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FINANCE 10. Capital Asset Pricing Model

FINANCE 10. Capital Asset Pricing Model. Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007. Capital asset pricing model (CAPM). Sharpe (1964) Lintner (1965) Assumptions Perfect capital markets Homogeneous expectations

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FINANCE 10. Capital Asset Pricing Model

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  1. FINANCE10. Capital Asset Pricing Model Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007

  2. Capital asset pricing model (CAPM) • Sharpe (1964) Lintner (1965) • Assumptions • Perfect capital markets • Homogeneous expectations • Main conclusions: Everyone picks the same optimal portfolio • Main implications: • 1. M is the market portfolio : a market value weighted portfolio of all stocks • 2. The risk of a security is the beta of the security: • Beta measures the sensitivity of the return of an individual security to the return of the market portfolio • The average beta across all securities, weighted by the proportion of each security's market value to that of the market is 1 MBA 2007 CAPM

  3. Risk premium and beta • 3. The expected return on a security is positively related to its beta • Capital-Asset Pricing Model (CAPM) : • The expected return on a security equals: the risk-free rate plus the excess market return (the market risk premium) times Beta of the security MBA 2007 CAPM

  4. CAPM - Illustration Expected Return 1 Beta MBA 2007 CAPM

  5. CAPM - Example • Assume: Risk-free rate = 6% Market risk premium = 8.5% • Beta Expected Return (%) • American Express 1.5 18.75 • BankAmerica 1.4 17.9 • Chrysler 1.4 17.9 • Digital Equipement 1.1 15.35 • Walt Disney 0.9 13.65 • Du Pont 1.0 14.5 • AT&T 0.76 12.46 • General Mills 0.5 10.25 • Gillette 0.6 11.1 • Southern California Edison 0.5 10.25 • Gold Bullion -0.07 5.40 MBA 2007 CAPM

  6. Measuring the risk of an individual asset • The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification. • The standard deviation is not an correct measure for the risk of an individual security in a portfolio. • The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification. • Remember: the optimal portfolio is the market portfolio. • The risk of an individual asset is measured by beta. • The definition of beta is: MBA 2007 CAPM

  7. Beta • Several interpretations of beta are possible: • (1) Beta is the responsiveness coefficient of Rito the market • (2) Beta is the relative contribution of stock i to the variance of the market portfolio • (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified MBA 2007 CAPM

  8. Beta as a slope MBA 2007 CAPM

  9. A measure of systematic risk : beta • Consider the following linear model • RtRealized return on asecurity during period t • Aconstant :areturn that the stock will realize in any period • RMtRealized return on the market as awhole during period t • Ameasure of the response of the return on the security to thereturn on the market • utAreturn specific to the security for period t(idosyncratic returnor unsystematic return)- arandom variable with mean 0 • Partition of yearly return into: • Market related part ßRMt • Company specific part a+ut MBA 2007 CAPM

  10. Measuring Beta • Data: past returns for the security and for the market • Do linear regression : slope of regression = estimated beta MBA 2007 CAPM

  11. Beta and the decomposition of the variance • The variance of the market portfolio can be expressed as: • To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio MBA 2007 CAPM

  12. Inside beta • Remember the relationship between the correlation coefficient and the covariance: • Beta can be written as: • Two determinants of beta • the correlation of the security return with the market • the volatility of the security relative to the volatility of the market MBA 2007 CAPM

  13. Properties of beta • Two importants properties of beta to remember • (1) The weighted average beta across all securities is 1 • (2) The beta of a portfolio is the weighted average beta of the securities MBA 2007 CAPM

  14. Beta of portfolios: examples Sources of funds: Equity = €100 (Beta = ?) Asset allocation: Bonds = €60 (Beta = 0)Stocks = €40 (Beta = 1) Example 1 MBA 2007 CAPM

  15. Home made leverage Sources of funds: Debt = €50 (Beta = 0) Asset allocation: Stocks = €150 (Beta = 1) Example 2 Equity = €100 (Beta = ?) MBA 2007 CAPM

  16. Arbitrage Pricing Model Professeur André Farber

  17. Market Model • Consider one factor model for stock returns: • Rj = realized return on stock j • = expected return on stock j • F = factor – a random variable E(F) = 0 • εj = unexpected return on stock j – a random variable • E(εj) = 0 Mean 0 • cov(εj ,F) = 0 Uncorrelated with common factor • cov(εj ,εk) = 0 Not correlated with other stocks MBA 2007 CAPM

  18. Diversification • Suppose there exist many stocks with the same βj. • Build a diversified portfolio of such stocks. • The only remaining source of risk is the common factor. MBA 2007 CAPM

  19. Created riskless portfolio • Combine two diversified portfolio i and j. • Weights: xiand xjwith xi+xj =1 • Return: • Eliminate the impact of common factor  riskless portfolio • Solution: MBA 2007 CAPM

  20. Equilibrium • No arbitrage condition: • The expected return on a riskless portfolio is equal to the risk-free rate. At equilibrium: MBA 2007 CAPM

  21. Risk – expected return relation Linear relation between expected return and beta For market portfolio, β= 1 Back to CAPM formula: MBA 2007 CAPM

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