Advanced Idiosyncratic Risk and Multi-Factor Models

Abstract

A common scenario risk analysis employs a multiple factor model with assumed changes in the factors to obtain changes in non-factor variables. This analysis is sometimes designated as a “predictive stress scenario”. We choose to designate the factor model as a multifactor “CAPM” model, also related to APT models. We are not concerned with the details of the factor model. The risks of the scenarios are evaluated for a portfolio of instruments depending on changes of the variables. A systemic problem with this factor model approach is that the empirical correlations between pairs of non-factor variables are not maintained. We designate “advanced idiosyncratic risk” or “AI-Risk” as the correction to the risk of a predictive stress CAPM model scenario that includes a better approximation to the physical correlations. We have developed a new, effective, and economical AI-Risk formalism. There are two parts to AI-Risk. The first part, which is actually standard, has independent normal random variables with no correlations; we designate that as part of the CAPM model to get the variances correctly reproduced; we believe this is standard practice. The second part contains the corrections to the correlations; this is new. We evaluate AI-Risk for some stock portfolios. We find that the AI-Risk can be important, typically between 10%-50% of the predictive stress risk. The amount depends on the ratio of long/short (“L/S”) position values, with larger AI-Risk for higher L/S ratios.A technical remark is that we use correlations as the metric for model improvement. Therefore we work with z-scores or “unit” returns with the empirical volatility divided out.Finally, we indicate the generalization to cross-sectional regression factor models.In summary this work achieves two results: 1. A parsimonious formalism incorporating correlated idiosyncratic risk that gives better agreement with given empirical correlations (so risk is calculated closer to the physical world). 2. Indications of substantial addition to risk due to correlated idiosyncratic contributions, with respect to the smaller risk obtained in the standard model with uncorrelated idiosyncratic risk.