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

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 riskmeasures for a portfolio of n 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.


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 of  loss random variables  1 ,  2 , . . .,   .The risk concentration based on the risk measure [⋅] is defined as Here, 1 −   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  1 ,  2 , . . .,   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  identically distributed but correlated deflated risks   =   ,  = 1, . . .,  under assumptions of secondorder regular variation on the survival functions of the risk  1 , . . .,   and deflator .
In the present paper we study mathematical properties of diversification effects under the different risk measures  [⋅].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 Valueat-Risk (VaR) of  at the level  is defined as VaR  [] = inf { ∈ R :  () ≥ } ,  ∈ (0, 1) , (2) and the conditional tail expectation (CTE) of  at the level  is defined as The Haezendonck-Goovaerts risk measure, which was introduced by Haezendonck and Goovaerts [17], is defined via an increasing and convex Young function  and a parameter  ∈ (0, 1) representing the confidence level.More precisely, let  be a nonnegative and convex function on [0, ∞) with (0) = 0, (1) = 1, and (∞) = ∞.This function is called a normalized Young function.Assume we have a realvalued random variable  with distribution function  such that and let   [, ] be the unique solution ℎ to the equation if () > 0 and 0 if () = 0, where  + = max{, 0} for any real number .Then the Haezendonck-Goovaerts risk measure of  at the confidence level  is defined as Some important properties and connections with other risk measures are given in Goovaerts et al. [18].with () =   for  ≥ 1 as  ↑ 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  : [0, 1] → [0, 1] such that (0) = 0 and (1) = 1.Then for any risk  with distribution function , the corresponding distortion risk measure   [⋅] is defined as follows: (7) where () = 1 − () denotes the survival function of .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  at the level  corresponds to the distortion function , where 1  is the indicator function of ; the conditional tail expectation (CTE) of  at level  corresponds to the distortion function () = min{/(1 − ), 1},  ∈ (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 , the tail distortion risk measure at level  of a loss variable  is defined as    [] =    [],  ∈ (0, 1), where Since the risk is always heavy-tailed and often obeys a law of regular variation, we choose [⋅] as VaR  [⋅], HG  [⋅], and    [⋅] at the level 0 <  < 1, respectively, in (1).We denote risk concentration   at the level  by   ().
Because risk managers become more and more concerned with tail area of risk, we will focus on the secondorder approximations of the risk concentrations based on the different risk measures as  ↑ 1, such as  VaR (),  HG (),  CTE (), and    () as  ↑ 1 for a portfolio of  loss random variables  1 ,  2 , . . .,   .In this paper, we assume that random variables  1 ,  2 , . . .,   are identically distributed but not independent; that is, where  1 ,  2 , . . .,   are  i.i.d random variables with a common survival function possessing the property of secondorder regular variation, and the deflator  is a random variable which is independent of  1 ,  2 , . . .,   .The first-order approximations of  VaR () as  ↑ 1 were studied by Embrechts et al. [8,9] under the model assumption that the underlying risks  1 ,  2 , . . .,   have identically distributed and regularly varying margins and have two forms of dependent structure, respectively.Degen et al. [10] derived second-order approximations of  VaR () for  independent and identically distributed (i.i.d) loss variables with a common survival function possessing the property of second-order regular variation (2RV).Secondorder approximations of the risk concentrations  CTE () and    () as  ↑ 1 for  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  i.i.d.risks, the second-order approximations of the risk concentrations  VaR (),  CTE () as  ↑ 1 have been discussed by Hashorva et al. [24], while Mao and Yang [25] consider the case with a portfolio of  dependent risks under FGM copula.Ling and Peng [26] derived higher-order approximations under some conditions.
The paper is organized as follows.In Section 2, 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  VaR (),  HG (), and    () as  ↑ 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 () and (),

Preliminaries
Regular variation is one of the basic concepts which appears in different contexts of applied probability.A function ℎ is said to be of regular variation with index  ∈ R, denoted by holds for any  > 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 − for some  ∈ R; then ℎ is said to be of second-order regular variation with first-order parameter  and secondorder parameter  ≤ 0, denoted by ℎ ∈ 2RV − , , if there exists some ultimately positive or negative function () with Here, () is referred to as an auxiliary function of ℎ and || ∈ RV  .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 secondorder regular variation (2RV) plays an important role in this article.The following proposition gives a characterization of any function ℎ ∈ 2RV − ,  with auxiliary function (),  ∈ R and  < 0, which is from Hua and Joe [29].
The next two propositions give first-and second-order approximations of Haezendonck-Goovaerts risk measure HG  [] of  at the confidence level  and tail distortion risk measure    [] of  at confidence level  for a distortion function , which will be used in the proofs of our main results.

Main Results and Their Proofs
3.1.Main Results.In this section, we give some results establishing the second-order approximations of the risk concentration   () as  → 1 for a portfolio of  random variables that satisfy (9).The first one is about the risk concentration  VaR ().If  ∈ 2 −, ,  > 0,  < 0, with auxiliary function () and  −+ < ∞ for some  > 0, then ) ,  ↑ 1 with  = lim →∞ () ∈ (0, ∞), when  = 1; In the following theorem, we derive the second-order asymptotic of risk concentration for Haezendonck-Goovaerts risk measure  HG () at level .
First, we introduce two definitions.Let  be a distribution function of a nonnegative random variable.We introduce the truncated mean of : Obviously, if the mean of ,   , exists, then   () →   as  → ∞.For 0 <  < 1, define The following lemma from Mao and Hu [12] states that the 2RV property is preserved by the formation of sum of  i.i.d random variables.

Lemma 1 .
Let  be the distribution function of a nonnegative random variable satisfying  ∈ 2 −, ,  ≤ 0, with auxiliary function ().We denoted by  *  the -fold convolution of .Then  *  ∈ 2 −, with auxiliary function (), where and () is given by The last lemma from Mao et al. [4] establishes the secondorder asymptotic of the risk concentration  VaR () for  i.i.d random variables with the underlying distribution possessing the 2RV property.
From Lemma 11, it follows that In view of | 1 | ∈ RV  and Theorem B.1.4 of de Haan and Ferreira [27], we have where we use the fact that () is ultimately positive or negative.Thus, Next, we consider three cases.

Proof of
and, hence, Therefore, the result is an immediate consequence of Theorem 6.

Examples
In this section, two examples are given to illustrate applications of our main results.(77)