Univariate and Multivariate Volatility Models for Portfolio Value at Risk

Jingyi Xiao¹, Siqi Mao^1, , Xufeng Niu¹ and Yixin Kang¹

¹Department of Statistics, Florida State University, Tallahassee, USA

Cite this paper:
Jingyi Xiao, Siqi Mao, Xufeng Niu and Yixin Kang. Univariate and Multivariate Volatility Models for Portfolio Value at Risk. Journal of Finance and Economics. 2024; 12(1):1-14. doi: 10.12691/jfe-12-1-1

Abstract

In modern day financial risk management, modeling and forecasting stock return movements via their conditional volatilities, particularly predicting the Value at Risk (VaR), became increasingly more important for a healthy economical environment. In this paper, we evaluate and compare two main families of models for the conditional volatilities - GARCH - in terms of their VaR prediction performance of 5 major US stock indices. We calculate GARCH-type model parameters via Quasi Maximum Likelihood Estimation (QMLE). Since financial volatilities are moving together across assets and markets, it becomes apparent that modeling the volatilities in a multivariate framework of modeling is more appropriate. However, existing studies in the literature do not present compelling evidence for a strong preference between univariate and multivariate models. In this paper we also address the problem of forecasting portfolio VaR via multivariate GARCH models versus univariate GARCH models. We construct 3 portfolios with stock returns of 3 major US stock indices, 6 major banks and 6 major technical companies respectively. For each portfolio, we model the portfolio conditional covariances with GARCH, EGARCH and MGARCH-BEKK, MGARCH-DCC, and GO-GARCH models. For each estimated model, the forecast portfolio volatilities are further used to calculate (portfolio) VaR. The ability to capture the portfolio volatilities is evaluated by MAE and RMSE; the VaR prediction performance is tested through a two-stage backtesting procedure and compared in terms of the loss function. The results of our study indicate that even though MGARCH models are better in predicting the volatilities of some portfolios, GARCH models could perform as well as their multivariate (and computationally more demanding) counterparts.

Keywords:
Value at Risk (VaR) GARCH-type Model Multivariate GARCH models

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