Testing CAPM with a Large Number of Assets (Updated 28th March 2012)

Abstract

(DISCLAIMER: Not all mathematical symbols in the abstract will display properly - please see the abstract in the pdf). This paper is concerned with testing the time series implications of the capital asset pricing model (CAPM) due to Sharpe (1964) and Lintner (1965), when the number of securities, N, is large relative to the time dimension, T, of the return series. Two new tests of CAPM are proposed that exploit recent advances on the analysis of large panel data models, and are valid even if N is much larger than T. When the errors are Gaussian and cross sectionally independent, a test, denoted by Ĵα,ı, is proposed which is N(0; 1) as N→∞, with T fixed. Even when the errors are non-Gaussian we are still able to show that Ĵα,ı, tends to N(0; 1) so long as the errors are cross-sectionally independent and N/T³→0, with N and T ! 1, jointly. In the case of cross sectionally correlated errors, using a threshold estimator of the average squares of pair-wise error correlations, a modified version of Ĵα,ı,, denoted by Ĵα,2, is proposed. Small sample properties of the tests are compared using Monte Carlo experiments designed specifically to match the correlations, volatilities, and other distributional features of the residuals of Fama-French three factor regressions of individual securities in the Standard Poor 500 index. Overall, the proposed tests perform best in terms of power, with empirical sizes very close to the chosen nominal value even in cases where N is much larger than T. The Ĵα,2 test (which allows for non-Gaussian and weakly cross correlated errors) is applied to all securities in the S&P 500 index with 60 months of return data at the end of each month over the period September 1989-September 2011. Statistically significant evidence against Sharpe-Lintner CAPM is found mainly during the recent financial crisis. Furthermore, a strong negative correlation is found between a twelve-month moving average p-values of the Ĵα,2 test and the returns of long/short equity strategies relative to the return on S&P 500 over the period December 2006 to September 2011, suggesting that abnormal profits are earned during episodes of market inefficiencies