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CHAPTER 9. The Capital Asset Pricing Model. It is the 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.

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chapter 9

CHAPTER 9

The Capital Asset Pricing Model

capital asset pricing model capm
It is the 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

Capital Asset Pricing Model (CAPM)

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assumptions investors
Individual investors are price takers

Single-period investment horizon

Investments are limited to traded financial assets

There are homogeneous expectations

Assumptions: Investors

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assumptions assets
Information is costless and available to all investors

No taxes and transaction costs

Risk-free rate available to all

Investors are rational mean-variance optimizers

Assumptions: Assets

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resulting equilibrium conditions
All investors will hold the same portfolio for risky assets – market portfolio, which contains all securities and the proportion of each security is its market value as a percentage of total market value

held by all investors

includes all traded assets

suppose not: then price… -> included

is on the efficient frontier

asset weights: for each $ in risky assets, how much is in IBM?

for stock i: market cap of stock i / market cap of all stocks

Resulting Equilibrium Conditions

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resulting equilibrium conditions continued
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

Resulting Equilibrium Conditions Continued

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figure 9 1 the efficient frontier and the capital market line
Figure 9.1 The Efficient Frontier and the Capital Market Line

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market risk premium
Market Risk Premium

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  • The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:
return and risk for individual securities
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Return and Risk For Individual Securities

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using ge text example
Using GE Text Example

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Covariance of GE return with the market portfolio:

Therefore, the reward-to-risk ratio for investments in GE would be:

using ge text example continued
Using GE Text Example Continued

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Reward-to-risk ratio for investment in market portfolio:

Reward-to-risk ratios of GE and the market portfolio:

And the risk premium for GE:

expected return beta relationship
Expected Return-Beta Relationship

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CAPM holds for the overall portfolio because:

This also holds for the market portfolio:

figure 9 2 the security market line
Figure 9.2 The Security Market Line

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figure 9 3 the sml and a positive alpha stock
Figure 9.3 The SML and a Positive-Alpha Stock

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the index model and realized returns
The Index Model and Realized Returns

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To move from expected to realized returns—use the index model in excess return form:

The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship

figure 9 4 estimates of individual mutual fund alphas 1972 1991
Figure 9.4 Estimates of Individual Mutual Fund Alphas, 1972-1991

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the capm and reality
The CAPM and Reality

Is the condition of zero alphas for all stocks as implied by the CAPM met

Not perfect but one of the best available

Is the CAPM testable

Proxies must be used for the market portfolio

CAPM is still considered the best available description of security pricing and is widely accepted

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econometrics and the expected return beta relationship
Econometrics and the Expected Return-Beta Relationship

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  • It is important to consider the econometric technique used for the model estimated
  • Statistical bias is easily introduced
    • Miller and Scholes paper demonstrated how econometric problems could lead one to reject the CAPM even if it were perfectly valid
extensions of the capm
Extensions of the CAPM

Zero-Beta Model

Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks

Consideration of labor income and non-traded assets

Merton’s Multiperiod Model and hedge portfolios

Incorporation of the effects of changes in the real rate of interest and inflation

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extensions of the capm continued
Extensions of the CAPM Continued

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  • A consumption-based CAPM
    • Models by Rubinstein, Lucas, and Breeden
      • Investor must allocate current wealth between today’s consumption and investment for the future
liquidity and the capm
Liquidity and the CAPM

Liquidity

Illiquidity Premium

Research supports a premium for illiquidity.

Amihud and Mendelson

Acharya and Pedersen

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figure 9 5 the relationship between illiquidity and average returns
Figure 9.5 The Relationship Between Illiquidity and Average Returns

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three elements of liquidity
Three Elements of Liquidity

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Sensitivity of security’s illiquidity to market illiquidity:

Sensitivity of stock’s return to market illiquidity:

Sensitivity of the security illiquidity to the market rate of return:

capm examples of practical problems 1
CAPM: Examples of Practical Problems 1

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capm examples of practical problems 2
CAPM: Examples of Practical Problems 2

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capm examples of practical problems 3
CAPM: Examples of Practical Problems 3

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capm examples of practical problems 4
CAPM: Examples of Practical Problems 4

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capm examples of practical problems 5
CAPM: Examples of Practical Problems 5

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capm examples of practical problems 6
CAPM: Examples of Practical Problems 6

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capm examples of practical problems 7
CAPM: Examples of Practical Problems 7

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capm examples of practical problems 8
CAPM: Examples of Practical Problems 8

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index model vs capm
Index model vs. CAPM

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  • Risk
      • CAPM (theoretical, unobservable portfolio)
      • Index model (observable, “proxy” portfolio)
index model vs capm 2
Index model vs. CAPM 2

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  • Beta Relationship
      • CAPM (no expected excess return for any security)
      • Index model (average realized alpha is 0)
          • Fig 10.3
market model
Market Model

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  • Idea
      • use realized excess returns
  • Equivalence
      • CAPM + Market model = Index model
summary
Summary

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CAPM

Factor model

Index model

Market model

chapter 10

CHAPTER 10

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

single factor model
Single Factor Model

Returns on a security come from two sources

Common macro-economic factor

Firm specific events

Possible common macro-economic factors

Gross Domestic Product Growth

Interest Rates

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single factor model equation
Single Factor Model Equation

ri= Return for security I

= Factor sensitivity or factor loading or factor beta

F = Surprise in macro-economic factor

(F could be positive, negative or zero)

ei = Firm specific events

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multifactor models 1
Multifactor Models 1

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  • Necessity
      • CAPM
          • not practical
      • Index model
          • practical
          • unique factor is unsatisfactory
          • example: Table 10.2 (very small R2)
  • Solution
      • multiple factors
multi factor models 2
Multi-factor Models 2

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  • Factors in practice
      • business cycles factors
        • examples (Chen Roll Ross)
          • industrial production % change
          • expected inflation % change
          • unanticipated inflation % change
          • LT corporate over LT gvt. bonds
          • LT gvt. bonds over T-bills
        • interpretation
          • residual variance = firm specific risk
multi factor models 3
Multi-factor Models 3

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  • Factors in practice
      • firm characteristics (Fama and French)
        • firm size
          • difference in return
          • between firms with low vs. high equity market value
          • proxy for business cycle sensitivity?
        • market to book
          • difference in return
          • between firms with low vs. high BTM ratio
          • proxy for bankruptcy risk?
multifactor models 4
Multifactor Models 4

Use more than one factor in addition to market return

Examples include gross domestic product, expected inflation, interest rates etc.

Estimate a beta or factor loading for each factor using multiple regression.

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multifactor model equation
Multifactor Model Equation

ri= E(ri) + GDPGDP + IRIR + ei

ri= Return for security I

GDP= Factor sensitivity for GDP

IR= Factor sensitivity for Interest Rate

ei= Firm specific events

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multifactor sml models
Multifactor SML Models

E(r) = rf + GDPRPGDP + IRRPIR

GDP = Factor sensitivity for GDP

RPGDP = Risk premium for GDP

IR = Factor sensitivity for Interest Rate

RPIR= Risk premium for Interest Rate

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arbitrage pricing theory apt
Arbitrage Pricing Theory (APT)

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  • Nature of arbitrage
  • APT
      • well-diversified portfolios
      • individual assets
  • APT vs. CAPM
  • APT vs. Index models
      • single factor
      • multi-factor
arbitrage pricing theory
Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit

Since no investment is required, an investor can create large positions to secure large levels of profit

In efficient markets, profitable arbitrage opportunities will quickly disappear

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apt well diversified portfolios
APT & Well-Diversified Portfolios

rP = E (rP) + bPF + eP

F = some factor

For a well-diversified portfolio:

eP approaches zero

Similar to CAPM,

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figure 10 2 returns as a function of the systematic factor an arbitrage opportunity
Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

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figure 10 3 an arbitrage opportunity
Figure 10.3 An Arbitrage Opportunity

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figure 10 4 the security market line
Figure 10.4 The Security Market Line

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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 - not lie on the SML

APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio

APT can be extended to multifactor models

APT and CAPM Compared

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multifactor apt
Multifactor APT

Use of more than a single factor

Requires formation of factor portfolios

What factors?

Factors that are important to performance of the general economy

Fama-French Three Factor Model

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two factor model
Two-Factor Model

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  • The multifactor APR is similar to the one-factor case
    • But need to think in terms of a factor portfolio
      • Well-diversified
      • Beta of 1 for one factor
      • Beta of 0 for any other
example of the multifactor approach
Example of the Multifactor Approach

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  • Work of Chen, Roll, and Ross
    • Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy
another example fama french three factor model
Another Example:Fama-French Three-Factor Model

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  • The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums

Where:

    • SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks
    • HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio
the multifactor capm and the apm
The Multifactor CAPM and the APM

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A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge

The APT is largely silent on where to look for priced sources of risk

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