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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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... One often finds its use in insurance, finance, and risk management when extreme risks are concerned. See some recent works such as [13,22,15,29] and [28]. ...

... The various parameters are set as follows. For (16), the parameter τ = 1.1 of the proportional hazards distortion function in (14), the Pareto tail index α = 1.9 in (25), and the parameter r = 1.6 of the Gumbel copula (22). For (17), τ = 0.91, α = 1.46, and the parameter β = 1 of the Frank copula (23). ...

... For (17), τ = 0.91, α = 1.46, and the parameter β = 1 of the Frank copula (23). For (18), τ = 0.9, α = 1.6, the Pareto tail index α 2 = 5 of X 2 larger than α, and r = 1.2 of the Gumbel copula (22) or γ = 1 of the Clayton copula (24). The asymptotic and simulated values of T g,q (S 2 ) as well as their RAEs are listed in the Tables 3-5. ...

Consider a portfolio of \begin{document}$ n $\end{document} losses accompanied with \begin{document}$ n $\end{document} stochastic loss adjustment factors. This paper establishes some asymptotic formulas of the tail distortion risk measure for aggregate weight-adjusted heavy-tailed losses under the framework of multivariate regular variation, pairwise quasi-asymptotic independence or arbitrary dependence. As an application, the corresponding results on the asymptotics for the risk concentration based on tail distortion risk measure are also derived. Several examples and simulation studies are provided to better illustrate the obtained results.

... 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. ...

... Two classic benchmarks are equally weighted (EW) portfolio and minimum variance (MV) portfolio. The other two benchmarks are the extreme risk index (ERI) strategy and most diversified portfolio (MDP) strategy, whose formal definitions are presented in Section 2. The ERI 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. ...

... 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. ...

Heavy tailedness and interconnectedness widely exist in stock returns and large insurance claims, which contributes to huge losses for financial institutions. Diversification ratio (DR) measures the degree of diversification using the Value-at-Risk, which is known to capture extreme risks better than variance. The portfolio optimization strategy based on DR maximizes the effect of diversification for extreme risks. In this paper, we empirically examine the DR strategy by using more than 350 S&P 500 stocks under the assumption that the stock losses are modeled with a flexible multivariate heavy-tailed model. This assumption is verified empirically. The performance of DR strategy is compared with four benchmark strategies: equally weighted portfolio, minimum-variance portfolio, extreme risk index portfolio, and most diversified portfolio. The performance of comparison includes annualized portfolio return, modified Sharpe ratio, maximum drawdown, portfolio concentration, portfolio turnover, and the degree of diversification. DR outperforms other strategies. In particular, DR shows the highest return and maintains the highest level of diversification during the global financial crisis of 2007–2009.

... ; see Tasche (2007). Other studies on DR can be found in e.g., Choueifaty and Coignard (2008), Bürgi et al. (2008), Mainik and Rüschendorf (2010) and Embrechts et al. (2015). For a literature review on diversification indices, see Koumou (2020). ...

... 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)). ...

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. ...

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.

... 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. ...

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.

... Complementing the rich literature on tail asymptotics for sums and maxima of independent heavytailed variables, we have [72,65,64] treating tail asymptotics of the form P (L(X) > u) in specific linear portfolio and reinsurance risk settings involving dependent heavy-tailed random variables. Specifically, these tail asymptotics invoke the structure present in multivariate regularly varying distributions to derive sharper tail asymptotics compared to (32). ...

Motivated by the increasing adoption of models which facilitate greater automation in risk management and decision-making, this paper presents a novel Importance Sampling (IS) scheme for measuring distribution tails of objectives modelled with enabling tools such as feature-based decision rules, mixed integer linear programs, deep neural networks, etc. Conventional efficient IS approaches suffer from feasibility and scalability concerns due to the need to intricately tailor the sampler to the underlying probability distribution and the objective. This challenge is overcome in the proposed black-box scheme by automating the selection of an effective IS distribution with a transformation that implicitly learns and replicates the concentration properties observed in less rare samples. This novel approach is guided by a large deviations principle that brings out the phenomenon of self-similarity of optimal IS distributions. The proposed sampler is the first to attain asymptotically optimal variance reduction across a spectrum of multivariate distributions despite being oblivious to the underlying structure. The large deviations principle additionally results in new distribution tail asymptotics capable of yielding operational insights. The applicability is illustrated by considering product distribution networks and portfolio credit risk models informed by neural networks as examples.

In this paper, we investigate the optimal portfolio construction aiming at extracting the most diversification benefit. We employ the diversification ratio based on the Value-at-Risk as the measure of the diversification benefit. With modeling the dependence of risk factors by the multivariate regularly variation model, the most diversified portfolio is obtained by optimizing the asymptotic diversification ratio. Theoretically, we show that the asymptotic solution is a good approximation to the finite-level solution. Our theoretical results are supported by extensive numerical examples. By applying our portfolio optimization strategy to real market data, we show that our strategy provides a fast algorithm for handling a large portfolio, while outperforming other peer strategies in out-of-sample risk analyses.

This article reviews methods from extreme value analysis with applications to risk assessment in finance. It covers three main methodological paradigms: the classical framework for independent and identically distributed data with application to risk estimation for market and operational loss data, the multivariate framework for cross-sectional dependent data with application to systemic risk, and the methods for stationary serially dependent data applied to dynamic risk management. The article is addressed to statisticians with interest and possibly experience in financial risk management who are not familiar with extreme value analysis.

Extremes Values, Regular Variation and Point Processes is a readable and efficient account of the fundamental mathematical and stochastic process techniques needed to study the behavior of extreme values of phenomena based on independent and identically distributed random variables and vectors. It presents a coherent treatment of the distributional and sample path fundamental properties of extremes and records. It emphasizes the core primacy of three topics necessary for understanding extremes: the analytical theory of regularly varying functions; the probabilistic theory of point processes and random measures; and the link to asymptotic distribution approximations provided by the theory of weak convergence of probability measures in metric spaces.
The book is self-contained and requires an introductory measure-theoretic course in probability as a prerequisite. Almost all sections have an extensive list of exercises which extend developments in the text, offer alternate approaches, test mastery and provide for enjoyable muscle flexing by a reader. The material is aimed at students and researchers in probability, statistics, financial engineering, mathematics, operations research, civil engineering and economics who need to know about:
* asymptotic methods for extremes;
* models for records and record frequencies;
* stochastic process and point process methods and their applications to obtaining distributional approximations;
* pervasive applications of the theory of regular variation in probability theory, statistics and financial engineering.
"This book is written in a very lucid way. The style is sober, the mathematics tone is pleasantly conversational, convincing and enthusiastic. A beautiful book!"
---Bulletin of the Dutch Mathematical Society
"This monograph is written in a very attractive style. It contains a lot of complementary exercises and practically all important bibliographical reference."
---Revue Roumaine de Mathématiques Pures et Appliquées

We estimate Value-at-Risk for sums of dependent random variables. We model multivariate dependent random variables using archimedean copulas. This structure allows one to calculate the asymptotic behaviour of extremal events. An important application of such results are Value-at-Risk estimates for sums of dependent random variables.

We survey multivariate extreme value distributions. These are limiting distributions of maxima and/or minima, componentwise, after suitable normalization. A distribution is extreme value stable if and only if its margins are stable and its dependence function is stable. Thus it is possible, without loss of generality, to choose any stable marginal distribution deemed convenient. We use the negative exponential one. The class of multivariate stable negative exponential distributions is characterized by the fact that weighted minima of components have negative exponential distributions. We examine several representations and the relationships among them and we consider, in terms of them, joint densities and scalar measures of dependence. We also consider estimation.

We establish the equivalence between the multivariate regular variation of a random vector and the univariate regular variation of all linear combinations of the components of such a vector. According to a classical result of Kesten [Acta Math. 131 (1973) 207-248], this result implies that stationary solutions to multivariate linear stochastic recurrence equations are regularly varying. Since GARCH processes can be embedded in such recurrence equations their finite-dimensional distributions are regularly varying.

The asymptotic theory of sample extremes has been developed in parallel with the central limit theory, and in fact the two theories bear some resemblance.

Due to published statistical analyses of operational risk data, methodological approaches to the "advanced measurement approach" modeling of operational risk can be discussed in more detail. In this paper we raise some issues concerning correlation (or diversification) effects, the use of extreme value theory and the overall quantitative risk management consequences of extremely heavy-tailed data. We especially highlight issues around infinite-mean models. In addition to methodological examples and simulation studies, the paper contains indications for further research.

This book is a comprehensive account of the theory and applications of regular variation. It is concerned with the asymptotic behaviour of a real function of a real variable x which is 'close' to a power of x. Such functions are much more than a convenient extension of powers. In many limit theorems regular variation is intrinsic to the result, and exactly characterises the limit behaviour. The book emphasises such characterisations, and gives a comprehensive treatment of those applications where regular variation plays an essential (rather then merely convenient) role. The authors rigorously develop the basic ideas of Karamata theory and de Haan theory including many new results and 'second-order' theorems. They go on to discuss the role of regular variation in Abelian, Tauberian, and Mercerian theorems. These results are then applied in analytic number theory, complex analysis, and probability, with the aim above all of setting the theory in context. A widely scattered literature is thus brought together in a unified approach. With several appendices and a comprehensive list of references, analysts, number theorists, and probabilists will find this an invaluable and complete account of regular variation. It will provide a rigorous and authoritative introduction to the subject for research students in these fields.