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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.
... Elliptical distributions, which encompass multivariate normal and t-distributions as specific instances, serve as fundamental tools in quantitative risk management(McNeil et al. (2015)). Meanwhile, the MRV model plays a significant role in Extreme Value Theory, particularly for analyzing portfolio diversification, as illustrated in works such asMainik and Rüschendorf (2010),Embrechts (2013), andBignozzi et al. (2016). For DQ based on ES and VaR, these two classes of distributions are studied byHan et al. (2023). ...
A diversification quotient (DQ) quantifies diversification in stochastic portfolio models based on a family of risk measures. We study DQ based on expectiles, offering a useful alternative to conventional risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). The expectile-based DQ admits simple formulas and has a natural connection to the Omega ratio. Moreover, the expectile-based DQ is not affected by small-sample issues faced by VaR-based or ES-based DQ due to the scarcity of tail data. The expectile-based DQ exhibits pseudo-convexity in portfolio weights, allowing gradient descent algorithms for portfolio selection. We show that the corresponding optimization problem can be efficiently solved using linear programming techniques in real-data applications. Explicit formulas for DQ based on expectiles are also derived for elliptical and multivariate regularly varying distribution models. Our findings enhance the understanding of the DQ's role in financial risk management and highlight its potential to improve portfolio construction strategies.
... Elliptical distributions, which encompass multivariate normal and t-distributions as specific instances, serve as fundamental tools in quantitative risk management(McNeil et al. (2015)). Meanwhile, the MRV model plays a significant role in Extreme Value Theory, particularly for analyzing portfolio diversification, as illustrated in works such asMainik and Rüschendorf (2010),Mainik and Embrechts (2013),and Bignozzi et al. (2016). For DQ based on ES and VaR, these two classes of distributions are studied byHan et al. (2023). ...
This paper investigates the diversification quotient (DQ), a recently introduced index quantifying diversification in stochastic portfolio models. We specifically examine the DQ constructed using expectiles, offering an alternative to conventional risk measures such as Value-at-Risk and Expected Shortfall. The expectile-based DQ admits simple formulas and has strong connections to the Omega ratio. Moreover, the expectile-based DQ is not affected by small sample issues faced by VaR or ES-based DQs due to the scarcity of tail data. Also, DQ based on expectiles exhibits pseudo-convexity in portfolio weights, allowing gradient descent algorithms for portfolio selection. Furthermore, we show that the corresponding optimization problem can be efficiently solved using linear programming techniques in real-data applications. Explicit formulas for DQ based on expectiles are also derived for elliptical and multivariate regularly varying distribution models. Our findings enhance the understanding of the DQ's role in financial risk management and highlight its potential to improve portfolio construction strategies.
... For example, in finance, widely used models for asset returns, such as the ARCH and GARCH models, have finitedimensional distributions following the MRV model, see, e.g., Davis and Mikosch (1998); and, in risk management, multiple underlying risk factors are usually assumed to be MRV as well, see, e.g., Hauksson et al. (2001), Barbe et al. (2006), and Embrechts et al. (2009). Some related applications of MRV can be referred to Mainik and Rüschendorf (2010), Zhou (2010), and Mainik and Embrechts (2013), among others. Recently, Einmahl et al. (2021) and Einmahl and (2023) propose two formal hypothesis tests for the MRV structure, and examine its validity by using the two real datasets: exchange rates (Yen-Dollar, Pound-Dollar), and stock indices (S&P, FTSE, Nikkei), which are also discussed in Cai et al. (2011) and He and Einmahl (2017). ...
Consider a credit portfolio with the investments in various sectors and exposed to an external stochastic environment. The portfolio loss due to defaults is of critical importance for social and economic security particularly in times of financial distress. We argue that the dependences among obligors within sectors (intradependence) and across sectors (interdependence) may coexist and influence the portfolio loss. To quantify the portfolio loss, we develop a multi-sector structural model in which a multivariate regular variation structure is employed to model the intradependence within sectors, and the interdependence across sectors is implied in the arbitrarily dependent macroeconomic factors, although, given them, obligors in different sectors are conditionally independent. We establish some sharp asymptotic formulas for the tail probability and the (tail) distortion risk measures of the portfolio loss. Our results show that the portfolio loss is mainly driven by the latent variables and the recovery rate function, and is also potentially affected by the macroeconomic factors and the intradependence within sectors. Moreover, we implement intensive numerical studies to examine the accuracy of the obtained approximations and conduct some sensitivity analysis.
... 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.
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 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.
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.