Modeling risk for CVaR-based decisions in risk aggregation

Title
Modeling risk for CVaR-based decisions in risk aggregation.
Abstract
Measuring the risk aggregation is an important exercise for any risk bearing carrier. It is not restricted to evaluation of the known portfolio risk position only, and could include complying with regulatory requirements, diversification, etc. The main difficulty of risk aggregation is creating an underlying robust probabilistic model. It is an irrefutable fact that the uncertainty of the individual risks is much lower in its complexity, as compared to modeling the dependence amongst the risks. As a result, it is often reasonable to assume that individual risks are modeled in a robust fashion, while the exact dependence remains unknown, yet some of its traits may be made available due to empirical evidence or "good practice". Our main contribution is to propose a numerical procedure that enables to identify the worst possible dependence scenario, when the risk preferences are modeled by the Conditional Value-at-Risk in the presence of dependence uncertainty.
For portfolios with two risks, it is known that CVaR ordering coincides with the lower-orthant stochastic ordering of the underlying bi-variate distributions. As a bi-product of our analysis, we show that no such extensions are possible to higher dimensions.
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Keywords
Risk management; Conditional Value-at-Risk; Uncertainty modeling; Bilinear optimization; Linear Programming; Risk aggregation

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