Bayesian Value-at-Risk and Expected Shortfall for a Large Portfolio (Multi- and Univariate Approaches)

Abstract

A. Pajor and J. Osiewalski Department of Econometrics and Operations Research, Cracow University of Economics Rakowicka 27, 31-510 Krakow, Poland Bayesian assessments of value-at-risk and expected shortfall for a given portfolio of dimension n can be based either on the n-variate predictive distribution of future returns of individual assets, or on the univariate model for portfolio volatility. In both cases, the Bayesian VaR and ES fully take into account parameter uncertainty and non-linear relationship between ordinary and logarithmic returns. We use the n-variate type I MSF-SBEKK(1,1) volatility model proposed specially to cope with large n. We compare empirical results obtained using this (more demanding) multivariate approach and the much simpler univariate approach based on modelling volatility of the whole portfolio (of a given structure).