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CHAPTER 9 The Capital Asset Pricing Model

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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) BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 2

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Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets There are homogeneous expectations ASSUMPTIONS: INVESTORS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 3

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 4

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 5

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 6

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FIGURE 9.1 THE EFFICIENT FRONTIER AND THE CAPITAL MARKET LINE BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 7

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MARKET RISK PREMIUM The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor: BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 8

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The risk premium on individual securities is a function of the individual securitys contribution to the risk of the market portfolio An individual securitys 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 9

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USING GE TEXT EXAMPLE Covariance of GE return with the market portfolio: Therefore, the reward-to-risk ratio for investments in GE would be: BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 10

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USING GE TEXT EXAMPLE CONTINUED 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: BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 11

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EXPECTED RETURN-BETA RELATIONSHIP CAPM holds for the overall portfolio because: This also holds for the market portfolio: BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 12

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FIGURE 9.2 THE SECURITY MARKET LINE BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 13

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FIGURE 9.3 THE SML AND A POSITIVE-ALPHA STOCK BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 14

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THE INDEX MODEL AND REALIZED RETURNS To move from expected to realized returnsuse 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 15

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FIGURE 9.4 ESTIMATES OF INDIVIDUAL MUTUAL FUND ALPHAS, 1972-1991 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 16

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 17

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ECONOMETRICS AND THE EXPECTED RETURN-BETA RELATIONSHIP 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 18

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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 Mertons Multiperiod Model and hedge portfolios Incorporation of the effects of changes in the real rate of interest and inflation BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 19

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EXTENSIONS OF THE CAPM CONTINUED A consumption-based CAPM Models by Rubinstein, Lucas, and Breeden Investor must allocate current wealth between todays consumption and investment for the future BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 20

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LIQUIDITY AND THE CAPM Liquidity Illiquidity Premium Research supports a premium for illiquidity. Amihud and Mendelson Acharya and Pedersen BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 21

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FIGURE 9.5 THE RELATIONSHIP BETWEEN ILLIQUIDITY AND AVERAGE RETURNS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 22

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THREE ELEMENTS OF LIQUIDITY Sensitivity of securitys illiquidity to market illiquidity: Sensitivity of stocks return to market illiquidity: Sensitivity of the security illiquidity to the market rate of return: BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 23

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CAPM: EXAMPLES OF PRACTICAL PROBLEMS 1 BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 24

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 25 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 2

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 26 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 3

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 27 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 4

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 28 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 5

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 29 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 6

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 30 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 7

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BAHATTIN BUYUKSAHIN, JHU INVESTMENTS1/16/2010 CAPM: EXAMPLES OF PRACTICAL PROBLEMS 8

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INDEX MODEL VS. CAPM BAHATTIN BUYUKSAHIN, JHU INVESTMENTS Risk CAPM (theoretical, unobservable portfolio) Index model (observable, proxy portfolio) 1/16/2010 32

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INDEX MODEL VS. CAPM 2 BAHATTIN BUYUKSAHIN, JHU INVESTMENTS Beta Relationship CAPM (no expected excess return for any security) Index model (average realized alpha is 0) Fig 10.3 1/16/2010 33

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MARKET MODEL BAHATTIN BUYUKSAHIN, JHU INVESTMENTS Idea use realized excess returns Equivalence CAPM + Market model = Index model 1/16/2010 34

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SUMMARY BAHATTIN BUYUKSAHIN, JHU INVESTMENTS CAPM Factor model Index model Market model 1/16/2010 35

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CHAPTER 10 Arbitrage Pricing Theory and Multifactor Models of Risk and Return

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 37

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SINGLE FACTOR MODEL EQUATION r i = 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) e i = Firm specific events BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 38

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MULTIFACTOR MODELS 1 BAHATTIN BUYUKSAHIN, JHU INVESTMENTS Necessity CAPM not practical Index model practical unique factor is unsatisfactory example: Table 10.2 (very small R 2 ) Solution multiple factors 1/16/2010 39

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MULTI-FACTOR MODELS 2 BAHATTIN BUYUKSAHIN, JHU INVESTMENTS 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 1/16/2010 40

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MULTI-FACTOR MODELS 3 BAHATTIN BUYUKSAHIN, JHU INVESTMENTS 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? 1/16/2010 41

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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. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 42

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MULTIFACTOR MODEL EQUATION r i = E(r i ) + GDP GDP + IR IR + e i r i = Return for security I GDP = Factor sensitivity for GDP IR = Factor sensitivity for Interest Rate e i = Firm specific events BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 43

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MULTIFACTOR SML MODELS E(r) = r f + GDP RP GDP + IR RP IR GDP = Factor sensitivity for GDP RP GDP = Risk premium for GDP IR = Factor sensitivity for Interest Rate RP IR = Risk premium for Interest Rate BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 44

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ARBITRAGE PRICING THEORY (APT) BAHATTIN BUYUKSAHIN, JHU INVESTMENTS Nature of arbitrage APT well-diversified portfolios individual assets APT vs. CAPM APT vs. Index models single factor multi-factor 1/16/2010 45

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 46

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APT & WELL-DIVERSIFIED PORTFOLIOS r P = E (r P ) + P F + e P F = some factor For a well-diversified portfolio: e P approaches zero Similar to CAPM, BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 47

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FIGURE 10.1 RETURNS AS A FUNCTION OF THE SYSTEMATIC FACTOR BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 48

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FIGURE 10.2 RETURNS AS A FUNCTION OF THE SYSTEMATIC FACTOR: AN ARBITRAGE OPPORTUNITY BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 49

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FIGURE 10.3 AN ARBITRAGE OPPORTUNITY BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 50

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FIGURE 10.4 THE SECURITY MARKET LINE BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 51

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 52

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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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 53

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TWO-FACTOR MODEL 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 54

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EXAMPLE OF THE MULTIFACTOR APPROACH 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 55

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ANOTHER EXAMPLE: FAMA-FRENCH THREE-FACTOR MODEL 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 56

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THE MULTIFACTOR CAPM AND THE APM 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 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 57