Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory McGraw Hill Irwin Copyright 2010 by The McGraw Hill Companies Inc All rights reserved 7 1 The Capital Asset Pricing Model 7 2 Simplifying Assumptions Individual investors are price takers Single period investment horizon Investments are limited to traded financial assets No taxes and no transaction costs 7 3 Simplifying Assumptions cont Information is costless and available to all investors Investors are rational mean variance optimizers Homogeneous expectations 7 4 Resulting Equilibrium Conditions All investors will hold the same portfolio for risky assets the market portfolio Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value Market price of risk or return per unit of risk depends on the average risk aversion of all market participants 7 5 Capital Market Line E r M The value weighted Market Portfolio of all risky assets E rM rf M m CML Efficient Frontier 7 6 Known Tangency Portfolio of CML Equilibrium conditions All investors will hold the same portfolio for risky assets the market portfolio Capital Market Line E rE r Market M The value weighted Market M The value weighted Portfolio of all risky assets Portfolio of all risky assets E rE rMM rrff MM CMLCML mm Pricing of individual securities is therefore related to the risk that individual securities have when they are included in the market portfolio 7 7 Slope and Market Risk Premium M rf Market portfolio Risk free rate Excess return on the market portfolio Optimal Market price of risk Slope of the CML E rM rf E rM rf MM Capital Market Line E rE r Market M The value weighted Market M The value weighted Portfolio of all risky assets Portfolio of all risky assets MM CMLCML E rE rMM rrff mm 7 8 Expected Return and Risk on Individual Securities The risk premium on individual securities is a function of the individual security s contribution to the risk of THE market portfolio What type of individual security risk will matter systematic or unsystematic risk An individual security s total risk 2 i can be partitioned into systematic and unsystematic risk i i 2 M market portfolio of all risky securities 2 2 ei 2 M 7 9 Expected Return and Risk on Individual Securities Individual security s contribution to the risk of the market portfolio is a function of the of the stock s returns with the market portfolio s returns and is measured by BETA covariance With respect to an individual security systematic risk can be measured by i COV ri rM 2 M 7 10 Individual Stocks Security Market Line E r E r Slope SML E rM rf M price of risk for market Equation of the SML CAPM E ri rf i E rM rf SMLSML E rE rMM rrff 1 0 1 0 MM 7 11 Sample Calculations for SML Equation of the SML E ri rf i E rM rf E rm rf 08 rf 03 Return per unit of systematic risk 8 the return due to the TVM 3 x 1 25 E rx 0 03 1 25 08 13 or 13 0 03 0 6 0 08 0 078 or 7 8 y 6 E ry If 1 If 0 7 12 Graph of Sample Calculations E r E r RRxx 13 13 RRMM 11 11 RRyy 7 8 7 8 3 3 SMLSML 08 08 If the CAPM is correct only risk matters in determining the risk premium for a given slope of the SML 6 6 yy 1 01 0 MM 1 251 25 xx 7 13 Disequilibrium Example E rE r 15 15 RRmm 11 11 rrff 3 3 SMLSML 13 Suppose a security with a of is offering an expected 1 25 15 return of 1 01 0 1 251 25 According to the SML the E r 13 should be E r 0 03 1 25 08 13 Is the security under or overpriced Underpriced It is offering too high of a rate of return for its level of risk The difference between the return required for the risk level as measured by the CAPM in this case and the actual return is called the stock s alpha denoted by What is the in this case 2 Positive is good negative is bad gives the buyer a abnormal return 7 14 More on Alpha and Beta 14 1 5 5 E rM S rf Required return rf S E rM rf 5 1 5 14 5 18 5 If you believe the stock will actually provide a return of 17 what is the implied alpha 17 18 5 1 5 A stock with a negative alpha plots below the SML gives the buyer a negative abnormal return 7 15 Portfolio Betas Wi i P If you put half your money in a stock with a beta of and 30 of your money in a stock with a beta of and the rest in T bills what is the portfolio beta 1 5 0 9 P 0 50 1 5 0 30 0 9 0 20 0 1 02 All portfolio beta expected return combinations should also fall on the SML All E ri rf i should be the same for all stocks 7 16 Measuring Beta Concept Method We need to estimate the relationship between the security and the Market portfolio Can calculate the Security Characteristic Line or SCL using historical time series excess returns of the security and unfortunately a proxy for the Market portfolio 7 17 7 2 The CAPM and Index Models 7 18 Security Characteristic Line SCL Excess Returns i Dispersion of the points around the line measures unsystematic risk The statistic is called e SCL Slope Excess returns on market index Ri i iRM ei What should equal 7 19 7 3 The CAPM and the Real World 7 20 Evaluating the CAPM The CAPM is false based on the validity of its assumptions The CAPM could still be a useful predictor of expected returns That is an empirical question Huge measurability problems because the market portfolio is unobservable Conclusion As a theory the CAPM is untestable 7 21 Evaluating the CAPM The we learn from the CAPM are still principles entirely valid Investors should diversify Systematic risk is the risk that matters A well diversified risky portfolio can be suitable for a wide range of investors The risky portfolio would have to be adjusted for tax and liquidity differences Differences in risk tolerances can be handled by changing the asset allocation decisions in the complete portfolio Even if the CAPM is false the markets can still be efficient 7 22 Arbitrage Pricing Model The result For a well diversified portfolio Rp pRS Excess returns rp i rf p rS i rf and for an individual security rp i rf p rS i rf ei RS is the excess return on a portfolio with a beta of 1 relative to systematic factor S Advantage of the APT over the CAPM No particular role for the Market Portfolio which can t be measured anyway Easily extended …