The Empirical Distribution of UK and US Stock Returns

Abstract

There is now substantial evidence that daily equity returns are not normally distributed but instead display significant leptokurtosis and, in many cases, skewness. Considerable effort has been made in order to capture these empirical characteristics using a range of ad hoc statistical distributions. In this paper, we investigate the distribution of daily, weekly and monthly equity returns in the UK and US using two very flexible families of distributions that have been recently introduced: the exponential generalised beta (EGB) and the skewed generalised‐t (SGT). These distributions permit very diverse levels of skewness and kurtosis and, between them, nest many of the distributions previously considered in the literature. Both the EGB and the SGT provide a very substantial improvement over the normal distribution in both markets. Moreover, for daily returns, we strongly reject the restrictions on the EGB and SGT implied by most of the distributions that are commonly used for modelling equity returns, including the student‐t, the power exponential and the logistic distributions. Instead, our preferred distributions for daily returns are the generalised‐t for the US and the skewed‐t for the UK, both of which are members of the SGT family. For weekly returns, our preferred distributions are the student‐t for the UK and the skewed‐t for the US, while for monthly returns, our preferred distributions are the EBR12 for the UK and the logistic for the US. We consider the implications of our findings for the implementation of value‐at‐risk, a risk management methodology that rests heavily on the distributional characteristics of returns.