Estimation and forecasting of covariance matrices with applications to financial risk management

Abstract

This thesis is about covariance matrix adjustment and forecasting which are two important areas in finance.

Covariance matrix of returns on a set of financial assets in a portfolio could change drastically and generate huge loss in the portfolio value under extreme conditions such as market interventions and financial crises. Estimation of the covariance matrix under a chaotic market is often a call to action in financial risk management. Nowadays, stress testing has become a standard procedure for many financial institutions to estimate the capital requirement for their portfolio holdings under various stress scenarios. A possible stress scenario is to adjust the covariance matrix to mimic market conditions under an underlying stress event. When some covariances are altered, it is natural that the other covariances should vary as well. Ng et al. (2014) proposed a unified approach to construct a proper correlation matrix which reflects the subjective views of correlations. However, high correlations often go hand in hand with high standard deviations during the crisis period. To cater this phenomenon, a new unified approach based on the Black-Litterman framework is proposed to adjust the covariance matrix according to the stressed market condition. Unlike Ng et al. (2014), this proposed method does not require matrix vectorization and hence, it is more computationally efficient, even for matrix of a higher dimension, say 20.

Modeling and forecasting the covariance matrices of asset returns play a crucial role in finance. Because of the available high frequency intraday data, the daily covariance matrices can now be directly determined using the notion of the realized covariance matrices. Nevertheless, many existing models may suffer from the curse of dimensionality and thus do not work well in high dimension. Tao et al. (2011) proposed to firstly factorize the high dimensional covariance matrices into low dimensional factor covariance matrices through Matrix Factor Analysis (MFA) model and then model the low dimensional factor covariance matrices. However, the factor loading matrix is constant over time and hence the dynamic correlation structure of the realized covariance matrices is completely controlled by the dynamic of the low dimensional factor covariance matrices. Therefore, the forecasting performance of this model for high dimensional covariance matrices is likely to be poor. To tackle this problem, the tree-based matrix factor models, which allow the daily factor loading matrices to vary according to the market conditions in factorizing the high dimensional realized covariance matrices, are proposed. Hence the dynamic of dependence structure is not only controlled by the dynamic of the low dimensional factor covariance matrices but also the dynamic of the changing factor loading matrices. The proposed tree-based matrix factor models are applied to a real-world dataset and their forecasting performance is compared with some existing models.

The Value-at-Risk (VaR), which summarizes the market risk exposure into a single number, is a useful tool for measuring the market risk. The covariance matrices forecasted by the proposed tree-based matrix factor models are used to derive the VaR. The backtesting result is compared with the VaR derived using the covariance matrices forecasted by some existing models.