Multivariate Extreme Value Theory Research Articles

On the estimation and application of max-stable processes

The theory of max-stable processes generalizes traditional univariate and multivariate extreme value theory by allowing for processes indexed by a time or space variable. We consider a particular class of max-stable processes, known as M4 processes, that are particularly well adapted to modeling the extreme behavior of multiple time series. We develop procedures for determining the order of an M4 process and for estimating the parameters. To illustrate the methods, some examples are given for modeling jumps in returns in multivariate financial time series. We introduce a new measure to quantify and predict the extreme co-movements in price returns.

Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

Consider a random sample from a bivariate distribution function F in the max-domain of attraction of all extreme-value distribution function G. This G is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of F. A major issue in multivariate extreme-value theory is the estimation of the spectral measure (1)p with respect to the L-p norm. For every p is an element of [1, infinity], a nonparametric maximum empirical likelihood estimator is proposed for Phi(p). The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case Studies illustrate how to implement the methods in practice.

Thresholding Events of Extreme in Simultaneous Monitoring of Multiple Risks

This article develops a threshold system for monitoring airline performance. This threshold system divides the sample space into regions with increasing levels of risk and allows instant assessments of risk level of any observed airline performance. Of particular concern is the performance with extreme risk. In this article, a multivariate extreme value theory approach is used to establish thresholds for signaling varying levels of extremeness in the context of simultaneous monitoring of multiple risk measures. The threshold system is justified in terms of multivariate extreme quantiles, and its sample estimator is shown to be consistent. This threshold system applies to general extreme risk management. Finally, a simulation and comparison study demonstrates the good performance of the proposed multivariate extreme quantile estimator. Supplemental materials providing technical details are available online.

Dependence Structure of Risk Factors and Diversification Effects

In this paper, we study the aggregated risk from dependent risk factors under the multivariate Extreme Value Theory (EVT) framework. We consider the heavy-tailness of the risk factors as well a non-parametric tail dependence structure. This allows a large scope of models on the dependency. We assess the Value-at-Risk of a diversified portfolio constructed from dependent risk factors. Moreover, we examine the diversification effects under this setup.

Multivariate extremes and the aggregation of dependent risks: examples and counter-examples

Properties of risk measures for extreme risks have become an important topic of research. In the present paper we discuss sub- and superadditivity of quantile based risk measures and show how multivariate extreme value theory yields the ideal modeling environment. Numerous examples and counter-examples highlight the applicability of the main results obtained.

Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution

Consider a random sample from a bivariate distribution function F in the max-domain of attraction of an extreme value distribution function G. This G is characterized by the two extreme value indices and its spectral measure, which determines the tail dependence structure of F. A major issue in multivariate extreme value theory is the estimation Φp, the spectral measure obtained by using the Lp norm in the definition. For every pe [1,∞], a nonparametric maximum empirical likelihood estimator is proposed for Φp. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under easily verifiable conditions that allow for tail independence. Some examples are discussed and a simulation study shows substantially improved performance of the new estimators.

Estimating the Probability of a Rare Event via Elliptical Copulas

A rare event happens with an extremely small probability but may cost billions of dollars. How to model and estimate the small probability of such an event is of importance to the insurance industry. Based on multivariate extreme value theory, methods have been proposed to extrapolate data into a far tail region. However, questions still remain open, such as the direction of extrapolation for a multivariate distribution and threshold selection for both marginals and the tail dependence function. In this paper we provide a way to estimate the probability of a rare event via modeling marginals and dependence by heavy tailed distributions and elliptical copulas, respectively. Hence, the direction of extrapolation becomes irrelevant. Moreover we employ recent threshold selection procedures to choose tuning parameters automatically.

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7–45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307–335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197–217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.

It was 30 years ago today when Laurens de Haan went the multivariate way

Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate) sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.

The extreme-value dependence of Asia-Pacific equity markets

In this paper we study the dependence structure of extreme realization of returns between seven Asia-Pacific stock markets and the U.S. Methodologically we apply the multivariate extreme value theory that best suits to the problem under investigation. The evidence we obtain indicates that extreme correlations are not substantially different from the unconditional ones or from those obtained from multivariate GARCH models. A clustering analysis shows that the Asia-Pacific countries do not belong to a distinct block of countries on the basis of the extreme correlations we have estimated. The policy implications of our study are that the benefits from portfolio diversification with assets from the Asia-Pacific stock markets are not eroded during crisis periods, in the sense that no correlation breakdown has been observed.

On the Tail Behavior of Sums of Dependent Risks

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

On the Tail Behavior of Sums of Dependent Risks

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

Multivariate Extreme Value Theory And Its Usefulness In Understanding Risk

This paper gathers recent results in the analysis of multivariate extreme values and illustrates their actuarial application. We review basic and essential background on univariate extreme value theory and stochastic dependence and then provide an introduction to multivariate extreme value theory. We present important concepts for the analysis of multivariate extreme values and collect research results in this area. We draw particular attention to issues related to extremal dependence and show the importance of model selection when fitting an upper tail copula to observed joint exceedances. These ideas are illustrated on four data sets: loss amount and allocated loss adjustment expense under insurance company indemnity claims, lifetimes of pairs of joint and lastsurvivor annuitants, hurricane losses in two states, and returns on two stocks. In each case the extremal dependence structure has an important financial impact.

Use of a Gaussian copula for multivariate extreme value analysis: Some case studies in hydrology

Risk assessment requires a description of the probabilistic properties of hydrological variables. In a number of cases, this description is made on a single variable, whereas most hydrological events are intrinsically multivariate. In this context, copulas have recently received attention in order to derive a multivariate frequency analysis. After a reminder of the general results in the field of multivariate extreme value theory, the paper gives a description of a very simple copula, the Gaussian copula. Four case studies demonstrate its usefulness in the contexts of field significance determination, regional risk analysis, discharge-duration-frequency (QdF) models with design hydrograph derivation and regional frequency analysis. The limitations and potential errors related to this statistical tool are also highlighted.

Analysis of multivariate extreme intakes of food chemicals

A recently published multivariate Extreme Value Theory (EVT) model [Heffernan, J.E., Tawn, J.A., 2004. A conditional approach for multivariate extreme values (with discussion). Journal of the Royal Statistical Society Series B 66 (3), 497–546] is applied to the estimation of population risks associated with dietary intake of pesticides. The objective is to quantify the acute risk of pesticide intake above a threshold and relate it to the consumption of specific primary food products. As an example daily intakes of a pesticide from three foods are considered. The method models and extrapolates simultaneous intakes of pesticide, and estimates probability of exceeding unobserved large intakes. Multivariate analysis was helpful in identifying whether the avoidance of certain food combinations would reduce the likelihood of exceeding a threshold. We argue that the presented method can be an important contribution to exposure assessment studies.

Banking System Stability: A Cross-Atlantic Perspective

This paper derives indicators of the severity and structure of banking system risk from asymptotic interdependencies between banks%u2019 equity prices. We use new tools available from multivariate extreme value theory to estimate individual banks%u2019 exposure to each other (%u201Ccontagion risk%u201D) and to systematic risk. Moreover, by applying structural break tests to those measures we study whether capital markets indicate changes in the importance of systemic risk over time. Using data for the United States and the euro area, we can also compare banking system stability between the two largest economies in the world. Finally, for Europe we assess the relative importance of cross-border bank spillovers as compared to domestic bank spillovers. The results suggest, inter alia, that systemic risk in the US is higher than in the euro area, mainly as cross-border risks are still relatively mild in Europe. On both sides of the Atlantic systemic risk has increased during the 1990s.

Fat Tail Risk in Portfolios of Hedge Funds and Traditional Investments

We analyse the risk of portfolios mixing hedge funds, stocks and bonds. The risk of the portfolios is quantified by the Value-at-Risk and the Expected Shortfall derived from the Extreme Value Theory. This approach enables us to take the impact of higher moments into account. We show that the risk of a traditional portfolio is reduced by the inclusion of hedge funds. An optimal weight of 50% hedge funds is found when the traditional portfolio is mostly composed of bonds. In equity dominated portfolios, investors should incorporate as much hedge funds as possible. Furthermore we examine the extremal dependence between funds of hedge funds and stocks or bonds using multivariate Extreme Value Theory. We do not find any significant extremal dependence between hedge funds and bonds. The evidence is more mixed between stocks and funds of hedge funds. Funds of hedge funds without a significant investment in Managed Futures exhibit significant dependence in the extreme with the stock market. The August 1998 event linked to the Russian crisis and the LTCM failure is the cause of this dependence.

European stock market dependencies when price changes are unusually large

This article studies dependencies between European stock markets when returns are unusually large ‘extreme’, using daily data on stock market indices for Germany, the UK, France, The Netherlands and Italy from 1973 to 2001. Dependency is measured by the conditional probability of an unusually large return in one market given an unusually large return in another and is estimated using an approach from multivariate extreme value theory. It finds the following. First, dependencies between markets in situations of unusually large returns have become closer over time. Second, they are generally higher for large negative returns than for large positive ones. Third, dependencies differ depending on the country pair considered. For example, stock markets in the Netherlands and France are more closely and those in the UK and Italy less closely linked to the German market. Fourth, overall dependencies are quite symmetric, in the sense that the conditional probability for an unusually large change given a large chan...

Tail Dependence from a Distributional Point of View

The dependence structure in the tails of bivariate random variables is studied by means of appropriate copulae. Weak convergence results show that these copulae are natural dependence structures for joint tail events. The results obtained apply to particular types of copulae such as archimedean copulae and the Gaussian copula. Further, connections to multivariate extreme value theory are investigated and a two-dimensional Pickands–Balkema–de Haan Theorem type is derived. Finally, a counterexample showing that the tail dependence coefficients do not completely determine the dependence structure of bivariate rare events is provided.

Dependencies between European Stock Markets When Price Changes are Unusually Large

The present paper studies dependencies between European stock markets when returns are unusually large, using daily data on stock market indices for Germany, the United Kingdom, France, the Netherlands and Italy from 1973 to 2001. Dependency is measured by the conditional probability of an unusually large return in one market given an unusually large return in another and is estimated using an approach from multivariate extreme value theory. The paper finds the following. First, dependencies between markets in situations of unusually large returns have become closer over time. Second, they are generally higher for large negative returns than for large positive ones. Third, dependencies differ depending on the country pair considered. For example, stock markets in the Netherlands and France are more closely and those in the United Kingdom and Italy less closely linked to the German market. Fourth, overall dependencies are quite symmetric, in the sense that the conditional probability for an unusually large change given a large change in the other country is similar irrespective of which of the two countries the probability is conditioned on.