# Capital Asset Pricing Model Part 2: The Empirics.

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Capital Asset Pricing Model Part 2: The Empirics

RECAP: Preference Expected Return E(R p ) Standard Deviation σ(R p ) Increasing Utility Ingredients 1)5 Axioms for Expected Utility Theorem 2)Prefer more to less (Greedy) 3)Risk aversion 4)Assets jointly normally distributed

RECAP: Min-Variance opp. set Efficient frontier Min-variance opp. set Individual risky assets σ(R p ) E(R p ) - Portfolios along the efficient set/frontier are referred to as “mean-variance” efficient

RECAP: Capital Market Line (a.k.a Linear efficient set) E(R M ) RfRf σMσM M E(R p ) σ(R p ) CML Ingredients 1)Homogenous Belief 2)Unlimited Lending/borrowing

RECAP: 2-fund separation RfRf A (M) Market Portfolio B CML σMσM E(R M ) E(R p ) σ(R p ) Everyone’s U-maximizing portfolio consists of a combination of 2 assets only: Risk-free asset and the market portfolio. This is true irrespective of the difference of their risk-preferences CML Equation: E(R p ) = R f + [(E(R M )- R f )/σ M ]σ(R p )

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

Empirical Studies of CAPM Is CAPM useful? –Given many unrealistic assumptions, how good does the model fit into the reality? Think about the following questions: [1] What exactly are the predictions of the CAPM? [2] Are they testable? [3] What is a regression? [4] How to test hypothesis? What is t-test?

[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?

[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 restriction. 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. 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 efficient or not.