A Bayesian approach to measuring risk on portfolios with many assets

Abstract

Hedge fund companies typically deal with huge liquid multi-asset portfolios, and modelling the risk of these investments can be challenging. Furthermore, their susceptibility to global market crashes makes modelling their risk even more important. Fitting multivariate models to such portfolios can be challenging given their size, while modelling them univariately runs the risk of ignoring dependencies between the different assets. In this study, a three-stage method for measuring risk on a hedge fund portfolio with many assets is proposed. The first step is that of performing dimension reduction using dynamic principal component analysis which yields orthogonal components that can then be modeled separately avoiding the need to consider multivariate models. This is followed by volatility modelling and forecasting of the individual principal components using a Bayesian generalized autoregressive conditional heteroscedastic (GARCH) model with t-distributed innovations. This allows one to construct a posterior predictive distribution for the whole portfolio. Finally, from this posterior predictive distribution, direct estimation of the risk of the portfolio is obtained using value at risk and expected shortfall. To determine the optimal balance between dimension reduction and accurate forecasts, this method is applied on 4, 11, and 36 dynamic principal components cut-off points determined by the elbow method and the total variation accounted for. Cross-validation over 135 trading days of the different modelling approaches is performed using log pseudo-maximum likelihood as measure of predictive ability. In this case study, it is found that the model with 11 dynamic principal components yields the most accurate forecasts, while the model with 4 principal components yields the least favourable ones.