|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
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.