Chapter 14 The CAPM -Applications and testsFan LongzhenPredictions and applicationsCAPM: 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 CAPMCross-sectional test of the model: Douglas (1969); Miller and Scholes (1972); Black, Jensen and Scholes (1972); Fama and Macbeth (1973)fmfiiifmifiRRRnieRRRERRE=+=+=?1?010,.,2 , 1,)()(continuedDouglas (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 betas Low stocks positive High stocks negative )(2 iei)(2 ie0)(? =+=iitfmtiifiteRRRRsisiHypothesis 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: ftmtmtftitititmtiiit RRXRRXeXX+=,0=i)1 (222mme iTVVar+=), 0(VNiSome numbers for monthly U.s. data,1985-1989 S -expected returns are unobservable, and could be time- varying; -volatility is not directly observable, and is time-varying.An ideal test of the CAPMIn 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?fR)(MRE)(iRE)var(/ ),cov(MiM iRRR=)(fiRREiA linear relation between risk and reward)()(fM ifiRRERRE=fR1=fMRRE)(Some practical compromiseThe market portfolio is unobservable:Use a proxy, e.g. the S Expected returns are unobservable:use sample average: Unobservable risk exposureUse sample estimates:MR)(),(iMRERE=ti ti tM tMRNmRNm1;1)var(/ ),cov(MiM iRRR=MiMiVC/ ,=()=tii tMM tiM tMM tMmRmRNCmRNV)(1,1,2Testing the linear relationPick 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: NtRM t,.,2 , 1: =NtandIiRit,.,2 , 1,.,2 , 1:=iMimm,Ii,.,2 , 1=ifiRm10+=Implication of the CAPM and testingImplication 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 betas; 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 ifiRm10+= 00=fMRm= 101= 00=%9 . 51=fMRmRegression in actionSet up a regression -the dependent variable: -the independent variable: -add noise. Feed beta to the regression package:iiieXY+=10fiiRmY= iiX=estimate tabdard errort-statisticR-square gamm06%1.80%3.50.02% gamm10.17%1.70%0.1A summary of the CAPM testsIn 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 explanations1. 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 e