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)
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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
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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
?
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continued
? Alternative Hypothesis:
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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?
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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:
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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:
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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
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Regression in action
? Set up a regression
?---the dependent variable:
? ---the independent variable:
? ---add noise.
? Feed beta to the regression package:
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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?