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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 financial theory

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 financial theoryinvestors

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 Line

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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 the individual security’s contribution to the risk of the market portfolio

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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 the individual security’s contribution to the risk of the market portfolio

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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 the individual security’s contribution to the risk of the market portfolio

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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 the individual security’s contribution to the risk of the market portfolio

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Figure 9 3 the sml and a positive alpha stock
Figure 9.3 The SML and a Positive-Alpha Stock the individual security’s contribution to the risk of the market portfolio

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The index model and realized returns
The Index Model and Realized Returns the individual security’s contribution to the risk of the market portfolio

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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 1972-1991

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 1972-1991

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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 1972-1991

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 1972-1991

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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 1972-1991

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 Returns

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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 ReturnsProblems 1

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Capm examples of practical problems 2
CAPM: Examples of Practical ReturnsProblems 2

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Capm examples of practical problems 3
CAPM: Examples of Practical ReturnsProblems 3

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Capm examples of practical problems 4
CAPM: Examples of Practical ReturnsProblems 4

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Capm examples of practical problems 5
CAPM: Examples of Practical ReturnsProblems 5

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Capm examples of practical problems 6
CAPM: Examples of Practical ReturnsProblems 6

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Capm examples of practical problems 7
CAPM: Examples of Practical ReturnsProblems 7

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Capm examples of practical problems 8
CAPM: Examples of Practical ReturnsProblems 8

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Index model vs capm
Index model Returnsvs. CAPM

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  • Risk

    • CAPM (theoretical, unobservable portfolio)

    • Index model (observable, “proxy” portfolio)


Index model vs capm 2
Index model Returnsvs. 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 Returns

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  • Idea

    • use realized excess returns

  • Equivalence

    • CAPM + Market model = Index model


  • Summary
    Summary Returns

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    CAPM

    Factor model

    Index model

    Market model


    Chapter 10

    CHAPTER 10 Returns

    Arbitrage Pricing Theory and Multifactor Models of Risk and Return


    Single factor model
    Single Factor Model Returns

    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 Returns

    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 ReturnsModels 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 Returns

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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 Returns

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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 ReturnsModels 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 Returns

    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 Returns

    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 ReturnsTheory (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 Returns

    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 Returns

    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 1 returns as a function of the systematic factor
    Figure 10.1 Returns as a Function of the Systematic Factor Returns

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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 An Arbitrage Opportunity

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    Figure 10 4 the security market line
    Figure 10.4 The Security Market Line An Arbitrage Opportunity

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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 necessarily to individual stocks

    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 necessarily to individual stocks

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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 necessarily to individual stocks

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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: necessarily to individual stocksFama-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 necessarily to individual stocks

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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