Risk contagion under regular variation and asymptotic tail independence
RRisk contagion under regular variation and asymptotictail independence
BIKRAMJIT DAS ∗ and VICKY FASEN Singapore University of Technology and Design8 Somapah Road, Singapore 487372E-mail: [email protected]
Karlsruhe Institute of TechnologyEnglerstraße 2, 76131 KarlsruheE-mail: [email protected]
Risk contagion concerns any entity dealing with large scale risks. Suppose Z = ( Z , Z ) denotes a risk vector pertainingto two components in some system. A relevant measurement of risk contagion would be to quantify the amount ofinfluence of high values of Z on Z . This can be measured in a variety of ways. In this paper, we study two suchmeasures: the quantity E [( Z − t ) + | Z > t ] called Marginal Mean Excess (MME) as well as the related quantity E [ Z | Z > t ] called Marginal Expected Shortfall (MES). Both quantities are indicators of risk contagion and useful invarious applications ranging from finance, insurance and systemic risk to environmental and climate risk. We workunder the assumptions of multivariate regular variation, hidden regular variation and asymptotic tail independencefor the risk vector Z . Many broad and useful model classes satisfy these assumptions. We present several examplesand derive the asymptotic behavior of both MME and MES as the threshold t → ∞ . We observe that although weassume asymptotic tail independence in the models, MME and MES converge to ∞ under very general conditions;this reflects that the underlying weak dependence in the model still remains significant. Besides the consistency of theempirical estimators we introduce an extrapolation method based on extreme value theory to estimate both MMEand MES for high thresholds t where little data are available. We show that these estimators are consistent andillustrate our methodology in both simulated and real data sets. AMS 2000 subject classifications:
Primary 62G32, 62-09, 60G70; secondary 62G10, 62G15, 60F05.
Keywords: asymptotic tail independence, consistency, expected shortfall, heavy-tail, hidden regular variation, meanexcess, multivariate regular variation, systemic risk.
1. Introduction
The presence of heavy-tail phenomena in data arising from a broad range of applications spanning hydrology[2], finance [37], insurance [16], internet traffic [8, 34], social networks and random graphs [5, 14] and riskmanagement [11, 24] is well-documented. Since heavy-tailed distributions often entail non-existence of somehigher order moments, measuring and assessing dependence in jointly heavy-tailed random variables posesa few challenges. Furthermore, one often encounters the phenomenon of asymptotic tail independence in theupper tails; which means given two jointly distributed heavy-tailed random variables, joint occurrence ofvery high (positive) values is extremely unlikely.In this paper, we look at heavy-tailed random variables under the paradigm of multivariate regular vari-ation possessing asymptotic tail independence in the upper tails and we study the average behavior of oneof the variables given that the other one is large in an asymptotic sense. The presence of asymptotic tailindependence might intuitively indicate that high values of one variable will have little influence on theexpected behavior of the other; we observe that such a behavior is not always true. In fact, under a quitegeneral set of conditions, we are able to calculate the asymptotic behavior of the expected value of a variablegiven that the other one is high. ∗ B. Das gratefully acknowledges support from MOE Tier 2 grant MOE-2013-T2-1-158. B. Das also acknowledges hospitalityand support from Karlsruhe Institute of Technology during a visit in June 2015. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 a r X i v : . [ m a t h . S T ] A p r B. Das and V. Fasen
A major application of assessing such a behavior is in terms of computing systemic risk, where one wantsto assess risk contagion among two risk factors in a system. Proper quantification of systemic risk has been atopic of active research in the past few years; see [1, 3, 6, 15, 17, 28] for further details. Our study concentrateson two such measures of risk in a bivariate set-up where both factors are heavy-tailed and possess asymptotictail independence. Note that our notion of risk contagion refers to the effect of one risk on another and viceversa. Risk contagion has other connotations which we do not address here; for example, it appears in causalmodels with time dependencies; see [19] for a brief discussion.First recall that for a random variable X and 0 < u < u is the quantilefunction VaR u ( X ) := inf { x ∈ R : Pr( X > x ) ≤ − u } = inf { x ∈ R : Pr( X ≤ x ) ≥ u } . Suppose Z = ( Z , Z ) denotes risk related to two different components of a system. We study the behaviorof two related quantities which capture the expected behavior of one risk, given that the other risk is high. Definition 1.1 (Marginal Mean Excess)
For a random vector Z = ( Z , Z ) with E | Z | < ∞ the Marginal Mean Excess (MME) at level p where 0 < p < p ) = E (cid:104) ( Z − VaR − p ( Z )) + (cid:12)(cid:12) Z > VaR − p ( Z ) (cid:105) . (1.1)We interpret the MME as the expected excess of one risk Z over the Value-at-Risk of Z at level (1 − p )given that the value of Z is already greater than the same Value-at-Risk. Definition 1.2 (Marginal Expected Shortfall)
For a random vector Z = ( Z , Z ) with E | Z | < ∞ the Marginal Expected Shortfall (MES) at level p where 0 < p < p ) = E [ Z | Z > VaR − p ( Z )] . (1.2)We interpret the MES as the expected shortfall of one risk given that the other risk is higher than its Value-atrisk at level (1 − p ). Note that smaller values of p lead to higher values of VaR − p .In the context of systemic risk, we may think of the conditioned variable Z to be the risk of the entiresystem (for example, the entire market) and the variable Z as one component of the risk (for example, onefinancial institution). Hence, we are interested in the average or expected behavior of one specific componentwhen the entire system is in distress. Although the problem is set up in a systemic risk context, the asymptoticbehaviors of MME and MES are of interest in scenarios of risk contagion in a variety of disciplines.Clearly, we are interested in computing both MME( p ) and MES( p ) for small values of p , which translatesto Z being over a high threshold t . In other words we are interested in estimators of E [( Z − t ) + | Z > t ](for the MME) and E [ Z | Z > t ] (for the MES) for large values of t . An estimator for MES( p ) has beenproposed by [7] which is based on the asymptotic behavior of MES( p ); if Z ∼ F and Z ∼ F , define R ( x, y ) := lim t →∞ t Pr (cid:16) − F ( Z ) ≤ xt , − F ( Z ) ≤ yt (cid:17) (1.3)for ( x, y ) ∈ [0 , ∞ ) . It is shown in [7] thatlim p → − p ( Z ) MES( p ) = (cid:90) ∞ R ( x − α ,
1) d x (1.4)if Z has a regularly varying tail with tail parameter α . In [25] a similar result is presented under the furtherassumption of multivariate regular variation of the vector Z = ( Z , Z ); see [21, 40] as well in this context.Under the same assumptions, we can check thatlim p → − p ( Z ) MME( p ) = (cid:90) ∞ c R ( x − α ,
1) d x (1.5)where c = lim p → VaR − p ( Z )VaR − p ( Z ) , imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 if c exists and is finite. For c to be finite we require that Z and Z are (right) tail equivalent ( c >
0) or Z has a lighter (right) tail than Z ( c = 0). Note that in both (1.4) and (1.5), the rate of increase of therisk measure is determined by the tail behavior of Z ; the tail behavior of Z has no apparent influence.However, these results make sense only when the right hand sides of (1.4) and (1.5) are both non-zero andfinite. Thus, we obtain that as p ↓ p ) ∼ const. VaR − p ( Z ) , and MES( p ) ∼ const. VaR − p ( Z ) . Unfortunately, if Z , Z are asymptotically upper tail independent then R ( x, y ) ≡ Z has positive upper tail dependence,which means that, Z and Z take high values together with a positive probability; examples of multivariateregularly varying random vectors producing such strong dependence can be found in [22]. A classical examplefor asymptotic tail independence, especially in financial risk modeling, is when the risk factors Z and Z are both Pareto-tailed with a Gaussian copula and any correlation ρ < R ≡
0. The results in (1.4) and (1.5) respectively, and hence, in [7]provide a null estimate which is not very informative. Hence, in such a case one might be inclined tobelieve that E ( Z | Z > t ) ∼ E ( Z ) and E (( Z − t ) + | Z > t ) ∼ Z and Z are asymptotically tailindependent. However, we will see that depending on the Gaussian copula parameter ρ we might even havelim t →∞ E (( Z − t ) + | Z > t ) = ∞ . Hence, in this case it would be nice if we could find the right rate ofconvergence of MME( p ) to a non-zero constant.In this paper we investigate the asymptotic behavior of MME( p ) and MES( p ) as p ↓ Z exhibiting asymptotic upper tailindependence. We will see that for a very general class of models MME( p ) and MES( p ), respectively behavelike a regularly varying function with negative index for p ↓
0, and hence, converge to ∞ although the tailsare asymptotically tail independent. However, the rate of convergence is slower than in the asymptoticallytail dependent case as presented in [7]. This result is an interplay between the tail behavior and the strengthof dependence of the two variables in the tails. The behavior of MES in the asymptotically tail independentcase has been addressed to some extent in [22, Section 3.4] for certain copula structures with Pareto margins.We address the asymptotically tail independent case in further generality. For the MME, we can provideresults with fewer technical assumptions than for the case of MES and hence, we cover a broader class ofasymptotically tail independent models. The knowledge of the asymptotic behavior of the MME and the MEShelps us in proving consistency of their empirical estimators. However, in a situation where data are scarceor even unavailable in the tail region of interest, an empirical estimator is clearly unsuitable. Hence, we alsoprovide consistent estimators using methods from extreme value theory which work when data availablilityis limited in the tail regions.The paper is structured as follows: In Section 2 we briefly discuss the notion of multivariate and hiddenregular variation. We also list a set of assumptions that we impose on our models in order to obtain limitsof the quantities MME and MES under appropriate scaling. The main results of the paper regarding theasymptotic behavior of the MME and the MES are discussed in Section 3. In Section 3.3, we illustrate a fewexamples which satisfy the assumptions under which we can compute asymtptoic limits of MME and MES;these include additive models, the Bernoulli mixture model for generating hidden regular variation and afew copula models. Estimation methods for the risk measures MME and MES are provided in Section 4.Consistency of the empirical estimators are the topic of Section 4.1, whereas, we present consistent estimatorsbased on methods from extreme value theory in Section 4.2. Finally, we validate our method on real andsimulated data in Section 5 with brief concluding remarks in Section 6.In the following we denote by v → vague convergence of measures, by ⇒ weak convergence of measures andby P → convergence in probability. For x ∈ R , we write x + = max(0 , x ). imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
2. Preliminaries
For this paper we restrict our attention to non-negative random variables in a bivariate setting. We discussmultivariate and hidden regular variation in Section 2.1. A few technical assumptions that we use throughoutthe paper are listed in Section 2.2. A selection of model examples that satisfy our assumptions is relegatedto Section 3.3.
First, recall that a measurable function f : (0 , ∞ ) → (0 , ∞ ) is regularly varying at ∞ with index ρ ∈ R iflim t →∞ f ( tx ) f ( t ) = x ρ for any x > f ∈ RV ρ . If the index of regular variation is 0 we call the function slowly varyingas well. Note that in contrast, we say f is regularly varying at 0 with index ρ if lim t → f ( tx ) /f ( t ) = x ρ forany x >
0. In this paper, unless otherwise specified, regular variation means regular variation at infinity. Arandom variable X with distribution function F has a regularly varying tail if F = 1 − F ∈ RV − α for some α ≥
0. We often write X ∈ RV − α by abuse of notation.We use the notion of M -convergence to define regular variation in more than one dimension; for furtherdetails see [12, 23, 27]. We restrict to two dimensions here since we deal with bivariate distributions in thispaper, although the definitions provided hold in general for any finite dimension. Suppose C ⊂ C ⊂ [0 , ∞ ) where C and C are closed cones containing { (0 , } ∈ R . By M ( C \ C ) we denote the class of Borel measureson C \ C which are finite on subsets bounded away from C . Then µ n M → µ in M ( C \ C ) if µ n ( f ) → µ ( f )for all continuous and bounded functions on C \ C whose supports are bounded away from C . Definition 2.1 (Multivariate regular variation)
A random vector Z = ( Z , Z ) ∈ C is (multivariate)regularly varying on C \ C , if there exist a function b ( t ) ↑ ∞ and a non-zero measure ν ( · ) ∈ M ( C \ C ) suchthat as t → ∞ , ν t ( · ) := t Pr( Z /b ( t ) ∈ · ) M → ν ( · ) in M ( C \ C ). (2.1)Moreover, we can check that the limit measure has the homogeneity property: ν ( cA ) = c − α ν ( A ) for some α >
0. We write Z ∈ MRV ( α, b, ν, C \ C ) and sometimes write MRV for multivariate regular variation.In the first stage, multivariate regular variation is defined on the space E = [0 , ∞ ) \ { (0 , } = C \ C where C = [0 , ∞ ) and C = { (0 , } . But sometimes we need to define further regular variation on subspacesof E , since the limit measure ν as obtained in (2.1) turns out to be concentrated on a subspace of E . Themost likely way this happens is through asymptotic tail independence of random variables. Definition 2.2 (asymptotic tail independence)
A random vector Z = ( Z , Z ) ∈ [0 , ∞ ) is called asymptotically independent (in the upper tail) iflim p ↓ Pr( Z > F ← (1 − p ) | Z > F ← (1 − p )) = 0 , where Z i ∼ F i , i = 1 , Z as well.Assume (w.l.o.g.) that F , F are strictly increasing continuous distribution functions with unique survivalcopula (cid:98) C (see [30]) such thatPr( Z > x, Z > y ) = (cid:98) C ( F ( x ) , F ( y )) for ( x, y ) ∈ R . Hence, in terms of the survival copula, asymptotic upper tail independence of Z implieslim p ↓ (cid:98) C ( p, p ) p = lim p ↓ Pr( Z > F ← ( p ) , Z > F ← ( p ))Pr( Z > F ← ( p )) = lim p ↓ Pr( Z > F ← ( p )) | Z > F ← (1 − p )) = 0 . (2.2) imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 Independent random vectors are trivially asymptotically tail independent. Note that asymptotic uppertail independence of Z ∈ MRV ( α, b, ν, E ) implies ν ((0 , ∞ ) × (0 , ∞ )) = 0 for the limit measure ν . On theother hand, for the converse, if Z and Z are both marginally regularly varying in the right tail withlim t →∞ Pr( Z > t ) / Pr( Z > t ) = 1, then ν ((0 , ∞ ) × (0 , ∞ )) = 0 implies asymptotic upper tail independenceas well (see [32, Proposition 5.27]). However, this implication does not hold true in general, e.g., for a regularlyvarying random variable X ∈ RV − α the random vector ( X, X ) is multivariate regularly varying with limitmeasure ν ((0 , ∞ ) × (0 , ∞ )) = 0; but of course ( X, X ) is asymptotically tail-dependent. Remark 2.3
Asymptotic upper tail independence of ( Z , Z ) implies that R ( x, y ) = lim t →∞ t Pr (1 − F ( Z ) ≤ x/t, − F ( Z ) ≤ y/t )= lim t →∞ t (cid:98) C (cid:16) xt , yt (cid:17) ≤ max( x, y ) lim s → (cid:98) C ( s, s ) s = 0 (using (2.2)) . Hence, the estimator presented in [7] for MES provides a trivial estimator in this setting.Consequently, in the asymptotically tail independent case where the tails are equivalent we would ap-proximate the joint tail probability by Pr( Z > x | Z > x ) ≈ x and conclude that riskcontagion between Z and Z is absent. This conclusion may be naive; hence the notion of hidden regularvariation on E = [0 , ∞ ) \ ( { } × [0 , ∞ ) ∪ [0 , ∞ ) × { } ) = (0 , ∞ ) was introduced in [33]. Note that we donot assume that the marginal tails of Z are necessarily equivalent in order to define hidden regular variation,which is usually done in [33]. Definition 2.4 (Hidden regular variation)
A regularly varying random vector Z on E possesses hiddenregular variation on E = (0 , ∞ ) with index α ( ≥ α >
0) if there exist scaling functions b ( t ) ∈ RV /α and b ( t ) ∈ RV /α with b ( t ) /b ( t ) → ∞ and limit measures ν, ν such that Z ∈ MRV ( α, b, ν, E ) ∩ MRV ( α , b , ν , E ) . We write Z ∈ HRV ( α , b , ν ) and sometimes write HRV for hidden regular variation.For example, say Z , Z are iid random variables with distribution function F ( x ) = 1 − x − , x > Z = ( Z , Z ) possesses MRV on E , asymptotic tail independence and HRV on E . Specifically, Z ∈MRV ( α = 1 , b ( t ) = t, ν, E ) ∩ MRV ( α = 2 , b ( t ) = √ t, ν , E ) where for x > , y > ν (([0 , x ] × [0 , y ]) c ) = 1 x + 1 y and ν ([ x, ∞ ) × [ y, ∞ )) = 1 xy . Lemma 2.5. Z ∈ MRV ( α, b, ν, E ) ∩ HRV ( α , b , ν , E ) implies that Z is asymptotically tail independent. Proof.
Let b i ( t ) = (1 / (1 − F i )) ← ( t ) where Z i ∼ F i , i = 1 ,
2. Due to the assumptions we havelim t →∞ max( b ( t ) , b ( t )) b ( t ) = ∞ and lim inf t →∞ min( b ( t ) , b ( t )) b ( t ) ≥ . Without loss of generality b ( t ) /b ( t ) → ∞ . Then for any M > t = t ( M ) so that b ( t ) ≥ M b ( t ) for any t ≥ t . Hence, for x, y > t →∞ t Pr (1 − F ( Z ) ≤ x/t, − F ( Z ) ≤ y/t ) = lim t →∞ t Pr ( Z ≥ b ( t/x ) , Z ≥ b ( t/y )) ≤ lim t →∞ t Pr (cid:0) Z ≥ M b ( t/x ) , Z ≥ − b ( t/y ) (cid:1) ≤ C x,y ν ([ M, ∞ ) × (cid:2) − , ∞ (cid:1) ) M →∞ → , so that Z is asymptotically tail independent (here C x,y is some fixed constant). Remark 2.6
The assumption Z ∈ MRV ( α, b, ν, E ) ∩ MRV ( α , b , ν , E ) and Z is asymptotic upper tailindependent already implies that Z ∈ HRV ( α , b , ν ); see [29, 33]. Consequently lim t →∞ b ( t ) /b ( t ) = ∞ aswell. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
In this section we list assumptions on the random variables for which we show consistency of relevantestimators in the paper. Parts of the assumptions are to fix notations for future results.
Assumption A (A1) Let Z = ( Z , Z ) ∈ [0 , ∞ ) such that Z ∈ MRV ( α, b, ν, E ) where b ( t ) = (1 / Pr(max( Z , Z ) > · )) ← ( t ) = F ← max( Z ,Z ) (1 /t ) ∈ RV /α . (A2) E | Z | < ∞ .(A3) b ( t ) := F ← Z (1 /t ) for t ≥ Z is [1 , ∞ ). A constant shift would not affectthe tail properties of MME or MES.(A5) Z ∈ MRV ( α , b , ν , E ) with α ≥ α ≥
1, where b ( t ) = (1 / Pr(min( Z , Z ) > · )) ← ( t ) = F ← min( Z ,Z ) (1 /t ) ∈ RV /α , and b ( t ) /b ( t ) → ∞ . Lemma 2.7.
Let F Z ∈ RV − β , β > . Then Assumption A implies α ≤ β ≤ α . Proof.
First of all, β ≥ α since otherwise Z ∈ MRV ( α, b, ν, E ) cannot hold. Moreover,1 ∼ t Pr( Z > b ( t ) , Z > b ( t )) ≤ t Pr( Z > b ( t )) ∈ RV − βα . (2.3)Thus, if α < β then lim t →∞ t Pr( Z > b ( t )) = 0 which is a contradiction to (2.3). Remark 2.8
In general, we see from this that under Assumption A, lim inf t →∞ t Pr( Z > b ( t )) ≥ (cid:15) > C ( (cid:15) ) > C ( (cid:15) ) > x ( (cid:15) ) > C ( (cid:15) ) x − α − (cid:15) ≤ Pr( Z > x ) ≤ C ( (cid:15) ) x − α + (cid:15) for any x ≥ x ( (cid:15) ).We need a couple of more conditions, especially on the joint tail behavior of Z = ( Z , Z ) in order totalk about the limit behavior of MME( p ) and MES( p ) as p ↓
0. We impose the following assumptions on thedistribution of Z . Assumption (B1) is imposed to find the limit of MME in (1.1) whereas both (B1) and(B2) (which are clubbed together as Assumption B) are imposed to find the limit in (1.2), of course, bothunder appropriate scaling. Assumption B (B1) lim M →∞ lim t →∞ (cid:90) ∞ M Pr( Z > xt, Z > t )Pr( Z > t, Z > t ) d x = 0.(B2) lim M →∞ lim t →∞ (cid:90) /M Pr( Z > xt, Z > t )Pr( Z > t, Z > t ) d x = 0.Assumption (B1) and Assumption (B2) deal with tail integrability near infinity and near zero for a specificintegrand, respectively that comes up in calculating limits of MME and MES. The following lemma triviallyprovides a sufficient condition for (B1). Lemma 2.9.
If there exists an integrable function g : [0 , ∞ ) → [0 , ∞ ) with sup t ≥ t Pr( Z > y, Z > t ) t Pr( Z > t, Z > t ) ≤ g ( y ) for y > and some t > then (B1) is satisfied. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 Lemma 2.10.
Let Assumption A hold.(a) Then (B2) implies lim t →∞ Pr( Z > t ) t Pr( Z > t, Z > t ) = 0 . (2.4) (b) Suppose F Z ∈ RV − β with α ≤ β ≤ α . Then α ≤ β + 1 is a necessary and α < β + 1 is a suffi-cient condition for (2.4) to hold. Hence, α ≤ β +1 is a necessary condition for Assumption (B2) as well. Proof. (a) Since the support of Z is [1 , ∞ ) we get for large t ≥ M , by (B2),Pr( Z > t ) t Pr( Z > t, Z > t ) = (cid:90) /t Pr( Z > xt, Z > t )Pr( Z > t, Z > t ) d x ≤ (cid:90) /M Pr( Z > xt, Z > t )Pr( Z > t, Z > t ) d x t,M →∞ → . But the left hand side is independent of M so that the claim follows.(b) In this case Pr( Z > t ) t Pr( Z > t, Z > t ) ∈ RV − β − α from which the statement follows. Remark 2.11 If Z , Z are independent then under the assumptions of Lemma 2.10(b), α = α + β .Moreover if 1 < α ≤ β then clearly α = α + β > β and α ≤ β cannot hold. Hence, Assumption (B2)is not valid if Z and Z are independent. In other words, Assumption (B2) signifies that although Z , Z are asymptotically upper tail independent, there is an underlying dependence between Z and Z which isabsent in the independent case.
3. Asymptotic behavior of the MME and the MES
For asymptotically independent risks, from (1.5) and Remark 2.3 we have thatlim p → − p ( Z ) MME( p ) = 0 , which doesn’t provide us much in the way of identifying the rate of increase (or decrease) of MME( p ). Theaim of this section is to get a version of (1.5) for the asymptotically tail independent case which is presentedin the next theorem. Theorem 3.1.
Suppose Z = ( Z , Z ) ∈ [0 , ∞ ) satisfies Assumption A and (B1). Then lim p ↓ pb ← ( b (1 /p )) b (1 /p ) MME ( p ) = lim p ↓ pb ← ( VaR − p ( Z )) VaR − p ( Z ) MME ( p ) = (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x. (3.1) Moreover, < (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x < ∞ . Proof.
We know that for a non-negative random variable W , we have E W = (cid:82) ∞ Pr(
W > x ) d x. Let t = b (1 /p ). Also note that b ← ( t ) = 1 / Pr(min( Z , Z ) > t ) = 1 / Pr( Z > t, Z > t ). Then pb ← ( b (1 /p )) b (1 /p ) MME( p ) = F Z ( t ) b ← ( t ) t E (( Z − t ) + | Z > t ) imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen = Pr( Z > t ) b ← ( t ) t (cid:90) ∞ t Pr( Z > x, Z > t )Pr( Z > t ) d x = (cid:90) ∞ t Pr( Z > x, Z > t ) t Pr( Z > t, Z > t ) d x = (cid:90) ∞ Pr( Z > tx, Z > t )Pr( Z > t, Z > t ) d x =: (cid:90) ∞ ν t ( x ) d x. (3.2)Observe that for x ≥
1, by Assumption (A5), ν t ( x ) = Pr( Z > tx, Z > t )Pr( Z > t, Z > t ) = b ← ( t )Pr (cid:18) Z t ∈ ( x, ∞ ) × (1 , ∞ ) (cid:19) t →∞ → ν (( x, ∞ ) × (1 , ∞ )) . We also have ν t ( x ) = Pr( Z > tx, Z > t )Pr( Z > t, Z > t ) ≤ , x ≥ . Now, for x ≥
1, we have ν (( x, ∞ ) × (1 , ∞ )) ≤ ν ((1 , ∞ ) × (1 , ∞ )) = lim t →∞ ν t (1) = 1. Hence, for any M ≥ (cid:82) M ν (( x, ∞ ) × (1 , ∞ )) d x ≤ M . Therefore using Lebesgue’s Dominated Convergence Theorem,lim t →∞ (cid:90) M ν t ( x ) d x = (cid:90) M ν (( x, ∞ ) × (1 , ∞ )) d x. (3.3)Next we check that 0 < (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x < ∞ . Define for M ≥ ψ M := lim t →∞ (cid:90) ∞ M ν t ( x )d x. By Assumption (B1), we have lim M →∞ ψ M = 0 . (3.4)Hence, there exists M > | ψ M | ≤ M > M . Applying Fatou’s Lemma, we know that forany M > M , (cid:90) ∞ M ν (( x, ∞ ) × (1 , ∞ )) d x ≤ lim inf t →∞ (cid:90) ∞ M ν t ( x )d x ≤ ψ M ≤ . Therefore, for fixed
M > M , (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x = (cid:90) M ν (( x, ∞ ) × (1 , ∞ )) d x + (cid:90) ∞ M ν (( x, ∞ ) × (1 , ∞ )) d x ≤ M + 1 < ∞ . Moreover, ν (( x, ∞ ) × ( x, ∞ )) is homogeneous of order − α so that (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x ≥ (cid:90) ∞ ν (( x, ∞ ) × ( x, ∞ )) d x = ν ((1 , ∞ ) × (1 , ∞ )) (cid:90) ∞ x − α d x > . Hence 0 < (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x < ∞ . Therefore, since t = b (1 /p ) ↑ ∞ as p ↓
0, we havelim p ↓ pb ← ( b (1 /p )) b (1 /p ) MME( p ) = lim t →∞ (cid:90) ∞ ν t ( x ) d x = lim t →∞ (cid:34)(cid:90) M ν t ( x ) d x + (cid:90) ∞ M ν t ( x ) d x (cid:35) imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 = lim M →∞ (cid:34) lim t →∞ (cid:90) M ν t ( x ) d x + lim t →∞ (cid:90) ∞ M ν t ( x ) d x (cid:35) (since it is true for any M ≥ M →∞ (cid:90) M ν (( x, ∞ ) × (1 , ∞ )) d x + lim M →∞ ψ M (using (3.3))= (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x (using (3.4)) . Corollary 3.2.
Suppose Z = ( Z , Z ) satisfies Assumptions A, (B1) and F Z ∈ RV − β for some α ≤ β ≤ α . Then MME (1 /t ) ∈ RV (1+ β − α ) /β . For β > α we have lim p → MME ( p ) = ∞ with β − α β ∈ (cid:20) − α − β , α (cid:21) ⊆ (0 , and for β < α we have lim p → MME ( p ) = 0 . Remark 3.3
A few consequences of Corollary 3.2 are illustrated below.(a) When 1 + β > α , although the quantity MME( p ) increases as p ↓
0, the rate of increase is slower thana linear function.(b) Let Z ∈ MRV ( α, b, ν, E ). Suppose Z and Z are independent and F Z ∈ RV − α then by Karamata’sTheorem, MME( p ) ∼ α − − p ( Z )Pr( Z > VaR − p ( Z )) ( p ↓ . This is a special case of Theorem 3.1.
Example 3.4
In this example we illustrate the influence of the tail behavior of the marginals as well asthe dependence structure on the asymptotic behavior of the MME. Assume that Z = ( Z , Z ) ∈ [0 , ∞ ) satisfies Assumptions (A1)-(A4). We compare the following tail independent and tail dependent models:(D) Tail dependent model: Additionally Z is tail dependent implying R (cid:54) = 0 and satisfies (1.5). We denoteits Marginal Mean Excess by MME D .(ID) Tail independent model: Additionally Z is asymptotically tail independent satisfying (A5), (B1) and1 + β > α > α . Its Marginal Mean Excess we denote by MME I .(a) Suppose Z , Z are identically distributed. Since t/b ← ( b ( t )) ∈ RV − α /α and 1 − α /α < I ( p )MME D ( p ) ∼ p → Cpb ← ( b (1 /p )) → p → . This means in the asymptotically tail independent case the Marginal Mean Excess increases at a slowerrate to infinity, than in the asymptotically tail dependent case, as expected.(b) Suppose Z , Z are not identically distributed and for some finite constant C > Z > t ) ∼ C Pr( Z > t, Z > t ) ( t → ∞ ) . This means that not only Z ∈ MRV ( α , b , ν , E ) but also Z ∈ RV − α and Z is heavier tailed than Z . Then lim t →∞ b ← ( b ( t )) t = lim t →∞ t Pr( Z > b ( t ) , Z > b ( t )) = lim t →∞ Ct Pr( Z > b ( t )) = C. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
Thus, lim p → − p ( Z ) MME I ( p ) = lim p → MME I ( p ) b (1 /p ) = C (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x and MME I ( · ) is regularly varying of index − α at 0. In this example Z is lighter tailed than Z , andhence, once again we find that in the asymptotically tail independent case the Marginal Mean ExcessMME I increases at a slower rate to infinity than the Marginal Mean Excess MME D in the asymptoticallytail dependent case. Here we derive analogous results for the Marginal Expected Shortfall.
Theorem 3.5.
Suppose Z = ( Z , Z ) satisfies Assumptions A and B. Then lim p ↓ pb ← ( VaR − p ( Z )) VaR − p ( Z ) MES ( p ) = lim p ↓ pb ← ( b (1 /p )) b (1 /p ) MES ( p ) = (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x. (3.5) Moreover, < (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x < ∞ . The proof of Theorem 3.5 requires further condition (B2) which can be avoided in Theorem 3.1.
Proof.
The proof is similar to that of Theorem 3.1 which we discussed in detail. As in Theorem 3.1 werewrite pb ← ( b (1 /p )) b (1 /p ) MES( p ) = F Z ( t ) b ← ( t ) t E ( Z | Z > t ) = (cid:34)(cid:90) /M + (cid:90) M /M + (cid:90) ∞ M (cid:35) Pr( Z > tx, Z > t )Pr( Z > t, Z > t ) d x. We can then conclude the statement from (B2) and similar arguments as in the proof of Theorem 3.1.A similar comparison can be made between the asymptotic behavior of the Marginal Expected Shortfallfor the tail independent and tail dependent case as we have done in Example 3.4 for the Marginal MeanExcess.
Remark 3.6
Define a ( t ) := b ← ( b ( t )) t b ( t ) . Then lim t →∞ a ( t ) = 0 is equivalent to lim t →∞ Pr( Z > t ) t Pr( Z > t, Z > t ) = 0 . Hence, a consequence of (B2) and (2.4) is that lim t →∞ a ( t ) = 0 and finally, lim p ↓ MES( p ) = ∞ . Again asufficient assumption for lim t →∞ a ( t ) = 0 is F Z ∈ RV − β with α < β + 1 and a necessary condition is α ≤ β + 1 (see Lemma 2.10). Remark 3.7
In this study we have only considered a non-negative random variable Z while computingMES( p ) = E ( Z | Z > VaR − p ( Z )). For a real-valued random variable Z , we can represent Z = Z +1 − Z − where Z +1 = max( Z ,
0) and Z − = max( − Z , Z +1 and Z − are non-negative and hence can bedealt with separately. The limit results will depend on the separate dependence structure and tail behaviorsof ( Z +1 , Z ) and ( Z − , Z ). imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 We finish this section up with a few models and examples where we can calculate limits for MES and MME.In Sections 3.3.1 and 3.3.2 we discuss generative models with sufficient conditions satisfying Assumptions Aand B. In Section 3.3.3 we further discuss two copula models where Theorems 3.1 and 3.5 can be applied.
First we look at models that are generated in an additive fashion (see [10, 38]). We will observe that manymodels can be generated using the additive technique.
Model C
Suppose Z = ( Z , Z ) , Y = ( Y , Y ) , V = ( V , V ) are random vectors in [0 , ∞ ) such that Z = Y + V . Assume the following holds:(C1) Y ∈ MRV ( α, b, ν, E ) where α ≥ Y , Y are independent random variables.(C3) F Y ∈ RV − α ∗ , 1 ≤ α ≤ α ∗ .(C4) V ∈ MRV ( α , b , ν , E ) and does not possess asymptotic tail independence where α ≤ α andlim t →∞ Pr( (cid:107) V (cid:107) > t )Pr( (cid:107) Y (cid:107) > t ) = 0 . (C5) Y and V are independent.(C6) α ≤ α < α ∗ .(C7) E | Z | < ∞ .Of course, we would like to know, when Model C satisfies Assumptions A and B; moreover, when is Z ∈ HRV ( α , b , ν , E )? The next theorem provides a general result to answer these questions in certainspecial cases. Theorem 3.8.
Let Z = Y + V be as in Model C. Then the following statements hold:(a) Z ∈ MRV ( α, b, ν, E ) ∩ HRV ( α , b , ν , E ) and satisfies Assumption B.(b) Suppose Y = 0 . Then ( Z , Z + Z ) ∈ MRV ( α, b, ν, E ) ∩ HRV ( α , b , ν +0 , E ) with ν +0 ( A ) = ν ( { ( v , v ) ∈ E : ( v , v + v ) ∈ A } ) for A ∈ B ( E ) and satisfies Assumption B.(c) Suppose lim inf t →∞ Pr( Y > t ) / Pr( Y > t ) > . Then ( Z , min( Z , Z )) ∈ MRV ( α, b, ν min , E ) ∩HRV ( α , b , ν min , E ) with ν min ( A ) = ν ( { ( y , ∈ E : ( y , ∈ A } ) for A ∈ B ( E ) ,ν min ( A ) = ν ( { ( v , v ) ∈ E : ( v , min( v , v )) ∈ A } ) for A ∈ B ( E ) and satisfies Assumption B.(d) Suppose Y = 0 . Then ( Z , max( Z , Z )) ∈ MRV ( α, b, ν, E ) ∩ HRV ( α , b , ν max , E ) with ν max ( A ) = ν ( { ( v , v ) ∈ E : ( v , max( v , v )) ∈ A } ) for A ∈ B ( E ) and satisfies Assumption B. For a proof of this theorem we refer to [9].
Remark 3.9
Note that, in a systemic risk context where the entire system consists of two institutions withrisks Z and Z , the above theorem addresses the variety of ways a systemic risk model can be constructed.If risk is just additive we could refer to part (b), if the system is at risk when both institutions are at riskthen we can refer to part (c) and if the global risk is connected to any of the institutions being at risk thenwe can refer to the model in part (d). Hence, many kinds of models for calculating systemic risk can beobtained under such a model assumption. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen3.3.2. Bernoulli model
Next we investigate an example generated by using a mixture method for getting hidden regular variationin a non-standard regularly varying model (see [12]).
Example 3.10
Suppose X , X , X are independent Pareto random variables with parameters α , α and γ , respectively, where 1 < α < α < γ and α < α . Let B be a Bernoulli( q ) random variable with0 < q < X , X , X . Now define Z = ( Z , Z ) = B ( X , X ) + (1 − B )( X , X ) . This is a popular example, see [10, 29, 33]. Note thatPr(max( Z , Z ) > t ) ∼ qt − α and Pr(min( Z , Z ) > t ) ∼ Pr( Z > t ) ∼ (1 − q ) t − α ( t → ∞ ) , so that b (1 /p ) ∼ q α p − α , b (1 /p ) ∼ b (1 /p ) ∼ (1 − q ) α p − α as p ↓
0. We denote by (cid:15) x , the Dirac measureat point x . Note that the limit measure on E concentrates on the two axes. We will look at usual MRV whichis given on E by t Pr (cid:18)(cid:18) Z b ( t ) , Z b ( t ) (cid:19) ∈ d x d y (cid:19) M → αx − α − d x · (cid:15) (d y ) =: ν (d x d y ) ( t → ∞ ) in M ( E ) , where the limit measure lies on the x-axis. Hence, we seek HRV in the next step on E \{ x -axis } = [0 , ∞ ) × (0 , ∞ ) and get t Pr (cid:18)(cid:18) Z b ( t ) , Z b ( t ) (cid:19) ∈ d x d y (cid:19) M → α x − α − d x · (cid:15) x (d y ) =: ν (d x d y ) ( t → ∞ ) in M ( E \{ x -axis } ) . Here the limit measure lies on the diagonal where x = y . Thus, we have for any x ≥ ν (( x, ∞ ) × (1 , ∞ )) = x − α . Now, we can explicitly calculate the values of MME and MES. For 0 < p < p ) = 1 q VaR − p ( Z ) − γ + (1 − q )VaR − p ( Z ) − α (cid:20) qαα − − p ( Z ) − γ + (1 − q ) α α − − p ( Z ) − α +1 (cid:21) , MME( p ) = 1 q VaR − p ( Z ) − γ + (1 − q )VaR − p ( Z ) − α (cid:20) qα − − p ( Z ) − γ − α +1 + (1 − q ) α − − p ( Z ) − α +1 (cid:21) . Therefore, pb ← ( b (1 /p )) b (1 /p ) MME( p ) ∼ b (1 /p ) MME( p ) ∼ α − (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x ( p ↓ , and pb ← ( b (1 /p )) b (1 /p ) MES( p ) ∼ b (1 /p ) MES( p ) ∼ α α − (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x ( p ↓ . The next two examples constructed by well-known copulas (see [30]) are illustrative of the limits which weare able to compute using Theorems 3.1 and 3.5.
Example 3.11
In financial risk management, no doubt the most famous copula model is the
Gaussiancopula : C Φ ,ρ ( u, v ) = Φ (Φ ← ( u ) , Φ ← ( v )) for ( u, v ) ∈ [0 , , imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 is a bivariate normal distribution function withstandard normally distributed margins and correlation ρ . Then the survival copula satisfies: (cid:98) C Φ ,ρ ( u, u ) = C Φ ,ρ ( u, u ) ∼ u ρ +1 (cid:96) ( u ) ( u → , for some function l which is slowly varying at 0, see [26, 31]. Suppose ( Z , Z ) has identical Pareto marginaldistributions with common parameter α > C Φ ,ρ ( u, v ) with ρ ∈ ( − , Z , Z ) ∈ MRV ( α, b, ν, E ) with asymptotic tail in-dependence and ( Z , Z ) ∈ MRV ( α , b , ν , E ) with α = 2 α ρ and ν (( x, ∞ ) × ( y, ∞ )) = x − α ρ y − α ρ , x, y > . Hence, for ρ ∈ (1 − / ( α + 1) ,
1) we have lim p → MME( p ) = ∞ . In this model, Assumptions A and (B1) aresatisfied when α > ρ and α >
1. We can also check that Assumption (B2) is not satisfied. Consequently,we can find estimates for MME but not for MES in this example.
Example 3.12
Suppose ( Z , Z ) has identical Pareto marginal distributions with parameter α > Marshall-Olkin survival copula : (cid:98) C γ ,γ ( u, v ) = uv min( u − γ , v − γ ) for ( u, v ) ∈ [0 , , for some γ , γ ∈ (0 , Z , Z ) ∈ MRV ( α, b, ν, E ) with asymptotictail independence and ( Z , Z ) ∈ MRV ( α , b , ν , E ) with α = α max(2 − γ , − γ ) and ν (( x, ∞ ) × ( y, ∞ )) = x − α (1 − γ ) y − α , γ < γ ,x − α y − α max( x, y ) − αγ , γ = γ ,x − α y − α (1 − γ ) , γ > γ , x, y > . Then min( γ , γ ) ∈ (1 − /α,
1) implies lim p → MME( p ) = ∞ . Moreover this model satisfies Assumptions Aand (B1) when γ ≥ γ . Unfortunately again, (B2) is not satisfied.
4. Estimation of MME and MES
Suppose ( Z (1)1 , Z (2)1 ) , . . . , ( Z (1) n , Z (2) n ) are iid samples with the same distribution as ( Z , Z ). We denote by Z (2)(1: n ) ≥ . . . ≥ Z (2)( n : n ) the order statistic of the sample Z (2)1 , . . . , Z (2) n in decreasing order. We begin by lookingat the behavior of the empirical estimator (cid:92) MME emp ,n ( k/n ) := 1 k n (cid:88) i =1 ( Z (1) i − Z (2)( k : n ) ) + { Z (2) i >Z (2)( k : n ) } of the quantity MME( k/n ) = E (( Z − b ( n/k )) + | Z > b ( n/k )) with k < n . The following theorem showsthat the empirical estimator is consistent in probability. Proposition 4.1.
Let the assumptions of Theorem 3.1 hold, and let F Z ∈ RV − β for some α ≤ β ≤ α .Furthermore, let k = k ( n ) be a sequence of integers satisfying k → ∞ , k/n → and b ← ( b ( n/k )) /n → as n → ∞ (note that this is trivially satisfied if b = b ). imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen(a) Then, as n → ∞ , b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 ( Z (1) i − Z (2)( k : n ) ) + { Z (2) i >Z (2)( k : n ) } P → (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x. (b) In particular, we have (cid:92) MME emp ,n ( k/n ) MME ( k/n ) P → as n → ∞ . To prove this theorem we use the following lemma.
Lemma 4.2.
Let the assumptions of Proposition 4.1 hold. Define for y > , E n ( y ) := b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 ( Z (1) i − b ( n/k ) y ) + { Z (2) i >b ( n/k ) y } ,E ( y ) := (cid:90) ∞ y ν (( x, ∞ ) × ( y, ∞ )) d x. Then E ( y ) = y − α E (1) and as n → ∞ , ( E n ( y )) y ≥ / P → ( E ( y )) y ≥ / in D ([1 / , ∞ ) , (0 , ∞ )) , where by D ( I, E ∗ ) we denote the space of c`adl`ag functions from I → E ∗ . Proof.
We already know from [34, Theorem 5.3(ii)], Z ∈ MRV ( α , b , ν , E ) and b ← ( b ( n/k )) /n → n → ∞ , ν ( n )0 := b ← ( b ( n/k )) n n (cid:88) i =1 (cid:15) (cid:32) Z (1) ib n/k ) , Z (2) ib n/k ) (cid:33) ⇒ ν in M + ( E ) . (4.1)Note that E n ( y ) = (cid:90) ∞ y ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x = b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 ( Z (1) i − b ( n/k ) y ) + { Z (2) i >b ( n/k ) y } . Hence, the statement of the lemma is equivalent to (cid:18)(cid:90) ∞ y ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x (cid:19) y ≥ P → ( E ( y )) y ≥ / in D ([1 / , ∞ ) , (0 , ∞ )) . (4.2)We will prove (4.2) by a convergence-together argument. Step 1.
First we prove that E ( y ) = y − α E (1). Note that b ← ( b ( n/k )) b ( n/k ) E (( Z − b ( n/k ) y ) + { Z >b ( n/k ) y } )= (cid:90) ∞ Pr( Z > xb ( n/k ) , Z > b ( n/k ) y )Pr( Z > b ( n/k ) , Z > b ( n/k )) d x = y Pr( Z > b ( n/k ) y, Z > b ( n/k ) y )Pr( Z > b ( n/k ) , Z > b ( n/k )) (cid:90) ∞ Pr( Z > x ( b ( n/k ) y ) , Z > b ( n/k ) y )Pr( Z > b ( n/k ) y, Z > b ( n/k ) y ) d x = y · Pr( Z > b ( n/k ) y, Z > b ( n/k ) y )Pr( Z > b ( n/k ) , Z > b ( n/k )) · (cid:90) ∞ ν t ( x ) d x imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 ν t is as defined in (3.2) with t = b ( n/k ) y ) n →∞ → y · y − α · (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x = y − α E (1) . (4.3)The final limit follows from the definition of hidden regular variation and Theorem 3.1. On the other hand,in a similar manner as in Theorem 3.1, we can exchange the integral and the limit such that using (2.1) weobtain b ← ( b ( n/k )) b ( n/k ) E (( Z − b ( n/k ) y ) + { Z >b ( n/k ) y } ) = (cid:90) ∞ Pr( Z > xb ( n/k ) , Z > b ( n/k ) y )Pr( Z > b ( n/k ) , Z > b ( n/k )) d x n →∞ → (cid:90) ∞ ν (( x, ∞ ) × ( y, ∞ )) d x = E ( y ) . (4.4)Since (4.3) and (4.4) must be equal, we have E ( y ) = y − α E (1). Step 2.
Now we prove that for any y ≥ / M >
0, as n → ∞ , E ( M ) n ( y ) := (cid:90) M ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x P → (cid:90) M ν (( x, ∞ ) × ( y, ∞ )) d x =: E ( M ) ( y ) . (4.5)Define the function f M,y : E → [0 , M ] as f M,y ( z , z ) = (min ( z , M ) − y ) { z >y,z >y } which is continuous,bounded and has compact support on E and for any y ≥ / F M,y : M + ( E ) → R + as m (cid:55)→ (cid:90) E f M,y ( z , z ) m ( dz , dz ) . Here m is a continuous map on M + ( E ) under the vague topology. Hence, using a continuous mappingtheorem and (4.1) we get, as n → ∞ , (cid:90) M ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x = F M,y ( ν ( n )0 ) ⇒ F M,y ( ν ) = (cid:90) M ν (( x, ∞ ) × ( y, ∞ )) d x (4.6)in R + . Since the right hand side is deterministic, the convergence holds in probability as well. Step 3.
Using Assumption (B1), E (cid:32) sup y ≥ (cid:90) ∞ M ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x (cid:33) = E (cid:18)(cid:90) ∞ M ν ( n )0 (( x, ∞ ) × (1 / , ∞ )) d x (cid:19) = b ← ( b ( n/k )) (cid:90) ∞ M Pr( Z > xb ( n/k ) , Z > b ( n/k ) /
2) d x = (cid:90) ∞ M Pr( Z > xb ( n/k ) , Z > b ( n/k ) / Z > b ( n/k ) , Z > b ( n/k )) d x n →∞ ,M →∞ → . Step 4.
Hence, a convergence-together argument (see [34, Theorem 3.5]), Step 2, Step 3 and E ( M ) ( y ) → E ( y )as M → ∞ result in E n ( y ) P → E ( y ) as n → ∞ . Step 5.
From Step 1, the function E : [1 / , ∞ ) → (0 , E (1 / E − denote its inverse and define for m ∈ N and k = 1 , . . . , m , y m,k := E − (cid:18) E (1 / km (cid:19) . As in the proof of the Glivenko-Cantelli-Theorem (see [4, Theorem 20.6]) we havesup y ≥ / | E n ( y ) − E ( y ) | ≤ E (1 / m + sup k =1 ,...,m | E n ( y m,k ) − E ( y m,k ) | . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
Let (cid:15) >
0. Choose m ∈ N such that m > E (1 / /(cid:15) . ThenPr (cid:32) sup y ≥ / | E n ( y ) − E ( y ) | > (cid:15) (cid:33) ≤ Pr (cid:32) sup k =1 ,...,m | E n ( y m,k ) − E ( y m,k ) | > E (1 / m − (cid:33) ≤ m (cid:88) k =1 Pr (cid:0) | E n ( y m,k ) − E ( y m,k ) | > E (1 / m − (cid:1) n →∞ → , where we used E n ( y m,k ) P → E ( y m,k ) as n → ∞ for any k = 1 , . . . , m, m ∈ N by Step 4. Hence, we canconclude the statement. Proof of Proposition 4.1. (a) By assumption, F Z ∈ RV − β . From [34, p. 82] we know that Z (2)( (cid:100) ky (cid:101) : n ) b ( n/k ) y> P → (cid:16) y − β (cid:17) y> in D ((0 , ∞ ] , (0 , ∞ ))and in particular, this and Lemma 4.2 result in ( E n ( y )) y ≥ , Z (2)( (cid:100) ky (cid:101) : n ) b ( n/k ) y> P → (cid:16) ( E ( y )) y ≥ , ( y − β ) y> (cid:17) in D ([1 / , ∞ ) , (0 , ∞ )) × D ((0 , ∞ ] , (0 , ∞ )) . Let D ↓ ( (cid:0) , β (cid:3) , [1 / , ∞ )) be a subfamily of D ( (cid:0) , β (cid:3) , [1 / , ∞ )) consisting of non-increasing functions. Letus similarly define C ↓ ( (cid:0) , β (cid:3) , [1 / , ∞ )). Define the map ϕ : D ([1 / , ∞ ) , (0 , ∞ )) × D ↓ ( (cid:0) , β (cid:3) , [1 / , ∞ ))with ( f, g ) (cid:55)→ f ◦ g . From [39, Theorem 13.2.2], we already know that ϕ restricted to D ([1 / , ∞ ) , (0 , ∞ )) × C ↓ ( (cid:0) , β (cid:3) , [1 / , ∞ )) is continuous. Thus, we can apply a continuous mapping theorem and obtain as n → ∞ , E n Z (2)( (cid:100) ky (cid:101) : n ) b ( n/k ) y ∈ (0 , β ] P → (cid:16) E ( y − β ) (cid:17) y ∈ (0 , β ] in D ( (cid:0) , β (cid:3) , (0 , ∞ )) . As a special case we get the marginal convergence as n → ∞ , b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 Z (1) i { Z (2) i >Z (2)( k : n ) } = E n Z (2)( k : n ) b ( n/k ) P → E (1) = (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x. (b) Finally, from part (a) and Theorem 3.1 we have k (cid:80) ni =1 Z (1) i { Z (2) i >Z (2)( k : n ) } MME( k/n ) = b ← ( b ( n/k )) b ( n/k ) 1 n (cid:80) ni =1 Z (1) i { Z (2) i >Z (2)( k : n ) } kn b ← ( b ( n/k )) b ( n/k ) MME( k/n ) P → (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x (cid:82) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x = 1 , which is what we needed to show. (cid:50) An analogous result holds for the empirical estimator (cid:91)
MES emp ,n ( k/n ) := 1 k n (cid:88) i =1 Z (1) i { Z (2) i >Z (2)( k : n ) } of MES( k/n ) = E ( Z | Z > b ( n/k )) where k < n . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 Proposition 4.3.
Let the assumptions of Theorem 3.5 hold, and let F Z ∈ RV − β for some α ≤ β ≤ α .Furthermore, let k = k ( n ) be a sequence of integers satisfying k → ∞ , k/n → and b ← ( b ( n/k )) /n → as n → ∞ .(a) Then, as n → ∞ , b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 Z (1) i { Z (2) i >Z (2)( k : n ) } P → (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x. (b) In particular, (cid:91) MES emp ,n ( k/n ) MES ( k/n ) P → , as n → ∞ The proof of the theorem is analogous to the proof of Proposition 4.1 based on the following version ofLemma 4.2. Hence, we skip the details.
Lemma 4.4.
Let the assumptions of Proposition 4.3 hold. Define for y > , E ∗ n ( y ) := b ← ( b ( n/k )) b ( n/k ) 1 n n (cid:88) i =1 Z (1) i { Z (2) i >b ( n/k ) y } ,E ∗ ( y ) := (cid:90) ∞ ν (( x, ∞ ) × ( y, ∞ )) d x. Then E ∗ ( y ) = y − α E ∗ (1) and as n → ∞ , ( E ∗ n ( y )) y ≥ / P → ( E ∗ ( y )) y ≥ / in D ([1 / , ∞ ) , (0 , ∞ )) . Proof.
The only differences between the proofs of Lemma 4.2 and Lemma 4.4 are that in the proof ofLemma 4.4 we use E ∗ ( M ) n ( y ) := (cid:82) M M ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x and that in Step 3, we havelim M →∞ lim n →∞ E (cid:32) sup y ≥ (cid:34)(cid:90) M + (cid:90) ∞ M (cid:35) ν ( n )0 (( x, ∞ ) × ( y, ∞ )) d x (cid:33) = 0where Assumption (B2) has to be used. In certain situations we might be interested in estimating MME( p ) or MES( p ) in a region where no dataare available. Since empirical estimators would not work in such a case we can resort to extrapolation viaextreme value theory. We start with a motivation for the definition of the estimator before we provide its’asymptotic properties. For the rest of this section we make the following assumption. Assumption D F Z ∈ RV − β for α ≤ β ≤ α < β + 1.Assumption D guarantees that lim t →∞ a ( t ) = 0 (see Remark 3.6). The idea here is that for all p ≥ k/n , weestimate MME( p ) empirically since sufficient data are available in this region; on the other hand for p < k/n we will use an extrapolating extreme-value technique. For notational convenience, define the function a ( t ) := b ← ( b ( t )) t b ( t ) . Since b ← ∈ RV α and b ∈ RV /β , we have a ∈ RV α − β − β . Now, let k := k ( n ) be a sequence of integers sothat k/n → n → ∞ . From Theorem 3.1 we already know thatlim p ↓ a (1 /p )MME( p ) = (cid:90) ∞ ν (( x, ∞ ) × (1 , ∞ )) d x = lim n →∞ a ( n/k )MME( k/n ) . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
Hence, MME( p ) ∼ a ( n/k ) a (1 /p ) MME( k/n ) ∼ (cid:18) knp (cid:19) β − α β MME( k/n ) ( p ↓ . (4.7)If we plug in the estimators (cid:98) α ,n , (cid:98) β n and (cid:92) MME n ( k/n ) for α , β and MME( k/n ) respectively in (4.7) weobtain an estimator for MME( p ) given by (cid:92) MME n ( p ) = (cid:18) knp (cid:19) (cid:98) βn − (cid:98) α ,n +1 (cid:98) βn (cid:92) MME emp ,n ( k/n ) . Similarly, we may obtain an estimator of MES( p ) given by (cid:91) MES n ( p ) = (cid:18) knp (cid:19) (cid:98) βn − (cid:98) α ,n +1 (cid:98) βn (cid:91) MES emp ,n ( k/n ) . If β > α then the parameter α , the index of regular variation of Z , is surprisingly not necessary for theestimation of either Marginal Mean Excess or Marginal Expected Shortfall. Theorem 4.5.
Let Assumptions A, (B1) and D hold. Furthermore, let k = k ( n ) be a sequence of integerssatisfying k → ∞ , k/n → as n → ∞ . Moreover, p n ∈ (0 , is a sequence of constants with p n ↓ and np n = o ( k ) as n → ∞ . Let (cid:98) α ,n and (cid:98) β n be estimators for α and β , respectively such that ln (cid:18) knp n (cid:19) ( (cid:98) α ,n − α ) P → and ln (cid:18) knp n (cid:19) (cid:16) (cid:98) β n − β (cid:17) P → n → ∞ ) . (4.8) (a) Then (cid:92) MME n ( p n ) MME ( p n ) P → as n → ∞ .(b) Additionally, if Assumption (B2) is satisfied then (cid:91) MES n ( p n ) MES ( p n ) P → as n → ∞ . Proof. (a) Rewrite (cid:92)
MME n ( p n )MME( p n ) = (cid:16) knp n (cid:17) (cid:98) βn − (cid:98) α ,n +1 (cid:98) βn (cid:92) MME emp ,n ( k/n )MME( p n )= (cid:92) MME emp ,n ( k/n )MME( k/n ) a ( n/k )MME( k/n ) a (1 /p n )MME( p n ) a (1 /p n ) a ( n/k ) (cid:0) np n k (cid:1) β − α β (cid:16) knp n (cid:17) (cid:98) βn − (cid:98) α ,n +1 (cid:98) βn (cid:16) knp n (cid:17) β − α β =: I ( n ) · I ( n ) · I ( n ) · I ( n ) . An application of Proposition 4.1 implies I ( n ) = (cid:92) MME emp ,n ( k/n )MME( k/n ) P → n → ∞ . For the second term I ( n ), using Theorem 3.1 we get I ( n ) = kn b ← ( b ( n/k )) b ( n/k ) MME( k/n ) p n b ← ( b (1 /p n )) b (1 /p n ) MME( p n ) P → n → ∞ ) . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 a ∈ RV ( α − β − /β , k/n → np n = o ( k ), we obtain lim n →∞ I ( n ) = 1 as well. For the last term I ( n ) we use the representation (cid:16) knp n (cid:17) (cid:98) βn − (cid:98) α ,n +1 (cid:98) βn (cid:16) knp n (cid:17) β − α β = exp (cid:18)(cid:18) − (cid:98) α ,n (cid:98) β n − − α β (cid:19) ln (cid:18) knp n (cid:19)(cid:19) , and 1 − (cid:98) α ,n (cid:98) β n − − α β = ( β − (cid:98) β n ) 1 − (cid:98) α ,n (cid:98) β n β + ( α − (cid:98) α ,n ) 1 β . Since by assumption (4.8) we have (cid:98) α ,n P → α , (cid:98) β n P → β , using a continuous mapping theorem we get,ln (cid:18) knp n (cid:19) ( β − (cid:98) β n ) 1 − (cid:98) α ,n (cid:98) β n β + ln (cid:18) knp n (cid:19) ( α − (cid:98) α ,n ) 1 β P → . Hence, we conclude that I ( n ) P → n → ∞ which completes the proof.(b) This proof is analogous to (a) and hence is omitted here.
5. Simulation study
In this section, we study the developed estimators for different models. We simulate from models describedin Section 2 and Section 3, estimate MME and MES values from the data and compare them with the actualvalues from the model. We also compare our estimator with a regular empirical estimator and observe thatour estimator provides a smaller variance in most simulated examples. Moreover our estimator is scalable tosmaller p < /n where n is the sample size, which is infeasible for the empirical estimator. As an estimator of β , the index of regular variation of Z we use the Hill-estimator based on the data Z (2)1 , . . . , Z (2) n whose order statistics is given by Z (2)(1: n ) ≥ . . . ≥ Z (2)( n : n ) . The estimator is (cid:98) β n = 1 k k (cid:88) i =1 [ln( Z (2)( i : n ) ) − ln( Z (2)( k : n ) )]for some k := k ( n ) ∈ { , . . . , n } . Similarly, we use as estimator for α , the index of hidden regularvariation, the Hill-estimator based on the data min( Z (1)1 , Z (2)1 ) , . . . , min( Z (1) n , Z (2) n ). Therefore, define Z min i =min( Z (1) i , Z (2) i ) for i ∈ N . The order statistics of Z min1 , . . . , Z min n are denoted by Z min(1: n ) ≥ . . . ≥ Z min( n : n ) . TheHill-estimator for α is then (cid:98) α ,n = 1 k k (cid:88) i =1 [ln( Z min( i : n ) ) − ln( Z min( k : n ) )]for some k := k ( n ) ∈ { , . . . , n } . Corollary 5.1.
Let Assumptions A and D hold. Furthermore, suppose the following conditions are satisfied:1. min( k, k , k ) → ∞ , max( k, k , k ) /n → as n → ∞ . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen2. p n ∈ (0 , such that p n ↓ , np n = o ( k ) and ln( k/ ( np n )) = o (min( √ k , √ k )) as n → ∞ .3. The second order conditions lim t →∞ b ( tx ) b ( t ) − x /α A ( t ) = x /α x ρ − ρ and lim t →∞ b ( tx ) b ( t ) − x /β A ( t ) = x /β x ρ − ρ , x > , where ρ , ρ ≤ are constants and A , A are positive or negative functions hold.4. lim t →∞ A ( t ) = lim t →∞ A ( t ) = 0 .5. There exist finite constants λ , λ such that lim n →∞ (cid:112) k A (cid:18) nk (cid:19) = λ and lim n →∞ (cid:112) k A (cid:18) nk (cid:19) = λ . Then (4.8) is satisfied.
Proof.
From [13, Theorem 3.2.5] we know that √ k ( β − (cid:98) β n ) D → N as n → ∞ where N is a normallydistributed random variable. In particular, (cid:98) β n P → β as n → ∞ . The analogous result holds for (cid:98) α ,n as well.Since by assumption ln( k/np n ) = o ( √ k i ) ( i = 0 , (cid:112) k ( (cid:98) α ,n − α ) ln (cid:16) knp n (cid:17) √ k P → (cid:112) k ( (cid:98) β n − β ) ln (cid:16) knp n (cid:17) √ k P → , ( n → ∞ )which is condition (4.8). Remark 5.2
In our simulation study in Section 5.2 we take k = k = k . In the study of extreme values,the choice of k plays an important role and much work goes on in this area; see [35] for a brief overview. Wechoose k to be 10% of n , which is an ad-hoc choice but often used in practice. Remark 5.3
An alternative to the Hill estimator is the probability weighted moment estimator based onthe block maxima method which is under some regularity condition consistent and asymptotically normallydistributed as presented in [18, Theorem 2.3] and hence, satisfies (4.8). Moreover, the peaks-over-threshold(POT) method is a further option to estimate α , β which satisfies as well under some regularity conditions(4.8); for more details on the asymptotic behavior of estimators based on the POT method see [36]. First we use our methods on a few simulated examples.
Example 5.4 (Gaussian copula)
Suppose ( Z , Z ) has identical Pareto marginal distributions with com-mon parameter α > C Φ ,ρ ( u, v ) with ρ ∈ ( − , ρ ∈ (1 − α +1 , p → MME( p ) = ∞ .In the Gaussian copula model, we can numerically compute the value of MME( p ) for any specific 0 < p < α = 2 , ρ = 0 .
9. Hence α = 2 . α = 2 , ρ = 0 .
5. Hence α = 2 . α = 2 . , ρ = 0 .
8. Hence α = 2 . α = 1 . , ρ = 0 .
8. Hence α = 2 . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =2, r =0.9 llllllllllllllllllllllll lllllllllllllllllllllllllll lllllllllllllllllllllll llllllllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =2, r =0.9 lllll lllllllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =2, r =0.5 llllllllllllllllllllllll lllllllllllllllllllllllllll lllllllllllllllllllllllllllll lllllllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =2, r =0.5 llllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =2.3, r =0.8 lllllllllllll lllllllllllllllllll llllllllllllllllllll llllllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =2.3, r =0.8 lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =1.9, r =0.8 llllllllllllllllllllllll lllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =1.9, r =0.8 Figure 5.1 . Box plots of (cid:92)
MME emp ( p ) / MME( p ) with p / , p / (cid:92) MME( p ) / MME( p ) with p / , p / , p / , p / α = 2 , ρ = 0 . α = 2 .
1; (b) topright: α = 2 , ρ = 0 . α = 2 .
67; (c) bottom left: α = 2 . , ρ = 0 . α = 2 .
55, (d) bottom right: α = 1 . , ρ = 0 . α = 2 . The parameters α and α are estimated using the Hill estimator which appears to estimate the parametersquite well; see [34] for details. The estimated values (cid:98) α and α are used to compute estimated values of MME.In order to check the performance of the estimator when p (cid:28) /n we create box-plots for (cid:92) MME / MMEfrom 500 samples in each of the four models, where n = 1000 , k = 100 and we restrict to 4 values of p givenby 1 / , / , / , / Example 5.5 (Marshall-Olkin copula)
Suppose ( Z , Z ) has identical Pareto marginal distributionswith parameter α > Marshall-Olkin survival copula with parameters γ , γ ∈ (0 ,
1) as given in Example 3.12.We note that a parameter restriction from Example 3.12 is given by min( γ , γ ) ∈ (1 − /α, γ ≥ γ case but not for MES in this example. For γ ≥ γ , we can explicitlycompute MME( p ) = 1 α − p − γ − /α . imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen llllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =2, g =0.8, g =0.7 llllllllllllllllllll llllllllllllllllllll llllllllllllllllllllllllll lllllllllllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =2, g =0.8, g =0.7 llllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) Empirical vs. actual: a =2.5, g =0.8, g =0.8 lllllllllllllllll lllllllllllllllll lllllllllllllllllllll llllllllllllllllllll MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) Estimated vs. actual: a =2.5, g =0.8, g =0.8 Figure 5.2 . Box plots of (cid:92)
MME emp ( p ) / MME( p ) with p / , p / (cid:92) MME( p ) / MME( p ) with p / , p / , p / , p / α = 2 , γ = 0 . , γ = 0 . α = 2 .
6; (b) right two plots: α = 2 . , γ = 0 . , γ = 0 . α = 3. In our study we generate the above distribution for two sets of choice of parameters:(a) α = 2 , γ = 0 . , γ = 0 .
7. Hence α = 2 . α = 2 . , γ = 0 . , γ = 0 .
8. Hence α = 3.In Figure 5.2, we create box-plots for (cid:92) MME / MME from 500 samples in each of the four models, where n = 1000 , k = 100 and we restrict to 4 values of p given by 1 / , / , / , / llllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllll MME emp ( p1 ) MME ( p1 ) MME emp ( p2 ) MME ( p2 ) MME: Empirical vs. actual lllllllllllllllllllllllllll llllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll
MME ( p1 ) MME ( p1 ) MME ( p2 ) MME ( p2 ) MME ( p3 ) MME ( p3 ) MME ( p4 ) MME ( p4 ) MME: Estimated vs. actual llllllllllllllllllllllllll
MES emp ( p1 ) MES ( p1 ) MES emp ( p2 ) MES ( p2 ) MES: Empirical vs. actual llllllllllllllllllll lllllllllllllllllllllll lllllllllllllllllllllllllll llllllllllllllllllllllllllllll
MES ( p1 ) MES ( p1 ) MES ( p2 ) MES ( p2 ) MES ( p3 ) MES ( p3 ) MES ( p4 ) MES ( p4 ) MES: Estimated vs. actual
Figure 5.3 . (a) Left two plots: Box plots of (cid:92)
MME emp ( p ) / MME( p ) with p / , p / (cid:92) MME( p ) / MME( p )with p / , p / , p / , p / α = 1 . , α = 2. (b) Right twoplots: Analog plots for MES. Example 5.6 (Model C)
We look at Model C where Y = ( Y , Y ) and Y , Y are iid Pareto ( α ) randomvariables, V = ( V , V ) with V = V following Pareto ( α ) and Z = Y + V . Using Theorem 3.8 we can imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 Z ∈ MRV ( α, b, ν ) ∩ HRV ( α , b , ν ) if α < α < α + 1 and all conditions (A), (B1) and (B2)are satisfied. Thus, we can find limits for both MME( p ) and MES( p ) for p going to 0. It is also possible tocalculate MME and MES explicitly. We do so for α = 1 . α = 2 here.We found that the Hill plots were not that stable, hence we used an L-moment estimator (a probabilityweighted moment estimator could be used as well) to estimate α and α ; see [13, 20] for details. Theestimates of the tail parameters are not shown here. In Figure 5.3, we create box-plots for (cid:92) MME / MMEand (cid:91)
MES / MES where n = 1000 , k = 100 with 500 samples and we restrict to 4 values of p given by1 / , / , / , / p = 1 / , p = 1 / In this section we use the method we developed in order to estimate MME and MES from a real dataset. We observe a data set which exhibits asymptotic tail independence and we compare estimates of bothstatistics (MME and MES) under this assumption versus a case when we use a formula that does not assumeasymptotic independence (similar to estimates obtained in [7]). lll ll l l lll llll l ll l llll l ll lll l ll llll l ll l ll ll l lll l ll l lll ll lllll ll ll l ll lllll l ll ll ll lll ll llll ll lll ll ll l l lll llll ll lllll ll ll ll lll llll llll llll l lll llll ll ll ll llll ll ll ll lll lll lllllll ll lll lll lll lll ll ll ll llll l lllll lll l ll ll lll ll lll ll ll lllll lll ll ll ll l lllll lll llll ll l llll ll ll ll ll lll lll ll ll lll ll ll ll ll ll ll ll lllll ll lllllll lll llll l ll ll l ll ll l lll ll l ll lll ll l ll ll lll ll lll l ll llll lll lll lll ll llll lll ll lll ll lll lll l l ll l lll llll lll l lll lll llllll l lll l ll l llll lll ll ll lllllll lll lllll l l lll lll ll llll l lll ll ll lll l lll lll ll l ll lll lll lll llll lll ll ll ll ll l ll lll ll ll lll ll lll ll l lll lll l ll l llllll ll l ll lll l lll ll l l lll llllll ll ll ll llll ll l ll ll ll lll lll llll lll ll l ll ll ll ll l ll l l llll lll lll l lll lll ll l lll l ll l llll ll llll ll llll llll ll ll l ll ll lll llllll l l . . . . . . Netflix returns S n P r e t u r n s llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll −0.5 0.0 0.5 1.0 1.5 2.0 . . . . angular density for Netflix vs SnP theta angu l a r m ea s u r e den s i t y Figure 5.4 . Left plot: Scatter plot of (NFLX, SNP). Right plot: angular density plot to of the rank-transformed returns data.
We observe return values from daily equity prices of Netflix (NASDAQ:NFLX) as well as daily returnvalues from S&P 500 index for the period January 1, 2004 to December 31, 2013. The data was downloadedfrom
Yahoo Finance ( http://finance.yahoo.com/ ). The entire data set uses 2517 trading days out of which687 days exhibited negative returns in both components and we used these 687 data points for our study.A scatter plot of the returns data shows some concentration around the axes but the data seems to exhibitsome positive dependence of the variables too; see leftmost plot in Figure 5.4. Since the scatterplot doesn’tclearly show whether the data has asymptotic tail independence or not, we create an angular density plot ofthe rank-transformed data. Under asymptotic independence we should observe two peaks in the density, oneconcentrating around 0 and the other around π/
2, which is what we see in the right plot in Figure 5.4; see [34]for further discussion on the angular density. Hence, we can discern that our data exhibits asymptotic tailindependence and proceed to compute the hidden regular variation tail parameter using min(NFLX, SNP)as the data used to get a Hill estimate of α . The left two plots in Figure 5.5 show Hill plots of both theNetflix negative returns (NFLX) and the S&P 500 negative returns (SNP). A QQ plot (not shown) suggeststhat both margins are heavy-tailed and by choosing k = 50 for the Hill-estimator we obtain as estimate of imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017 B. Das and V. Fasen
Netflix number of order statistics H ill e s t i m a t e SnP number of order statistics H ill e s t i m a t e min( Netflix , SnP ) number of order statistics H ill e s t i m a t e Figure 5.5 . Hill plots of the tail parameters of the two negative returns (NFLX,SNP) and that of hidden tail parameter α estimated using min(NFLX,SNP). the tail parameters (cid:98) α NFLX = 2 . , (cid:98) α SNP = 2 .
46 (indicated by blue horizontal lines in the plot). Again usinga Hill-estimator with k = 50, the estimate (cid:98) α = 2 .
86 is obtained; see the rightmost plot in Figure 5.5.Now, we use the values of (cid:98) α SNP = 2 .
46 and (cid:98) α = 2 .
86 to compute estimated values of MME and MES.In Figure 5.6 we plot the empirical estimates of MME and MES (dotted lines), the extreme value estimatewithout assuming asymptotic independence (blue bold line) and the extreme value estimate assuming asymp-totic independence (black bold line). We observe that both MME and MES values are smaller under theassumption of asymptotic independence than in the case where we do not assume asymptotic independence.Hence, without an assumption of asymptotic independence, the firm might over-estimate its’ capital shortfallif the systemic returns tend to show an extreme loss. . . . . . . quantiles MM E EmpiricalEstimated (asy. dep.)Estimated (asy. ind.)0.01 0.02 0.03 0.04 0.05 0.06 0.07 . . . . . . . . . . . . quantiles M ES EmpiricalEstimated (asy. dep.)Estimated (asy. ind.)Expected Shortfall0.01 0.02 0.03 0.04 0.05 0.06 0.07 . . . . . . Figure 5.6 . MME and MES plots under the tail dependence model as well as the asymptotic independent model. imsart-bj ver. 2014/10/16 file: eshrv_jmva_resub.tex date: April 26, 2017
6. Conclusion
In this paper we study two measures of systemic risk, namely
Marginal Expected Shortfall and
MarginalMean Excess in the presence of asymptotic independence of the marginal distributions in a bivariate set-up.We specifically observe that the very useful Gaussian copula model with Pareto-type tails satisfies our modelassumptions for the MME and we can find the right rate of increase (decrease) of MME in this case. Moreoverwe observe that if the data exhibit asymptotic tail independence , then we can provide an estimate of MMEthat is closer to the empirical estimate (and possibly smaller) than the one that would be obtained if we didnot assume asymptotic tail independence .In a companion paper, [9], we investigate various copula models and mixture models which satisfy ourassumptions under which we can find asymptotic limits of MME and MES. A further direction of work wouldinvolve finding the influence of multiple system-wide risk events (for example, multiple market indicators)on a single or group of components (for example, one or more financial institutions).
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