)= 3.46%. 

A higher Sharpe ratio indicates better portfolio performance. Sharpe ratios can be increased either by increasing returns or by decreasing risk. 

As we know, a portfolio can achieve higher returns by taking on additional risks. Using the Sharpe Ratio one can determine the source of higher returns: better performance or from additional risks.

Historically, Sharpe ratios over long periods of time for most major asset classes have ranged from 0.3 to 2.


Sharpe Ratio for Measuring Return on Risk

In a business context, <a title="Sharpe Ratio for Measuring Return on Risk" href="https://financetrain.com/sharpe-ratio-for-measuring-return-on-risk/">Sharpe ratios</a> can be used for:
<ul>
	<li>Setting targets for results</li>
	<li>Evaluating performance after the fact</li>
	<li>Benchmarking</li>
</ul>
<strong>Sharpe Ratio is a standard benchmark</strong>

Sharpe ratios are easy to calculate and can be used to compare the performance of a variety of different asset classes, or businesses. For budgeting and planning purposes, we can preset Sharpe ratio performance targets for business units based on historically observed return on risk for comparable asset classes or businesses.

<strong>Allocating Sharpe ratios</strong>

In translating a firm's target Sharpe ratio down to business units, we must be mindful of the effect of diversification. In addition to taking into account the type of business (for example, proprietary trading), it is important to take into account a business group's size. Smaller groups that have less opportunity for diversification (for example, a wheat futures trading group with three traders) should be given lower Sharpe ratio targets than larger businesses (for example, global fixed income with 50 traders).

<strong>Performance evaluation cycle</strong>

Sharpe ratio measurement can be integrated into the overall business performance evaluation cycle of goal setting, monitoring, and evaluation. Although fixed Sharpe Ratio targets are useful for planning purposes and set a clear return on risk objective, it's best to define a dynamic Sharpe ratio benchmark (for example, the Sharpe ratio of an index of hedge funds) for evaluating performance. In evaluating performance, superior return on risk should be rewarded.

Sharpe Ratio as Performance Benchmark

Jensen’s Alpha, also known as the Jensen’s Performance Index, is a measure of the excess returns earned by the portfolio compared to returns suggested by the CAPM model. It represents by the symbol &alpha;.

The value of the excess return may be positive, negative, or zero. The CAPM model itself provides risk-adjusted returns, i.e., it takes into account the risk of the security. So, if the security is fairly priced, its actual returns will be same as CAPM. The Alpha in this case will be 0. If, however, the security earns even more than the risk-adjusted returns, it will have a positive Alpha. Negative alpha indicates that the portfolio has not earned its required return. A higher Alpha is always desirable by portfolio managers.

Jensen’s alpha focuses only on non-diversifiable, relevant risk by using beta and CAPM. It assumes that the portfolio has been adequately diversified.  

The Jensen’s Alpha can be calculated using the following formula:

<div class="formula1">
[latex]\alpha =R_{P}-\left ( R_{f}+\beta \left ( R_{M} -R_{f}\right ) \right ) [/latex]

Where:
<ul>
<li>Rp = Returns of the Portfolio</li>
<li>Rf = Risk-free rate</li>
<li>&beta; = Stock’s beta</li>
<li>Rm = Market return</li>
</ul>
</div>
Jensen’s alphas are reported in a percentage format, indicating the percentage by which the portfolio over-or-under-performed the market on a risk-adjusted basis.  

Let’s take the example of XYZ stock. If the daily return based on CAPM is 0.15% and the actual stock return is 0.20%, then Jensen’s alpha is 0.05%, which is a good indicator.


Jensen’s Alpha

The Single Index Model (SIM) is an asset pricing model, according to which the returns on a security can be represented as a linear relationship with any economic variable relevant to the security.

In case of stocks, this single factor is the market return.

The SIM for stock returns can be represented as follows:
<div class="formula1">
[latex]r_{s}-r_{f}=\alpha +\beta \left ( r_{m} - r_{f}\right )+\varepsilon [/latex]

Where:
<ul>
<li><a href="https://financetrain.com/jensens-alpha/" title="Jensen’s Alpha">Alpha (&alpha;)</a> represents the abnormal returns for the stock</li>
<li>&beta;(r<sub>m</sub> − r<sub>f</sub>) represents the movement of the market modified by the stock's <a href="https://financetrain.com/how-to-calculate-stock-beta-in-excel/" title="How to Calculate Stock Beta in Excel">beta</a></li>
<li>&epsilon; represents the unsystematic risk of the security due to firm-specific factors. </li>
</ul>
</div>
According to this equation, asset’s returns is influenced by the market (reflected in beta), it has firm specific excess returns (reflected in alpha) and also has firm-specific risk (the residual).

Single Index Model

The risk arising from the volatility in the stock prices is referred to as Equity Price Risk.

While talking about price volatility, it is important to differentiate between systematic risk and unsystematic risk.

Systematic risk refers to the risk due to general market factors and affects the entire industry. It cannot be diversified away.

Unsystematic risk, or specific risk is the risk specific to a company that arises due to the company specific characteristics. According to portfolio theory, this risk can be eliminated through diversification.

Systematic and Specific Risk

We had seen earlier that CAPM, which is essentially a Single Index Model, considers that the returns of a stock are affected only by a single factor, which is the excess market returns.

The Arbitrage Pricing Theory takes a more complex approach and allows the returns of a stock to be influenced by multiple factors. These factors could be interest rates, inflation, exchange rates, etc.

The sensitivity of the returns to each factor is represented by the factor-specific beta coefficient.

This will help in pricing the asset more accurately, and if the actual price differs from the theoretical price, the arbitrage opportunities will exist which will eventually bring the price to its correct level.

Using APT, the returns of the risky asset can be represented as follows:
<div class="formula1">[latex]r_{i}=\alpha_{i} +\beta_{i1} F_{1}+\beta_{i2} F_{2}+\cdot\cdot\cdot+\beta_{i2} F_{n}+\varepsilon _{i}[/latex]

Where:
<ul>
	<li>R<sub>i</sub> = Asset returns</li>
	<li>α<sub>i</sub> = The asset specific factor</li>
	<li>β<sub>in</sub> = Sensitivity of the i<sub>th</sub> asset to the factor n.</li>
	<li>F<sub>n</sub> = Systematic (Macroeconomic) factor</li>
</ul>
</div>
<h2>Some Observations</h2>
<ul>
	<li>The factors are macroeconomic aggregates rather than firm-specific characteristics.</li>
	<li>It’s a regression equation. The model is based on multivariate regression analysis.</li>
	<li>It assumes that the returns are generated by a factor model</li>
</ul>

Arbitrage Pricing Theory (APT)

CAPM and Multi-factor Models

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COURSE

CAPM and Multi-factor Models

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