Multivariate Extreme Value Theory Research Articles

Clustering of Maxima: Spatial Dependencies among Heavy Rainfall in France

Abstract One of the main objectives of statistical climatology is to extract relevant information hidden in complex spatial–temporal climatological datasets. To identify spatial patterns, most well-known statistical techniques are based on the concept of intra- and intercluster variances (like the k-means algorithm or EOFs). As analyzing quantitative extremes like heavy rainfall has become more and more prevalent for climatologists and hydrologists during these last decades, finding spatial patterns with methods based on deviations from the mean (i.e., variances) may not be the most appropriate strategy in this context of studying such extremes. For practitioners, simple and fast clustering tools tailored for extremes have been lacking. A possible avenue to bridging this methodological gap resides in taking advantage of multivariate extreme value theory, a well-developed research field in probability, and to adapt it to the context of spatial clustering. In this paper, a novel algorithm based on this plan is proposed and studied. The approach is compared and discussed with respect to the classical k-means algorithm throughout the analysis of weekly maxima of hourly precipitation recorded in France (fall season, 92 stations, 1993–2011).

Spatial assessment of precipitation deficits in the Duero basin (central Spain) with multivariate extreme value statistics

[1] Nonirrigated agriculture on the Iberian Peninsula is regularly affected by dry periods that can cause important losses. This paper focuses on the comparison of the classical Standard Precipitation Index (SPI) with a fragility index developed by the multivariate extreme value theory community, which is used to describe monthly precipitation deficits below 30.5 mm (about 1 mm/d) in the Spanish Duero basin. The multivariate extreme value model allows to capture relevant information concerning the dependence structure among extreme precipitation deficits. Maps of those extremal dependence summaries and of loadings of principal components of the SPI provide quantitative information for water management. In addition, jointly analyzing data from several stations improves the inference of uncertainty. Spatial patterns of extremal dependence emerged with respect to orographic features. Most severe dry spells occur in the southeast of the Duero basin. In central plain of the Duero basin, a predominantly agricultural area, a strong fragility index for severity of dry spells is particularly found in eastern regions. Results of the MEVT and SPI analysis point in the same direction. Beyond this, the MEVT assessment gives a quantitative measure of the dependence between stations and regions. Estimates of return periods for extreme dry spell severity are discussed. Deficits below 42.7 mm are also analyzed.

Two case studies on NARCCAP precipitation extremes

We introduce novel methodology to examine the ability of six regional climate models (RCMs) in the North American Regional Climate Change Assessment Program (NARCCAP) ensemble to simulate past extreme precipitation events seen in the observational record over two different regions and seasons. Our primary objective is to examine the strength of daily correspondence of extreme precipitation events between observations and the output of both the RCMs and the driving reanalysis product. To explore this correspondence, we employ methods from multivariate extreme value theory. These methods require that we account for marginal behavior, and we first model and compare climatological quantities which describe tail behavior of daily precipitation for both the observations and model output before turning attention to quantifying the correspondence of the extreme events. Daily precipitation in a West Coast region of North America is analyzed in two seasons, and it is found that the simulated extreme events from the reanalysis‐driven NARCCAP models exhibit strong daily correspondence to extreme events in the observational record. Precipitation over a central region of the United States is examined, and we find some daily correspondence between winter extremes simulated by reanalysis‐driven NARCCAP models and those seen in observations, but no such correspondence is found for summer extremes. Furthermore, we find greater discrepancies among the NARCCAP models in the tail characteristics of the distribution of daily summer precipitation over this region than seen in precipitation over the West Coast region. We find that the models which employ spectral nudging exhibit stronger tail dependence to observations in the central region.

Estimating the basis risk of index-linked hedging strategies using multivariate extreme value theory

This paper studies the empirical quantification of basis risk in the context of index-linked hedging strategies. Basis risk refers to the risk of non-payment of the index-linked instrument, given that the hedger’s loss exceeds some critical level. The quantification of such risk measures from empirical data can be done in various ways and requires special consideration of the dependence structure between the index and the company’s losses as well as the estimation of the tails of a distribution. In this context, previous literature shows that extreme value theory can be superior to traditional methods with respect to estimating quantile risk measures such as the value at risk. Thus, the aim of this paper is to conduct an empirical analysis of basis risk using multivariate extreme value theory and extreme value copulas to estimate the underlying risk processes and their dependence structure in order to obtain a more adequate picture of basis risk associated with index-linked hedging strategies. Our results emphasize that the application of extreme value theory leads to better fits of the tails of the marginal distributions in the considered stock price sample and that traditional methods in regard to estimating marginal distributions tend to overestimate basis risk, while basis risk can in contrast be higher when taking into account extreme value copulas.

Measuring systemic importance of financial institutions: An extreme value theory approach

This paper proposes a set of market-based measures on the systemic importance of a financial institution or a group of financial institutions, each designed to capture different aspects of systemic importance of financial institutions. Multivariate extreme value theory approach is used to estimate these measures. Using six big Canadian banks as the proxy for Canadian banking sector, we apply these measures to identify systemically important banks in Canadian banking sector and major risk contributors from international financial institutions to Canadian banking sector. The empirical evidence reveals that (i) the top three banks, RBC Financial Group, TD Bank Financial Group, and Scotiabank, are more systemically important than other banks, while we also find that the size of a financial institution should not be considered as a proxy of systemic importance; (ii) compared to the European and Asian banks, the crashes of the U.S. banks, on average, are the most damaging to Canadian banking sector, while the risk contribution to the Canadian banking sector from Asian banks is quite lower than that from banks in the U.S. and euro area; (iii) the risk contribution to Canadian banking sector exhibits “home bias”, that is, cross-country risk contribution tends to be smaller than domestic risk contribution.

Bayesian model averaging for multivariate extremes

The main framework of multivariate extreme value theory is well-known in terms of probability, but inference and model choice remain an active research field. Theoretically, an angular measure on the positive quadrant of the unit sphere can describe the dependence among very high values, but no parametric form can entirely capture it. The practitioner often makes an assertive choice and arbitrarily fits a specific parametric angular measure on the data. Another statistician could come up with another model and a completely different estimate. This leads to the problem of how to merge the two different fitted angular measures. One natural way around this issue is to weigh them according to the marginal model likelihoods. This strategy, the so-called Bayesian Model Averaging (BMA), has been extensively studied in various context, but (to our knowledge) it has never been adapted to angular measures. The main goal of this article is to determine if the BMA approach can offer an added value when analyzing extreme values.

Systemic Contingent Claims Analysis: Estimating Market-Implied Systemic Risk

The recent global financial crisis has forced a re-examination of risk transmission in the financial sector and how it affects financial stability. Current macroprudential policy and surveillance (MPS) efforts are aimed establishing a regulatory framework that helps mitigate the risk from systemic linkages with a view towards enhancing the resilience of the financial sector. This paper presents a forward-looking framework (Systemic CCA) to measure systemic solvency risk based on market-implied expected losses of financial institutions with practical applications for the financial sector risk management and the system-wide capital assessment in top-down stress testing. The suggested approach uses advanced contingent claims analysis (CCA) to generate aggregate estimates of the joint default risk of multiple institutions as a conditional tail expectation using multivariate extreme value theory (EVT). In addition, the framework also helps quantify the individual contributions to systemic risk and contingent liabilities of the financial sector during times of stress.

Systemic Risk Allocation for Systems with a Small Number of Banks

This paper provides a new estimation method for the marginal expected shortfall (MES) based on multivariate extreme value theory. In contrast to previous studies, the method does not assume specific dependence structure among bank equity returns and is applicable to both large and small systems. Furthermore, our MES estimator inherits the theoretical additive property. Thus, it serves as a tool to allocate systemic risk. We apply the proposed method to 29 global systemically important financial institutions (G-SIFIs) to evaluate the cross sections and dynamics of the systemic risk allocation. We show that allocating systemic risk according to either size or individual risk is imperfect and can be unfair. Between the allocation with respect to individual risk and that with respect to size, the former is less unfair. On the time dimension, both allocation fairness across all the G-SIFIs has decreased since 2008.

Systemic Contingent Claims Analysis – Estimating Market-Implied Systemic Risk

The recent global financial crisis has forced a re-examination of risk transmission in the financial sector and how it affects financial stability. Current macroprudential policy and surveillance (MPS) efforts are aimed establishing a regulatory framework that helps mitigate the risk from systemic linkages with a view towards enhancing the resilience of the financial sector. This paper presents a forward-looking framework (Systemic CCA) to measure systemic solvency risk based on market-implied expected losses of financial institutions with practical applications for the financial sector risk management and the system-wide capital assessment in top-down stress testing. The suggested approach uses advanced contingent claims analysis (CCA) to generate aggregate estimates of the joint default risk of multiple institutions as a conditional tail expectation using multivariate extreme value theory (EVT). In addition, the framework also helps quantify the individual contributions to systemic risk and contingent liabilities of the financial sector during times of stress.

Study of the Tail Dependence Structure in Global Financial Markets Using Extreme Value Theory

Abstract: The presence of tail dependencies invalidates the multivariate normality assumptions in portfolio risk management. The identification of tail (in)dependencies has drawn major attention in empirical financial studies. Yet it is still a challenging issue both theoretically and practically. Previous studies based on either a restrictive model or the null hypothesis of tail (perfect) dependence does not well describe or interpret extreme co-movements in financial markets. This paper examines tail dependence structures underlying a broad range of financial asset classes employing the newly developed tail quotient correlation coefficients. In theory, the original tail quotient correlation coefficient proposed in (Zhang 2008) is adapted to incorporate cases with varying data driven random thresholds. Our empirical results demonstrate different tail dependence structures underlying various global financial markets. Either omission or unanimous treatment of the tail dependence structures for different financial markets will lead to erroneous conclusions or suboptimal investment choices. The multivariate extreme value theory framework in this study has the potential to serve as an useful tool in exploiting arbitrage opportunities, optimizing asset allocations, and building robust risk management strategies.

Assessment dynamic ratio for traffic loading on highway bridges

The determination of characteristic bridge load effect is a complex problem. Usually, statistical extrapolation of simulated static load effects is used to derive a lifetime characteristic static load effect. However, when a vehicle crosses a bridge, dynamic interactions occur which often cause greater total load effect. This total load effect is related to the static load effect through a dynamic amplification factor (DAF). Specifications often recommend a conservative level for DAF, based on bridge length, number of lanes, and type of load effect only. Therefore significant improvements in the accuracy of this calculation are possible if a DAF, specific to the considered bridge, is applied. In this paper, the authors develop a novel method that considers site-specific bridge and traffic load conditions and allows for the reduced probability of both high static loading and high dynamic interaction occurring simultaneously. This approach utilises multivariate extreme value theory, in conjunction with static simulations and finite element vehicle–bridge dynamic interaction models. It is found that the dynamic allowance for the sample bridge and traffic considered is significantly less than recommended by bridge codes. This finding can have significant implications for the assessment of existing bridge stock.

Fragility index of block tailed vectors

Financial crises are a recurrent phenomenon with important effects on the real economy. The financial system is inherently fragile and it is therefore of great importance to be able to measure and characterize its systemic stability. Multivariate extreme value theory provide us such a framework through the fragility index (Geluk et al., 2007; Falk and Tichy, to appear-a, to appear-b). Here we generalize this concept and contribute to the modeling of the stability of a stochastic system divided into blocks. We will find several relations with well-known tail dependence measures in the literature, which will provide us immediate estimators. We end with an application to financial data.

Extreme value analysis of actuarial risks: estimation and model validation

We give an overview of several aspects arising in the statistical analysis of extreme risks with actuarial applications in view. In particular it is demonstrated that empirical process theory is a very powerful tool, both for the asymptotic analysis of extreme value estimators and to devise tools for the validation of the underlying model assumptions. While the focus of the paper is on univariate tail risk analysis, the basic ideas of the analysis of the extremal dependence between different risks are also outlined. Here we emphasize some of the limitations of classical multivariate extreme value theory and sketch how a different model proposed by Ledford and Tawn can help to avoid pitfalls. Finally, these theoretical results are used to analyze a data set of large claim sizes from health insurance.

On tail dependence: A characterization for first-order max-autoregressive processes

In this paper, we consider first-order MARMA or ARMAX processes and a modified version of these involving a power transformation, denoted pARMAX. We assume Pareto-type tails, the most interesting case for inference within these processes. Some well-known dependence measures of multivariate extreme value theory are considered in a time series framework. In calculating these measures, we find that ARMAX and pARMAX have opposite behavior in concomitant extremes, covering all types of tail dependence. This characterization will serve modeling purposes.

Spatial modeling of extreme snow depth

The spatial modeling of extreme snow is important for adequate risk management in Alpine and high altitude countries. A natural approach to such modeling is through the theory of max-stable processes, an infinite-dimensional extension of multivariate extreme value theory. In this paper we describe the application of such processes in modeling the spatial dependence of extreme snow depth in Switzerland, based on data for the winters 1966--2008 at 101 stations. The models we propose rely on a climate transformation that allows us to account for the presence of climate regions and for directional effects, resulting from synoptic weather patterns. Estimation is performed through pairwise likelihood inference and the models are compared using penalized likelihood criteria. The max-stable models provide a much better fit to the joint behavior of the extremes than do independence or full dependence models.

The Simple Econometrics of Tail Dependence

The aim of this paper is to show that measures on tail dependence can be estimated in a convenient way by regression analysis. This yields the same estimates as the non-parametric method within the multivariate Extreme Value Theory framework. The advantage of the regression approach is contained by its straightforward extension to the estimation of higher dimensional tail dependence. We provide an example on international stock markets. The regression approach to tail dependence can be applied to estimate several measures of systemic importance of financial institutions in the literature.

Conditioning on an extreme component: Model consistency with regular variation on cones

Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector. This necessitates that each component satisfies a marginal domain of attraction condition. An approximation of the joint distribution of a random vector obtained by conditioning on one of the components being extreme was developed by Heffernan and Tawn [12] and further studied by Heffernan and Resnick [11]. These papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we here provide clarification of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory and standard regular variation on cones of the form $[0,\infty]\times(0,\infty]$.

COMOVEMENTS OF DIFFERENT ASSET CLASSES DURING MARKET STRESS

This paper assesses the linkages between the most important U.S. financial asset classes (stocks, bonds, T-bills and gold) during periods of financial turmoil. Our results have potentially important implications for strategic asset allocation and pension fund management. We use multivariate extreme value theory to estimate the exposure of one asset class to extreme movements in the other asset classes. By applying structural break tests to those measures we study to what extent linkages in extreme asset returns and volatilities are changing over time. Univariate results andch bivariate comovement results exhibit significant breaks in the 1970s and 1980s corresponding to the turbulent times of e.g. the oil shocks, Volcker’s presidency of the Fed or the stock market crash of 1987.

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-tailedness of the risk factors as well as the non-parametric tail dependence structure. This allows a large range of models on the dependence. We assess the Value-at-Risk of a diversified portfolio constructed from dependent risk factors. Moreover, we examine the diversification effects under this setup.

Are Banks Too Big To Fail? Measuring Systemic Importance of Financial Institutions

We consider three measures on the systemic importance of a financial institution within a interconnected financial system. Based on the measures, we study the relation between the size of a financial institution and its systemic importance. From both theoretical model and empirical analysis, we find that in analyzing the systemic risk posed by one financial institution to the system, size should not be considered as a proxy of systemic importance. In other words, the too big to fail argument is not always valid, and alternative measures on systemic importance should be considered. We provide the estimation methodology of systemic importance measures under the multivariate Extreme Value Theory (EVT) framework.