beta and capm lecture4 2014

8/10/2019 Beta and CAPM Lecture4 2014

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Beta and CAPM

Lecture 4, FM 2.2


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The Historical Tradeoff

Between Risk and Return

Excess Returns

The difference between the average return for an

investment and the average return for risk-free

investment (DTS, T-bills,)


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Risk-Free Rate

Determining the Risk-Free Rate

The yield on U.S. Treasury securities or

government bonds in the country of the

project/company (DTC in The Netherlands)

LIBOR (ended up in tears)

Currently: overnight swap rate


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Volatility Versus Excess Return of U.S. Small Stocks, Large Stocks

(S&P 500), Corporate Bonds, and Treasury Bills, 19262008


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The Historical Tradeoff Between Risk and Return in Large

Portfolios, 19262005

Source: CRSP, Morgan Stanley Capital International


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The Returns of Individual Stocks

Is there a positive relationship betweenvolatility and average returns also forindividual stocks?

As shown on the next slide, there is no preciserelationship between volatility and average returnfor individual stocks.

Larger stocks tend to have lower volatility thansmaller stocks.

All stocks tend to have higher risk and lower returnsthan large portfolios.


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Historical Volatility and Return for 500 Individual Stocks,

by Size, Updated Quarterly, 19262005


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Common Versus Independent Risk

Common Risk

Risk that is perfectly correlated Risk that affects all securities

Independent Risk

Risk that is uncorrelated Risk that affects a particular security

Diversification

The averaging out of independent risks in alarge portfolio


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Diversification

in Stock Portfolios (cont'd)

Firm-Specific Versus Systematic Risk

Independent Risks

Due to firm-specific news

Also known as:

Firm-Specific Risk

Idiosyncratic Risk

Unique Risk Unsystematic Risk

Diversifiable Risk


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Diversification



Common Risks

Due to market-wide news

Also known as:

Systematic Risk

Undiversifiable Risk

Market Risk


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Diversification



When many stocks are combined in a large

portfolio, the firm-specific risks for each stock will

average out and be diversified.

The systematic risk, however, will affect all firms

and will not be diversified.


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No Arbitrage and the Risk Premium

The risk premium for diversifiable risk is zero,

so investors are not compensated for holding

firm-specific risk.

The risk premium of a security is determined

by its systematic risk and does not depend onits diversifiable risk.


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No Arbitrage

and the Risk Premium (cont'd)

A stocks volatility, which is a measure of total risk

(systematic risk plus diversifiable risk), is not an

appropriate measure of risk for an individual

security. (It is for portfolios).

There is no clear relationship between volatility

and average returns for individual securities.

To estimate a securitys expected return, we needto find a measure of a securitys systematic risk.


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Measuring Systematic Risk

To measure the systematic risk of a stock, we

need to determine how much of the variability

of its return is due to systematicrisk vs

idiosyncratic risk.

How to do that?Concept of beta


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Market Portfolio

An (idealized) portfolio that contains all shares and

securities in the market

The S&P 500 (500 biggest stocks in US) is often used as a

proxy for the market portfolio in the US. AEX in The Netherlands

MSCI World Index: for the whole developed world (>1600

stocks)

Most of these indices are value-weighted(according to

stocks market capitalization)

Some of them areprice-weighted(DJIA)


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Concept of Beta

Diversified portfolio: only MARKET RISK

Contribution of an individual asset to portfolio risk is NO LONGERrelated toits volatility

it is related to how sensitivean asset is to market movements.

This sensitivity is called betaof an asset

Interpretation: On average, if the market moves by 1%, the asset price willmove by beta%. ON AVERAGE!

Bottom line: In a diversified portfolio context, risk of an asset is measuredby its beta!


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Definition of Beta

Sensitivity to Systematic Risk: Beta ()

The expected percent change in the excess return of a security for a 1%

change in the excess return of the market portfolio.

Beta differs from volatility. Volatility measures total risk (systematic plus

unsystematic risk), while beta is a measure of only systematic risk. Beta of asset iis defined as

where is the covariance between asset is return and market

return, and is the variance of the market return.

,),(

2

market

i

imarketCov

),( imarketCov2

market


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Another way of calculating Beta:

The beta is calculated as:

Volatility of that is common with the market

i

( ) ( , ) ( , )

( ) ( )

i

Mkt i i Mkt i Mkt

iMkt Mkt

SD R Corr R R Cov R R

SD R Var R


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Trivial examples

What is beta of a risk-free asset?

What is beta of the market portfolio?

What is beta of a portfolio consisting for x% of

market portfolio and (1-x)% of risk free asset?


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Linear regression for calculating beta

Monthly Returns for Cisco Stock and for the S&P 500, 19962009


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Scatterplot of Monthly Excess Returns for Cisco Versus the

S&P 500, 19962009


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Estimating Beta from Historical

Returns

Beta corresponds to the slope of the best-fitting

line in the plot of the securitys excess returns

versus the market excess return.


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Using Linear Regression

Linear Regression:

iis the intercept term

i represents the sensitivity of the stock to

market risk. When the markets return increases by 1%,

the securitys return increases by i%

iis the error term and represents the deviation from the

best-fitting line and is zero on average.

(Ri r

f)

i

i(R

Mkt r

f)

i


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Using Linear Regression

Given data forrf, Ri, and RMkt, statistical

packages for linear regression can estimatei.

A regression for Cisco using the monthly returns for

19962009 indicates the estimated beta is 1.80.

The estimate of Ciscos alpha from the regression is1.2%.


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Practical Considerations When

Forecasting Beta

1. Time Horizon

For stocks, common practice is to use at least two

years of weekly return data or five years of

monthly return data.

2. The Market Proxy

In practice the S&P 500 is used as the US market

proxy. Other proxies include the MSCI World or

regional indices (FTSE, AEX, DAX, ).


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Forecasting Beta

3. Beta Variation and Extrapolation

Betas vary over time, so many practitioners prefer

to use average industry betas rather than

individual stock betas.

In addition, evidence suggests that betas tend to

regress toward the average beta of 1.0 over time.


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Estimated Betas for Cisco Systems,

19992009


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Forecasting Beta

Beta Extrapolation

Adjusted Betas 2 1Adjusted Beta of Security (1.0)

3 3

ii

Estimation Methodologies Used by Selected Data

Providers


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Betas with Respect to the

S&P 500 for Individual

Stocks (based on

monthly data for 2004

2008)


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Forecasting Beta

4. Outliers

The beta estimates obtained from linear

regression can be very sensitive to outliers, which

are returns of unusually large magnitude.


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Beta Estimation with and without Outliers for Genentech

Using Monthly Returns for 20022004


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Determining the Risk Premium

Capital Asset Pricing Model (CAPM):

Given a market portfolio, the expected return of

an investment is:

Risk premium for security

[ ] ( [ ] ) Mkti i f i Mkt f

i

E R r r E R r


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Example

Assume the risk-free return is 5% and the

market portfolio has an expected return of

12% and a volatility of 44%.

ATP Oil and Gas has a volatility of 68% and a

correlation with the market of 0.91.

What is ATPs beta with the market?

Under the CAPM assumptions, what is its

expected return?


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Solution

i

( ) ( , ) (.68)(.91)1.41

( ) .44

i i Mkt

Mkt

SD R Corr R R

SD R

[ ] ( [ ] ) 5% 1.41(12% 5%) 14.87% Mkti f i Mkt f E R r E R r


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The Security Market Line

CAPM says that there is a linear relationship between a stocks beta and its

expected return:

This is the equation for the so-called security market line (SML).

According to the CAPM, if the expected return and beta for individual

securities are plotted, they should all fall along the SML. They dont!

Risk premium for security

[ ] ( [ ] ) Mkti i f i Mkt f

i

E R r r E R r


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The Security Market Line

The SML shows the

expected return for each

security as a function of

its beta with the market.

According to the CAPM,

all stocks and portfolios

should lie on the SML.

In reality: low-beta

stocks perform better

than CAPM predicts;

high-beta stocks perform

worse than CAPM

predicts.


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What about reality?

Do all stocks lie on SML? NO!

Testing CAPM: plot estimates of betas versus average premium on an asset ( average return

risk-free return)

Turns out that: high-beta assetsgenerate lower returnsthan predicted by CAPM and low-beta assetsgenerated higher returns.

Also: some zero-beta assets (e.g. catastrophe bonds) offer returns higher than risk-free rate

of return !

Why? CAPM has many assumptions: does not take into account borrowing, behavioralaspects, no default by the state, etc etc etc

Beta is not estimated very well for a single company


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Identifying a Stocks Alpha

To improve the performance of their portfolios,

investors will compare the expected (or historical

average) return of a security with its requiredreturn from the security market line.

( [ ] )s f s Mkt fr r E R r


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Identifying a Stocks Alpha

The difference between a stocks expected return

and its required return according to the security

market line is called the stocks alpha.

If the market portfolio was efficient, all stockswould be on the security market line and have an

alpha of zero.

[ ]s s sE R r


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Deviations from the Security Market Line

W l d fi b f f li (i d f i di id l


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We can also define beta of a portfolio (instead of individual

asset) and use CAPM to estimate the required return on it.

The beta of a portfolio is the weighted average beta of the

securities in the portfolio:

,( , ) ( , )( ) ( ) ( )

i i Mkt iP Mkt i Mkt

P i i ii iMkt Mkt Mkt

Cov x R RCov R R Cov R Rx xVar R Var R Var R


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Example

Problem

Suppose the stock of the 3M Company (MMM)has a beta of 0.69 and the beta of Hewlett-

Packard Co. (HPQ) stock is 1.77.

Assume the risk-free interest rate is 5% and theexpected return of the market portfolio is 12%.

What is the expected return of a portfolio of 40%of 3M stock and 60% Hewlett-Packard stock,according to the CAPM?


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Example (contd)

Solution

[ ] ( [ ] ) Mkti f i Portfolio f E R r E R r

(.40)(0.69) (.60)(1.77) 1.338 P i iix

[ ] 5% 1.338(12% 5%) 14.37% iE R


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Summary of the Capital Asset

Pricing Model

The expected return on any security is

proportional to beta and is given by:

Risk premium for security[ ] ( [ ] )

Mkt

i i f i Mkt f

iE R r r E R r

beta and capm lecture4 2014

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