Extremal Dependence Structure Research Articles

Modeling the dependence structure and systemic risk of all listed insurance companies in the Chinese insurance market

AbstractAs a developing country, China's insurance market is already the second largest insurance market in the world, and its development is inseparable from the comprehensive development of insurance companies' business. At present, insurance companies are engaged in both traditional and nontraditional insurance businesses. Thus, to avoid extreme financial events, the insurance market requires more risk management. Hence, examining systemic risk in the insurance market in China is an interesting research topic. This paper studies the comovement between all listed insurance companies and the insurance market in China by modeling the average and extreme dependence structure using different copula specifications. Furthermore, it analyzes both downside and upside risk spillover effects from insurance companies to the insurance market by quantifying the following three market risk measures: the value‐at‐risk (VaR), the conditional VaR (CoVaR) and the delta CoVaR (ΔCoVaR). Finally, we test the significance of risk spillover effects and provide a formal ranking of the listed insurance companies with respect to their contribution to both the downside and upside systemic risk of the insurance market.

Bootstrapping Hill estimator and tail array sums for regularly varying time series

In the extreme value analysis of stationary regularly varying time series, tail array sums form a broad class of statistics suitable to analyze their extremal behavior. This class includes for example, the Hill estimator or estimators of the extremogram and the tail dependence coefficient. General asymptotic theory for tail array sums has been developed by Rootzén et al. (Ann. Appl. Probab. 8 (1998) 868–885) under mixing conditions and in Kulik et al. (Stochastic Process. Appl. 129 (2019) 4209–4238) for functions of geometrically ergodic Markov chains. A more general framework of cluster functionals is presented in Drees and Rootzén (Ann. Statist. 38 (2010) 2145–2186). However, the resulting limiting distributions turn out to be very complex and cumbersome to estimate as they usually depend on the whole extremal dependence structure of the time series. Hence, a suitable bootstrap procedure is desired, but available bootstrap consistency results for tail array sums are scarce. In this paper, following Drees (Drees (2015)), we consider a multiplier block bootstrap to estimate the limiting distribution of tail array sums. We prove that, conditionally on the data, an appropriately constructed multiplier block bootstrap statistic converges to the correct limiting distribution. Interestingly, in contrast, it turns out that an apparently natural, but naïve application of the multiplier block bootstrap scheme does not yield the correct limit. In simulations, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed multiplier block bootstrap over some obvious competitors. The proposed bootstrap scheme proves to be computationally efficient in comparison to other approaches.

Conditional modelling approach to multivariate extreme value distributions: application to extreme rainfall events in South Africa

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. The results showed that the multivariate conditional modelling fitted to daily maximum rainfall events provided apparent benefits in terms of improved precision in the estimation of the marginal parameters of generalised Pareto distribution. The conditional modelling approach provided all forms of dependence using Laplace marginal transformations, for which all weather stations are not extreme equally. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors, and compared with the prediction obtained from the conditional model fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, and the extremal dependence structure described consistently for extreme daily maximum rainfall events between weather stations. The current study contributes towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with e.g., flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by responsible authorities.

Statistical Modelling of Extreme Rainfall Indices using Multivariate Extreme Value Distributions

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. Moreover, 1-day to 5-day indices of rainfall events were constructed to investigate the frequencies and intensities of rainfall events for selected weather stations. Then, the conditional multivariate modelling was fitted to investigate dependence between series of extreme rainfall events. The conditional multivariate modelling has provided all forms of dependence, using Laplace marginal transformations, for which all weather stations are not equally extreme. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors and compared with the prediction obtained from the conditional modelling that was fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, as well as the extremal dependence structure described consistently for indices of rainfall events between weather stations. The modelling framework and results of this study contribute towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with, e.g. flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by communities.

Spatial deformation for nonstationary extremal dependence

AbstractModeling the extremal dependence structure of spatial data is considerably easier if that structure is stationary. However, for data observed over large or complicated domains, nonstationarity will often prevail. Current methods for modeling nonstationarity in extremal dependence rely on models that are either computationally difficult to fit or require prior knowledge of covariates. Sampson and Guttorp (1992) proposed a simple technique for handling nonstationarity in spatial dependence by smoothly mapping the sampling locations of the process from the original geographical space to a latent space where stationarity can be reasonably assumed. We present an extension of this method to a spatial extremes framework by considering least squares minimization of pairwise theoretical and empirical extremal dependence measures. Along with some practical advice on applying these deformations, we provide a detailed simulation study in which we propose three spatial processes with varying degrees of nonstationarity in their extremal and central dependence structures. The methodology is applied to Australian summer temperature extremes and UK precipitation to illustrate its efficacy compared with a naive modeling approach.

Modeling Time-Varying Tail Dependence, with Application to Systemic Risk Forecasting

Abstract Empirical evidence for multivariate stock suggests that there are changes from asymptotic independence to asymptotic dependence and vice versa. Under asymptotic independence, the probability of joint extremes vanishes, whereas under asymptotic dependence, this probability remains positive. In this paper, we propose a dynamic model for bivariate extremes that allows for smooth transitions between regimes of asymptotic independence and asymptotic dependence. In doing so, we ignore the bulk of the distribution and only model the joint tail of interest. We propose a maximum-likelihood estimator for the model parameters and demonstrate its accuracy in simulations. An empirical application to losses on the CAC 40 and DAX 30 illustrates that our model provides a detailed description of changes in the extremal dependence structure. Furthermore, we show that our model issues adequate forecasts of systemic risk, as measured by CoVaR. Finally, we find some evidence that our CoVaR forecasts outperform those of a benchmark dynamic t-copula model.returns

Co-Movement between Tourist Arrivals of Inbound Tourism Markets in South Korea: Applying the Dynamic Copula Method Using Secondary Time Series Data

Tourism demand is severely affected by unpredicted events, which has prompted scholars to examine ways of predicting the effects of positive and negative shocks on tourism, to ensure a sustainable tourism industry. The purpose of this study was to investigate if non-linear dependence structures exist between tourist flows into South Korea from five major source countries, as South Korea has undergone fluctuations in tourist arrivals due to diverse circumstances and has complex relations with tourism source countries. Additionally, the study examines the structures of extreme tail dependence, which is indicated in the case of unexpected events, and identifies how co-movements vary over time through dynamic copula–GARCH (generalized autoregressive conditional heteroskedasticity) tests. The secondary time series data for the 2005–2019 period of tourist arrivals to Korea were derived from the Korea Tourism Knowledge and Information System for testing the copula models. The copula estimations indicate significant dependencies among all market pairs as well as the strongest dependence between China and Taiwan. Moreover, extreme tail dependence structures show co-movements for four pairs of tourism markets in only negative shocks, for five pairs in both positive and negative conditions, but no co-movement in the China–Taiwan pair. Finally, the dynamic dependence structures reveal that the China–Taiwan dependence is higher than the other time-varying dependence structures, implying that the two markets complement each other.

Evaluating the dependence structure of compound precipitation and wind speed extremes

Abstract. Estimating the likelihood of compound climate extremes such as concurrent drought and heatwaves or compound precipitation and wind speed extremes is important for assessing climate risks. Typically, simulations from climate models are used to assess future risks, but it is largely unknown how well the current generation of models represents compound extremes. Here, we introduce a new metric that measures whether the tails of bivariate distributions show a similar dependence structure across different datasets. We analyse compound precipitation and wind extremes in reanalysis data and different high-resolution simulations for central Europe. A state-of-the-art reanalysis dataset (ERA5) is compared to simulations with a weather model (Weather Research and Forecasting – WRF) either driven by observation-based boundary conditions or a global circulation model (Community Earth System Model – CESM) under present-day and future conditions with strong greenhouse gas forcing (Representative Concentration Pathway 8.5 – RCP8.5). Over the historical period, the high-resolution WRF simulations capture precipitation and wind extremes as well as their response to orographic effects more realistically than ERA5. Thus, WRF simulations driven by observation-based boundary conditions are used as a benchmark for evaluating the dependence structure of wind and precipitation extremes. Overall, boundary conditions in WRF appear to be the key factor in explaining differences in the dependence behaviour between strong wind and heavy precipitation between simulations. In comparison, external forcings (RCP8.5) are of second order. Our approach offers new methodological tools to evaluate climate model simulations with respect to compound extremes.

Dependence Structure between Tourist Arrivals of Inbound Tourism Markets inSouth Korea

Tourism demand is severely affected by unpredicted events. The purpose of this study was to investigate if non-linear dependence structures exist between tourist flows into South Korea from five major source countries, as South Korea has undergone fluctuations in tourist arrivals due to diverse circumstances and has complex relations with tourism source countries. Additionally, the study examines the structures of extreme tail dependence, which is indicated in the case of unexpected events, and identifies how co-movements vary over time through dynamic copula-GARCH (Generalized Autoregressive Conditional Heteroskedasticity) tests. The copula estimations indicate significant dependencies among all market pairs as well as the strongest dependence between China and Taiwan. Moreover, extreme tail dependence structures show co-movements for four pairs of tourism markets in only negative shocks, for five pairs in both positive and negative conditions, but no co-movement in the China-Taiwan pair. Finally, the dynamic dependence structures reveal that the China-Taiwan dependence is higher than the other time-varying dependence structures, implying that the two markets complement each other.

Multivariate extremes over a random number of observations

AbstractThe classical multivariate extreme‐value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max‐stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max‐stable distributions. We also connect the extremal dependence structure of classical max‐stable distributions and that of our new family of max‐stable distributions. Using an inversion method, we derive a semiparametric composite‐estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large‐sample theory of the composite‐estimator and illustrate its finite‐sample performance via a simulation study.

High-dimensional inference using the extremal skew-t process

Max-stable processes are a popular tool for the study of environmental extremes, and the extremal skew-t process is a general model that allows for a flexible extremal dependence structure. For inference on max-stable processes with high-dimensional data, exact likelihood-based estimation is computationally intractable. Composite likelihoods, using lower dimensional components, and Stephenson-Tawn likelihoods, using occurrence times of maxima, are both attractive methods to circumvent this issue for moderate dimensions. In this article we establish the theoretical formulae for simulations of and inference for the extremal skew-t process. We also incorporate the Stephenson-Tawn concept into the composite likelihood framework, giving greater statistical and computational efficiency for higher-order composite likelihoods. We compare 2-way (pairwise), 3-way (triplewise), 4-way, 5-way and 10-way composite likelihoods for models of up to 100 dimensions. Furthermore, we propose cdf approximations for the Stephenson-Tawn likelihood function, leading to large computational gains, and enabling accurate fitting of models in large dimensions in only a few minutes. We illustrate our methodology with an application to a 90-dimensional temperature dataset from Melbourne, Australia.

Determining the dependence structure of multivariate extremes

Summary In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies in determining which subsets of variables can take their largest values simultaneously while the others are of smaller order. Our approach to this problem exploits hidden regular variation properties on a collection of nonstandard cones, and provides a new set of indices that reveal aspects of the extremal dependence structure not available through existing measures of dependence. We derive theoretical properties of these indices, demonstrate their utility through a series of examples, and develop methods of inference that also estimate the proportion of extremal mass associated with each cone. We apply the methods to river flows in the U.K., estimating the probabilities of different subsets of sites being large simultaneously.

Modelling The Spatial Extent and Severity of Extreme European Windstorms

Summary Windstorms are a primary natural hazard affecting Europe that are commonly linked to substantial property and infrastructural damage and are responsible for the largest spatially aggregated financial losses. Such extreme winds are typically generated by extratropical cyclone systems originating in the North Atlantic and passing over Europe. Previous statistical studies tend to model extreme winds at a given set of sites, corresponding to inference in an Eulerian framework. Such inference cannot incorporate knowledge of the life cycle and progression of extratropical cyclones across the region and is forced to make restrictive assumptions about the extremal dependence structure. We take an entirely different approach which overcomes these limitations by working in a Lagrangian framework. Specifically, we model the development of windstorms over time, preserving the physical characteristics linking the windstorm and the cyclone track, the path of local vorticity maxima, and make a key finding that the spatial extent of extratropical windstorms becomes more localized as its magnitude increases irrespective of the location of the storm track. Our model allows simulation of synthetic windstorm events to derive the joint distributional features over any set of sites giving physically consistent extrapolations to rarer events. From such simulations improved estimates of this hazard can be achieved in terms of both intensity and area affected.

Space–time trends and dependence of precipitation extremes in North‐Western Germany

The assessment of long‐term trends in environmental extremes is a challenging and important subject in the current discussion on global climate change. We propose a new approach for evaluating temporal trends and spatial homogeneity in extremes accounting also for spatial dependence. Based on exceedances over a space–time threshold, we provide estimators for a so‐called scedasis function and the extreme value parameters. Two novel test statistics based on the scedasis function provide a way to assess space–time inhomogeneities. Finally, we propose a procedure to achieve stationarity in space and time and evaluate residual extremal dependence over space through a variogram analysis that includes anisotropy.The methods are applied to daily accumulated precipitation data observed at 64 weather stations in North‐Western Germany. The time period from 1931 to 2014 is considered, separately for northern hemispheric winter (November – March) and summer (May – September). The results reveal a clear seasonal differentiation on extremal behavior with the highest precipitation records and generally heavier tails during summer. Spatial inhomogeneities in the frequency of precipitation extremes are related to topography and are particularly significant during winter. During this season, we further observe significant deviations from time‐stationarity at a few stations, related to a period of a high frequency of extremes around 1992. Spatial anisotropy in the extremal dependence structure is stronger during winter than during summer and reveals an increased dependence in the east–west direction.

Nonparametric inference for distortion risk measures on tail regions

Suppose X is some interesting loss and Y is a benchmark variable. Given some extreme scenarios of Y, it is indispensable to measure the tail risk of X by applying a class of univariate risk measures to study the co-movement of the two variables. In this paper, we consider the extreme and nonparametric inference for the distortion risk measures on the tail regions when the extreme scenarios of some benchmark variable are considered. We derive the limit of the proposed risk measures based on Extreme Value Theory. The asymptotics of the risk measures shows the decomposition of the marginal extreme value index and the extreme dependence structure which implies how these two pieces of information have influences on the limit of the risk measures. Finally, for practical purpose, we develop a nonparametric estimation method for the distortion risk measures on tail regions and its asymptotic normality is derived.

Weak comonotonicity

The classical notion of comonotonicity has played a pivotal role when solving diverse problems in economics, finance, and insurance. In various practical problems, however, this notion of extreme positive dependence structure is overly restrictive and sometimes unrealistic. In the present paper, we put forward a notion of weak comonotonicity, which contains the classical notion of comonotonicity as a special case, and gives rise to necessary and sufficient conditions for a number of optimization problems, such as those arising in portfolio diversification, risk aggregation, and premium calculation. In particular, we show that a combination of weak comonotonicity and weak antimonotonicity with respect to some choices of measures is sufficient for the maximization of Value-at-Risk aggregation, and weak comonotonicity is necessary and sufficient for the Expected Shortfall aggregation. Finally, with the help of weak comonotonicity acting as an intermediate notion of dependence between the extreme cases of no dependence and strong comonotonicity, we give a natural solution to a risk-sharing problem.

Local Likelihood Estimation of Complex Tail Dependence Structures, Applied to U.S. Precipitation Extremes

To disentangle the complex nonstationary dependence structure of precipitation extremes over the entire contiguous United States (U.S.), we propose a flexible local approach based on factor copula models. Our subasymptotic spatial modeling framework yields nontrivial tail dependence structures, with a weakening dependence strength as events become more extreme; a feature commonly observed with precipitation data but not accounted for in classical asymptotic extreme-value models. To estimate the local extremal behavior, we fit the proposed model in small regional neighborhoods to high threshold exceedances, under the assumption of local stationarity, which allows us to gain in flexibility. By adopting a local censored likelihood approach, we make inference on a fine spatial grid, and we perform local estimation by taking advantage of distributed computing resources and the embarrassingly parallel nature of this estimation procedure. The local model is efficiently fitted at all grid points, and uncertainty is measured using a block bootstrap procedure. We carry out an extensive simulation study to show that our approach can adequately capture complex, nonstationary dependencies, in addition, our study of U.S. winter precipitation data reveals interesting differences in local tail structures over space, which has important implications on regional risk assessment of extreme precipitation events. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Bayesian Model Averaging Over Tree-based Dependence Structures for Multivariate Extremes

Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue, we develop a novel statistical method that describes extremal dependence taking advantage of the inherent tree-based dependence structure of the max-stable nested logistic distribution, and which identifies possible clusters of extreme variables using reversible jump Markov chain Monte Carlo techniques. Parsimonious representations are achieved when clusters of extreme variables are found to be completely independent. Moreover, we significantly decrease the computational complexity of full likelihood inference by deriving a recursive formula for the likelihood function of the nested logistic model. The method’s performance is verified through extensive simulation experiments which also compare different likelihood procedures. The new methodology is used to investigate the dependence relationships between extreme concentrations of multiple pollutants in California and how these concentrations are related to extreme weather conditions. Overall, we show that our approach allows for the representation of complex extremal dependence structures and has valid applications in multivariate data analysis, such as air pollution monitoring, where it can guide policymaking. Supplementary materials for this article are available online.

On additivity of tail comonotonic risks

ABSTRACTAs perceived from daily experience together with numerous empirical studies, the multivariate risks demonstrate a strong coherence in the extremal dependence structure especially over the course of financial turmoil or industrial accidents and outbreaks. Under this motivating paradigm, we show the universal asymptotic additivity under upper tail comonotonicity, as the probability level approaching to 1, for Value-at-Risk and Conditional Tail Expectation for a portfolio of fixed number of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three max domains of attraction. Our obtained results do not require the tail equivalence assumption as needed in the existing literature. This resolves a lasting problem in quantitative risk management and covers most distributions commonly encountered in practice.

Semiparametric bivariate modelling with flexible extremal dependence

Inference over multivariate tails often requires a number of assumptions which may affect the assessment of the extreme dependence structure. Models are usually constructed in such a way that extreme components can either be asymptotically dependent or be independent of each other. Recently, there has been an increasing interest on modelling multivariate extremes more flexibly, by allowing models to bridge both asymptotic dependence regimes. Here we propose a novel semiparametric approach which allows for a variety of dependence patterns, be them extremal or not, by using in a model-based fashion the full dataset. We build on previous work for inference on marginal exceedances over a high, unknown threshold, by combining it with flexible, semiparametric copula specifications to investigate extreme dependence, thus separately modelling marginals and dependence structure. Because of the generality of our approach, bivariate problems are investigated here due to computational challenges, but multivariate extensions are readily available. Empirical results suggest that our approach can provide sound uncertainty statements about the possibility of asymptotic independence, and we propose a criterion to quantify the presence of either extreme regime which performs well in our applications when compared to others available. Estimation of functions of interest for extremes is performed via MCMC algorithms. Attention is also devoted to the prediction of new extreme observations. Our approach is evaluated through simulations, applied to real data and assessed against competing approaches. Evidence demonstrates that the bulk of the data do not bias and improve the inferential process for extremal dependence in our applications.