Time varying betas and the unconditional distribution of asset returns

Abstract

This paper draws attention to the fact that under standard assumptions the time varying betas model cannot capture the dynamics in beta. Conversely, evidence of time variation in beta using this model is equivalent to non-normality in the unconditional distribution of asset returns. Using the multivariate normal as a model for the joint distribution of returns on market indices and predetermined information variables, it is shown how to capture skewness and kurtosis in the unconditional distributions of asset returns. Under the assumptions of the model, asset returns are unconditionally distributed as an extended quadratic form (EQF) in normal variables. Expressions are given for the moment generating function and for the computation of the distribution and density functions. The market-timing model is derived formally using this model. The properties of bias when the standard linear betas model is used to estimate alpha when the correct model is the EQF are also investigated. It is shown that a different time varying betas model can arise as a consequence of portfolio selection. It is also shown that the predetermined information variables have the potential to account for the time series properties of returns, including heterogeneity of variance. An empirical study applies the model to returns on 46 UK bond funds. An analysis of the residuals shows that the model described in this paper is able to capture the dynamics of alpha and beta and properly account for other features of the time series of returns for 28 of these funds, of which 15 exhibit time variation in beta. The study reports the effect of the EQF model on the computation of VaR and CVaR and bias in the estimation of alpha.