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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 . ...
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)). ...
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.
... 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) . ...
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.
... 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. ...
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). ...
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.
This paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value‐at‐Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.
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.
The notion of asymptotic portfolio loss order is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of spectral measures. This allows for analytical and numerical verification in applications. Worst and best case dependence structures with respect to the asymptotic portfolio loss order are determined. Comparison criteria in terms of further stochastic ordering notions are derived. The examples include elliptical distributions and multivariate regularly varying models with Gumbel, Archimedean, and Galambos copulas. Particular interest is paid to the inverse influence of dependence on the diversification of risks with infinite expectations.
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
A statistical analysis which provides a risk assessment of nuclear safety based on historical data is conducted. Classical probabilistic models from risk theory are used to analyze data on nuclear power accidents from 1952 to 2011. Findings are that the severities of nuclear power accidents should be modeled with an infinite mean model and, thus, cannot be insured by an unlimited cover.
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.
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.