Regressions

2.3 The Capital Asset Pricing Model (CAPM) .1 Introduction.1Introduction

2.3.4 Empirical Tests

2.3.4.2 Regressions

There are two different simple linear regressions which are used to test Eq. (2.37):

• Thecross-sectional regressionstest the CAPM across assets with different betas.

The returns of different assets are regressed over the betas.

• Thetime-series regressionstest the CAPM equations for each individual asset over time. The excess portfolio returns in the subperiods are regressed over the respective excess return of the market portfolio.

Cross-Sectional Regression

The cross-sectional regression is the main tool used to test Eq. (2.37) over a period Œ0; T ofT years. CAPM tests do not exactly test Eq. (2.37) because it includes expected returnswhich are not measureable. Instead, the question is if the relation holds forrealized returns, i.e., if the relation has been correct in the past. The cross- sectional regression tests the CAPM equation by regressing the (arithmetic) average annual returnrAof an assetAover beta.

Figure2.10shows a typical cross-sectional regression. For every assetA, the data point (ˇA; rA) is plotted withˇAits beta andrAas its (arithmetic) average annual return. We then get the regression line which has the form

39These are from Fama and French (2004, p. 30), and Fama and MacBeth (1973, p. 610 and p. 613).

40Literally, the beta premium is the premium per unit of beta. The CAPM implies that the beta premium is the excess market return, i.e., the difference between the expected return on the market portfolio and the risk-free rate. But this equality is equivalent to the fourth hypothesis, that zero- beta assets expect to earn the risk-free rate. The positive beta premiumdoes not test the equality with the excess market return, only the positivity.

Fig. 2.10 Cross-sectional regression. Annualized returns of asset returns are regressed over their respective betas.Source: Own, for illustrative purposes only

1 2 3 4 5 6 7 8 9 10

−1 0.5 1.0 1.5 2.0

Annual returnrA

BetaβA

a 1

b

Regression line

rADaCbˇA: (2.38)

This diagram looks very similar to Fig.2.9where the expected returns of assets are plotted against their betas. They lie on the security market line (SML) which is the linepredictedby the CAPM, whereas the regression line is theempirical one we observe, based on historical data. The empirical test of the CAPM examines if both lines are equal, or in other words, if the data fits the predicted line, the SML. The interceptain Eq. (2.38) and the slope coefficientb(the beta premium) correspond to the risk-free raterrf in Eq. (2.37) and the market risk premiumR^Mkt rrf, respectively. With this regression, we can check hypotheses (C3) and (C4) from the list on page123:

• Positive beta premium: The beta premiumbis positive.

• Risk-free return on zero-beta assets: The interceptaequals the risk-free raterrf. Given the market data, we still need the betas ˇA to plot a diagram like Fig.2.10and to perform our regression analysis. We get the betas from a time-series regression.

Time-Series Regression

The time-series regression tests the CAPM equation for each individual asset separately onN equidistant subperiods of a time horizonŒ0; T . The excess asset return in each subperiodkis regressed over the excess market return from the same subperiod. Figure2.11shows a typical time-series regression. Given an assetA, for every subperiodk, the data pointrQMkt^k ;rQ_A^k is plotted whererQMkt^k DrMkt^k r_rf^k is the excess return of the market portfolio in subperiodk, i.e., the difference between the market return and the return on a risk-free asset, andrQ_A^k D r_A^kr_rf^k is the excess return of the assetAin the subperiodk.

We get a regression line of the form

rQ_A^subperiod D aCb QrMkt^subperiod (2.39)

Fig. 2.11 Time-series regression. Excess returns of an assetAin subperiodiare regressed over the excess market return.Source: Own, for illustrative purposes only

4 8 12 16 20 24

−4

−8

−12

4 8 12 16 20 24 28

−4

−8

−12

−16

−20

Excess asset return ˜r^kA

Excess market return ˜r^kMkt

a

Regression line

1%

b%

wheresubperiod can be quarterly, monthly, weekly, daily, etc. and refers to the length of the subperiods. For example, if we choose monthly subperiods, then we have a regression on monthly returns:

rQ_A^monthly

„ ƒ‚ …

monthly excess return

DaCb QrMkt^monthly: (2.40)

Let us recall the CAPM equation using monthly returns (Eq. (2.36)):

EŒR_A^monthly D r_rf^monthlyCˇA.EŒRMkt^monthlyr_rf^monthly/:

Rewrite this as

EŒR^monthlyPf r_rf^monthly

„ ƒ‚ …

expected monthly excess return

D ˇA.EŒR^monthlyMkt r_rf^monthly/; (2.41)

and we can see that in the regression equation—Eq. (2.40)—ashould be zero and bis the betaˇA. For any assetA, we call the interceptafrom the above regression thealphaofA. In case of monthly subperiods, we write˛_A^monthly, similarly to other subperiods. The alpha represents the incremental rate of return which exceeds the theoretical rate of return implied by the CAPM. It is introduced in Jensen (1967) as the intercept of the linear time-series regression and as a performance measure to evaluate a portfolio manager who invests in portfolioA.⁴¹

41Jensen (1967, p. 8).

Regarding the CAPM hypotheses on page123, the time-series regression can be used to test hypothesis (C4):

Risk-free return on zero-beta assets: The alphas of all assets are zero.

Note that for our regressions, we do not use the real market portfolioMktbecause it is hardly observable. Instead, we use a proxy like a stock market index to mimic the market return.