New Class of Distortion Risk Measures and Their Tail Asymptotics with Emphasis on VaR

Chuancun Yin, Dan Zhu

School of Statistics, Qufu Normal University, Qufu, China.

DOI: 10.4236/jfrm.2018.71002                

Abstract

Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of copula. We also investigate the tail subadditivity for VaR and other distortion risk measures. In particular, we demonstrate that VaR is tail subadditive for the case where the support of risk is bounded. Various examples are also presented to illustrate the results.

Keywords

Coherent Risk Measure, Copula, Distortion Risk Measure, Extreme Value Theory, GlueVaR, Tail Subadditivity, Tail Distortion Risk Measure

1. Introduction

A risk measure $\rho$ is a mapping from the set of random variables $\mathcal{X}$ , standing for risky portfolios of assets and/or liabilities, to the real line R. In the subsequent discussion, positive values of elements of $\mathcal{X}$ will be considered to represent losses, while negative values will represent gains. Distortion risk measures are a particular and most important family of risk measures that have been extensively used in finance and insurance as capital requirement and principles of premium calculation for the regulator and supervisor. Several popular risk measures belong to the family of distortion risk measures. For example, the value-at-risk (VaR), the tail value-at-risk (TVaR) and the Wang distortion measure. Distortion risk measures satisfy a set of properties including positive homogeneity, translation invariance and monotonicity. When the associated distortion function is concave, the distortion risk measure is also subadditive (Denneberg, 1994; Wang & Dhaene, 1998) . VaR is one of the most popular risk measures used in risk management and banking supervision due to its computational simplicity and for some regularity reasons, despite it has some shortcomings as a risk measure. For example, VaR is not a subadditive risk measure (see, for instance, Artzner et al., 1999; Denuit et al., 2005 ), it only concerns about the frequency of risk, but not the size of risk. TVaR, although being coherent, concerns only losses exceeding the VaR and ignores useful information of the loss distribution below VaR. Clearly, it is difficult to believe that a unique risk measure could capture all characteristics of risk, so that an ideal measure does not exist. Moreover, since risk measures associate a single number to a risk, as a matter of fact, they cannot exhaust all the information of a risk. However, it is reasonable to search for risk measures which are ideal for the particular problem under investigation. As all the proposed risk measures have drawbacks and limited applications, the selection of the appropriate risk measures continues to be a hot topic in risk management.

Zhu & Li (2012) introduced and studied the tail distortion risk measure which was reformulated by Yang (2015) as follows. For a distortion function g, the tail distortion risk measure at level p of a loss variable X is defined as the distortion risk measure with distortion function

$g_p\left(x\right)=\{\begin{array}{ll}g\left(\frac{x}{1-p}\right)\mathrm{,}\hfill & \text{if}0\le x\le 1-p\mathrm{,}\hfill \\ \mathrm{1,}\hfill & \text{if}1-p<x\le 1.\hfill \end{array}$

Some properties and applications can be found in Mao, Lv, & Hu (2012) , Mao & Hu (2013) and Lv, Pan, & Hu (2013) .

As an extension of VaR and TVaR, Belles-Sampera et al. (2014a) proposed a new class of distortion risk measures called GlueVaR risk measures, which can be expressed as a combination of VaR and TVaR measures at different probability levels. They obtain the analytical closed-form expressions for the most frequently used distribution functions in financial and insurance applications, while a subfamily of these risk measures has been shown to satisfy the tail-subadditivity property which means that the benefits of diversification can be preserved, at least they hold in extreme cases. The applications of GlueVaR risk measures in capital allocation can be found in the recent paper Belles-Sampera et al. (2014b) .

Cherubini & Mulinacci (2014) propose a class of distortion measures based on contagion from an external “scenario” variable. The dependence between the scenario and the variable whose risk is modeled with a copula function with horizontal concave sections, they give conditions to ensure that coherence requirements be met, and propose examples of measures in this class based on copula functions.

The first purpose of this paper is to construct new risk measures following Zhu & Li (2012) , Belles-Sampera et al. (2014a) and Cherubini & Mulinacci (2014) . The newly introduced risk measures are included the tail distortion risk measure and the GlueVaR as specials. The second goal of the paper is to investigate the tail asymptotics of distortion risk measures for the sum of possibly dependent risks with emphasis on VaR. The rest of the paper is organized as follows. We review some basic definitions and notations such as distorted functions, distorted expectations and distortion risk measures in Section 2. In Section 3 several new distortion functions and risk measures are introduced. In Section 4 we investigate the tail asymptotics as well as subadditivity/superadditivity of VaR. In Section 5 we analyze the subadditivity properties of a class of distortion risk measures and Section 6 focuses on conclusion.

2. Distortion Risk Measures

2.1. Distorted Functions

A distortion function is a non-decreasing function $g\mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ such that $g\left(0\right)=0,g\left(1\right)=1$ . Since Yaari (1987) introduced distortion function in dual theory of choice under risk, many different distortions g have been proposed in the literature. Here we list some commonly used distortion functions. A summary of other proposed distortion functions can be found in Denuit et al. (2005) .

• $g\left(x\right)=1_{\left(x>1-p\right)}$ , where the notation $1_A$ to denote the indicator function, which equals 1 when A holds true and 0 otherwise.

• $g\left(x\right)=\min \left\{\frac{x}{1-p},1\right\}$ .

・ Incomplete beta function $g\left(x\right)=\frac{1}{\beta \left(a,b\right)}{\displaystyle {\int }_0^x}{t}^{a-1}{\left(1-t\right)}^{b-1}\text{d}t$ , where $a>0$ and $b>0$ are parameters and $\beta \left(a\mathrm{,}b\right)={\displaystyle {\int }_0^1}{t}^{a-1}{\left(1-t\right)}^{b-1}\text{d}t$ . Setting $b=1$ gives the power distortion $g\left(x\right)=x^a$ ; setting $a=1$ gives the dual-power distortion $g\left(x\right)=1-{\left(1-x\right)}^b$ .

・ The Wang distortion $g\left(x\right)=\Phi \left({\Phi }^{-1}\left(x\right)+{\Phi }^{-1}\left(p\right)\right),0<p<1$ , where $\Phi$ is the distribution function of the standard normal.

・ The lookback distortion $g\left(x\right)=x^p\left(1-p\ln x\right),p\in \left(0,1\right].$

Obviously, every concave distortion function is continuous on the interval $\left(\mathrm{0,1}\right]$ and can have jumps in 0. In contrast, every convex distortion function is continuous on the interval $\left[\mathrm{0,1}\right)$ and can have jumps in 1. The identity function is the smallest concave distortion function and also the largest convex distortion function; $g_0\left(x\right):=1_{\left(x>0\right)}$ is concave on $\left[\mathrm{0,1}\right]$ and is the largest distortion function. $g^0\left(x\right):=1_{\left(x=1\right)}$ is convex on $\left[\mathrm{0,1}\right]$ and is the smallest

distortion function. For $0<p<1$ , we remark that $g_1\left(x\right)\mathrm{<mo>\; :\; </mo>}=min\left\{\frac{x}{1-p}\mathrm{,1}\right\}$ is

the smallest concave distortion function such that $g_1\left(x\right)\ge 1_{\left(x>1-p\right)}$ . In fact, we consider a concave distortion function g such that $g\left(x\right)\ge 1_{\left(x>1-p\right)}$ , then $g\equiv 1$

on $\left(1-p\mathrm{,1}\right]$ . As g is concave, it follows that $g\left(x\right)\ge \frac{x}{1-p}$ for $x\le 1-p$ , and thus $g\left(x\right)\ge min\left\{\frac{x}{1-p}\mathrm{,1}\right\}$ for $0<x<1$ . Any concave distortion function g

gives more weight to the tail than the identity function $g\left(x\right)=x$ , whereas any convex distortion function g gives less weight to the tail than the identity function $g\left(x\right)=x$ .

2.2. Distorted Risk Measures

Let $\left(\Omega \mathrm{,}F\mathrm{,}P\right)$ be a probability space on which all random variables involved are defined. Let $F_X$ be the cumulative distribution function of random variable X and the decumulative distribution function is denoted by ${\bar{F}}_X$ , i.e. ${\bar{F}}_X\left(x\right)=1-F_X\left(x\right)=P\left(X>x\right)$ . Let g be a distortion function. The distorted expectation of the random variable X, notation ${\rho }_g\left[X\right]$ , is defined as

${\rho }_g\left[X\right]={\displaystyle {\int }_0^{+\infty }}g\left({\bar{F}}_X\left(x\right)\right)\text{d}x+{\displaystyle {\int }_{-\infty }^0}\left[g\left({\bar{F}}_X\left(x\right)\right)-1\right]\text{d}x,$

provided at least one of the two integrals above is finite. If X a non-negative random variable, then ${\rho }_g$ reduces to

${\rho }_g\left[X\right]={\displaystyle {\int }_0^{+\infty }}g\left({\bar{F}}_X\left(x\right)\right)\text{d}x.$

From a mathematical point of view, a distortion expectation is the Choquet integral (see Denneberg (1994) ) with respect to the nonadditive measure $\mu =g\circ P$ . That is ${\rho }_g\left[X\right]={\displaystyle \int }X\text{d}\mu$ . In view of Dhaene et al. (2012: Theorems 4 and 6) we know that, when the distortion function g is right continuous on $\left[\mathrm{0,1}\right)$ , then ${\rho }_g\left[X\right]$ may be rewritten as

${\rho }_g\left[X\right]={\displaystyle {\int }_{\left[0,1\right]}}Va{R}_{1-q}^{+}\left[X\right]\text{d}g\left(q\right),$

where $Va{R}^{+}p\left[X\right]=sup\left\{x\mathrm{<mo>\; |\; </mo>}{F}_X\left(x\right)\le p\right\}$ , and when the distortion function g is left continuous on $\left(\mathrm{0,1}\right]$ , then ${\rho }_g\left[X\right]$ may be rewritten as

${\rho }_g\left[X\right]={\displaystyle {\int }_{\left[0,1\right]}}Va{R}_{1-q}\left[X\right]\text{d}g\left(q\right)={\displaystyle {\int }_{\left[0,1\right]}}Va{R}_q\left[X\right]\text{d}\bar{g}\left(q\right),$

where $Va{R}_p\left[X\right]=inf\left\{x\mathrm{<mo>\; |\; </mo>}{F}_X\left(x\right)\ge p\right\}$ and $\bar{g}\left(q\right)\mathrm{<mo>\; :\; </mo>}=1-g\left(1-q\right)$ is the dual distortion of g. Obviously, $\overline{\stackrel{¯}{g}}=g$ , g is left continuous if and only if $\bar{g}$ is right continuous; g is concave if and only if $\bar{g}$ is convex. The distorted expectation ${\rho }_g\left[X\right]$ is called a distortion risk measure with distortion function g. Distortion risk measures are a particular class of risk measures which as premium principles were introduced by Denneberg (1994) and further developed by Wang (1996, 2000) among others.

Distortion risk measures satisfy a set of properties including positive homogeneity, translation invariance and monotonicity. A risk measure is said to be coherent if it satisfies the following set of four properties (see, e.g., Artzner et al., 1997, 1999 ):

(M) Monotonicity: $\rho \left(X\right)\le \rho \left(Y\right)$ provided that $P\left(X\le Y\right)=1$ .

(P) Positive homogeneity: For any positive constant $c>0$ and loss X, $\rho \left(cX\right)=c\rho \left(X\right)$ .

(S) Subadditivity: For any losses $X\mathrm{,}Y$ , then $\rho \left(X+Y\right)\le \rho \left(X\right)+\rho \left(Y\right)$ .

(T) Translation invariance: If c is a constant, then $\rho \left(X+c\right)=\rho \left(X\right)+c$ .

It is furthermore shown by Artzner et al. (1999) that all mappings satisfying the above properties allow a representation:

$\rho \left(X\right)=\underset{p\in \mathcal{P}}{sup}{E}_p\left[X\right]\mathrm{,}$

where $\mathcal{P}$ is a collection of “generalised scenarios”. A risk measure $\rho$ is called a convex risk measure if it satisfies monotonicity, translation invariance and the following convexity (C):

$\rho \left(\lambda X+\left(1-\lambda \right)Y\right)\le \lambda \rho \left(X\right)+\left(1-\lambda \right)\rho \left(Y\right)\mathrm{,}0\le \lambda \le 1.$

Clearly, under the assumption of positive homogeneity, monotonicity and translation invariance, the convexity of a risk measure is equivalent subadditivity.

The most well-known examples of distortion risk measures are the above-mentioned VaR and TVaR, corresponding to the distortion functions,

respectively, are $g\left(x\right)=1_{\left(x>1-p\right)}$ and $g\left(x\right)=min\left\{\frac{x}{1-p}\mathrm{,1}\right\}$ . Notice that

$TVa{R}_p\left[X\right]$ can be alternatively expressed as the weighted average of VaR and losses exceeding VaR:

$TVa{R}_p\left[X\right]=Va{R}_p\left[X\right]+\frac{1-F_X\left(Va{R}_p\left[X\right]\right)}{1-p}E\left[X-Va{R}_p\left[X\right]|X>Va{R}_p\left[X\right]\right].$ (2.1)

For continuous distributions, TVaR coincide with the expected loss exceeding p-Value-at Risk, i.e., the mean of the worst $\left(1-p\right)\mathrm{100\; <mi>\; \%\; </mi>}$ losses in a specified time period which defined by

$CT{E}_p\left[X\right]=E\left[X|X>Va{R}_p\left[X\right]\right].$

Detailed studies of distortion risk measures and their relation with orderings of risk and the concept of comonotonicity can be found in, for example, Wang (1996) , Wang & Young (1998) , Hua & Joe (2012) and the references therein. The following lemma will be used in proofs of later results, which characterizes an ordering of distortion risk measures in terms of their distortion functions.

Lemma 2.1 (Belles-Sampera et al., 2014b) . If $g\left(x\right)\le g^{\mathrm{<mo>\; *\; </mo>}}\left(x\right)$ for $x\in \left[\mathrm{0,1}\right]$ , then ${\rho }_g\left[X\right]\le {\rho }_{g^{\mathrm{<mo>\; *\; </mo>}}}\left[X\right]$ for any random variable X.

3. Generating New Distortion Functions and Measures

Distortion functions can be considered as a starting point for constructing families of distortion risk measures. Thus, constructions of distortion functions play an important role in producing various families of risk measures. Using the technique of mixing, composition and copula allow the construction of new class of distortion functions and measures.

3.1. Composting Methods

The first approach to construct distortion functions is the composition of distortion functions.

Let $h_1\mathrm{,}{h}_2\mathrm{,}\cdots$ be distortion functions, define $f_1\left(x\right)=h_1\left(x\right)$ and composite functions $f_n\left(x\right)=f_{n-1}\left(h_n\left(x\right)\right),n=1,2,\cdots$ . It is easy to check that $f_n\left(x\right),n=1,2,\cdots$ are all distortion functions. If $h_1\mathrm{,}{h}_2\mathrm{,}\cdots$ are concave distortion functions, then each $f_n\left(x\right)$ is concave and satisfies that

$f_1\le f_2\le f_3\le \cdots$

and

$\underset{n\to \infty }{\lim }{f}_n\left(x\right)=1_{\left(x>0\right)},x\in \left[0,1\right].$

The associated risk measures satisfy (by Lemma 2.1)

${\rho }_{f_1}\left[X\right]\le {\rho }_{f_2}\left[X\right]\le {\rho }_{f_3}\left[X\right]\le \cdots$

and

$\underset{n\to \infty }{\lim }{\rho }_{f_n}\left[X\right]=Va{R}_1\left[X\right]=\text{esssup}\left(X\right)\mathrm{.}$

If $h_1\mathrm{,}{h}_2\mathrm{,}\cdots$ are convex distortion functions, then each $f_n\left(x\right)$ is convex and satisfies that

$f_1\ge f_2\ge f_3\ge \cdots$

and

$\underset{n\to \infty }{\lim }{f}_n\left(x\right)=1_{\left(x=1\right)},x\in \left[0,1\right].$

The associated risk measures satisfy (by Lemma 2.1)

${\rho }_{f_1}\left[X\right]\ge {\rho }_{f_2}\left[X\right]\ge {\rho }_{f_3}\left[X\right]\ge \cdots$

and

$\underset{n\to \infty }{\lim }{\rho }_{f_n}\left[X\right]=Va{R}_0\left[X\right]=\text{essinf}\left(X\right)\mathrm{.}$

Consider two distortion functions $g_1$ and $g_2$ . If

$g_2\left(x\right)=\{\begin{array}{ll}\frac{x}{1-p},\hfill & \text{if}0\le x\le 1-p,\hfill \\ 1,\hfill & \text{if}1-p<x\le 1,\hfill \end{array}$

then we get

$g_p\left(x\right):=g_1\left(g_2\left(x\right)\right)=\{\begin{array}{ll}{g}_1\left(\frac{x}{1-p}\right),\hfill & \text{if}0\le x\le 1-p,\hfill \\ 1,\hfill & \text{if}1-p<x\le 1.\hfill \end{array}$

The corresponding risk measure ${\rho }_{g_p}\left[X\right]$ is the tail distortion risk measure which was first introduced by Zhu & Li (2012) , and was reformulated by Yang (2015) . In particular, on the space of continuous loss random variables X,

${\rho }_{g_p}\left[X\right]={\displaystyle {\int }_0^{\infty }}{g}_p\left(1-P\left(X\le x\mathrm{<mo>\; |\; </mo>}X>Va{R}_p\left[X\right]\right)\right)\text{d}x\mathrm{.}$

If $g_1\left(x\right)=x^r,0<r<1$ and

$g_2\left(x\right)=\{\begin{array}{ll}\frac{x}{1-p},\hfill & \text{if}0\le x\le 1-p,\hfill \\ 1,\hfill & \text{if}1-p<x\le 1,\hfill \end{array}$

then

$g_{12}\left(x\right):=g_1\left(g_2\left(x\right)\right)=\{\begin{array}{ll}{\left(\frac{x}{1-p}\right)}^r,\hfill & \text{if}0\le x\le 1-p,\hfill \\ 1,\hfill & \text{if}1-p<x\le 1,\hfill \end{array}$

and

$g_{21}\left(x\right):=g_2\left(g_1\left(x\right)\right)=\{\begin{array}{ll}\frac{x^r}{1-p},\hfill & \text{if}0\le x\le {\left(1-p\right)}^{\frac{1}{r}},\hfill \\ 1,\hfill & \text{if}{\left(1-p\right)}^{\frac{1}{r}}<x\le 1.\hfill \end{array}$

Clearly, $g_1<g_{21}$ and $g_2<g_{12}$ , so that, by Lemma 2.1, ${\rho }_{g_1}\left[X\right]<{\rho }_{g_{21}}\left[X\right]$ and ${\rho }_{g_2}\left[X\right]<{\rho }_{g_{12}}\left[X\right]$ .

In practice, sometimes one needs distort the initial distribution more than one times.

Example 3.1 Consider two risks X and Y with distributions, respectively, are:

$F_X\left(x\right)=\{\begin{array}{ll}0,\hfill & \text{if}x<0,\hfill \\ 0.6,\hfill & \text{if}0\le x<100,\hfill \\ 0.975,\hfill & \text{if}100\le x<500,\hfill \\ 1,\hfill & \text{if}x\ge 500,\hfill \end{array}$

and

$F_Y\left(x\right)=\{\begin{array}{ll}0,\hfill & \text{if}x<0,\hfill \\ 0.6,\hfill & \text{if}0\le x<100,\hfill \\ 0.99,\hfill & \text{if}100\le x<1100,\hfill \\ 1,\hfill & \text{if}x\ge 1100.\hfill \end{array}$

Then $EX=EY=50$ , $Va{R}_{0.95}\left[X\right]=Va{R}_{0.96}\left[X\right]=100$ , $Va{R}_{0.95}\left[Y\right]=Va{R}_{0.96}\left[Y\right]=100$ .

TVaR can be calculated by formula (2.1):

$TVa{R}_{0.95}\left[X\right]=TVa{R}_{0.95}\left[Y\right]=300$ , $TVa{R}_{0.96}\left[X\right]=TVa{R}_{0.96}\left[Y\right]=350$ . So that when $\alpha =0.95$ and $\beta =0.96$ , according to the measures of VaR and TVaR, both X and Y bear the same risk! However, the maximal loss for Y (1100) is more than double than for loss X (500), clearly, risk Y is more risky than risk X. Now we consider distortion expectation ${\rho }_{g_p}$ with

$g_1\left(x\right)=g_2\left(x\right)=\{\begin{array}{ll}\frac{x}{1-p},\hfill & \text{if}0\le x\le 1-p,\hfill \\ 1,\hfill & \text{if}1-p<x\le 1.\hfill \end{array}$

One can easily find that, with $p=0.95$ , ${\rho }_{g_p}\left[X\right]=500$ and ${\rho }_{g_p}\left[Y\right]=1100$ .

3.2. Mixing Methods

One of the easiest ways to generate distortion functions is to use the method of mixing along with finitely distortion functions or infinitely many distortion functions. Specifically, if $g_w$ ( $w\in \langle a\mathrm{,}b\rangle$ ) is a one-parameter family of distortion functions, $\psi$ is an increasing function on $\langle a\mathrm{,}b\rangle$ such that

${\displaystyle {\int }_{\langle a\mathrm{,}b\rangle }}\text{d}\psi \left(w\right)=1$ , then the function $g={\displaystyle {\int }_{\langle a\mathrm{,}b\rangle }}{g}_w\text{d}\psi \left(w\right)$ is a distortion function,

the associated risk measure is given by

${\rho }_g\left[X\right]={\displaystyle {\int }_{\langle a\mathrm{,}b\rangle }}{\rho }_{g_w}\left[X\right]\text{d}\psi \left(w\right)\mathrm{.}$ (3.1)

In particular, if $\psi$ is discrete distribution, then (3.1) can be written as the form of convex linear combination $g={\displaystyle {\sum }_i}{w}_i{g}_i$ $\left(w_i\ge 0,{\displaystyle {\sum }_i}{w}_i=1\right)$ , the associated risk measure is given by

${\rho }_g\left[X\right]={\displaystyle \underset{i}{\sum }}{w}_i{\rho }_{g_i}\left[X\right].$ (3.2)

Further studies on this line can be found recent papers He et al. (2015) and Wei (2017) .

The following lemma is well known (cf. Kriele & Wolf (2014: Theorem 2.1, p. 33) ).

Lemma 3.1 If all ${\rho }_{g_w}$ ( $w\in \langle a\mathrm{,}b\rangle$ ) are monotone, positively homogeneous, subadditive and translation invariant, then ${\rho }_g\left[X\right]$ also has the corresponding properties. That is, if all $g_w$ ( $w\in \langle a\mathrm{,}b\rangle$ ) are coherent, then ${\rho }_g\left[X\right]$ is also coherent.

Now we list three interesting special cases:

Ÿ If $\left[a\mathrm{,}b\right)=\left[\mathrm{0,}\infty \right)$ , $g_i\left(x\right)=1-{\left(1-x\right)}^i\mathrm{,}i\ge 1$ and $w_i\ge 0,{\displaystyle {\sum }_i}{w}_i=1$ , then v in (3.2) is coherent since $g_i\left(x\right)$ is concave. As in Tsukahara (2009) , if we take $w_i$ from Bin v ( $0<\theta <1$ ), then $g_{\theta }\left(u\right)=u+u\theta -u^2\theta$ . If we take

$w_i=\frac{{\theta }^i}{\left({\text{e}}^{\theta }-1\right)i!},\theta >0,$

then

$g_{\theta }\left(u\right)=\frac{{\text{e}}^{\theta }\left(1-{\text{e}}^{-\theta u}\right)}{{\text{e}}^{\theta }-1}\mathrm{.}$

Also, if we take $w_i={\left(1-\theta \right)}^{i-1}\theta$ ( $0<\theta <1$ ), the geometric distribution, then

$g_{\theta }\left(u\right)=\frac{u}{u+\theta \left(1-u\right)}\mathrm{,}$

which is the proportional odds distortion; see Example 2.1 in Cherubini & Mulinacci (2014) .

Ÿ If $\left[a,b\right]=\left[0,1\right]$ , ${\rho }_{g_w}=Va{R}_w\left[X\right]$ and $\text{d}\psi \left(w\right)=\varphi \left(w\right)\text{d}w$ , then ${\rho }_g\left[X\right]$ in (3.1) reduces to

${\rho }_{\varphi }\left[X\right]={\displaystyle {\int }_0^1}Va{R}_w\left[X\right]\varphi \left(w\right)\text{d}w\mathrm{,}$ (3.3)

which is spectral risk measure (see Acerbi, 2002 ). Here $\varphi$ is called a

weighting function satisfies the following properties: $\varphi \ge 0$ , ${\displaystyle {\int }_0^1}\varphi \left(w\right)\text{d}w=1$ . The

following lemma gives a sufficient condition for ${\rho }_{\varphi }\left[X\right]$ to be a coherent risk measure (cf. Kriele & Wolf (2014) ).

Lemma 3.2 Spectral risk measure ${\rho }_{\varphi }\left[X\right]$ is coherent if $\varphi$ is (almost everywhere) monotone increasing.

Clearly, there exists a one-to-one correspondence between distortion function g and weighting function $\varphi$ , namely, $g\left(1-t\right)=1-{\displaystyle {\int }_0^t}\varphi \left(s\right)\text{d}s\mathrm{.}$

Ÿ If $\left[a,b\right]=\left[0,1\right]$ , ${\rho }_{g_w}=TVa{R}_w\left[X\right]$ and $\psi =\mu$ is a probability measure on $\left[\mathrm{0,1}\right]$ , then ${\rho }_g\left[X\right]$ in (3.1) reduces to

${\rho }_{\mu }\left[X\right]={\displaystyle {\int }_0^1}TVa{R}_w\left[X\right]\text{d}\mu \left(w\right)\mathrm{,}$ (3.4)

which is the weighted TVaR (see Cherny (2006) ). $TVa{R}_p$ is a special weighted TVaR with $\mu \left(w\right)=1\left(w\ge p\right)$ . According to Lemma 3.1, since each $TVa{R}_w\left[X\right]$ is coherent risk measure, the weighted TVaR is coherent risk measure. The weighted TVaR can be rewritten as the form of spectral risk measure as following:

$\begin{array}{c}{\rho }_{\mu }\left[X\right]={\displaystyle {\int }_0^1}TVa{R}_w\left[X\right]\text{d}\mu \left(w\right)={\displaystyle {\int }_0^1}\left(\frac{1}{1-w}{\displaystyle {\int }_w^1}Va{R}_q\left[X\right]\text{d}q\right)\text{d}\mu \left(w\right)\\ ={\displaystyle {\int }_0^1}\left(Va{R}_q\left[X\right]{\displaystyle {\int }_0^q}\frac{1}{1-w}\text{d}\mu \left(w\right)\right)\text{d}q\left(\text{by}\text{the}\text{Fubini}\text{theorem}\right)\\ ={\displaystyle {\int }_0^1}Va{R}_q\left[X\right]\varphi \left(q\right)\text{d}q={\displaystyle {\int }_0^1}Va{R}_{1-q}\left[X\right]\text{d}g\left(q\right)\mathrm{,}\end{array}$

where g is a function with $g\left(0\right)=0$ and satisfies

$g^{\prime }\left(1-q\right)=\varphi \left(q\right)={\displaystyle {\int }_0^q}\frac{1}{1-w}\text{d}\mu \left(w\right)\mathrm{.}$

Because $\varphi \left(q\right)$ is increasing function of q, it follows from Lemma 3.2 that the weighted TVaR ${\rho }_{\mu }\left[X\right]$ is coherent. Or, equivalently, $g^{\prime }\left(q\right)$ is decreasing function of q, i.e. g is a concave function, moreover, g is increasing and

$\begin{array}{c}g\left(1\right)={\displaystyle {\int }_0^1}{g}^{\prime }\left(1-w\right)\text{d}w={\displaystyle {\int }_0^1}\text{d}q{\displaystyle {\int }_0^q}\frac{1}{1-w}\text{d}\mu \left(w\right)\\ ={\displaystyle {\int }_0^1}\frac{1}{1-w}\text{d}\mu \left(w\right){\displaystyle {\int }_w^1}\text{d}q={\displaystyle {\int }_0^1}\text{d}\mu \left(w\right)=1.\end{array}$

so that g is a concave distortion function, and hence the weighted TVaR is coherent.

Conversely, the distortion measure with concave distortion function g can be expressed by the weighted TVaR. In fact, note that $\varphi \left(q\right)=g^{\prime }\left(1-q\right)$ is monotone increasing, we define a measure $\nu \left(\left[\mathrm{0,}q\right]\right)=\varphi \left(q\right)$ . As in the proof of Theorem 2.4 in Kriele & Wolf (2014) we have

${\rho }_g\left[X\right]={\displaystyle {\int }_0^1}TVa{R}_w\left[X\right]\text{d}\mu \left(w\right)\mathrm{,}$

where

$\text{d}\mu \left(w\right)=\left(1-w\right)\text{d}\nu \left(w\right)\mathrm{.}$

It can be shown that $\mu$ is a probability measure. In fact,

${\displaystyle {\int }_0^1}\text{d}\mu \left(w\right)={\displaystyle {\int }_0^1}\nu \left(\left[\mathrm{0,}w\right]\right)\text{d}w={\displaystyle {\int }_0^1}\varphi \left(w\right)\text{d}w={\displaystyle {\int }_0^1}{g}^{\prime }\left(w\right)\text{d}w=1.$

We now give some examples of interesting distortion functions and risk measures.

Example 3.2 If $w_1,w_2,w_3,w_4\ge 0,{\displaystyle {\sum }_{i=1}^4}{w}_i=1$ , then

$g_{\alpha \beta }\left(x\right)=w_1{\nu }_{\beta }\left(x\right)+w_2{\nu }_{\alpha }\left(x\right)+w_3{\psi }_{\beta }\left(x\right)+w_4{\psi }_{\alpha }\left(x\right),$

is a distortion function, where ${\nu }_{\beta }\mathrm{,}{\nu }_{\alpha }\mathrm{,}{\psi }_{\beta }\mathrm{,}{\psi }_{\alpha }$ are the distortion functions of TVaR and VaR at confidence levels and, respectively. Then the corresponding risk measure

is called the GlueVaR risk measure, which were initially defined by Belles-Sampera et al. (2014a) (in the case) and the closed-form expressions of GlueVaR for Normal, Log-normal, Student’s t and Generalized Pareto distributions are provided. Two new proportional capital allocation principles based on GlueVaR risk measures are studied in Belles-Sampera et al. (2014b) .

Although GlueVaR has superior mathematical properties than VaR and TVaR, however, the GlueVaR risk measure may also fails to recognize the differences between two risks. For example, consider two risks X and Y in Example 3.1, we have computed that, ., . So that when and, we have. Thus according to, both X and Y bear the same risk! However, the maximal loss for Y (1100) is more than double than for loss X (500), clearly, risk Y is more risky than risk X.

Example 3.3 Let, define a distortion function

where and g is an arbitrary distortion function. Note that can be rewritten as

In particular, if, then we get the esssup-expectation convex combination distortion function with weight on the essential supremum, which was introduced in Bannör & Scherer (2014) . The corresponding risk measure

which is a convex combination of the essential supremum of X and the ordinary expectation of X w.r.t. P.

If

where, are constants, then we get

where

As illustration, we consider the risks X and Y in Example 3.1, if, then. It follows that

and

Taking, then and. Taking, then and. Thus can measure the differences between two risks X and Y.

3.3. A Copula-Based Approach

If F is a distribution function on, then F can be used as a distortion function. The well-known examples are the PH transform and the dual power transform and, more generally, the beta transform; see Wirch & Hardy (1999) for details. Similarly, we use this technique to a distribution function on. We first introduce the notion of copula in the two-dimensional case.

Definition 3.1. A two-dimensional copula is a bivariate distribution on the square having uniform margins. That is a function is right-continuous in each variable such that, , and for,

For an introduction to copula theory and some of its applications, we refer to Joe (1997) , Denuit et al. (2005) and Nelsen (1999) .

The well-known examples of copulas are, and describing, respectively, comonotone dependence, independence and countermonotone dependence between two random variables X and Y. The copula version of the Fréchet-Hoeffding bounds inequality tells us

Any copula has the following decomposition (cf. Yang et al. (2006) )

where,. Here G is a copula which called the indecomposable part.

For a given two-dimensional copula, define one-parameter family

by or. Clearly, for each p, is a right

continuous distortion function. For example,

• is continuous and both convex and concave, the associated risk measure is;

• is continuous and concave, the corresponding risk measure is;

• is continuous and convex, the corresponding risk measure is.

Conversely, if is a family of distortion functions, then, however, is not a copula in general; A sufficient condition can be found in Cherubini & Mulinacci (2014) .

We give below the most common bivariate copulas and the corresponding distortion functions.

• The Archimedean copulas:

for some generator with such that is convex. The pseudo-inverse of is the function with and given by

If is twice differentiable and, then is componentwise

concave if, and only if is concave, where is the derivative of (see

Dolati & Nezhad (2014) ). Aa a consequence, we have

Theorem 3.1 For each, the distortion function

is concave if, and only if is concave.

Some examples of the Archimedean copulas and the corresponding distortion functions:

a) The Clayton copula with parameter is generated by and takes the form

The limit of for and leads to independence and comonotonicity respectively (Nelsen, 1999) . The corresponding distortion functions are

In particular, if, we get the proportional odds distortion which is found by Cherubini & Mulinacci (2014) :

Since, is concave.

b) In case, we get the Frank copulas:

The corresponding distortion functions are

Since, is convex if and concave if.

c) In case, we get the Pareto survival copulas:

The corresponding distortion functions are

Since, is concave.

d) In case, we get the Ali-Mikhail-Haq copulas:

The corresponding distortion functions:

Since, is convex if and concave if.

e) In case, we get the Gumbel-Hougaard copulas:

The corresponding distortion functions:

The value gives independence and the limit for leads to comonotonicity. Since

is concave if and, if, is convex on and concave on.

Among other copulas, which do not belong to Archimedean family, it is worth to mention the following three copulas, given in the bivariate case as:

• The Farlie-Gumbel-Morgenstern copulas:

The corresponding distortion functions are

which is convex if and concave if.

• The Marshall-Olkin copulas:

Note that this copula is not symmetric for. The corresponding distortion functions are

which is concave. In particular, ,

• The normal copulas:

where is a bivariate normal distribution with standard normal marginal distributions and the correlation coefficient, is the inverse function of the standard normal distribution. The corresponding distortion functions:

4. Tail-Asymptotics for VaR

Subadditivity is an appealing property when aggregating risks in order to preserve the benefits of diversification. Subadditivity of two risks is not only dependent on their dependence structure but also on the marginal distributions. Value at risk is one of the most popular risk measures, but this risk measure is not always subadditive, nor convex, exception of elliptically distributed risks. This family consists of many symmetric distributions such as the multivariate normal family, the multivariate Student-t family, the multivariate logistic family and the multivariate exponential power family, and so on. A recent development in the VaR literature concerns the subadditivity in the tails (see Daníelsson et al. (2013) ) which demonstrate that VaR is subadditive in the tails of all fat tailed distributions, provided the tails are not super fat. However, in most practical models of interest the support of loss is bounded so that the maximum loss is simply finite. We will also show that for this class losses VaR is subadditive in the tail. We can illustrate the ideas here with three simple examples. In Examples 4.1 and 4.3, X and Y are independent, while in Example 4.2, X and Y are dependent.

Example 4.1 Let X and Y be i.i.d. random variables which are Bernoulli (0.02) distributed, i.e.. Then

Dhaene et al. (2006) verified that VaR is not subadditive since and. However, for, and

Thus for,

Example 4.2 Suppose we have losses X and Y, both dependent on the same underlying Uniform (0,1) random variable U as follows.

Note that

Hardy (2006) found that,. Thus

However, for any, , . Thus,

Example 4.3 Let X and Y be i.i.d. random variables which are Uniform (0,1) distributed. Then

and for, ,

Thus for,

Generally, we have the following result.

Theorem 4.1 If the risks have finite upper endpoints, then

Proof The proof is very simple. Denote by. Then and, which lead to

Hence

and the result follows.

Next theorem consider the random variables that are not necessarily has finite upper endpoint, we first recall the notion of (extended) regularly varying function:

Definition 4.1 A function f is called regularly varying at some point (or, respectively) with index if for all,

(or, respectively). We write (, respectively). For we say f is slowly varying; for rapidly varying.

Definition 4.2 Assume that F is the distribution function of a nonnegative random. We say F belongs to the extended regular variation class, if there are some such that

or equivalently

We write.

A standard reference to the topic of (extended) regular variation is Bingham et al. (1987) while main results are summarized by Embrechts et al. (1997) .

Theorem 4.2 We assume that have the same absolutely continuous marginal distributions F with infinite upper endpoint.

1) If

(4.1)

then

(4.2)

(2) If

then

3) If

then

Proof We prove (1) only since the other cases follow immediately in the same way. Because all the marginal distributions are absolutely continuous, so we have for any,

This, together with (4.1), implies that

(4.3)

The absolute continuity of F implies that is continuous and strictly monotone decreasing. Then from (4.3) we have

which is (4.2). This completes the proof.

Example 4.4 Suppose that each is regularly varying with index. When the are mutually independent, it follows from (Feller, 1971: p. 279) that

Thus we get

Suppose that the are comonotonic, i.e., then

So that in the case the result for the independent and the comonotonic case are the same.

The following result generalizes the result in Jang & Jho (2011) in which all ‘s are assumed identically distributed.

Theorem 4.3 Suppose are nonnegative random variables (but not necessarily independent or identically distributed.) If has distribution F satisfying, where is slowly varying at

infinity. If and, as, , then

Proof It follows from Lemma 2.1 in Davis & Resnick (1996) that

This leads to

(4.4)

Because

Thus from (4.4) that

which is equivalent to

This implies that

since is continuous and strictly monotone decreasing. Note that, (as) and

completing the proof.

Remark 4.1 The above result is obtained by Embrechts et al. (2009) for identically distributed and Archimedean copula dependent ‘s. However, our result can not obtained from their’s due to the following fact: The famous Farlie-Gumbel-Morgenstern family, does not belong to Archimedean family, which has the form

where are two distributions and is a constant. When

, it satisfying as.

In the next theorem we consider the extended regularly varying instead of regularly varying.

Theorem 4.4 Suppose are nonnegative random variables with the common identical distribution function F. If and

, as, , then

1) If,

2) If,

3) If,

Proof It follows from Lemma 2.2 in Zhang et al. (2009) that

This leads to

from which and using the same argument as that in the proof of Theorem 4.3 leads to

(4.51)

If, then

This and (4.5) imply that

It follows that

(4.6)

Thus

Similarly, if,

(4.7)

and hence

If, then by (4.6) and (4.7) one has

(4.8)

This ends the proof of Theorem 4.4.

To give applications of our results we employ extreme value theory techniques. A distribution function F (or the rv X) is said to belong to the Maximum Domain of Attraction (MDA) of the extreme value distribution H if there exist

constants such that. We

write or. According to the Fisher-Tippett theorem (see Theorem 3.2.3 in Embrechts et al. (1997) ) H belongs to one of the three standard extreme value distributions:

Let denote the right-endpoint of the support of F:. Then we have the following results (see Embrechts et al. (1997: pp. 132-157) ).

• Fréchet case: For some,.

Examples are Pareto, Cauchy, Burr, Loggamma and Stable with index.

• Weibull case: For some, , .

Examples are Uniform and Beta distribution.

• Gumbel case: and there exists a positive measurable function a such that for

(4.9)

Examples are Exponential-like, Weibull-like, Gamma, Normal, Lognormal, Benktander-type-I and Benktander-type-II.

Remark 4.2 1) For, if, in view of Weibull case above they are all have finite supports. It follows from Theorem 4.1, is subadditive for p is sufficiently close to 1.

2) For, if and are identically distributed, has an Archimedean copula with generator, which is regularly varying at 0 with index. We apply (2.2) in Alink et al. (2004) and Definition 4.1 to obtain

where in the last step we have used Lemma 3.1(d) in Embrechts et al. (2009) which states that

This, together with Theorem 4.2 we recover the result Theorem 2.5 in Embrechts et al. (2009) .

3) If have common distribution F, has an Archimedean copula with generator, which is regularly varying at 0 with index. We apply (2.6) in Alink et al. (2004) to obtain

where

The constant is strictly increasing in with

For more details, see Alink et al. (2004) for the case and Chen et al. (2012) for general case. Thus by Theorem 4.2,

In particular, when,

5. Tail-Subadditivity for Distortion Risk Measures

The tail-subadditivity property for GlueVaR risk measures were initially defined by Belles-Sampera et al. (2014a) and the milder condition of subadditivity in the tail region is investigated. Furthermore, they verified that a GlueVaR risk measure is tail-subadditive if its associated distortion function is concave in, where parameters is confidence level and is an extra confidence level such that and,

where and are two distorted survival probabilities at levels and, respectively. Here. We note, however, from their proof to Theorem 6.1 that the result will hold for any distortion function that is concave in, not restricted to. In this section we state the corresponding result without proof. As in Belles-Sampera et al. (2014a) , for a given confidence level, the tail region of a random variable Z is defined as, where is the a-quantile. For simplicity, we use the notation.

Theorem 5.1 For a confidence level and two risks defined on the same probability space. If and g is a concave distortion function in, then the distortion risk measure is tail-subadditive. That is

where

Example 5.1 Consider the distortion functions associated with the Gumbel-Hougaard copulas (cf. Section 3.3):

If, then is concave on and convex on. Thus the distortion risk measure is tail-subadditive.

6. Conclusion

In this paper, we give three methods to construct new class of distortion functions and distortion risk measures and then we investigate the tail asymptotics of distortion risk measures for the sum of possibly dependent risks with emphasis on VaR. We study the concept of tail subadditivity for distortion risk measures and give sufficient conditions for a distortion risk measure to be tail subadditive. The multivariate tail distortion risk measure and more applications of the risk measure will be investigated in the coming research.

Acknowledgements

We thank the anonymous reviewers for their comments. The research was supported by the National Natural Science Foundation of China (11171179, 11571198) and the Research Fund for the Doctoral Program of Higher Education of China (20133705110002).

Author Contributions

Two authors contributed equally to this work.

Conflicts of Interest

The authors declare no conflict of interest.

Conflicts of Interest

The authors declare no conflicts of interest.

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