# Second-Order Asymptotics of the Risk Concentration of a Portfolio with Deflated Risks

Authors: Yu Chen; Yu Gao; Wenxue Gao; Weiping Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. In this paper, we establish second-order asymptotics of the risk concentration based on several risk measures for a portfolio ofn identically distributed but dependent deflated risks Xj=RjS, j=1,2,…,n under the assumptions of second-order regular variation on the survival functions of the risks Rj and the deflator S, where R1,R2,…,Rn are n independent and identically distributed random variables with a common survival function and S is a random variable being independent of R1,R2,…,Rn. Examples are also given to illustrate our main results.


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## Body

## 1. Introduction

The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. Measuring a risk and quantifying its diversification benefits have become an important task. Especially when the underlying risk factors show a heavy-tailed pattern, many papers discussed diversification benefits; see, for instance, Degen et al. [1] (2010), Ibragimov and Walden [2], Ibragimov et al. [3], Mao et al. [4], Lv et al. [5, 6], Hashorva et al. [7], and references therein.Risk measure is understood as a function that can assign a nonnegative real number to a risk. Consider a portfolio ofn loss random variables X1,X2,…,Xn. The risk concentration based on the risk measure ϱ[·] is defined as(1)Cϱ=ϱ∑i=1nXi∑i=1nϱXi.Here, 1-Cϱ refers to the diversification benefit. In recent years, empirical work has argued that financial variables often exhibit stronger dependence, while the existing work usually assumes that the risks X1,X2,…,Xn are independent and identically distributed; see Embrechts et al. [8, 9], Degen et al. [10], Mao and Hu [11], Mao and Hu [12], Lv et al. [6], and so on. We focus on the asymptotic of risk concentration for a portfolio of n identically distributed but correlated deflated risks Xj=RjS, j=1,…,n under assumptions of second-order regular variation on the survival functions of the risk R1,…,Rn and deflator S.In the present paper we study mathematical properties of diversification effects under the different risk measuresϱ[·]. Several popular risk measures have been introduced to measure tail risk, such as the Value-at-Risk (VaR), the conditional tail expectation (CTE), and the Haezendonck-Goovaerts risk measure. These risk measures have been used extensively in insurance and finance as a tool of risk management; see Denuit et al. [13], Artzner et al. [14], Cheung and Lo [15], Zhu et al. [16], and references therein. The Value-at-Risk (VaR) of X at the level p is defined as (2)VaRpX=inf⁡x∈R:Fx≥p,p∈0,1,and the conditional tail expectation (CTE) of X at the level p is defined as (3)CTEpX=EX∣X>VaRpX,p∈0,1.The Haezendonck-Goovaerts risk measure, which was introduced by Haezendonck and Goovaerts [17], is defined via an increasing and convex Young function ϕ and a parameter p∈(0,1) representing the confidence level. More precisely, let ϕ be a nonnegative and convex function on [0,∞) with ϕ(0)=0, ϕ(1)=1, and ϕ(∞)=∞. This function is called a normalized Young function. Assume we have a real-valued random variable X with distribution function F such that(4)EϕcX<∞∀c>0,and let Hp[X,t] be the unique solution h to the equation(5)EϕX-t+h=1-p,p∈0,1,if  F¯(t)>0 and 0 if  F¯(t)=0, where x+=max⁡{x,0} for any real number x. Then the Haezendonck-Goovaerts risk measure of X at the confidence level p is defined as(6)HGpX=inft∈Rt+HpX,t.Some important properties and connections with other risk measures are given in Goovaerts et al. [18]. It is well known that the simplest case of the Haezendonck-Goovaerts risk measure HGp[X] with ϕ(x)=x reduces to CTEp[X]. Even for a power Young function, the explicit solution to (5) is generally not available. Now, in this paper, we instead considered the asymptotic behavior of risk concentration based on the Haezendonck-Goovaerts risk measure HGp[X] with ϕ(t)=tk for k≥1 as p↑1.Another family of risk measures, introduced by Wang [19], is defined by using the concept of the distortion function. A distortion function is an increasing function g:[0,1]→[0,1] such that g(0)=0 and g(1)=1. Then for any risk X with distribution function F, the corresponding distortion risk measure Dg[·] is defined as follows: (7)DgX=∫0∞gF¯tdt-∫-∞01-gF¯tdt,where F¯(t)=1-F(t) denotes the survival function of X. The distortion risk measure has several useful properties such as positive homogeneity, translation invariance, additivity for comonotonic risks, and monotonicity. For more details, see Denuit et al. [13], Dhaene et al. [20], and Balbás et al. [21]. Several popular risk measures belong to the family of distortion risk measures. For example, the Value-at-Risk (VaR) of X at the level p corresponds to the distortion function g(x)=1(1-p,1](x), x∈[0,1], where 1E is the indicator function of E; the conditional tail expectation (CTE) of X at level p corresponds to the distortion function g(x)=min⁡{x/(1-p),1}, x∈(0,1).The tail distortion risk measure, first introduced by Zhu and Li [22], was reformulated by Yang [23] as follows: for a distortion function g, the tail distortion risk measure at level p of a loss variable X is defined as Tgp[X]=Dgp[X], p∈(0,1), where (8)gpu=gu1-p,0≤u≤1-p,1,1-p<u≤1.Since the risk is always heavy-tailed and often obeys a law of regular variation, we choose ϱ[·] as VaRp[·], HGp[·], and Tgp[·] at the level 0<p<1, respectively, in (1). We denote risk concentration Cϱ at the level p by Cϱ(p).Because risk managers become more and more concerned with tail area of risk, we will focus on the second-order approximations of the risk concentrations based on the different risk measures asp↑1, such as CVaR(p), CHG(p), CCTE(p), and CTg(p) as p↑1 for a portfolio of n loss random variables X1,X2,…,Xn. In this paper, we assume that random variables X1,X2,…,Xn are identically distributed but not independent; that is,(9)X1,X2,…,Xn=R1S,R2S,…,RnS,where R1,R2,…,Rn are n i.i.d random variables with a common survival function possessing the property of second-order regular variation, and the deflator S is a random variable which is independent of R1,R2,…,Rn.The first-order approximations ofCVaR(p) as p↑1 were studied by Embrechts et al. [8, 9] under the model assumption that the underlying risks X1,X2,…,Xn have identically distributed and regularly varying margins and have two forms of dependent structure, respectively. Degen et al. [10] derived second-order approximations of CVaR(p) for n independent and identically distributed (i.i.d) loss variables with a common survival function possessing the property of second-order regular variation (2RV). Second-order approximations of the risk concentrations CCTE(p) and CTg(p) as p↑1 for n i.i.d loss random variables were derived by Mao et al. [4], Mao and Hu [12], Lv et al. [6], and Hashorva et al. [7]. For a portfolio of n i.i.d. risks, the second-order approximations of the risk concentrations CVaR(p), CCTE(p) as p↑1 have been discussed by Hashorva et al. [24], while Mao and Yang [25] consider the case with a portfolio of n dependent risks under FGM copula. Ling and Peng [26] derived higher-order approximations under some conditions.The paper is organized as follows. In Section2, we describe the definition of the second-order regular variation and some useful propositions of it. In Section 3, we obtain our main results, that is, the second-order approximations of the risk concentrations CVaR(p), CHG(p), and CTg(p) as p↑1, and present their proofs. In Section 4, some examples are provided to illustrate the performance of our approximations. Throughout, the notation “~” means asymptotic equivalence, that is, for functions f(x) and g(x), (10)fx~gx,x⟶x0⟺limx→x0fxgx=1.


## 2. Preliminaries

Regular variation is one of the basic concepts which appears in different contexts of applied probability. A functionh is said to be of regular variation with index α∈R, denoted by h∈RV-α, if(11)limx→∞hxyhx=y-αholds for any y>0. Next we recall the definition of the second-order regular variation from de Haan and Ferreira [27] and de Haan and Stadtmüller [28]. Suppose that h∈RV-α for some α∈R; then h is said to be of second-order regular variation with first-order parameter α and second-order parameter ρ≤0, denoted by h∈2RV-α, ρ, if there exists some ultimately positive or negative function A(x) with A(x)→0 as x→∞ such that(12)limx→∞hxy/hx-y-αAx=y-α∫1yuρ-1du,∀x>0.Here, A(x) is referred to as an auxiliary function of h and |A|∈RVρ. Several classes of parametric survival functions are shown to possess 2RV properties; see Hashorva et al. [7]. For more details on RV and 2RV, see Hua and Joe [29] and Lv et al. [5].The function which possesses the property of second-order regular variation (2RV) plays an important role in this article. The following proposition gives a characterization of any functionh∈2RV-α, ρ with auxiliary function A(x), α∈R and ρ<0, which is from Hua and Joe [29].Proposition 1.
Letα∈R, ρ<0, and A(x)∈RVρ. Then h∈2RV-α, ρ with auxiliary function A(x) if and only if(13)hx=cx-α1+Axρ+oAx,x⟶∞,where c=limx→∞xF¯(x)∈(0,∞).The next two propositions give first- and second-order approximations of Haezendonck-Goovaerts risk measureHGp[X] of X at the confidence level p and tail distortion risk measure Tgp[X] of X at confidence level p for a distortion function g, which will be used in the proofs of our main results.Proposition 2.
LetX be a random variable with survival function F¯∈RV-α, α>0, and let ϕ(t)=tk for some α>k≥1. Then one has the following: (i)
The first-order asymptotic (see [30]; Mao and Hu, 2012a): (14)HGpX~CαF←p,p↑1.(ii)
The second-order asymptotic (see Mao and Hu, 2012a): ifF¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x), then (15)HGpX=CαF←p1+Hα,ρ,kAF←p1+o1,p↑1,where (16)Cα=αα-kk/α-1kk-1/αBα-k,k1/α,Hα,ρ,k=1αρα-kρk/αkρ/α1-kξk,0ρ/α-1ξk,ρ/α-1with (17)ξk,t=Bα1-t-k,kand B(·,·) is the Beta function as usual; that is, (18)Ba,b=∫01xa-11-xb-1dx,a,b>0.Proposition 3.
LetX be a random variable with survival function F¯∈RV-α, α>0, and let g be any distortion function with(19)∫01x-1/α-δdgx<∞for  someδ>0.We have the follwoing:(i)
The first-order asymptotic (see [22, 23]): (20)TgpX~JαF←p,p↑1.(ii)
The second-order asymptotic (see [23]): if F¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x), then (21)TgpX=JαF←p+F←pAF←pIα,ρ+o1,p↑1,where (22)Jα=∫01x-1/αdgx,Iα,ρ=∫01x-1/αx-ρ/α-1αρdgx.Propositions2(ii) and 3(ii) are, respectively, modified from Theorem 4.5 in Mao and Hu (2012a) and Corollary 4.1 in Yang [23] by using the fact that F¯∈2RV-α,ρ with auxiliary function A(t) if and only if its tail quantile function U∈2RV1/α,ρ/α with auxiliary function α-2A(U(t)) (see Theorem 2.3.9 in de Haan and Ferreira [27]).


## 3. Main Results and Their Proofs

### 3.1. Main Results

In this section, we give some results establishing the second-order approximations of the risk concentrationCϱ(p) as p→1 for a portfolio of n random variables that satisfy (9). The first one is about the risk concentration CVaR(p).Theorem 4.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0, then (i)
forρ<-α and 0<α<1, (23)CVaRp=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p,p↑1;(ii)
forρ<-1, (24)CVaRp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←p+o1F←p,p↑1whenα>1, (25)CVaRp=1+c1-1nESα+1ESα1+1/α·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞), when α=1;(iii)
forρ>-(1∧α), (26)CVaRp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.In the following theorem, we derive the second-order asymptotic of risk concentration for Haezendonck-Goovaerts risk measureCHG(p) at level p.Theorem 5.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0 and if ϕ(t)=tk for some α>k≥1, then (i)
forρ<-α and 0<α<1, (27)CHGp=n1/α-11+n-12nα·ES2αESα2·1-α2Hα,-α,kξα1-p+o1-p,p↑1;(ii)
forρ<-1, (28)CHGp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αHα,-1,k·μFF←p+o1F←p,p↑1whenα>1, and (29)CHGp=1+c1-1nESα+1ESα1+1/α·1-H1,-1,k·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞) when α=1;(iii)
forρ>-(1∧α), (30)CHGp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρHα,ρ,kAF←p+oAF←p,p↑1.The last theorem gives the second-order asymptotic of risk concentration for tail distortion risk measureCTg(p) at level p.Theorem 6.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F, and S is a nonnegative random variable independent of {R1,R2,…,Rn}. Further assume that F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0. Let g be a distortion function with (31)∫01x-1/α-δdgx<∞for  someδ>0.Then(i)
forρ<-α and 0<α<1, (32)CTgp=n1/α-11+n-12nα·ES2αESα2·1-α2Iα,-αJαξα1-p+o1-p,p↑1;(ii)
forρ<-1, (33)CTgp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αIα,-1Jα·μFF←p+o1F←p,p↑1whenα>1, and (34)CTgp=1+c1-1nESα+1ESα1+1/α·1-I1,-1J1·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞) when α=1;(iii)
forρ>-(1∧α), (35)CTgp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρIα,ρJαAF←p+oAF←p,p↑1.Thus, we immediately obtain the following corollary which establishes the second-order asymptotic of risk concentration for conditional tail expectationCCTE(p). And this corollary can also be obtained easily by Lemma 8.Corollary 7.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS) be a continuous random vector, where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>1,  ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0. Then (i)
forρ<-1, (36)CCTEp=n1/α-1+n-1n1-1α·ESα+1ESα1+1/α·μFF←p+o1F←p,p↑1;(ii)
forρ>-1, (37)CCTEp=n1/α-11+nρ/α-1ρ/α·α-1α2α-1-ρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.


### 3.2. Proofs

Before proving the above results, we introduce some lemmas. The first one gives a second-order form of Breiman’s theorem (see Breiman [31]), which is from Hashorva et al. [7].Lemma 8.
LetR be a random variable with survival function F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x), and let S be a nonnegative random variable satisfying ESα-ρ+ϵ<∞ for some ϵ>0, independent of X. Then (38)PRS>xF¯x=ESα1+εx,where ε(x)=1/ρ(ESα-ρ/ESα-1)A(x)(1+o(1)) as x→∞, and thus RS∈2RV-α,ρ with auxiliary function (39)A∗x=ESα-ρESαAx.The second lemma talks about the first- and second-order asymptotic of Value-at-Risk of the productRS at the level p, which was proved by Hashorva et al. [7].Lemma 9.
LetR be a random variable with survival function F¯∈RV-α, α>0, and let S be a nonnegative random variable satisfying ESα+ϵ<∞ for some ϵ>0, independent of X. Then one has the following: (i)
The first-order asymptotic:(40)VaRpRS~ESα1/αF←p,p↑1.(ii)
The second-order asymptotic: ifF¯∈2RV-α,ρ, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞, then (41)VaRpRS=ESα1/αF←p1+ζα,ρAF←p1+o1,p↑1,whereζα,ρ=1/αρ(ESα-ρ/ESα1-ρ/α-1).First, we introduce two definitions. LetF be a distribution function of a nonnegative random variable. We introduce the truncated mean of F: (42)μFt=∫0txdFx,t>0.Obviously, if the mean of F, μF, exists, then μF(t)→μF as t→∞. For 0<α<1, define (43)ξα=22α-2α+1+2α∫011-x-α-1x-α-1dx.The following lemma from Mao and Hu [12] states that the 2RV property is preserved by the formation of sum of n i.i.d random variables.Lemma 10.
LetF be the distribution function of a nonnegative random variable satisfying F¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x). We denoted by F∗n the n-fold convolution of F. Then F∗n¯∈2RV-α,γ with auxiliary function B(x), where(44)γ=-min⁡1,α,-ρ,and B(x) is given by(45)Bx=-n-12αξαF¯x,ρ<-α,0<α<1,-n-1αx-1μFx,ρ<-1,α≥1,Ax,ρ>-1∧α.The last lemma from Mao et al. [4] establishes the second-order asymptotic of the risk concentration CVaR(p) for n i.i.d random variables with the underlying distribution possessing the 2RV property.Lemma 11.
LetR1,R2,…,Rn be i.i.d. nonnegative random variables with common continuous distribution function F, and assume that F¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x). Then (i)
forρ<-α and 0<α<1, (46)VaRp∑i=1nRinVaRpRi=n1/α-11+n-12nαξα1-p+o1-p,p↑1;(ii)
forρ<-1 and α≥1, (47)VaRp∑i=1nRinVaRpRi=n1/α-11+n-1/αn-1μFF←pF←p+oμFF←pF←p,p↑1;(iii)
forρ>-(1∧α), (48)VaRp∑i=1nRinVaRpRi=n1/α-11+nρ/α-1αρAF←p+oAF←p,p↑1.Now we turn to prove our theorems.Proof of Theorem4.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45); we have (49)VaRpYS=ESα1/αFY←p1+ζα,γBFY←p1+o1=ESα1/αFY←p1+η1FY←p1+o1,p↑1,where η1(t)=ζα,γB(t) and |η1|∈RVγ.
Similarly,(50)VaRpRiS=ESα1/αF←p1+ζα,ρAF←p1+o1=ESα1/αF←p1+η2F←p1+o1,p↑1,for i=1,2,…,n, where η2(t)=ζα,ρA(t) and |η2|∈RVρ. From Lemma 11, it follows that (51)FY←pF←p⟶n1/α,p↑1.In view of |η1|∈RVγ and Theorem  B.1.4 of de Haan and Ferreira [27], we have (52)η1FY←p=nγ/αη1F←p·1+o1,p↑1,where we use the fact that B(x) is ultimately positive or negative. Thus, (53)CVaRp=FY←pnF←p·1+η1FY←p1+o11+η2F←p1+o1=VaRpYnVaRpRi·1+nγ/αη1F←p1+o11+η2F←p1+o1,p↑1.Next, we consider three cases.
Case  1(ρ<-α and 0<α<1). In this case, from (44) and (45), it follows that γ=-α and (54)η1F←p=-n-1αζα,-α2ξα1-p=n-12αES2αESα2-1ξα1-p.Note that A(F←(p))=o(1-p) as p↑1. So, by Lemma 11, we have (55)CVaRp=n1/α-11+n-12nαξα1-p+o1-p1+n-12nαES2αESα2-1ξα1-p+o1-p1+o1=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p,p↑1.
Case  2(ρ<-1 and α≥1). In this case, γ=-1. By Karamata’s theorem, it can be proved that μF(t)∈RV0; see (2.7) and (2.8) in Mao and Hu (2012a). Hence, A(t)=o(μF(t)/t) as t→∞. We have (56)η1F←p=ζα,-1BF←p=n-1ESα+1ESα1+1/α-1μFF←pF←p,η2F←p=ζα,-1AF←p=oμFF←pF←p.Thus, (57)CVaRp=n1/α-11+n-1/αn-1μFF←pF←p1+o11+n-1/αn-1EYα+1EYα1+1/α-1·μFF←pF←p1+o1=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←pF←p1+o1,p↑1.For α>1, μF(t)→μF as t→∞. For α=1, by Proposition 1, (58)μFt~∫0tF¯xdx~∫1tcx-11+Axρdx~clog⁡t+cρ-1∫1tx-1Axdx~clog⁡t,t⟶∞,where the first equation follows from (28) in Mao and Hu [12], and the last equation follows since ∫1∞‍x-1|A(x)|dx<∞. Thus, we prove the case.
Case  3(ρ>-(1∧α)). In this case, γ=ρ, we have (59)η1F←p=ζα,ρAF←p,η2F←p=ζα,ρAF←p.Thus, (60)CVaRp=n1/α-11+nρ/α-1αρAF←p1+o11+nρ/αζα,ρAF←p1+o11+ζα,ρAF←p1+o1=n1/α-11+nρ/α-1αρAF←p1+o1·1+nρ/α-1αρESα-ρESα1-ρ/α-1AF←p1+o1=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.Proof of Theorem5.
From Proposition2 and Lemmas 8 and 9, we can get (61)HGpRiS=CαVaRpRiS1+Hα,ρ,k·ESα-ρESαAVaRpRiS1+o1=CαVaRpRiS1+Hα,ρ,k·ESα-ρESα1-ρ/αAF←p1+o1,p↑1,for i=1,2,…,n, where we use the fact that |A(x)|∈RVρ and A(x) is ultimately positive or negative.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45). So, similarly, from Lemma 10, we can get (62)HGpYS=CαVaRpYS1+Hα,γ,k·ESα-γESαBVaRpYS1+o1=CαVaRpYS1+Hα,γ,k·ESα-γESαBESα1/αFY←p1+o1=CαVaRpYS1+nγ/αHα,γ,k·ESα-γESα1-γ/αBF←p1+o1,p↑1,where we use the fact that FY←(p)/F←(p)→n1/α, as p↑1, |B(x)|∈RVγ and B(x) is ultimately positive or negative.
Thus,(63)CHGp=HGpYS∑i=1nHGpRiS=VaRpYSnVaRpRiS·1+nγ/αHα,γ,k·ESα-γ/ESα1-γ/αBF←p1+o11+Hα,ρ,k·ESα-ρ/ESα1-ρ/αAF←p1+o1=CVaRp·1+nγ/αη1F←p1+o11+η2F←p1+o1,where (64)η1x=ESα-γESα1-γ/αHα,γ,kBx,η2x=ESα-ρESα1-ρ/αHα,ρ,kAx.Next, we consider three cases. (i)
Forρ<-α and 0<α<1. In this case, γ=-α. From (45), it follows that (65)η1F←p=-n-1αξα2·ES2αESα2Hα,-α,k1-p.Note thatA(F←(p))=o(1-p) as p↑1. So, by Lemma 10, we have (66)CHGp=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p1-n-1αξα2n·ES2αESα21-p+o1-p1+o1=n1/α-11+n-12nα·ES2αESα2·1-α2Hα,-α,kξα1-p+o1-p,p↑1.(ii)
ρ<-1 and α≥1. In this case, γ=-1. By Karamata’s theorem, it can be proved that μF(t)∈RV0; see (2.7) and (2.8) in Mao and Hu (2012a). Hence, A(t)=o(μF(t)/t) as t→∞. Thus, (67)CHGp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←pF←p1+o1·1-n-1/αn-1αHα,-1,kESα+1ESα1+1/α·μFF←pF←p1+o1=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αHα,-1,k·μFF←pF←p+oμFF←pF←p,p↑1.ConsideringμF(t) as t→∞ for α>1 and α=1 as Theorem 4, we can get the result easily.(iii)
ρ>-(1∧α). In this case, γ=ρ. Thus, (68)CHGp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p1+o1·1+nρ/α-1Hα,ρ,kESα-ρESα1-ρ/αAF←p1+o1=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρHα,ρ,kAF←p+oAF←p,p↑1.Proof of Theorem6.
From Proposition3 and Lemmas 8 and 9, we can get (69)TgpRiS=JαVaRpRiS+VaRpRiSA∗VaRpRiSIα,ρ+o1=JαVaRpRiS1+Iα,ρJα·ESα-ρESα1-ρ/αAF←p1+o1,p↑1,for i=1,2,…,n, where we use the fact that |A(x)|∈RVρ and A(x) is ultimately positive or negative.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45). So, similarly, from Lemma 10, we can get (70)TgpYS=JαVaRpYS+VaRpYS·ESα-ρESαBVaRpYSIα,γ+o1=JαVaRpYS1+nγ/αIα,γJα·ESα-γESα1-γ/αBF←p1+o1,p↑1,where we use the fact that FY←(p)/F←(p)→n1/α, as p↑1, |B(x)|∈RVγ and B(x) is ultimately positive or negative. Thus, (71)CTgp=TgpYS∑i=1nTgpRiS=VaRpYSnVaRpRiS·1+nγ/αIα,γ/Jα·ESα-γ/ESα1-γ/αBF←p1+o11+Iα,ρ/Jα·ESα-ρ/ESα1-ρ/αAF←p1+o1=CVaRp·1+nγ/αη1F←p1+o11+η2F←p1+o1,where (72)η1x=Iα,γJα·ESα-γESα1-γ/αBx,η2x=Iα,ρJα·ESα-ρESα1-ρ/αAx.Next, similar to Theorems 4 and 5, we consider three cases: (i) for ρ<-α and 0<α<1; (ii) for ρ<-1 and α≥1; (iii) for ρ>-(1∧α) to obtain the result. Thus, we complete the proof.Proof of Corollary7.
Note that if the distortion functiong(x)=x, then Tg,p[X] reduces to CTEp[X] for continuous risk variables X, and CTg(p) reduces to CCTE(p). It is easy to see that, for α>1 and ρ<0, (73)Jα=αα-1,Iα,ρ=1α-ρ-1α-1,and, hence, (74)αIα,-1Jα=1α,αρIα,ρJα=α-1α-1-ρ.Therefore, the result is an immediate consequence of Theorem 6.


## 3.1. Main Results

In this section, we give some results establishing the second-order approximations of the risk concentrationCϱ(p) as p→1 for a portfolio of n random variables that satisfy (9). The first one is about the risk concentration CVaR(p).Theorem 4.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0, then (i)
forρ<-α and 0<α<1, (23)CVaRp=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p,p↑1;(ii)
forρ<-1, (24)CVaRp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←p+o1F←p,p↑1whenα>1, (25)CVaRp=1+c1-1nESα+1ESα1+1/α·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞), when α=1;(iii)
forρ>-(1∧α), (26)CVaRp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.In the following theorem, we derive the second-order asymptotic of risk concentration for Haezendonck-Goovaerts risk measureCHG(p) at level p.Theorem 5.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0 and if ϕ(t)=tk for some α>k≥1, then (i)
forρ<-α and 0<α<1, (27)CHGp=n1/α-11+n-12nα·ES2αESα2·1-α2Hα,-α,kξα1-p+o1-p,p↑1;(ii)
forρ<-1, (28)CHGp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αHα,-1,k·μFF←p+o1F←p,p↑1whenα>1, and (29)CHGp=1+c1-1nESα+1ESα1+1/α·1-H1,-1,k·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞) when α=1;(iii)
forρ>-(1∧α), (30)CHGp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρHα,ρ,kAF←p+oAF←p,p↑1.The last theorem gives the second-order asymptotic of risk concentration for tail distortion risk measureCTg(p) at level p.Theorem 6.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS), where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F, and S is a nonnegative random variable independent of {R1,R2,…,Rn}. Further assume that F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0. Let g be a distortion function with (31)∫01x-1/α-δdgx<∞for  someδ>0.Then(i)
forρ<-α and 0<α<1, (32)CTgp=n1/α-11+n-12nα·ES2αESα2·1-α2Iα,-αJαξα1-p+o1-p,p↑1;(ii)
forρ<-1, (33)CTgp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αIα,-1Jα·μFF←p+o1F←p,p↑1whenα>1, and (34)CTgp=1+c1-1nESα+1ESα1+1/α·1-I1,-1J1·log⁡F←pF←p1+o1,p↑1withc=limx→∞xF¯(x)∈(0,∞) when α=1;(iii)
forρ>-(1∧α), (35)CTgp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρIα,ρJαAF←p+oAF←p,p↑1.Thus, we immediately obtain the following corollary which establishes the second-order asymptotic of risk concentration for conditional tail expectationCCTE(p). And this corollary can also be obtained easily by Lemma 8.Corollary 7.
Let(X1,X2,…,Xn)=(R1S,R2S,…,RnS) be a continuous random vector, where {R1,R2,…,Rn} are i.i.d. nonnegative random variables with common continuous distribution function F and S is a nonnegative random variable independent of {R1,R2,…,Rn}. If F¯∈2RV-α,ρ, α>1,  ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞ for some ϵ>0. Then (i)
forρ<-1, (36)CCTEp=n1/α-1+n-1n1-1α·ESα+1ESα1+1/α·μFF←p+o1F←p,p↑1;(ii)
forρ>-1, (37)CCTEp=n1/α-11+nρ/α-1ρ/α·α-1α2α-1-ρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.


## 3.2. Proofs

Before proving the above results, we introduce some lemmas. The first one gives a second-order form of Breiman’s theorem (see Breiman [31]), which is from Hashorva et al. [7].Lemma 8.
LetR be a random variable with survival function F¯∈2RV-α,ρ, α>0, ρ<0, with auxiliary function A(x), and let S be a nonnegative random variable satisfying ESα-ρ+ϵ<∞ for some ϵ>0, independent of X. Then (38)PRS>xF¯x=ESα1+εx,where ε(x)=1/ρ(ESα-ρ/ESα-1)A(x)(1+o(1)) as x→∞, and thus RS∈2RV-α,ρ with auxiliary function (39)A∗x=ESα-ρESαAx.The second lemma talks about the first- and second-order asymptotic of Value-at-Risk of the productRS at the level p, which was proved by Hashorva et al. [7].Lemma 9.
LetR be a random variable with survival function F¯∈RV-α, α>0, and let S be a nonnegative random variable satisfying ESα+ϵ<∞ for some ϵ>0, independent of X. Then one has the following: (i)
The first-order asymptotic:(40)VaRpRS~ESα1/αF←p,p↑1.(ii)
The second-order asymptotic: ifF¯∈2RV-α,ρ, ρ<0, with auxiliary function A(x) and ESα-ρ+ϵ<∞, then (41)VaRpRS=ESα1/αF←p1+ζα,ρAF←p1+o1,p↑1,whereζα,ρ=1/αρ(ESα-ρ/ESα1-ρ/α-1).First, we introduce two definitions. LetF be a distribution function of a nonnegative random variable. We introduce the truncated mean of F: (42)μFt=∫0txdFx,t>0.Obviously, if the mean of F, μF, exists, then μF(t)→μF as t→∞. For 0<α<1, define (43)ξα=22α-2α+1+2α∫011-x-α-1x-α-1dx.The following lemma from Mao and Hu [12] states that the 2RV property is preserved by the formation of sum of n i.i.d random variables.Lemma 10.
LetF be the distribution function of a nonnegative random variable satisfying F¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x). We denoted by F∗n the n-fold convolution of F. Then F∗n¯∈2RV-α,γ with auxiliary function B(x), where(44)γ=-min⁡1,α,-ρ,and B(x) is given by(45)Bx=-n-12αξαF¯x,ρ<-α,0<α<1,-n-1αx-1μFx,ρ<-1,α≥1,Ax,ρ>-1∧α.The last lemma from Mao et al. [4] establishes the second-order asymptotic of the risk concentration CVaR(p) for n i.i.d random variables with the underlying distribution possessing the 2RV property.Lemma 11.
LetR1,R2,…,Rn be i.i.d. nonnegative random variables with common continuous distribution function F, and assume that F¯∈2RV-α,ρ, ρ≤0, with auxiliary function A(x). Then (i)
forρ<-α and 0<α<1, (46)VaRp∑i=1nRinVaRpRi=n1/α-11+n-12nαξα1-p+o1-p,p↑1;(ii)
forρ<-1 and α≥1, (47)VaRp∑i=1nRinVaRpRi=n1/α-11+n-1/αn-1μFF←pF←p+oμFF←pF←p,p↑1;(iii)
forρ>-(1∧α), (48)VaRp∑i=1nRinVaRpRi=n1/α-11+nρ/α-1αρAF←p+oAF←p,p↑1.Now we turn to prove our theorems.Proof of Theorem4.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45); we have (49)VaRpYS=ESα1/αFY←p1+ζα,γBFY←p1+o1=ESα1/αFY←p1+η1FY←p1+o1,p↑1,where η1(t)=ζα,γB(t) and |η1|∈RVγ.
Similarly,(50)VaRpRiS=ESα1/αF←p1+ζα,ρAF←p1+o1=ESα1/αF←p1+η2F←p1+o1,p↑1,for i=1,2,…,n, where η2(t)=ζα,ρA(t) and |η2|∈RVρ. From Lemma 11, it follows that (51)FY←pF←p⟶n1/α,p↑1.In view of |η1|∈RVγ and Theorem  B.1.4 of de Haan and Ferreira [27], we have (52)η1FY←p=nγ/αη1F←p·1+o1,p↑1,where we use the fact that B(x) is ultimately positive or negative. Thus, (53)CVaRp=FY←pnF←p·1+η1FY←p1+o11+η2F←p1+o1=VaRpYnVaRpRi·1+nγ/αη1F←p1+o11+η2F←p1+o1,p↑1.Next, we consider three cases.
Case  1(ρ<-α and 0<α<1). In this case, from (44) and (45), it follows that γ=-α and (54)η1F←p=-n-1αζα,-α2ξα1-p=n-12αES2αESα2-1ξα1-p.Note that A(F←(p))=o(1-p) as p↑1. So, by Lemma 11, we have (55)CVaRp=n1/α-11+n-12nαξα1-p+o1-p1+n-12nαES2αESα2-1ξα1-p+o1-p1+o1=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p,p↑1.
Case  2(ρ<-1 and α≥1). In this case, γ=-1. By Karamata’s theorem, it can be proved that μF(t)∈RV0; see (2.7) and (2.8) in Mao and Hu (2012a). Hence, A(t)=o(μF(t)/t) as t→∞. We have (56)η1F←p=ζα,-1BF←p=n-1ESα+1ESα1+1/α-1μFF←pF←p,η2F←p=ζα,-1AF←p=oμFF←pF←p.Thus, (57)CVaRp=n1/α-11+n-1/αn-1μFF←pF←p1+o11+n-1/αn-1EYα+1EYα1+1/α-1·μFF←pF←p1+o1=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←pF←p1+o1,p↑1.For α>1, μF(t)→μF as t→∞. For α=1, by Proposition 1, (58)μFt~∫0tF¯xdx~∫1tcx-11+Axρdx~clog⁡t+cρ-1∫1tx-1Axdx~clog⁡t,t⟶∞,where the first equation follows from (28) in Mao and Hu [12], and the last equation follows since ∫1∞‍x-1|A(x)|dx<∞. Thus, we prove the case.
Case  3(ρ>-(1∧α)). In this case, γ=ρ, we have (59)η1F←p=ζα,ρAF←p,η2F←p=ζα,ρAF←p.Thus, (60)CVaRp=n1/α-11+nρ/α-1αρAF←p1+o11+nρ/αζα,ρAF←p1+o11+ζα,ρAF←p1+o1=n1/α-11+nρ/α-1αρAF←p1+o1·1+nρ/α-1αρESα-ρESα1-ρ/α-1AF←p1+o1=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p+oAF←p,p↑1.Proof of Theorem5.
From Proposition2 and Lemmas 8 and 9, we can get (61)HGpRiS=CαVaRpRiS1+Hα,ρ,k·ESα-ρESαAVaRpRiS1+o1=CαVaRpRiS1+Hα,ρ,k·ESα-ρESα1-ρ/αAF←p1+o1,p↑1,for i=1,2,…,n, where we use the fact that |A(x)|∈RVρ and A(x) is ultimately positive or negative.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45). So, similarly, from Lemma 10, we can get (62)HGpYS=CαVaRpYS1+Hα,γ,k·ESα-γESαBVaRpYS1+o1=CαVaRpYS1+Hα,γ,k·ESα-γESαBESα1/αFY←p1+o1=CαVaRpYS1+nγ/αHα,γ,k·ESα-γESα1-γ/αBF←p1+o1,p↑1,where we use the fact that FY←(p)/F←(p)→n1/α, as p↑1, |B(x)|∈RVγ and B(x) is ultimately positive or negative.
Thus,(63)CHGp=HGpYS∑i=1nHGpRiS=VaRpYSnVaRpRiS·1+nγ/αHα,γ,k·ESα-γ/ESα1-γ/αBF←p1+o11+Hα,ρ,k·ESα-ρ/ESα1-ρ/αAF←p1+o1=CVaRp·1+nγ/αη1F←p1+o11+η2F←p1+o1,where (64)η1x=ESα-γESα1-γ/αHα,γ,kBx,η2x=ESα-ρESα1-ρ/αHα,ρ,kAx.Next, we consider three cases. (i)
Forρ<-α and 0<α<1. In this case, γ=-α. From (45), it follows that (65)η1F←p=-n-1αξα2·ES2αESα2Hα,-α,k1-p.Note thatA(F←(p))=o(1-p) as p↑1. So, by Lemma 10, we have (66)CHGp=n1/α-11+n-1ξα2nα·ES2αESα21-p+o1-p1-n-1αξα2n·ES2αESα21-p+o1-p1+o1=n1/α-11+n-12nα·ES2αESα2·1-α2Hα,-α,kξα1-p+o1-p,p↑1.(ii)
ρ<-1 and α≥1. In this case, γ=-1. By Karamata’s theorem, it can be proved that μF(t)∈RV0; see (2.7) and (2.8) in Mao and Hu (2012a). Hence, A(t)=o(μF(t)/t) as t→∞. Thus, (67)CHGp=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·μFF←pF←p1+o1·1-n-1/αn-1αHα,-1,kESα+1ESα1+1/α·μFF←pF←p1+o1=n1/α-11+n-1/αn-1ESα+1ESα1+1/α·1-αHα,-1,k·μFF←pF←p+oμFF←pF←p,p↑1.ConsideringμF(t) as t→∞ for α>1 and α=1 as Theorem 4, we can get the result easily.(iii)
ρ>-(1∧α). In this case, γ=ρ. Thus, (68)CHGp=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/αAF←p1+o1·1+nρ/α-1Hα,ρ,kESα-ρESα1-ρ/αAF←p1+o1=n1/α-11+nρ/α-1αρ·ESα-ρESα1-ρ/α·1+αρHα,ρ,kAF←p+oAF←p,p↑1.Proof of Theorem6.
From Proposition3 and Lemmas 8 and 9, we can get (69)TgpRiS=JαVaRpRiS+VaRpRiSA∗VaRpRiSIα,ρ+o1=JαVaRpRiS1+Iα,ρJα·ESα-ρESα1-ρ/αAF←p1+o1,p↑1,for i=1,2,…,n, where we use the fact that |A(x)|∈RVρ and A(x) is ultimately positive or negative.
DefineY=∑i=1nRi, and denote by FY the distribution function of Y. By Lemma 10, F¯Y∈2RV-α,γ with auxiliary function B(x) with γ and B(x) given by (44) and (45). So, similarly, from Lemma 10, we can get (70)TgpYS=JαVaRpYS+VaRpYS·ESα-ρESαBVaRpYSIα,γ+o1=JαVaRpYS1+nγ/αIα,γJα·ESα-γESα1-γ/αBF←p1+o1,p↑1,where we use the fact that FY←(p)/F←(p)→n1/α, as p↑1, |B(x)|∈RVγ and B(x) is ultimately positive or negative. Thus, (71)CTgp=TgpYS∑i=1nTgpRiS=VaRpYSnVaRpRiS·1+nγ/αIα,γ/Jα·ESα-γ/ESα1-γ/αBF←p1+o11+Iα,ρ/Jα·ESα-ρ/ESα1-ρ/αAF←p1+o1=CVaRp·1+nγ/αη1F←p1+o11+η2F←p1+o1,where (72)η1x=Iα,γJα·ESα-γESα1-γ/αBx,η2x=Iα,ρJα·ESα-ρESα1-ρ/αAx.Next, similar to Theorems 4 and 5, we consider three cases: (i) for ρ<-α and 0<α<1; (ii) for ρ<-1 and α≥1; (iii) for ρ>-(1∧α) to obtain the result. Thus, we complete the proof.Proof of Corollary7.
Note that if the distortion functiong(x)=x, then Tg,p[X] reduces to CTEp[X] for continuous risk variables X, and CTg(p) reduces to CCTE(p). It is easy to see that, for α>1 and ρ<0, (73)Jα=αα-1,Iα,ρ=1α-ρ-1α-1,and, hence, (74)αIα,-1Jα=1α,αρIα,ρJα=α-1α-1-ρ.Therefore, the result is an immediate consequence of Theorem 6.


## 4. Examples

In this section, two examples are given to illustrate applications of our main results.Example 1 (Burr distribution and Beta distribution).
LetR be a random variable with Burr distribution function F given by (75)F¯x=1+x-ρα/ρ,x>0,α>0,ρ<0denoted by F~Burr(α,ρ). It is known from Example 1 in Mao et al. [4] and Example 2 in Degen et al. [10] that F¯∈2RV-α,ρ with auxiliary function A(x)=αxρ. Suppose that S~Beta(a,b), where Beta(a,b) stands for the Beta distribution with positive parameters a and b and density function (76)gx=1Ba,bxa-11-xb-1,0<x<1,a,b>0.It is obvious that ESκ=B(a+κ,b)/B(a,b) for all κ>0.
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Similarly, we can get risk concentration based on other risk measures. We setn=2 and compare the second-order approximations with the actual true value of CVaR(p) for Burr distribution with different parameters α and ρ and Beta distribution with a=b=1 in Figure 1.Figure 1
Empirical risk concentration (full, based on107 simulations) together with second-order approximation (dotted) for two i.i.d. Burr(α,ρ) and Beta(1,1) random variables based on CVaR(p).Example 2 (absolute studenttα distribution and Beta distribution).
LetR be a random variable having the standard Student tα distribution with density function (78)fRr=Γα+1/2απΓα/21+r2α-α+1/2,r>0.Denote by F the distribution function of R. Suppose that S~Beta(a,b), where Beta(a,b) stands for the Beta distribution with positive parameters a and b and density function (79)gx=1Ba,bxa-11-xb-1,0<x<1,a,b>0.From Example 3 in Hua and Joe [29], we know that F¯∈2RV-α,-2 and the mean of F, μF=α/(α-1) for α>1 and ESκ=B(a+κ,b)/B(a,b) for all κ>0.
For0<α<1, by Theorems 4(i) and 3.2(i), we have (80)CVaRp=n1/α-11+n-1ξα2nα·Ba,b·Ba+2α,bB2a+α,b1-p+o1-p,p↑1,CHGp=n1/α-11+n-12nα·Ba,b·Ba+2α,bB2a+α,b·1-α2Hα,-α,kξα1-p+o1-p,p↑1.Choose distortion function g(x)=x1/β with 0<β<α. It is easy to see that (81)Jα=αα-β,Iα,-1=βαα-β,Iα,-α=βα-βα-β+αβ.By Theorem 6(i), we have (82)CTgp=n1/α-11+n-12nα·Ba,b·Ba+2α,bB2a+α,b·α-βα-β+αβξα1-p+o1-p,p↑1.
Forα>1, since F←(p)=tα←((1+p)/2), the quantile function of tα distribution is at the level (p+1)/2. By Theorems 4(i) and 3.2(i), we have that, as p↑1, (83)CVaRp=n1/α-11+n-1/αn-1Ba,b1/α·Ba+α+1,bBa+α,b1+1/α·αα-1·1tα←1+p/2+o1tα←1+p/2,CHGp=n1/α-11+n-1/αn-1Ba,b1/α·Ba+α+1,bBa+α,b1+1/α·α1-αHα,-1,kα-1·1tα←1+p/2+o1tα←1+p/2.Considering the distortion function g(x)=x1/β with 0<β<α, by Theorem 6(ii), we have that, as p↑1, (84)CTgp=n1/α-11+n-1/αn-1Ba,b1/α·Ba+α+1,bBa+α,b1+1/α·α-βα-1·1tα←1+p/2+o1tα←1+p/2.Choosing β=1, CTg(p) reduces to (85)CCTEp=n1/α-11+n-1/αn-1Ba,b1/α·Ba+α+1,bBa+α,b1+1/α·1tα←1+p/2+o1tα←1+p/2,p↑1.
We setn=2 and compare the second-order approximations with the actual true value of CVaR(p) for tα distribution with different parameters α and Beta distribution with a=b=1 in Figure 2.Figure 2
Empirical risk concentration (full, based on107 simulations) together with second-order approximation (dotted) for two i.i.d. tα and Beta(1,1) random variables based on CVaR(p) with α=0.5 for (a) and α=1.4 for (b).
(a) (b)



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*Source: 4689479-2018-04-17.xml*