Portfolio optimization with systemic risk aversion

Abstract

We consider several optimization problems in the portfolio selection problem subject to systemic risk aversion. The systemic risk has been a big concern in the financial systems since the global financial crisis 2007-2008. The goal of this thesis is to study the portfolio selection problems which reduce the systemic risk and to measure the systemic risk contributions to cope with unpredictable crises. We present the mathematical formulations of the problems and propose several algorithms for the portfolio selection problem.

First, We propose extensions of the general mean-risk portfolio selection models, with the aim of limiting the spillover effect of risk. We use the Conditional Value at Risk(CoVaR) as a measure to identify the tail risk contribution of securities. The goal is to find a portfolio which minimizes general risk measures while the tail risk contribution is limited. Based on the frameworks of the general mean-risk models, we additionally guarantee that a portfolio has limited tail risk contributions. Extensions of the mean-risk models can be formulated as mixed integer nonlinear programs. To solve these problems efficiently, we propose a Dantzig-Wolfe decomposition reformulation and the column generation algorithm in which the subproblem reduces the spillover effect of tail risk.

Next, We propose the quantile regression and the support vector regression models to measure the systemic risk exposure, that is, Exposure CoES(Conditional Expected Shortfall). The proposed methods are nonparametric in the sense that they do not require assumption on the distribution of data, and they are less sensitive to outliers. We use both the original definition and the modified definition of CoVaR and CoES. We test the performance of systemic risk exposure for the companies of various industrial sectors. The empirical analysis shows that the companies of Energy and Industrials sector are highly correlated with the system risk like the financial institutions. Also, we provide a way of using Exposure CoES for the portfolio selection problem. The results show that the portfolios obtained from our approach perform well and are less affected by future crisis.

Lastly, We develop a Bayesian network model to express the risk contagion in a financial system. To calculate the score of a Bayesian network, joint probability distributions or conditional probability tables are required in general. Instead of them, we propose an information criterion based scoring function which is based on the maximum likelihood estimation. We utilize the support vector machine to incorporate the information criterion into the model. Moreover, the scores of subgraphs should be calculated for all of exponentially many variables a priori. We propose a mathematical formulation for the Bayesian network and a column generation based algorithm to handle the huge number of variables efficiently. The goal is to obtain the Bayesian networks which explain the contagion effect of risk. In most studies about risk contagion in financial network, researches are limited to the financial institutions. We extend the area of risk contagion to other financial sectors.