dc.contributor.author Jia, Huameng dc.date.accessioned 2021-01-29 20:12:41 (GMT) dc.date.available 2021-01-29 20:12:41 (GMT) dc.date.issued 2021-01-29 dc.date.submitted 2020-12-11 dc.identifier.uri http://hdl.handle.net/10012/16768 dc.description.abstract In portfolio risk management, the main foci are to control the aggregate risk of the en 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. dc.language.iso en en dc.publisher University of Waterloo en dc.subject portfolio risk management en dc.subject capital allocation en dc.subject multivariate risk measures en dc.subject VaR en dc.subject CVaR en dc.subject aggregate risk en dc.subject distortion risk measures en dc.title Multivariate Risk Measures for Portfolio Risk Management en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Statistics and Actuarial Science en uws-etd.degree.discipline Actuarial Science en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws-etd.embargo.terms 0 en uws.contributor.advisor Cai, Jun uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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