Multivariate Risk Measures for Portfolio Risk Management

Abstract

In portfolio risk management, the main foci are to control the aggregate risk of the entire portfolio and to understand the contribution of each individual risk unit in the portfolio to the aggregate risk. When univariate risk measures are used to quantify the risks associated with a portfolio, there is usually a lack of consideration of correlations between individual risk units and the aggregate risk and of dependence among these risks. For this reason, multivariate risk measures defined by considering the joint distribution of risk units in the portfolio are more desirable. In this thesis, we define new multivariate risk measures by minimizing multivariate loss functions subject to various. constraints. With the proposed multivariate risk measures, we obtain risk measures for the entire portfolio and each individual risk unit in the portfolio at the same time. In Chapter 2, we introduce a multivariate extension of Conditional Value-at-Risk (CVaR) based on a multivariate loss function associated with different risks related to portfolio risk management. We prove that the defined multivariate risk measure satisfies many desirable properties such as positive homogeneity, translation invariance and subadditivity. Then, we provide numerical illustrations with multivariate normal distribution to show the effects of the parameters in the model. After that, we also perform a comparison between our multivariate CVaR and other traditional univariate risk measures such as VaR and CVaR. In Chapter 3, we define a multivariate risk measure for capital allocation purposes. Unlike most of the existing allocation principles that assume the total capital is exogenously given, we obtain the optimal total capital for the entire portfolio and the optimal capital allocation to all the individual risk units in the portfolio at the same time. In this chapter, we first discuss our model with a two-level organization/portfolio structure. Then, we move to a more complex three-level organization/portfolio structure. We find that many of the existing allocation principles can be seen as special or limiting cases of our model. In addition, our model can explain those allocation principles as solutions to optimization problems. Finally, we provide a numerical example for the two-level organization/portfolio structure model with two different error functions. In Chapter 4, we introduce a multivariate shortfall risk measure induced by cumulative prospect theory (CPT) and give the corresponding risk allocations under the multivariate shortfall risk measure. To obtain this risk measure, we make an extension of previously studied univariate generalized shortfalls induced by CPT and incorporate the idea of systemic risk. In this study, we discuss the properties of the risk measure and conditions for its existence and uniqueness. Also, we perform a simulation study and a comparison to a previously studied multivariate shortfall to show that our model can provide a more reasonable risk measure and allocation result.