VIEWS: 110 PAGES: 44 CATEGORY: Technology POSTED ON: 7/9/2010
Capital Asset Pricing Model and Arbitrage Theory Riccardo Colacito Capital Asset Pricing Model (CAPM) • 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 Foundations of Financial Markets 2 Link to Factor Models • What risk should be priced? – Idiosyncratic risk: no – Aggregate risk: yes • Only aggregate/macro risk commands a premium Foundations of Financial Markets 3 Why? Because: 1. idiosyncratic risk can be diversified away 2. Macro risk affects all assets and cannot be diversified Remember our example using factor models? Foundations of Financial Markets 4 Example: two assets Two assets: R1 1 1 Rm e1 and R2 2 2 Rm e2 Assume: 1 2 1 Var(e1 ) Var(e2 ) 1 cov(e1 , e2 ) 0 Invest equal portfolio shares in the two assets Foundations of Financial Markets 5 What is the portfolio variance? Var ( w1 R1 w2 R2 ) Var ( Rm ) 1 / 2 Systematic Idiosyncratic Risk Risk Foundations of Financial Markets 6 Example: three assets Two assets : Ri i i Rm ei , i 1,2,3 Assume : 1 2 3 1 Var(e1 ) Var(e2 ) Var(e3 ) 1 cov(e1 , e2 ) cov(e1 , e3 ) cov(e2 , e3 ) 0 Invest equal portfolio shares in the three assets Foundations of Financial Markets 7 What is the portfolio variance? Var ( w1 R1 w2 R2 w3 R3 ) Var ( Rm ) 1 / 3 Systematic Idiosyncratic Risk Risk •Systematic risk: unchanged •Idiosyncratic risk: decreased •Can you guess what would happen if we had an infinite number of assets? Foundations of Financial Markets 8 Infinite assets • For a well diversified portfolio 2 Var wi Ri i wi Var Rm , i i or R p p m • That is: we got rid of any idiosyncratic shock and we are left only with systematic risk Foundations of Financial Markets 9 What lesson did we learn? • The only source of risk that we are entitled to ask a compensation for is aggregate risk. • Idiosyncratic risk does not entitle to any compensation because it can be diversified away. Foundations of Financial Markets 10 Risk compensation on Any asset entitles to a compensati proportional to its contribution to aggregate risk : E ( R1 rf ) E ( R2 rf ) E ( Rm rf ) ... 1 m 2 m m or, equivalently E ( R1 ) rf 1 E ( Rm rf ) E ( R2 ) rf 2 E ( Rm rf ) ... Foundations of Financial Markets 11 The fundamental equation of the Capital Asset Pricing Model (CAPM) Any asset entitles to a risk premium that is proportional to the risk premium of the market portfolio. The of the asset is the coefficient of proportionality. E ( Ri ) rf i E ( Rm rf ), i 1,2,... Foundations of Financial Markets 12 Questions • What assumptions need to be satisfied for the CAPM to hold? • Why the market portfolio? What is it anyway? • Why b? What is b anyway? • What are the empirical predictions of the CAPM? • Can we measure it from the data? • If so, is it empirically accepted? Foundations of Financial Markets 13 Assumptions • Individual investors are price takers • Single-period investment horizon • Investments are limited to traded financial assets • No taxes, and transaction costs • Lending and borrowing can be done at same rate • Information is costless and available to all investors • Investors are rational mean-variance optimizers • Homogeneous expectations Foundations of Financial Markets 14 Assumptions… in other words! • Everybody knows how to solve the problem that we discussed during the last three classes • Everybody is forecasting returns, variances and correlations in the same way • … if this is the case we are all going to end up with same optimal risky portfolio! Foundations of Financial Markets 15 The Efficient Frontier and the Capital Market Line Foundations of Financial Markets 16 Some re-labeling • We call the optimal CAL, the Capital Market Line • We call the optimal risky portfolio, the Market Portfolio • Remember the separation property? Foundations of Financial Markets 17 Separation property • Aka mutual fund theorem • All investors desire the same portfolio of risky assets: the optimal risky portfolio or market portfolio. Foundations of Financial Markets 18 This answers: Why the market portfolio? • If everybody is holding this portfolio, then it is an excellent candidate to proxy for aggregate market risk. • We are now left with the question of why is the coefficient of proportionality! Foundations of Financial Markets 19 First things first: what is ? • Remember factor models Ri r f i i Rm r f ei • Remember how we computed covRm rf , Ri rf i VarRm rf Foundations of Financial Markets 20 in words • The more correlated the asset is with market the larger is. • What does it mean in terms of the CAPM? E ( Ri ) rf i E ( Rm rf ) Foundations of Financial Markets 21 and the risk premium • It means that the higher the correlation with the market, the higher the risk premium that an asset commands. • Why? Foundations of Financial Markets 22 and the risk premium: intuition Which asset is more appealing: 1. An asset whose return goes up when the market goes up 2. An asset whose return goes down when the market goes up Foundations of Financial Markets 23 Answer • Everything else equal you prefer an asset that co-varies negatively with the market, because it gives you an insurance against bad states of the world. • If an asset is perceived as a good asset because it provides an hedge against bad states, its demand will go up, increasing the price and decreasing the expected return. • How about an example? Foundations of Financial Markets 24 Example • Two assets, the market portfolio, a risk free rate. Two states of the world. R1 R2 Rm rf Boom -2 4 4 .5 Recession 4 -2 -2 .5 • Let’s compute ’s and required risk premia! Foundations of Financial Markets 25 Results • Some computations: Var(r1 ) Var(r2 ) Var(rm ) 9 cov(r1 , rm ) 9, cov(r2 , rm ) 9 1 1, 2 1, E (rm ) 1 • Hence E r1 rf 1 E rm rf .5 1 .5 0 E r2 rf 2 E rm rf .5 1 .5 1 Foundations of Financial Markets 26 What did we learn? • Asset 2 doesn’t do too much to protect us against market fluctuations: therefore it must `promise’ a higher return to convince us to buy it • Asset 1 can protect us against market fluctuations and therefore we are willing to buy it even if its return is low in expectation. Foundations of Financial Markets 27 What happens to the price of the first security? • The CAPM says that the expected return of asset 1 should be 0 • However at the current prices, asset 1 has an expected return of 1 • Looks like a great deal! • Investors will want to buy more of asset 1 • But if its current price goes up, its return will go down in expectation • When does the price increase stop? – When the expected return at the current prices equals the CAPM expected return • What happens to the current price of security 2? – Nothing! This security is price correctly! Foundations of Financial Markets 28 Empirical predictions of the CAPM • Remember the one factor model: Ri i i Rm ei E ( Ri ) i E ( Rm ) • The intercept should be equal to zero! Foundations of Financial Markets 29 The Security Market Line and Positive Alpha Stock Foundations of Financial Markets 30 Alpha • The abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as the CAPM • Can we test this prediction? Foundations of Financial Markets 31 Estimating the Index Model • Using historical data on T-bills, S&P 500 and individual securities • Regress risk premiums for individual stocks against the risk premiums for the S&P 500 • Slope is the beta for the individual stock Foundations of Financial Markets 32 Characteristic Line for GM Foundations of Financial Markets 33 Statistical analysis •In this example we cannot reject that is equal to zero •However in many cases a is significantly different from zero: what does it mean? Foundations of Financial Markets 34 Rejecting the CAPM? • Not necessarily! • We may have chosen an imprecise proxy for the market risk: after all the Market Portfolio is not directly observable. • We may be omitting some risk factors: research shows that this is possible. Foundations of Financial Markets 35 Fama-French Research • Returns are related to factors other than market returns • Size • Book value relative to market value • Three factor model better describes returns Foundations of Financial Markets 36 Why are these factors supposed to help? • Firms with high ratios of book to market value are more likely to be in financial distress • Small firms are more sensitive to changes in business conditions Foundations of Financial Markets 37 Fama-French applied • A Fama-French three factors regression for GM rGM rf GM M rm rf HMLrHML SMB rSMB eGM • where rHML differencein returns between small and large firms rSMB differencein returns between firms with a high vs low B/M Foundations of Financial Markets 38 Regression Statistics for the Single- index and FF Three-factor Model Foundations of Financial Markets 39 Resulting Equilibrium Conditions • All investors will hold the same portfolio for risky assets – market portfolio • Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value • Risk premium on an individual security is a function of its covariance with the market Foundations of Financial Markets 40 Arbitrage Pricing Theory (APT) Arbitrage - arises if an investor can construct a zero beta investment portfolio with a return greater than the risk-free rate • If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high-priced portfolio • In efficient markets, profitable arbitrage opportunities will quickly disappear Foundations of Financial Markets 41 APT and well diversified portfolios • A well diversified portfolio has no exposure to idiosyncratic risk RP rf P P Rm rf • Claim: P must equal zero. Why? Foundations of Financial Markets 42 P=0 to avoid arbitrage opportunities • Take two well-diversified portfolios Rv rf v v Rm rf , Ru rf u u Rm rf • Pick the following shares of investment u v wv , wu v u v u • The resulting portfolio is not subject to any risk and provides a non- zero return u v wv Rv wu Ru v u 0 v u v u • Hence ’s must be equal to zero Foundations of Financial Markets 43 Conclusion • The equilibrium condition is the same as the CAPM E ( R p rf ) p p E ( Rm ) r f • Only assumption needed is the absence of arbitrage opportunities • Can be extended beyond well diversified portfolios Foundations of Financial Markets 44