The Capital Asset Pricing Model CHAPTER 9

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 Bahattin Buyuksahin, JHU, Investment 2

Assumptions: Investors Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets There are homogeneous expectations Bahattin Buyuksahin, JHU, Investment

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 Bahattin Buyuksahin, JHU, Investment

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 Bahattin Buyuksahin, JHU, Investment

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 Bahattin Buyuksahin, JHU, Investment

Figure 9.1 The Efficient Frontier and the Capital Market Line Bahattin Buyuksahin, JHU, Investment

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

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 Bahattin Buyuksahin, JHU, Investment

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

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

Expected Return-Beta Relationship CAPM holds for the overall portfolio because: This also holds for the market portfolio: Bahattin Buyuksahin, JHU, Investment

Figure 9.2 The Security Market Line Bahattin Buyuksahin, JHU, Investment

Figure 9.3 The SML and a Positive-Alpha Stock Bahattin Buyuksahin, JHU, Investment

The Index Model and Realized Returns 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 Bahattin Buyuksahin, JHU, Investment

Figure 9.4 Estimates of Individual Mutual Fund Alphas, 1972-1991 Bahattin Buyuksahin, JHU, Investment

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

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

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 Bahattin Buyuksahin, JHU, Investment

Extensions of the CAPM Continued A consumption-based CAPM Models by Rubinstein, Lucas, and Breeden Investor must allocate current wealth between today’s consumption and investment for the future Bahattin Buyuksahin, JHU, Investment

Liquidity and the CAPM Liquidity Illiquidity Premium Research supports a premium for illiquidity. Amihud and Mendelson Acharya and Pedersen Bahattin Buyuksahin, JHU, Investment

Figure 9.5 The Relationship Between Illiquidity and Average Returns Bahattin Buyuksahin, JHU, Investment

Three Elements of Liquidity 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: Bahattin Buyuksahin, JHU, Investment

CAPM: Examples of Practical Problems 1 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 2 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 3 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 4 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 5 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 6 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 7 Bahattin Buyuksahin, JHU Investments 1/16/2010

CAPM: Examples of Practical Problems 8 Bahattin Buyuksahin, JHU Investments 1/16/2010

Index model vs. CAPM Risk CAPM (theoretical, unobservable portfolio) Index model (observable, “proxy” portfolio) Bahattin Buyuksahin, JHU Investments 1/16/2010

Index model vs. CAPM 2 Beta Relationship CAPM (no expected excess return for any security) Index model (average realized alpha is 0) Fig 10.3 Bahattin Buyuksahin, JHU Investments 1/16/2010

Market Model Idea Equivalence use realized excess returns CAPM + Market model = Index model Bahattin Buyuksahin, JHU Investments 1/16/2010

Summary CAPM Factor model Index model Market model Bahattin Buyuksahin, JHU Investments 1/16/2010

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

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

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 Bahattin Buyuksahin, JHU, Investment

Multifactor Models 1 Necessity Solution CAPM not practical Index model unique factor is unsatisfactory example: Table 10.2 (very small R2) Solution multiple factors Bahattin Buyuksahin, JHU Investments 1/16/2010

Multi-factor Models 2 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 Bahattin Buyuksahin, JHU Investments 1/16/2010

Multi-factor Models 3 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 between firms with low vs. high BTM ratio proxy for bankruptcy risk? Bahattin Buyuksahin, JHU Investments 1/16/2010

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

Multifactor Model Equation ri = E(ri) + GDP GDP + IR IR + ei ri = Return for security I GDP= Factor sensitivity for GDP IR = Factor sensitivity for Interest Rate ei = Firm specific events Bahattin Buyuksahin, JHU, Investment

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 Bahattin Buyuksahin, JHU, Investment

Arbitrage Pricing Theory (APT) Nature of arbitrage APT well-diversified portfolios individual assets APT vs. CAPM APT vs. Index models single factor multi-factor Bahattin Buyuksahin, JHU Investments 1/16/2010

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

APT & Well-Diversified Portfolios rP = E (rP) + bPF + eP F = some factor For a well-diversified portfolio: eP approaches zero Similar to CAPM, Bahattin Buyuksahin, JHU, Investment

Figure 10.1 Returns as a Function of the Systematic Factor Bahattin Buyuksahin, JHU, Investment

Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity Bahattin Buyuksahin, JHU, Investment

Figure 10.3 An Arbitrage Opportunity Bahattin Buyuksahin, JHU, Investment

Figure 10.4 The Security Market Line Bahattin Buyuksahin, JHU, Investment

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 Bahattin Buyuksahin, JHU, Investment

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

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

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

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

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