Self-consistent asset pricing models

Abstract

We discuss the foundations of factor or regression models in the light of the self-consistency condition that the market portfolio (and more generally the risk factors) is (are) constituted of the assets whose returns it is (they are) supposed to explain. As already reported in several articles, self-consistency implies correlations between the return disturbances. As a consequence, the alpha's and beta's of the factor model are unobservable. Self-consistency leads to renormalized beta's with zero effective alpha's, which are observable with standard OLS regressions. When the conditions derived from internal consistency are not met, the model is necessarily incomplete, which means that some sources of risk cannot be replicated (or hedged) by a portfolio of stocks traded on the market, even for infinite economies. Analytical derivations and numerical simulations show that, for arbitrary choices of the proxy which are different from the true market portfolio, a modified linear regression holds with a non-zero value α i at the origin between an asset i's return and the proxy's return. Self-consistency also introduces "orthogonality" and "normality" conditions linking the beta's, alpha's (as well as the residuals) and the weights of the proxy portfolio. Two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the S&P500 in the period from Jan. 1990 to Feb. 2005. These two diagnostics show interesting departures from dynamical self-consistency starting about 2 years before the end of the Internet bubble. Assuming that the CAPM holds with the selfconsistency condition, the OLS method automatically obeys the resulting orthogonality and normality conditions and therefore provides a simple way to self-consistently assess the parameters of the model by using proxy portfolios made only of the assets which are used in the CAPM regressions. Finally, the factor decomposition with the self-consistency condition derives a risk-factor decomposition in the multi-factor case which is identical to the principal components analysis (PCA), thus providing a direct link between model-driven and datadriven constructions of risk factors. This correspondence shows that PCA will therefore suffer from the same limitations as the CAPM and its multi-factor generalization, namely lack of out-of-sample explanatory power and predictability. In the multi-period context, the self-consistency conditions force the beta's to be time-dependent with specific constraints.

Index Cohesive Force Analysis Reveals That the US Market Became Prone to Systemic Collapses Since 2002

“…The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the systemic risk parameter beta ( β ) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. The correlation C ( i,m ) between the return of the given stock i and the daily return of the market index r m ( t ), is similar to β i - the systematic risk parameter of this stock which is defined within the security characteristic line (SCL) theory [32] , [34] , [35] . More specifically, using these parameters, the return of the asset on the return of the index is given by, Where is a random variable and the regression parameters and are given by: According to these definitions, the residual correlation can be viewed as the correlation between the residuals , after removing the dependency of the given stock on the index.…”

Risk evaluation with enhanced covariance matrix

“…An important outcome of studies basing on Markowitz approach is the Capital Asset Pricing Model (CAPM) [10,27,28,29] that relates risk to correlations within the market portfolio [10,27,28,29] of course the risk now is that all investments will collapse simultaneously. Furthermore it is assumed that risk that achieves premiums in the long term should not be reducible otherwise arbitrage is possible [28]. This is essentially the Arbitrage Pricing Theory (APT).…”