A Model-Independent Measure of Aggregate Idiosyncratic Risk

Abstract

This paper introduces a model-independent measure of aggregate idiosyncratic risk based on the mean-variance portfolio theory and the concept of gain from portfolio diversification. With the new approach, there is no need to estimate the covariance terms or the industry-level or firm-level beta coefficients when constructing the average idiosyncratic risk at the industry- or firm-level. Since there is no gain from diversification when the correlations among individual stocks equal one, the variance of the portfolio with perfectly correlated securities contains systematic risk and idiosyncratic risk of the securities in the portfolio. We also think that the stock market index can be viewed as a fully diversified portfolio, which does not contain any idiosyncratic risk. Since the market portfolio contains a large number of stocks, enough diversification gains are achieved and the idiosyncratic risk contributes nothing to the total risk of the market portfolio. That is, the risk of this well-diversified portfolio is due solely to the systematic risk of the securities in the portfolio. The new measure of average idiosyncratic volatility is defined as the difference between the variance of the non-diversified portfolio and the variance of the fully diversified portfolio. We present two versions of the new methodology; one decomposing total risk into firm and market variance, and the other decomposing total risk into firm, industry, and market variance. The statistical results and graphical analyses provide strong evidence that there are significant level and trend differences between the average idiosyncratic volatility measures of Campbell, Lettau, Malkiel, and Xu (2001, CLMX) and the new methodology. Although both approaches indicate a noticeable increase in the firm-level idiosyncratic risk, the volatility measure of CLMX is greater and has a stronger upward trend than the new idiosyncratic volatility measure. The analytical and empirical results show that the significant upward trend in the differences of the two idiosyncratic volatility measures is related to the increase in the cross-sectional dispersion of the volatility of individual stocks.