Chapter 14 The CAPM ---Applications and tests Fan Longzhen Predictions and applications ? CAPM: in market equilibrium, investors are only rewarded for bearing the market risk; ? APT: in the absence of arbitrage, investors are only rewarded for bearing the factor risk; ? Applications: ? ---professional portfolio managers: evaluating security returns and fund performance ? ---corporate manager: capital budgeting decisions. Early tests of CAPM ? Cross-sectional test of the model: ? Douglas (1969); ? Miller and Scholes (1972); ? Black, Jensen and Scholes (1972); ? Fama and Macbeth (1973) fm f iii fmifi RR R nieR RRERRE ? =++= ?+= = = ? 1 ? 0 10 ? ? ,...,2,1, ? )][()( γ γ βγγ β continued ? Douglas (1969) ? Adds own-variance to regression significant; ? Linter adds to regression significant; ? Miller and Scholes (1972) ? Measurement error in ‘s; ? Correlation between measurement error and ? Skewness of returns . ? Black, Jensen, and Scholes (1972) ? Time-series test ? Use portfolio to maximize dispersion of beta’s ? Low stocks positive ? High stocks negative )(? 2 i eσ i β ? )(? 2 i eσ 0 )( ? = +?+=? i itfmtiifit eRRRR α βα β ? s i '?α β ? s i '?α Hypothesis testing ? Definition of size and power ? H true H false ? Accept correct Type II error ? Reject type I error correct ? Size=Pr(Type I error); ? Power=1-Pr(type II error); ? Tradeoff between size and power; ? Fix size, find most powerful test. CAPM test ? CAPM holds ? ?H: ftmtmtftitit itmtiiit RRXRRX eXX ?≡?≡ ++= , βα 0= i α )1(] ? [ 2 22 m me i T VVar σ μσ α α +=≡ ),0(~ ? α α VN i Some numbers for monthly U.s. data,1985-1989 ? S&P500 T-bills: T=60, ? What is ? ? market model ? ? for typical NYSE stock ? for typical NYSE stock 2 2 01744.0)0465.01( 60 ? 051.0?,011.0? e e mm V σ σ σμ α =+= == 2 e σ itmtiiit eRR ++= βα 2222 emii σσβσ += 22 10.0? = i σ 25.0=ρ 0075.0)10.0)(25.01( ? 22 =?= e σ 01144.0 ? = α V continued ? Alternative Hypothesis: ? Does change? 01.0: = i K α 12% Annually! α V Testability of CAPM ? The wide acceptance of the CAPM and APT makes it all more important to test their predictions empirically. ? How does a product of abstract reasoning hold in reality? ? Unfortunately, the predictions of the CAPM and APT are hard to test empirically ? ---neither the market portfolio in CAPM nor the risk factor in APT is observable; ? ---expected returns are unobservable, and could be time- varying; ? ---volatility is not directly observable, and is time-varying. An ideal test of the CAPM ? In an idea situation, we have the following input: ? 1. Risk-free borrowing/lending rate ; ? 2. Expected returns on the market and on the risky asset ; ? 3. The exposure to market risk ; ? These input allow us to examine the relation between reward ? and risk : ? 1. More risk, more reward? ? 2. Do they line up? ? 3. What is the reward for a risk exposure of 1? ? 4. Zero risk, zero reward? f R )( M RE )( i RE )var(/),cov( MiM i RRR=β ))(( f i RRE ? i β A linear relation between risk and reward ))(()( f M if i RRERRE ?=? β f R 1=β fM RRE ?)( Some practical compromise ? The market portfolio is unobservable: ? Use a proxy, e.g. the S&P 500 index; ? Expected returns are unobservable:use sample average: ? ? Unobservable risk exposure ? Use sample estimates: M R )(),( iM RERE ∑∑ == t i ti t M tM R N mR N m 1 ?; 1 ? )var(/),cov( MiM i RRR=β MiMi VC ? / ?? , =β () ∑∑ ??=?= t i i tM M tiM t M M tM mRmR N CmR N V ) ? )( ? ( 1 ? , ? 1 ? , 2 Testing the linear relation ? Pick a proxy for the market portfolio, and record N monthly returns: ? For the same sample period, collect a sample of I firms, each with N monthly returns: ? Construct sample estimates ? For , test the linear relation: ? NtR M t ,...,2,1: = NtandIiR i t ,...,2,1,...,2,1: == iMi mm β ? , ? , ? Ii ,...,2,1= ifi Rm βγγ ? ? 10 +=? Implication of the CAPM and testing ? Implication of CAPM: with ? : zero exposure, zero reward; ? : one unit of exposure, the same reward as the market. ? With 43 industrial portfolios, the test tells us that this relation does not hold exactly. ? One possibility: our measures of the expected returns are contaminated by noises that are unrelated to the beta’s; ? What we still would like to know: ? ---on average, is reward related to risk at all? Or not? ? ---On average, does zero risk results in zero reward? ? Or not? ? ---on average, does one unit of risk exposure pay market return? ?or not ifi Rm βγγ ? ? 10 +=? 0 0 =γ fM Rm ?= ? 1 γ 0 1 =γ 0 0 =γ %9.5? 1 =?= fM Rmγ Regression in action ? Set up a regression ?---the dependent variable: ? ---the independent variable: ? ---add noise. ? Feed beta to the regression package: iii eXY ++= 10 γγ fii RmY ?= ? ii X β ? = estimate tabdard error t-statistic R-square gamm0 6% 1.80% 3.5 0.02% gamm1 0.17% 1.70% 0.1 A summary of the CAPM tests ? In general, the test results depend on the sample data, sample periods, statistical approaches, proxy for the market portfolio, ect. But the following findings remain robust: ? the relation between risk and reward is much flatter than that predicted by the CAPM; ? The risk beta can not even to begin to explain the cross-sectional variation in the expected returns; ? Contrary to the prediction of the CAPM, the intercept is significant different from zero. Some possible explanations ? 1. Is the stock market index a good proxy for the market portfolio? ? Only 1/3 non-governmental tangible assets are owned by the corporate sector; ? Among the corporate assets, only 1/3 are financed by equity ? What are about intangible assets, like human capital; ? What about international markets? ? 2. Measurement error in beta ? Except for the market portfolio, we never observe the true beta; ? To test CAPM, we use estimates for beta, which are measured with errors ? 3. Measurement error in expected returns ? We use sample mean as the unobservable expected returns; ? There are noises in our estimates; ? Is the noises are correlated, then we have a statistical problem(error-in- variables) ? 4. Borrowing restrictions ? Fisher black showed that borrowing restrictions might cause low-beta stocks to have higher expected returns than the CAPM predicts. Going beyond the CAPM ? Is beta a good measure of risk exposure? What about the risk associated with negative skewness? ? Could there be other risk factors? ? Time-varying volatility, ? Time-varying expected returns, ? Time –varying risk aversion, ? And time-varying beta?