Validity of CAPM, Roll's Critique

This is originally posted on  Sunday, 3 February 2013 at 23:19

2011 ZA 1b 

Critically assess the validity of the CAPM in light of Roll's critique and other anomalies.

2011 ZB 2b

"The CAPM is untestable." How far do you agree and disagree with this statement? Critically evaluate this statement.

The Capital Market Line and Mean-Variance Efficient Frontier meets at the Tangency Portfolio. The Tangency Portfolio is supposed to be the optimal weighting of all the risky and risk-free assets in the market. The investor can then construct an efficient portfolio with the highest expected return (given his/her risk appetite) by adjusting weights on the Risk Free Asset and Tangency Portfolio.

However it is not possible to calculate the Tangency Portfolio because means and convariances are generally unobservable. In addition, there are too many assets in the market; there are virtually infinite permutations of portfolio combinations even if all their means, variances and covariances are known. Since calculation is not possible, the next best alternative is to develop a theory which allows the investor to reasonably deduce the efficient portfolio. The CAPM is one such theory which facilitates this deduction.

The CAPM is based on the assumptions that investors care only about the mean and variances of their portfolio's return, holding homogeneous beliefs about means and variances of all feasible portfolios and frictionless markets. It concludes that the Tangency Portfolio must be the Market Portfolio. 

According to the CAPM, every investor should optimally hold a combination of the Risk Free Asset and Market Portfolio if a Risk Free asset exists. This has a key implication: although the Tangency Portfolio cannot be calculated, the CAPM suggests that it is not necessary to choose the Tangency Portfolio for optimal investment. An investor can simply choose a Market Portfolio which mimics the returns of a Tangency Portfolio, and still attain an optimal (mean-variance efficient) level of investment. 

However, the CAPM is subjected to various criticisms and empirical tests of the CAPM has yielded several anomalies. 

Roll (1977) pointed out the unobservability of the Market Portfolio makes it inherently untestable. There are many choices of market proxies but they may not accurately mimic the returns of the Market Portfolio. Since the Market Portfolio is the centerpiece of the CAPM's conclusions, empirical tests using market proxies will not provide clear evidence that leads one to accept or reject the CAPM. 

The expected returns of all assets are linearly related to their betas when calculated with respect to the Mean-Variance Efficient (Tangency) Portfolio. Even if the CAPM is actually wrong and the actual Market Portfolio is not mean-variance efficient, the market proxy may turn out to be mean-variance efficient. The test will fail to reject the CAPM (Type II error).  

If the CAPM turns out to be true, the market proxy may fail to mimic the returns of Market Portfolio- it willl not be mean-variance efficient. The test using the market proxy will therefore incorrectly reject the CAPM (Type I error).  

Therefore empirical tests which purports to test the "validity of the CAPM" are actually testing whether market proxies are mean-variance efficient. The CAPM's validity remains unconfirmed, though it is taken for granted that empirical fit implies theoretical validity (line of thought: if market proxies are mean-variance efficient, it implies that CAPM is true). 

However, the CAPM implies a one-factor model with the factor being the excess market return. The CAPM is tested using a 2 Stage Least Squares approach. In the first stage, we run a time series regression of each asset's excess returns on the market's excess return. The CAPM implies that the intercept term of this regression should be approximately zero for all assets. If we find many assets with the inctercept term not equal to zero, we can infer that the CAPM is not an accurate theory. 

In the second stage, we run a cross-sectional regression across all assets based on their estimated historical beta values from the first stage. The CAPM implies a linear relationship between an asset's excess return and its average historical beta. By regressing across various assets' excess returns against their corresponding average historical beta, we can test for the linearity implied by CAPM. The CAPM implies that the intercept term will be zero and coefficient will be the average market premium (the market's excess return).  

The data are generally not supportive of CAPM. Plotting the CAPM's predicted industry returns (multiple of average historical beta and average market premium) against their average historical betas will show the predicted relationship. Plotting the actual industry returns against their average historical betas will show the actual relationship. 

The assets with higher beta tend to have lower actual returns than the CAPM's predictions while the assets with lower beta tend to have higher actual returns than the CAPM's predictions. The relationship of an asset's actual returns to its average historical beta is usually positive as the CAPM suggests, but it tends to be flatter than predicted.     Furthermore, there are certain assets which consistently yield non-zero intercept terms for the first stage time-series regression of the CAPM. 

The too-flat relationship between average historical beta and predicted returns can be attributed to misspecification. The first misspecification may arise from measurement error. Suppose the true beta is not obseved. Instead the observed beta is the true beta plus a measurement error with zero mean. The assets with very high observed beta are likely to have very positive measurement errors- their observed betas are far higher than their true betas. Similarly, assets with very low observed beta are likely to have very negative measurement errors

CAPM suggests that the average market premium should be the sole significant determinant of the asset retusns, and that the differences in beta is sufficient to explain the variations of asset returns. However this may not be true. Another misspecification may include the omission of other factors (eg. firm size, dividend yields, etc) which are crucial in determining ex-post realized returns on assets. 

Taking the empirical evidence discussed above, the CAPM is not an acceptable theory even if Roll's Critique is ignored. However if we regard Roll's Critique, we can argue that the above discussion does little to reveal the validity of the CAPM. In conclusion, the CAPM can be either rejected or considered  untestable, but there are scarce evidence that supports its validity.