How to account for extreme co-movements between individual stocks and the market

ABSTRACT Using the framework of factor models, we study the extreme co-movements between two stocks and between a stock and the market. We establish the general expression of the coefficient of tail dependence between the market and a stock (that is, the probability that the stock incurs a large loss, assuming that the market has also undergone a large loss) and between two stocks as a function of the parameters of the underlying factor model and of the tail parameters of the distributions of the factor and of the idiosyncratic noise of each stock. Our formula holds for arbitrary marginal distributions and in addition does not require any parameterization of the multivariate distributions of the market and stocks. The determination of the tail dependence parameter, which is not accessible by a direct statistical inference, is made possible by the measurement of parameters whose estimation involves a significant part of the data. Our empirical tests find a good agreement between the calibration of the tail dependence coefficient and the realized large losses over the period from 1962 to 2000. Nevertheless, a bias is detected as well as the presence of an outlier in the form of the crash of October 1987.