Multiple Linear Regression 75
Let’s interpret the results. The t-statistics of the betas are statistically
significant for all levels of significance. The regression results suggest that
relative to the S&P 500, the average large-cap mutual fund makes statisti-
cally significant bets against the market, against value, and against size. The
adjusted R^2 is 0.63. This means that 63% of the variation in the average
large-cap mutual fund’s returns is explained by the regression model. The
intercept term, a, is –0.007 (–7 basis points) and is interpreted as the aver-
age active return after controlling for risk (i.e., net of market, value, and
size). Statistically, the intercept term is not significant. So, the average active
return is indistinguishable from zero. Given that the return indexes con-
structed by Lipper are net of fees and expenses, the conclusion of this simple
regression model is that the average large-cap mutual funds covers its costs
on a risk-adjusted basis.
Tests of the Capital Asset Pricing Model
The Capital Asset Pricing Model (CAPM) is an equilibrium model of asset
pricing. While portfolio managers do not devote time to testing the validity
of this model since few have to be convinced of its limitation, there has been
more than 40 years of empirical testing of the validity of this model and the
primary tool that has been used is regression analysis. While there have been
extensions of the CAPM first developed by William Sharpe in 1964, we will
only discuss the tests of the original model.
Based on the above assumptions, the CAPM is
E(Ri) − Rf = βi[E(RM) − Rf] (3.19)
where E(Ri)=expected return for asset i
Rf=risk-free rate
E(RM)=expected return for the market portfolio
βi=the index of systematic risk of asset iThe index of systematic risk of asset i, βi, popularly referred to as beta,
is the degree to which an asset covaries with the market portfolio and for
this reason is referred to as the asset’s systematic risk. More specifically, sys-
tematic risk is the portion of an asset’s variability that can be attributed to a
common factor. Systematic risk is the risk that results from general market
and economic conditions that cannot be diversified away. The portion of an
asset’s variability that can be diversified away is the risk that is unique to an
asset. This risk is called nonsystematic risk, diversifiable risk, unique risk,
residual risk, or company-specific risk. We calculated the beta for individual
securities in the previous chapter.