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Extreme losses of portfolios with heavy-tailed components are studied in the framework of multivariate regular variation. Asymptotic distributions of extreme portfolio losses are characterized by a functional γ ξ =γ ξ (α,Ψ) of the tail index α, the spectral measure Ψ, and the vector ξ of portfolio weights. Existence, uniqueness, and location of the optimal portfolio are analysed and applied to the minimization of risk measures. It is shown that diversification effects are positive for α>1 and negative for α<1. Strong consistency and asymptotic normality are established for a semiparametric estimator of the mapping ξ ↦ γ ξ . Strong consistency is also established for the estimated optimal portfolio.
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... It is crucial for financial institutions to manage these extreme risks. Diversification is a common strategy in managing portfolios and it has been studied in different contexts involving financial risks or insurance risks, for example in Schnieper (2000), Choueifaty and Coignard (2008), Choo and de Jong (2010), Mainik and Rüchendorf (2010), and Cui et al. (2021). In this paper, we investigate the performance of an optimal strategy aiming at maximizing the effect of diversification to mitigate extreme risks. ...
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... which was proposed in Mainik and Rüchendorf (2010) under the MRV structure. ERI uses VaR to measure risks and is essentially the "minimum-VaR" strategy, seen as the counterpart of MV. ...
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