Tail Risk Aggregation And Dependence Under Heavy Tailed Environment: Theory And Empirical Studies

It is a well-known fact in the literature that tail risk of the underlying portfolio is infu-enced both by the tail behaviors of the margins and their dependence structure. Pioneered byMandelbrot [113] and Fama [71], numerous papers in economics and fnance have indicatedthat time series encountered in these area are typically heavy-tailed distributed. On the otherhand, evidence of extreme co-movements between diferent fnancial markets (assets) hasalso been widely reported in the literature. In this thesis we are going to investigate tail riskaggregation and contagion under heavy tailed environment with the help of extreme valuetheory and copula theory.The main contents and conclusions of the thesis are summarized as follows.(1)Inthefrstpartofthethesis, westudytheimpactofheavytail(marginaltailrisk)andtail dependence (dependence structure risk) on the tail risk of an aggregated portfolio. Weestablish second-order approximation of risk concentration associated with an aggregatedportfolio in terms of Value at Risk (VaR) within the methodological framework of second-order regular variation and the theory of Archimedean copula. The result shows that therate of convergence of the frst-order approximation of risk concentration depends on theinterplay between the tail behavior of the marginal loss random variables and their depen-dence structure. Specifcally, we fnd that the rate of convergence is determined by either thesecond-order parameter of Archimedean copula generator or the second-order parameter ofthe tail margins, leading to either the so-called marginal dominated risk case or dependencerisk dominated case.(3) In light of the theory of regular variation(RV) in extreme value theory (EVT) andcopula, we investigate tail risk aggregation of operational risk under heavy-tailed environ-ment. Because the size of operational losses are usually large and scare, we employ regularvarying function to characterize the tail behavior of the loss severities. In the calculation ofthe operational risk capital for a single cell model, we obtain frst and second-order expan- sion for for operational risk quantifed with spectral risk measure (OpSRM) under second-order regular condition(2RV) as the confdence level converges to100%. Moreover, we in-vestigate a refned second-order approximation of operational risk quantifed with spectralrisk measures (OpSRMs) within the theory of second-order regular variation and second-order subexponentiality. The result show that asymptotically two cases arises depending onwhether ρ <1(fast convergence case) or1<ρ≤0(slow convergence case) holds.We also show that the second-order approximation under2RV is asymptotically equivalentto the slower convergence case. A large number of Monte Carlo simulations for a range ofempirically relevant frequency and severity distributions are employed to illustrate the per-formance of our second-order results. The simulation results indicate that our second-orderapproximations tend to reduce the estimation errors to a great degree, especially for the fastconvergence case, and are able to capture the sub-extremal behavior of OpSRMs better thanthe frst-order approximation.In the calculation of total regulatory capital charge for operational risk, the core prob-lem is to construct an appropriate model to characterize the dependence structure betweendiferent aggregate operational loss processes. Following B cker and Klüppelberg [34] andBiaginiandUlmer[30],weemployLévycopulastocharacterizethedependencestructurebe-tweendiferentaggregateoperationallossprocesses. Wederivefrst-orderasymptoticresultsfor operational spectral risk measure (OpSRM) in various dependence scenarios, includingmultivariate models with one dominating cell, completely dependent cells, and completelyindependent cells.(3)Traditionalmethodbasedon multivariate Gaussianassumptionisnotable tocharac-terize nonlinear dependence structures and extreme co-movements because Gaussian copulahaszerotaildependence. Weinvestigatethetaildependencestructuresbetweencrudeoilandrefned petroleum markets using copula-GARCH method. More precisely, we investigate t-wo types of asymmetries, i.e., the asymmetry in the lower and upper tail dependence andthe asymmetry in the propagation of crisis (bubble), between crude oil market and refnedpetroleum markets. Thirteen copula models with diferent types of dependence structuresand time-varying dependence parameters are considered, of which we fnd that in generalthe MALM copula with the ability to characterize two types of asymmetry is the most ap-propriate model to ft our sample data according to AIC criterion. We fnd evidence of bothpositive lower and upper tail dependence, which indicate that crude oil market and refnedmarkets tend to co-move together. Finally, although our data prefers the nonexchangeable MALM copula according to AIC, the evidence for the asymmetry in the propagation of crisis(bubble) between crude oil and refned products is very weak.(4) Up until now most of the papers focus on investigating the dependence structuresbetween fnancial variables of interest using close-to-close returns. The literature to studythe tail dependence structures between overnight returns and daytime returns respectivelyare few. Therefore, we study the dynamic dependence structures between the overnight re-turn series and daytime return series of four major bank stocks in China, respectively, usingcopula-GARCH models. Besides, to examine the impact of the creation of CSI300stockindex futures on the dependence structure, we use the date (April,16,2010) when CSI300index futures was launched to break the sample into two parts. Our results show that the de-pendence between the overnight returns and day-time return series are time-varying. More-over, the magnitude of the dependence decrease substantially after the creation of the CSI300index futures. Additionally, in general the correlation (dependence) between daytimereturn series are larger than the correlation (dependence) between overnight return series,especially for the period after the creation of the CSI300index futures.The main innovations of this thesis include:(1) We investigate tail risk aggregation under heavy tailed environment with thehelp of extreme value theory and the theory of copula.Second-order asymptotic result for risk concentration is obtained under second-orderregular condition (2RV). The result shows that the rate of convergence is determined byeither the second-order parameter of Archimedean copula generator or the second-order pa-rameter of the tail margins, leading to either the so-called marginal dominated risk case ordependence risk dominated case.(2) In light of the theory of regular variation(RV) in extreme value theory (EVT)and copula, we investigate tail risk aggregation of operational risk under heavy-tailedenvironment.Diferent from the previous studies which use the risk measure VaR or Expected Short-fall (ES) to quantify the aggregation of operational risk, this paper employ a more generalcoherent risk measure SRMs which allows international commercial banks (bank superviso-ry committee) to include their own risk attitudes in the estimation of capital charges to coverthepotentiallossofoperationalrisk. Weobtainseveralasymptoticresultsfortheaggregationof operational risk with the help of RV and copula theory.(3)Weinvestigatetaildependencestructuresbetweencrudeoilandrefnedpetroleum markets using copula method.Diferent from the previous literature which study the lead-lag relationship between oilprice and refned products prices, this paper aims to investigate their contemporaneous rela-tionship, or how they co-move using copula methodology. Besides, we employ a new typeof nonexchangeable copula with the ability to characterize two types of asymmetry to studytail risk contagion between crude oil and refned petroleum markets.(4) We investigate tail dependence structures between the overnight return se-ries and daytime return series of four major bank stocks in China, respectively, usingcopula-GARCH models.The results show that the dependence between the overnight returns and day-time returnseries are time-varying. Moreover, the magnitude of the dependence decrease substantial-ly after the creation of the CSI300index futures. Additionally, in general the correlation(dependence) between daytime return series are larger than the correlation (dependence) be-tweenovernight return series, especially for the period after the creation of the CSI300indexfutures.