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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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