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Chapter 7: Capital Asset Pricing Model and Arbitrage Pricing Theory

In Chapter 6, we showed how a single-factor model could be used to estimate the 2 components of risk (“market” and “firm-specific”) for a security.

In Chapter 7, we’ll discuss the most commonly used single-factor model

Capital Asset Pricing Model (CAPM): a model that relates the required rate of return for a security to its risk as measured by beta

Then we’ll discuss how CAPM can be used to calculate risk premiums, alpha, and underpriced and overpriced securities.

Then we’ll briefly introduce• Multi-factor pricing models• Arbitrage pricing theory (APT)

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Capital Asset Pricing Model (CAPM)

• Equilibrium model that underlies all modern financial theory

• Derived using principles of diversification with simplified assumptions

• Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development (early 1970’s)

CAPM is a theoretical economic model that requires these assumptions:

– Individual investors are price takers

– Single-period investment horizon

– Investments are limited to traded financial assets

– No taxes nor transaction costs

– Information is costless and available to all investors

– Investors are rational mean-variance optimizers

– Homogeneous expectations

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Capital Asset Pricing Model (CAPM)

Resulting Equilibrium Conditions

• All investors will hold the same portfolio for risky assets (market portfolio)

• Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

• Risk premium on the market depends on the average risk aversion of all market participants

• Risk premium on an individual security is a function of its covariance with the market

CAPM implies the expected-return-beta relationship

E(ri ) = rf + βi [E(rM ) – rf ] [Text, eqn. 7.2, p 194]

The rate of return on any asset exceeds the risk-free rate by a risk premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the (benchmark) market portfolio.

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Capital Asset Pricing Model (CAPM)

CAPM expected-return-beta relationship

E(ri ) = rf + βi [E(rM ) – rf ] [Text, eqn. 7.2, p 194]

The rate of return on any asset exceeds the risk-free rate by a risk premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the (benchmark) market portfolio.

Your text (page 194): The of the expected-return-beta relationship of the CAPM makes a powerful economic statement. Consider 2 stocks:

Stock A 40% 0.5

Stock B 15% 1.5

CAPM states that Stock B should be priced such that its risk premium will be 3x that of Stock A’s risk premium. Only systematic risk matters for prices.

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CAPM Portfolio Beta Coefficients

The beta of any set of securities is the weighted average of the individual securities’ betas.

 

 

 

 Example: What is the beta of a portfolio made up of:

• 25% of Stock H that has beta = 2

• 45% of Stock A that has beta = 1, and

• 30% of Stock L that has beta = 0.5

• Beta of the portfolio = (.25)(2) + (.45)(1) + (.30)(0.5) = __________

w

w ... w w

j j

n n2211p

N

1j

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Figure 7.2 The security market line (SML) and a positive alpha stock:- The SML is a graphical representation of the expected-return-beta relationship of the CAPM.- Alpha is the abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as the CAPM.

In this example, a stock with a beta of 1.2 has an E(r) = 6%+1.2(14%-6%)=15.6%.

If its actual E(r) is 17%, then the stock’s = 1.4%

How is the SML different from the CAL?

1) SML relates E(r) to 2) CAL relates E(r) to

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Figure 6.12 Scatter Diagram and Security Characteristic Line for Dell Stock

How is the SML different from the SCL in Ch 6?1) SML relates E(r) to 2) SCL relates security’s excess return to the market portfolio’s excess return

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Estimating the CAPM beta of individual stocks

• Use (monthly) historical data on T-bills, S&P 500 and individual securities

• Use Ordinary Least Squares (OLS) Regression to regress risk premiums for individual stocks against the risk premiums for the S&P 500

• Slope is the beta for the individual stock

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Table 7.1 Monthly Return Statistics for T-bills, S&P 500 and General Motors

*Note: In your text (9h ed. ) the firm in this example is Google

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Figure 7.3 Cumulative Returns for T-bills, S&P 500 and GM Stock

GM is a cyclical stock --- its returns tend to be above or below the S&P 500 depending on the business (economic) cycle. It looks like it moves more with the business cycle than does the S&P 500. What do you expect for GM’s beta?

*Note: In your text (9th ed. ) the firm in this example is Google

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Table 7.2 Security Characteristic Line for GM: Summary Output from Excel OLS

*Note: In your text (9th ed. ) the firm in this example is Google

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Limitations of the CAPMLimitations of the CAPM

• The CAPM is based on expected conditions, but we only have historical data to use to estimate beta.

• Timeframe (frequency of returns and historical time period used) for the regression of the historical data greatly impacts our estimate of beta.

• As conditions change, future volatility may differ from past volatility. Volatility here just means magnitude of changes in prices.

• Where does our forecast of the risk-free rate ( r RF ) and the required rate of return for the market ( r m ) come from?

• Alternative “multi-factor” models have been developed to try to address some of these limitations. “Multi-factor” means that there are more explanatory variables on the right-hand side of the regression.

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Fama French Three-Factor Model• Model developed in 1996 by 2 professors, Eugene Fama and

Kenneth French. • Kenneth French is still at Dartmouth but runs a multi-million dollar

consulting firm largely based on the model.• Data for the monthly factors for the model available for free at:

mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

• Model specification is:

ri - rf = i + βM (rM - rf) + βHML (rHML) + βSMB (rSMB) + ei

• First factor is the market’s return (like CAPM)• The 2nd Factor is Size aka Market Cap (Small minus Big, or “SMB”). • The 3rd Factor is Book value relative to Market value (High minus

Low, or “HML”) • The basic rational is that stocks of small firms historically earn

higher returns and carry higher risk. Similarly, stocks of companies with lower book to market values (“Value Stocks”) do better.

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Arbitrage Pricing Theory (APT)• Arbitrage – creation of riskless (aka risk-free) profits by taking

positions that take advantage of security “mispricing”

• If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high-priced portfolio

• Arbitrage arises if an investor can construct a zero beta investment portfolio with a return greater than the risk-free rate

• In efficient markets, profitable arbitrage opportunities will quickly disappear

• Arbitrage Pricing Theory is a theory of risk-return relationships derived from no-arbitrage considerations in large capital markets --- in other words, if there are arbitrage opportunities do not exist, what relationships between risk and return have to hold?

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The result: For a well diversified portfolio

Rp = βpRS (Excess returns)

(rp,i – rf) = βp(rS,i – rf)

and for an individual security

(rp,i – rf) = βp(rS,i – rf) + ei

Arbitrage Pricing Theory (APT) - continued

For example, for 4 systematic factors:

(rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + βp,4(r4,i – rf) + ei

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Arbitrage Pricing Theory posits a single-factor security market

Rp = i + βp RM + ep

• Rp is the excess return of a well-diversified portfolio. • i is the excess return of the portfolio when RM = 0.• βp is the portfolio’s responsiveness to a macroeconomic factor• RM is our “single factor”.• ep is the portion of excess return that results from nonsystematic risk

• By definition, well-diversified means the portfolio has negligible firm-specific risk.

• Therefore: Rp = i + βp RM

No arbitrage opportunities mean that i = 0. (If βp = 0, Then Rp = 0.)

Thus, APT implies the same expected-return-beta relationship as CAPM.

E(ri ) = rf + βi [E(rM ) – rf ]

Arbitrage Pricing Theory (APT) - continued

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APT and CAPM Compared• APT applies to well diversified portfolios and not necessarily to

individual stocks

• With APT it is possible for some individual stocks to be mispriced – i.e., they may not lie on the SML

• APT is more general in that it gets to an expected return and beta relationship without the assumptions of the CAPM theory

• APT can be extended to multifactor models.

• Hedge funds use multifactor models to search for arbitrage opportunities:– As they take positions, prices move toward equilibrium to

eliminate the arbitrage opportunity– Sometimes their models “get equilibrium wrong” and lose money