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# Diversification in heavy-tailed portfolios: properties and pitfalls

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We discuss risk diversification in multivariate regularly varying models and provide explicit formulas for Value-at-Risk asymptotics in this case. These results allow us to study the influence of the portfolio weights, the overall loss severity, and the tail dependence structure on large portfolio losses. We outline sufficient conditions for the sub- and superadditivity of the asymptotic portfolio risk in multivariate regularly varying models and discuss the case when these conditions are not satisfied. We provide several examples to illustrate the resulting variety of diversification effects and the crucial impact of the tail dependence structure in infinite mean models. These examples show that infinite means in multivariate regularly varying models do not necessarily imply negative diversification effects. This implication is true if there is no loss-gain compensation in the tails, but not in general. Depending on the loss-gain compensation, asymptotic portfolio risk can be subadditive, superadditive, or neither.
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... In particular, MRV models have been applied to DR based on VaR (e.g., Mainik and Rüschendorf (2010) and Mainik and Embrechts (2013)). Since VaR α (X)/ES α (X) → (γ − 1)/γ as α ↓ 0 for X ∈ RV γ with finite mean (see e.g., McNeil et al. (2015, p.154)), we only present the case of VaR. ...
... Proof. If X ∈ MRV γ (Ψ) with γ ∈ (0, 1), we have (Lemma 2.2 of Mainik and Embrechts (2013)) ...
... The random vectors X and Y are not elliptically distributed. Using the results in Mainik and Embrechts (2013), we have η w η 11 = (w 1 + w 2 r) ν + w 2 1 − r 2 ν , and η w η 12 = (w 1 + w 2 r) ν + w 2 √ 1 − r 2 ν r ν + √ 1 − r 2 ν . ...
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The diversification quotient (DQ) is recently introduced for quantifying the degree of diversification of a stochastic portfolio model. It has an axiomatic foundation and can be defined through a parametric class of risk measures. Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most prominent risk measures widely used in both banking and insurance , we investigate DQ constructed from VaR and ES in this paper. In particular, for the popular models of elliptical and multivariate regular varying (MRV) distributions, explicit formulas are available. The portfolio optimization problems for the elliptical and MRV models are also studied. Our results further reveal favourable features of DQ, both theoretically and practically, compared to traditional diversification indices based on a single risk measure.
... For a general treatment of elliptical models in risk management, see McNeil et al. (2015). Heavy-tailed models are known to exhibit complicated and even controversial phenomena in finance; see e.g., Ibragimov et al. (2011) and Mainik and Embrechts (2013). In this section, we study DQs based on VaR and ES for elliptical distributions and multivariate regularly varying (MRV) models. ...
... Extreme Value Theory is widely used for investigating tail risk measures such as VaR and ES at high levels (McNeil et al. (2015)). In particular, MRV models have been applied to DR based on VaR (e.g., Mainik and Rüschendorf (2010) and Mainik and Embrechts (2013)). ...
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The diversification quotient (DQ) is proposed as a new notion of diversification indices. Defined through a parametric family of risk measures, DQ satisfies three natural properties, namely, non-negativity, location invariance and scale invariance, which are shown to be conflicting for traditional diversification indices based on a single risk measure. We pay special attention to the two important classes of risk measures, Value-at-Risk (VaR) and Expected Shortfall (ES or CVaR). DQs based on VaR and ES enjoy many convenient technical properties, and they are efficient to optimize in portfolio selection. By analyzing the two popular multivariate models of elliptical and regular varying distributions, we find that DQ can properly distinguish tail heaviness and common shocks, which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive when contrasted with other diversification methods in portfolio optimization.
... These asymptotics can be applied in many areas of quantitative risk management such as portfolio diversification (e.g. Alink et al. 2004;Mao and Yang 2015;Mainik and Rüschendorf 2010;Mainik and Embrechts 2013), credit risk (e.g. Bassamboo et al. 2008;Tang et al. 2019), and efficient estimation of risk measures (e.g. ...
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We study the tail asymptotics of the sum of two heavy-tailed random variables. The dependence structure is modeled by copulas with the so-called tail order property. Examples are presented to illustrate the approach. Further for each example we apply the main results to obtain the asymptotic expansions for Value-at-Risk of aggregate risk.
... This topic has already been investigated in the literature; see e.g. Basrak et al. (2002), Barbe et al. (2006), Mainik and Embrechts (2013), Cuberos et al. (2015). For instance, assuming X multivariate regularly varying, X ∈ MRV − (b, ) implies that the sum S d ∈ RV − (b) . ...
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... For example, in the portfolio risk example, if one intends to estimate the Value-at-Risk or the Expected Shortfall of a portfolio loss L = d j=1 v j X j , where v 1 , . . . , v d represent weights of the stocks (see, e.g., Mainik and Embrechts, 2013), the failure region is of the form { j v j X j > x} with x large. In this case, we need to directly model the original vector X: a nonparametric transformation for the marginals will lead to untractable failure regions. ...
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An important problem in extreme-value theory is the estimation of the probability that a high-dimensional random vector falls into a given extreme failure set. This paper provides a parametric approach to this problem, based on a generalization of the tail pairwise dependence matrix (TPDM). The TPDM gives a partial summary of tail dependence for all pairs of components of the random vector. We propose an algorithm to obtain an approximate completely positive decomposition of the TPDM. The decomposition is easy to compute and applicable to moderate to high dimensions. Based on the decomposition, we obtain parameters estimates of a max-linear model whose TPDM is equal to that of the original random vector. We apply the proposed decomposition algorithm to industry portfolio returns and maximal wind speeds to illustrate its applicability.
... This topic has already been investigated in the literature; see e.g. Basrak et al. (2002), Barbe et al. (2006), Mainik and Embrechts (2013), and Cuberos et al. (2015). For instance, assuming X multivariate regularly varying, X ∈ MRV −α (b, ν) (as defined in Definition A.5, Appendix A.1), implies that the sum S d ∈ RV −α (b). ...
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