Capital Asset Pricing Model CAPM II: The Empirics

RECAP: CAPM & SML SML β i = COV(R i, R M )/Var(R M ) slope = [E(R M ) - R f ] = Eqm. Price of risk E(R i ) E(R M ) RfRf β M = 1 E(return) = Risk-free rate of return + Risk premium specific to asset i = R f + (Market price of risk)x(quantity of risk of asset i) E(R i ) = R f + [E(R M )-R f ] x [COV(R i, R M )/Var(R M )] E(R i ) = R f + [E(R M )-R f ] x β i

[1] What are the predictions ? [a] CAPM says: more risk, more rewards [b] HOWEVER, “reward-able” risk ≠ asset total risk, but = systematic risk (beta) [c] We ONLY need Beta to predict returns [d] return LINEARLY depends on Beta

[2] Testable ? E(R i ) = R f + [E(R M )-R f ] x [COV(R i, R M )/Var(R M )] E(R i ) = R f + [E(R M )-R f ] x β i Ideally, we need the following inputs: [a] Risk-free borrowing/lending rate {R f } [b] Expected return on the market {E(R M )} [c] The exposure to market risk {β i = cov(R i,R M )/var(R M )}

[2] Testable ? E(R i ) = R f + [E(R M )-R f ] x [COV(R i, R M )/Var(R M )] E(R i ) = R f + [E(R M )-R f ] x β i In reality, we make compromises: [a] short-term T-bill (not entirely risk-free) {R f } [b] Proxy of market-portfolio (not the true market) {E(R M )} [c] Historical beta {β i = cov(R i,R M )/var(R M )}

[2] Testable ? Problem 1: What is the market portfolio? We never truly observe the entire market. We use stock market index to proxy market, but: [i] Only 1/3 non-governmental tangible assets are owned by corporate sector. Among them, only 1/3 is financed by equity. [ii] What about intangible assets, like human capital? [iii] Which market index to use? Or how many to use? [iv] What about international market?

[2] Testable ? Problem 2: Without a valid market proxy, do we really observe the true beta? [i] suggesting beta is destined to be estimated with measurement errors. [ii] how would such measurement errors bias our estimation?

[2] Testable ? Problem 3: Borrowing restrictions. Short-selling restrictions. Problem 4: Expected return measurement. [i] are historical returns good proxies for future expected returns? Ex Ante VS Ex Post

[3] Regression E(R i ) = R f + [E(R M )-R f ] x [COV(R i, R M )/Var(R M )] E(R i ) = R f + [E(R M )-R f ] x β i E(R i ) – R f = [E(R M )-R f ] x β i With our compromises, we test : [R i – R f ] = [R M -R f ] x β i Using the following regression equation : [R it – R ft ] = γ 0 + γ 1 β i + ε it In words, Excess return of asset i at time t over risk-free rate is a linear function of beta plus an error (ε). Cross-sectional Regressions to be performed!!!

[3] Regression [R it – R ft ] = γ 0 + γ 1 β i + ε it CAPM predicts: [a] γ 0 should NOT be significantly different from zero. [b] γ 1 = (R Mt - R ft ) [c] Over long-period of time γ 1 > 0 [d] β should be the only factor that explains the return [e] Linearity

[4] Generally agreed results [R it – R ft ] = γ 0 + γ 1 β i + ε it [a] γ 0 > 0 [b] γ 1 < (R Mt - R ft ) [c] Over long-period of time, we have γ 1 > 0 [d] β may not be the ONLY factor that explains the return (firm size, p/e ratio, dividend yield, seasonality) [e] Linearity holds, β 2 & unsystematic risk become insignificant under the presence of β.

CAPM Predicts Actual γ 1 = (R Mt - R ft ) γ 0 = 0 [R it – R ft ] βiβi [4] Generally agreed results

Roll’s Critique Message: We aren’t really testing CAPM. Joint Hypothesis testing: 1) “CAPM is valid” and 2) “Market portfolio is ex post efficient” Argument: Quote from Fama & French (2004) “Market portfolio at the heart of the model is theoretically and empirically elusive. It is not theoretically clear which assets (e.g., human capital) can legitimately be excluded from the market portfolio, and data availability substantially limits the assets that are included. As a result, tests of CAPM are forced to use compromised proxies for market portfolio, in effect testing whether the proxies are on the min-variance frontier.” Viewpoint: essentially, implications from CAPM aren’t independently testable. We do not have the benchmark market to base on. Every implications are tested jointly with whether the proxy is ex post efficient or not.