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 of identically distributed but dependent deflated risks , under the assumptions of second-order regular variation on the survival functions of the risks and the deflator , where are independent and identically distributed random variables with a common survival function and is a random variable being independent of . Examples are also given to illustrate our main results.
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 of loss random variables . The risk concentration based on the risk measure is defined asHere, 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 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 , under assumptions of second-order regular variation on the survival functions of the risk 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 Value-at-Risk (VaR) of at the level is defined as and the conditional tail expectation 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 representing the confidence level. More precisely, let be a nonnegative and convex function on with , , and . This function is called a normalized Young function. Assume we have a real-valued random variable with distribution function such thatand let be the unique solution to the equationif and 0 if , where for any real number . Then the Haezendonck-Goovaerts risk measure of at the confidence level is defined asSome 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 with reduces to . 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 with for as .
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 such that and . Then for any risk with distribution function , the corresponding distortion risk measure is defined as follows: where 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 is the indicator function of ; the conditional tail expectation (CTE) of at level corresponds to the distortion function , .
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 , , where Since the risk is always heavy-tailed and often obeys a law of regular variation, we choose as , , and at the level , 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 second-order approximations of the risk concentrations based on the different risk measures as , such as , , , and as for a portfolio of loss random variables . In this paper, we assume that random variables are identically distributed but not independent; that is,where are i.i.d random variables with a common survival function possessing the property of second-order regular variation, and the deflator is a random variable which is independent of .
The first-order approximations of as were studied by Embrechts et al. [8, 9] under the model assumption that the underlying risks have identically distributed and regularly varying margins and have two forms of dependent structure, respectively. Degen et al. [10] derived second-order approximations of for 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 and as 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 , as 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 , , and as , 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 ,
2. Preliminaries
Regular variation is one of the basic concepts which appears in different contexts of applied probability. A function is said to be of regular variation with index , denoted by , ifholds for any . 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 for some ; then is said to be of second-order regular variation with first-order parameter and second-order parameter , denoted by , , if there exists some ultimately positive or negative function with as such thatHere, is referred to as an auxiliary function of and . 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 function , with auxiliary function , and , which is from Hua and Joe [29].
Proposition 1. Let , , and . Then , with auxiliary function if and only ifwhere .
The next two propositions give first- and second-order approximations of Haezendonck-Goovaerts risk measure 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.
Proposition 2. Let be a random variable with survival function , , and let for some . Then one has the following: (i)The first-order asymptotic (see [30]; Mao and Hu, 2012a): (ii)The second-order asymptotic (see Mao and Hu, 2012a): if , , with auxiliary function , then where with and is the Beta function as usual; that is,
Proposition 3. Let be a random variable with survival function , , and let be any distortion function withWe have the follwoing:(i)The first-order asymptotic (see [22, 23]): (ii)The second-order asymptotic (see [23]): if , , with auxiliary function , then where
Propositions 2(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 with auxiliary function if and only if its tail quantile function with auxiliary function (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 concentration as for a portfolio of random variables that satisfy (9). The first one is about the risk concentration .
Theorem 4. Let , where are i.i.d nonnegative random variables with common continuous distribution function and is a nonnegative random variable independent of . If , , , with auxiliary function and for some , then (i)for and , (ii)for , when , with , when ;(iii)for ,
In the following theorem, we derive the second-order asymptotic of risk concentration for Haezendonck-Goovaerts risk measure at level .
Theorem 5. Let , where are i.i.d. nonnegative random variables with common continuous distribution function and is a nonnegative random variable independent of . If , , , with auxiliary function and for some and if for some , then (i)for and , (ii)for , when , and with when ;(iii)for ,
The last theorem gives the second-order asymptotic of risk concentration for tail distortion risk measure at level .
Theorem 6. Let , where are i.i.d. nonnegative random variables with common continuous distribution function , and is a nonnegative random variable independent of . Further assume that , , , with auxiliary function and for some . Let be a distortion function with Then(i)for and , (ii)for , when , and with when ;(iii)for ,
Thus, we immediately obtain the following corollary which establishes the second-order asymptotic of risk concentration for conditional tail expectation . And this corollary can also be obtained easily by Lemma 8.
Corollary 7. Let be a continuous random vector, where are i.i.d. nonnegative random variables with common continuous distribution function and is a nonnegative random variable independent of . If , , , with auxiliary function and for some . Then (i)for , (ii)for ,
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. Let be a random variable with survival function , , , with auxiliary function , and let be a nonnegative random variable satisfying for some , independent of . Then where as , and thus with auxiliary function
The second lemma talks about the first- and second-order asymptotic of Value-at-Risk of the product at the level , which was proved by Hashorva et al. [7].
Lemma 9. Let be a random variable with survival function , , and let be a nonnegative random variable satisfying for some , independent of . Then one has the following: (i)The first-order asymptotic: (ii)The second-order asymptotic: if , , with auxiliary function and , then where
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 , 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 10. Let be the distribution function of a nonnegative random variable satisfying , , with auxiliary function . We denoted by the -fold convolution of . Then with auxiliary function , whereand is given by
The last lemma from Mao et al. [4] establishes the second-order asymptotic of the risk concentration for i.i.d random variables with the underlying distribution possessing the 2RV property.
Lemma 11. Let be i.i.d. nonnegative random variables with common continuous distribution function , and assume that , , with auxiliary function . Then (i)for and , (ii)for and , (iii)for ,
Now we turn to prove our theorems.
Proof of Theorem 4. Define , and denote by the distribution function of . By Lemma 10, with auxiliary function with and given by (44) and (45); we have where and .
Similarly, for , where and . From Lemma 11, it follows that In view of 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.
Case 1 ( and ). In this case, from (44) and (45), it follows that and Note that as . So, by Lemma 11, we have Case 2 ( and ). In this case, . By Karamata’s theorem, it can be proved that ; see and in Mao and Hu (2012a). Hence, as . We have Thus, For , as . For , by Proposition 1, where the first equation follows from (28) in Mao and Hu [12], and the last equation follows since . Thus, we prove the case.
Case 3 (). In this case, , we have Thus,
Proof of Theorem 5. From Proposition 2 and Lemmas 8 and 9, we can get for , where we use the fact that and is ultimately positive or negative.
Define , and denote by the distribution function of . By Lemma 10, with auxiliary function with and given by (44) and (45). So, similarly, from Lemma 10, we can get where we use the fact that , as , and is ultimately positive or negative.
Thus, where Next, we consider three cases. (i)For and . In this case, . From (45), it follows that Note that as . So, by Lemma 10, we have (ii) and . In this case, . By Karamata’s theorem, it can be proved that ; see and in Mao and Hu (2012a). Hence, as . Thus, Considering as for and as Theorem 4, we can get the result easily.(iii). In this case, . Thus,
Proof of Theorem 6. From Proposition 3 and Lemmas 8 and 9, we can get for , where we use the fact that and is ultimately positive or negative.
Define , and denote by the distribution function of . By Lemma 10, with auxiliary function with and given by (44) and (45). So, similarly, from Lemma 10, we can get where we use the fact that , as , and is ultimately positive or negative. Thus, where Next, similar to Theorems 4 and 5, we consider three cases: (i) for and ; (ii) for and ; (iii) for to obtain the result. Thus, we complete the proof.
Proof of Corollary 7. Note that if the distortion function , then reduces to for continuous risk variables , and reduces to . It is easy to see that, for and , and, hence, 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). Let be a random variable with Burr distribution function given by denoted by . It is known from Example 1 in Mao et al. [4] and Example 2 in Degen et al. [10] that with auxiliary function . Suppose that , where stands for the Beta distribution with positive parameters and and density function It is obvious that for all .
By Theorem 4, we haveSimilarly, we can get risk concentration based on other risk measures. We set and compare the second-order approximations with the actual true value of for Burr distribution with different parameters and and Beta distribution with in Figure 1.
Example 2 (absolute student distribution and Beta distribution). Let be a random variable having the standard Student distribution with density function Denote by the distribution function of . Suppose that , where stands for the Beta distribution with positive parameters and and density function From Example 3 in Hua and Joe [29], we know that and the mean of , for and for all .
For , by Theorems 4(i) and 3.2(i), we have Choose distortion function with . It is easy to see that By Theorem 6(i), we have For , since , the quantile function of distribution is at the level . By Theorems 4(i) and 3.2(i), we have that, as , Considering the distortion function with , by Theorem 6(ii), we have that, as , Choosing , reduces to We set and compare the second-order approximations with the actual true value of for distribution with different parameters and Beta distribution with in Figure 2.
(a)
(b)
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work is supported by the National Key Research and Development Plan (no. 2016YFC0800104) and NSFC (nos. 71771203 and 11671374).