Asymptotics for the conditional higher moment coherent risk measure with weak contagion

Abstract

Various measures have been introduced in the existing literature to evaluate extreme risk exposure under the effect of an observable factor. Due to the nice properties of the higher-moment (HM) coherent risk measure, we propose a conditional version of the HM (CoHM) risk measure by incorporating the information of an observable factor. We conduct an asymptotic analysis of this measure in the presence of extreme risks under the weak contagion at a high confidence level, which is further applied to the special case of the conditional Haezendonck–Goovaerts risk measure (CoHG). Numerical illustrations are also provided to examine the accuracy of the asymptotic formulas and to analyze the sensitivity of the risk contribution of the CoHG. Based on the asymptotic result in the Fréchet case, we propose an estimator for the CoHM via an extrapolation, supported by a simulation study.

1. Introduction

In the bank regulatory frameworks of Basel II and Basel III, as well as in the insurance regulatory regimes such as Solvency II and Swiss Solvency Test, capital requirements for all financial institutions and (re)insurance companies operating within the European Union and Switzerland are solely based on the Value at Risk (VaR) and the Expected Shortfall (ES). For these reasons and not only, risk measures are widely applied by financial and insurance institutions for pricing and decision-making. Generally, a risk measure is a mapping from some space of random risks to a set of real numbers, which quantifies risk exposure. However, in the broader literature of risk management, there is no answer to the question of which risk measure is the best. An axiomatic definition of coherent risk measures is introduced by Artzner (1999) to prescribe a set of reasonable risk measures, satisfying the properties of monotonicity, positive homogeneity, sub-additivity, and translation invariance.

The most popular coherent risk measure is the Tail Value at Risk introduced by Rockafellar and Uryasev (2002), which is similar to the ES in Acerbi and Tasche (2002). Many other typical coherent risk measures have been proposed from different perspectives, such as the worst conditional expectation of Artzner (1999), the distortion risk measure of Wang et al. (1997), the spectral risk measure of Acerbi (2002), and the entropic risk measure of Föllmer and Knispel (2011), Gini-type risk measures of Furman et al. (2017), among many others. These risk measures are usually categorized as loss risk measures and variability risk measures, which reflect a first-order quantification of risk but may overlook higher-order risk aversion properties of investors, as pointed out by Chen et al. (2016). Bauer and Zanjani (2016) argue that the crucial issue of constructing a suitable measure needs to be considered on institutional specifications and the economic environment.

Krokhmal (2007) constructs a new family of the higher-moment (HM) coherent risk measures in the context of stochastic optimization, which generalizes the ES. Similar constructs are studied by Ben-Tal and Teboulle (2007). Due to its nice properties, the HM risk measure is further investigated by Krokhmal and Soberanis (2010), Dentcheva et al. (2010), Krokhmal et al. (2011), Chen et al. (2011), Matmoura and Penev (2013), Vinel and Krokhmal (2017), Dentcheva et al. (2017), Kouri (2019), and Gómez et al. (2022) among many others, in portfolio optimization and risk management. The HM risk measure shares the features of both loss risk measures and variability risk measures, as noted by Gómez et al. (2022). Parallel to the HM risk measure, on Orlicz spaces, the Haezendonck–Goovaerts (HG) risk measure (Haezendonck and Goovaerts 1982; Goovaerts et al. 2004) is defined as the optimal value in a problem of minimization based on Orlicz norms. The HG risk measure has originated an active research trend in actuarial science, see Bellini and Rosazza Gianin (2008, 2012), Tang and Yang (2012, 2014), Ahn and Shyamalkumar (2014), Gao et al. (2020), Bellini et al. (2021), and Xun et al. (2021).

Assessing extreme risk exposure under a certain risk measure, especially in the most adverse scenario, is a common practice in the modern risk management framework. For example, academics, practitioners, and regulators have considered multiple approaches to measuring the sensitivity of a legal entity’s financial performance when an observable factor is located in an extreme region. The Marginal Expected Shortfall (MES) is a popular indicator of systemic risk in the financial and actuarial literature. Comprehensive discussions can be found in Idier et al. (2014). Specifically, the MES is a conditional expectation, $E(X\big|Y \gt t)$ for large values of t, where X is a risk from the insurer/investor portfolio of risks and Y is the common risk or reference risk of the portfolio specifying the adverse scenario. Various measures of systemic risk are proposed in the literature, such as the Conditional Value at Risk (CoVaR) and the $\Delta\mathrm{CoVaR}$ by Adrian and Brunnermeier (2016) and the Conditional Expected Shortfall (CoES) by Mainik and Schaanning (2014). To further generalize the CoES in the study of systemic risk, Xun et al. (2021) propose a conditional version of the HG (CoHG) risk measure of X given a catastrophic scenario that Y exceeds a high threshold. It is often hard to calculate these conditional risk measures analytically as it requires understanding how the risks are interplayed, which makes it meaningful and valuable to seek asymptotic estimates. Various asymptotic studies of the aforementioned conditional risk measures are conducted in the literature, including the MES and its variations by Li (2022) and Chen and Liu (2022), and the CoVaR by Nolde et al. (2022). Our main results provide asymptotic results for a conditional version of the HM (CoHM) risk measure, which generalizes the CoES.

Motivated by these, in this paper, to retrieve the nice properties of the HM risk measure, we propose the CoHM risk measure by incorporating the information of an observable factor. Specifically, we consider the CoHM risk measure of $\left(X\big|Y \gt t\right)$ , where t serves as a threshold. A common choice of t is the VaR of Y at level $p\in(0,1).$ Clearly, the CoHM shares many similarities with the CoHG. The purpose of this paper is to evaluate asymptotic approximations of the CoHM for the confidence level $q\in(0,1)$ in a reasonable range under a weak contagion over risk variables X and Y. We carry out intensive numerical studies to apply this research in approximating the CoHG and to examine the performance of these asymptotic formulas. An immediate application of parameter sensitivity analysis on the $\Delta \mathrm{CoHG}$ is provided to further illustrate the risk contribution of the CoHG in the study of systemic risk. As a further extension, we propose an extrapolated estimator for the CoHM with the Fréchet tail and perform simulations to show its good performance. It is apparent that our results may help regulators and decision-makers understand the downside risk better.

The rest of this paper is organized as follows: Section 2 introduces the CoHM risk measure and sets up the framework for our asymptotic study of the CoHM risk measure; Section 3 presents our main results; Section 4 gives numerical studies; Section 5 compares the performance of estimators with a simulation study; Section 6 makes some concluding remarks. Finally, the Appendix prepares several lemmas and collects all proofs.

2. Preliminaries

2.1. Notation conventions

Throughout the paper, we will derive various types of limit relations for which the limit procedure is either $q\uparrow1$ or $x \uparrow \infty$ unless otherwise stated. For two positive functions $f({\cdot})$ and $g({\cdot})$ , we write $f({\cdot}) \sim g({\cdot})$ if the ratio of the left-hand side and right-hand side converges to 1; that is, $\lim f/g = 1$ . We also write $f=o(g)$ if $\lim f/g = 0$ .

For $x, y \in \mathbb{R} = ({-}\infty, \infty)$ , we write $x\wedge y = \min\{x,y\}$ and $x \vee y = \max\{x,y\}$ . Moreover, we use $x_{+}=x\vee 0=x1_{(x\geq 0)}$ to denote the positive part of x, where $1_{D}$ is the indicator of an event D, which equals 1 if D occurs and 0 otherwise.

For a distribution function F of X, denote by $F^{\leftarrow}$ and $F^{\rightarrow}$ its left and right generalized inverses, respectively:

\begin{align*}F^{\leftarrow}(q)=\inf\left\{x\in\mathbb{R}\,:\,F(x)\geq q\right\},\quad F^{\rightarrow}(q)=\sup\left\{x\in\mathbb{R}\,:\,F(x)\leq q \right\},\quad q\in\left[0,1\right], \end{align*}

where, by convention, $\inf\emptyset$ is the right endpoint of its support set and $\sup\emptyset$ is the left endpoint of it. Also, $F^{\leftarrow}(q)$ is the VaR of X at level q.

2.2. The CoHM risk measure

Let X be a real-valued random variable, representing a risk variable in loss–profit style, with a distribution function $F=1-\overline{F}$ on $\mathbb{R}$ and the essential supremum $\hat{x}=F^{\leftarrow}(1)$ . Following Krokhmal (2007), for $X\in L^{k}$ for some $k\geq1,$ that is, $ \left\lVert X\right\rVert _{k}=\left(E\big|X\big|^k\right)^{\frac{1}{k}} \lt \infty,$ the HM risk measure of X for $q\in(0,1)$ is defined to be

\begin{eqnarray*}\mathrm{HM}_q(X)=\inf_{0 \lt x \lt \hat{x}}\left\{x+\frac{1}{1-q}\left\lVert(X-x)_{+}\right\rVert _{k}\right\}. \end{eqnarray*}

Now, we introduce a CoHM risk measure as a natural extension by incorporating the information from Y so that the dependence between $X \in L^{k}$ and Y is taken into account explicitly. The CoHM risk measure of X given $(Y \gt y_p)$ and for two confidence levels $p,q\in(0,1)$ is defined as:

(2.1) \begin{eqnarray}\mathrm{CoHM}_{p,q}\left(X\big|Y\right)=\inf_{0 \lt x \lt \hat{x}}\left\{x+\frac{1}{1-q}\left\lVert(X-x)_{+}\big|Y \gt y_p\right\rVert_{k}\right\},\end{eqnarray}

where $y_p=G^{\leftarrow}(p)$ is the VaR of the risk variable Y distributed by G at level p.

When $k=1$ , the infimum in (2.1) is attained at any point in a closed set $\left[F^{\leftarrow}_{X |Y \gt y_p}(q),F^{\rightarrow}_{X |Y \gt y_p}(q)\right]$ , which is a proper interval when q corresponds to the level of a flat piece of the conditional distribution $F_{X |Y \gt y_p}$ and is a single point otherwise. Thus,

(2.2) \begin{eqnarray}\mathrm{CoHM}_{p,q}\left(X\big|Y\right)=x^{\ast}+\frac{1}{1-q}E\left[(X-x^{\ast})_{+}\big|Y \gt y_p\right],\end{eqnarray}

where the minimizer is

\begin{align*}x^{\ast}=\mathrm{CoVaR}_{p,q}\left(X\big|Y\right)=F^{\leftarrow}_{X |Y \gt y_p}(q). \end{align*}

Following the discussions of the CoVaR and the CoES in Fang and Li (2018), we realize that (2.2) is an alternative expression for the CoES with confidence levels $p,q\in(0,1)$ :

\begin{eqnarray*}\mathrm{CoES}_{p,q}\left(X\big|Y\right)=\frac{1}{1-q}\int_{q}^{1}\mathrm{CoVaR}_{p,r}\left(X\big|Y\right)\mathrm{d}r.\end{eqnarray*}

For a risk variable X, denote by $\hat{p}=P(X=\hat{x})$ the probability assigned to the upper endpoint. Obviously, $\hat{p}=0$ holds automatically if X is unbounded from above or if F is continuous at $\hat{x}.$ For simplicity, we only consider the case with $\hat{p}=0.$ By transplanting Theorem 2.1 of Tang and Yang (2012) to the current context, when $k \gt 1,$ the CoHM risk measure is given by:

(2.3) \begin{eqnarray}\mathrm{CoHM}_{p,q}\left(X\big|Y\right)=x^{\ast}+\frac{1}{1-q}\left\lVert(X-x^{\ast})_{+}\big|Y \gt y_p\right\rVert_k,\end{eqnarray}

where the minimizer $x=x^{\ast} \lt \hat{x}$ is the unique solution to the equation:

(2.4) \begin{eqnarray}\frac{\left\lVert(X-x)_{+}\big|Y \gt y_p\right\rVert_{k-1}}{\left\lVert(X-x)_{+}\big|Y \gt y_p\right\rVert_{k}}=(1-q)^{\frac{1}{k-1}}.\end{eqnarray}

We further point out that the CoHG risk measure proposed in Xun et al. (2021) reduces to the CoHM risk measure (2.1) when taking a power Young function, that is, $\varphi(t)=t^k$ for $k\geq 1$ and the confidence level $\tilde{q}=1-(1-q)^{k}$ . See also Section 4.1. Hence, similar to Property 2.1 of Xun et al. (2021), we conclude the following lemma.

Lemma 2.1. Considering the $\mathrm{CoHM}_{p, q}\left(X\big|Y\right)$ risk measure above, we have

1. (i) $\mathrm{CoHM}_{p, q}\left(X\big|Y\right)$ increases in $q \in(0,1)$ ;

2. (ii) $\mathrm{CoVaR}_{p, q}\left(X\big|Y\right) \leq \mathrm{CoHM}_{p, q}\left(X\big|Y\right) \leq \hat{x}$ ;

3. (iii) The CoHM risk measure is coherent.

2.3. Regular variation

A positively measurable function $f({\cdot})$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}$ is said to be regularly varying at $x_0=0+$ or $\infty$ with a index $\alpha\in\mathbb{R}$ , denoted by $f({\cdot})\in\mathcal{R}_{\alpha}(x_{0})$ , if

\begin{equation*}\lim _{x \rightarrow x_{0}} \frac{f(x y)}{f(x)}=y^{\alpha},\quad y \gt 0.\end{equation*}

The class $\mathcal{R}_{0}(x_{0})$ consists of functions slowly varying at $x_0$ . Moreover, a positively measurable function $f({\cdot})$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}$ is said to be rapidly varying at $x_0=0+$ or $\infty$ , denoted by $f({\cdot})\in\mathcal{R}_{\infty}(x_{0})$ , if

\begin{equation*}\lim _{x \rightarrow x_{0}} \frac{f(x y)}{f(x)}= \infty, \quad y \gt 1.\end{equation*}

2.4. Extreme value theory and maximum domain of attraction

To develop the main results of this paper, we extensively employ the Extreme Value Theory. A distribution function F is said to belong to the Maximum Domain of Attraction (MDA) of a non-degenerate distribution function H, written as $F\in \mathrm{MDA}(H)$ , if there are some $c_{n} \gt 0$ and $d_{n} \in \mathbb{R}$ for $n \in \mathbb{N}$ such that

\begin{equation*}\lim _{n \rightarrow \infty} F^{n}\left(c_{n} x+d_{n}\right)=H(x).\end{equation*}

Due to the classical Fisher–Tippett theorem (see Fisher and Tippett, 1928 and Gnedenko, 1943), H has to be the generalized extreme value distribution whose standard version is given by:

A distribution function F belongs to $\mathrm{MDA}(\Phi_{\gamma})$ if and only if $\hat{x}$ is infinite and the relation

\begin{equation*}\lim _{x \uparrow \infty} \frac{\overline{F}(xy)}{\overline{F}(x)}=y^{-\gamma}, \qquad y \gt 0,\end{equation*}

holds; see Theorem 3.3.7 of Embrechts et al. (1997). Alternatively, the condition can be restated as $F\in \mathrm{MDA}(\Phi_{\gamma})$ if and only if $\overline{F}({\cdot})\in \mathcal{R}_{-\gamma}(\infty)$ .

A distribution function F belongs to $\mathrm{MDA}(\Lambda)$ with $\hat{x} \leq \infty$ if and only if the relation

\begin{equation*}\lim _{x \uparrow \hat{x}} \frac{\overline{F}(x+y a(x))}{\overline{F}(x)}=\mathrm{e}^{-y}, \qquad y \in \mathbb{R},\end{equation*}

holds for some positive auxiliary function $a({\cdot})$ on ( $-\infty$ , $\hat{x}$ ). For $F \in \mathrm{MDA}(\Lambda)$ , $\overline{F}({\cdot})\in \mathcal{R}_{-\infty}(\infty)$ provided $\hat{x}=\infty$ or $\overline{F}(\hat{x}-\cdot)\in \mathcal{R}_{\infty}(0{+})$ provided $\hat{x} \lt \infty$ . The mean excess loss function is one of the commonly used choices for the auxiliary function $a({\cdot})$ , that is, $a(x)=E(X-x\big|X \gt x)$ , for $x \lt \hat{x}$ . Moreover, as $x \rightarrow \hat{x}$ ,

(2.5) \begin{align} \begin{cases}a(x)=o(x), & \text { if } \hat{x}=\infty \\[3pt] a(x)=o(\hat{x}-x), & \text { if } \hat{x} \lt \infty\end{cases}\end{align}

see details in Resnick (1987) and Embrechts et al. (1997). The following gives a nice representation of $F \in \mathrm{MDA}(\Lambda)$ due to Balkema and de Haan (1972), Proposition 1.4 of Resnick (1987), or relation (3.35) Embrechts et al. (1997). For $F \in \mathrm{MDA}(\Lambda)$ with $\hat{x} \leq \infty$ , there is some $x_{0} \lt \hat{x}$ such that

(2.6) \begin{align}\overline{F}(x)=b(x) \exp \left\{-\int_{x_{0}}^{x} \frac{1}{a(y)} \mathrm{d}y \right\}, \qquad x_{0} \lt x \lt \hat{x},\end{align}

where $a({\cdot})$ denotes an auxiliary function, which is positive and absolutely continuous with $\lim _{x \uparrow \hat{x}} a^{\prime}(x)=0$ , and $b({\cdot})$ denotes a positive function with $\lim _{x \uparrow \hat{x}} b(x)= b \gt 0$ .

A distribution function F belongs to $\mathrm{MDA}(\Psi_{\gamma})$ if and only if $\hat{x}$ is finite and the relation

\begin{equation*}\lim _{x \downarrow 0} \frac{\overline{F}(\hat{x}-x y)}{\overline{F}(\hat{x}-x)}=y^{\gamma}, \qquad y \gt 0,\end{equation*}

holds; see Theorem 3.3.12 of Embrechts et al. (1997). The equivalent condition in terms of regular variation is that $F\in \mathrm{MDA}(\Psi_{\gamma})$ if and only if $\overline{F}(\hat{x}-\cdot)\in \mathcal{R}_{\gamma}(0{+})$ .

2.5. Copula and dependence

Now, we turn to the dependence structure between the primary risk X and the reference risk Y by a comprehensive treatment of copulas. Let (X, Y) be a random vector with continuous marginal distributions F(x) and G(y). Then the dependence structure between X and Y is characterized in terms of a unique bivariate copula by Sklar’s theorem, with its joint distribution $P(X \leq x, Y\leq y)=C\left(F(x), G(y)\right)$ . The reader is referred to Nelsen (2007).

Definition 2.1. The corresponding survival copula is defined as:

\begin{eqnarray*}\widehat{C}(u,v)=u+v-1+C(1-u,1-v), \qquad (u,v) \in [0,1]^{2}.\end{eqnarray*}

Clearly, such a bivariate function with respect to C(u, v) can also be described as:

\begin{eqnarray*}\widehat{C}\left(\overline{F}(x),\overline{G}(y)\right)=P(X \gt x,Y \gt y).\end{eqnarray*}

Assume that C(u, v) is absolutely continuous, denote by $C_{1}(u,v) \,:\!=\, \frac{\partial}{\partial u}C(u,v)$ , $C_{2}(u,v)\,:\!=\,\frac{\partial}{\partial v}C(u,v)$ , and $C_{12}(u,v)\,:\!=\,\frac{\partial ^{2}}{\partial u \partial v}C(u,v)$ , then

\begin{eqnarray*}\widehat{C}_{2}(u,v)\,:\!=\,\frac{\partial}{\partial v}\widehat{C}(u,v)=1-C_{2}(1-u,1-v), \\[-25pt]\end{eqnarray*}

\begin{eqnarray*}\widehat{C}_{12}(u,v)\,:\!=\,\frac{\partial ^{2}}{\partial u \partial v}\widehat{C}(u,v)=C_{12}(1-u,1-v).\end{eqnarray*}

Given the properties of copula that enable the separation of dependence from marginal distributions, we assume X and Y are mutually affected via a general dependence structure defined in terms of copula. This assumption, attributed to Asimit and Badescu (2010), has been widely studied in financial and insurance literature as it allows both positive and negative dependences and is satisfied by many commonly used bivariate copulas. For detailed discussions, we refer the reader to Yang and Konstantinides (2015) and Yang et al. (2016) in modeling dependent financial and insurance risks and to Asimit and Li (2018a, b), Li (2022), and Ling and Liu (2022) in dealing with conditional risk measures.

Assumption 2.1. Let (X,Y) be a bivariate risk with a survival copula $\widehat{C}$ . Assume that the relation

(2.7) \begin{eqnarray}\widehat{C}_{2}(u,v) \sim u\widehat{C}_{12}(0+,v),\qquad u \downarrow 0,\end{eqnarray}

holds uniformly for $v\in [0,1]$ .

Remark 2.1.

1. (i) We remark that the uniformity is understood as:

   \begin{align*}\lim_{u\downarrow 0}\sup_{v\in[0,1]}\left|\frac{\widehat{C}_2(u,v)}{u\widehat{C}_{12}(0+,v)}-1\right|=0. \end{align*}

2. (ii) Clearly, the asymptotic property is equivalent to

   \begin{eqnarray*}1-C_{2}(u,v) \sim (1-u)C_{12}(1-,v),\qquad u \uparrow 1,\end{eqnarray*}
   hold uniformly for $v\in [0,1]$ .

3. (iii) By the uniformity, integrating both sides of (2.7) with respect to $G(\mathrm{d}y)$ over the range $[0,\infty)$ with $v=\overline{G}(y)$ leads to $\int_{0}^{\infty} \widehat{C}_{12}\left(0+,\overline{G}(y)\right) G(\mathrm{d}y) =1.$ Thus, $G_h$ defined by

   \begin{align*}G_{h}(\mathrm{d}y) = C_{1}\left(1-,G(\mathrm{d}y)\right) = C_{12}\left(1-,G(y)\right)G(\mathrm{d}y) \end{align*}
   forms a proper distribution function of $Y_h$ . As $x \uparrow \hat{x}$ , by the uniformity we also have
   (2.8) \begin{eqnarray}\overline{F}_{X|Y \gt y_p}(x) &=& \frac{1}{\overline{G}(y_p)} \widehat{C}\left(\overline{F}(x),\overline{G}(y_p)\right) = \frac{1}{\overline{G}(y_p)} \left[ \int_{0}^{1-p} \widehat{C}_2\left(\overline{F}(x),v\right) \mathrm{d}v \right]_{v=\overline{G}(y)} \notag\\&\sim & \frac{\overline{F}(x)}{\overline{G}(y_p)} \left[\int_{0}^{1-p} \widehat{C}_{12}\left(0+,v\right) \mathrm{d}v \right]_{v=\overline{G}(y)} = C_p \overline{F}(x),\end{eqnarray}
   where $C_p=\overline{G}_h(y_p)/\overline{G}(y_p)$ .

4. (iv) Some examples of copulas satisfying Assumption 2.1 are listed here for reference.

   1. (a) The Ali–Mikhail–Haq copula is of the form:

      \begin{equation*}C(u, v)=\frac{u v}{1-r(1-u)(1-v)}, \qquad r \in({-}1,1),\end{equation*}
      with $\widehat{C}_{12}(0+,v)=1+r-2rv$ .

   2. (b) The Farlie–Gumbel–Morgenstern (FGM) copula is of the form:

      (2.9) \begin{align}C(u, v)=u v+ruv(1-u)(1-v), \qquad r\in({-}1,1),\end{align}
      with $\widehat{C}_{12}(0+,v)=1+r-2rv$ .

   3. (c) The Clayton copula is of the form:

      (2.10) \begin{align}C(u, v)=(u^{-r}+v^{-r}-1)^{-1/r}, \qquad r\in(0,\infty),\end{align}
      with $\widehat{C}_{12}(0+,v)=(1+r)(1-v)^r$ .

   4. (d) The independent copula is of the form:

      (2.11) \begin{align}C(u, v)=uv,\end{align}
      with $\widehat{C}_{12}(0+,v)=1$ .

2.6. On the weak contagion

The aforementioned dependence structure (2.7) is employed to model the weak contagion between risks. Bandt et al. (2012) define contagion risk as the component of systemic risk stemming from idiosyncratic shocks. It is also widely recognized that the interdependence between risk factors can facilitate the propagation of risk contagion. The magnitude of this interdependence under extreme adverse scenarios, also known as tail dependence, significantly impacts the degree of contagion. A common measure of tail dependence is the coefficient of upper tail dependence, which is defined to assess the probability of extreme co-movements. According to McNeil et al. (2015), two random variables X and Y, distributed by F and G, respectively, are said to be asymptotically independent if they have a zero coefficient of (upper) tail dependence:

\begin{align*}\lambda=\lim _{q \uparrow 1} P\left(X \gt F^{\leftarrow}(q) \mid Y \gt G^{\leftarrow}(q)\right)=0. \end{align*}

It can be verified that X and Y are asymptotically independent when following (2.7), see, for example, Lemma 3.1 of Asimit and Badescu (2010) or Section 1 of Chen and Yuen (2012). Liu and Yang (2021) observe that even the degree of contagion arising from certain asymptotically independent risks, that is, bivariate risks following a Gaussian copula, could be relatively strong. However, in this paper, we focus on the weak contagion driven by the dependence structure (2.7).

Our motivation for considering a weak contagion is several-fold. First, in good times or under normal market conditions, the contagion between risks may be a weak one; see, for example, Balla et al. (2014), Kiriliouk et al. (2016), and Abduraimova (2022) for the empirical evidence. Second, the contagion risk usually can be mitigated by proper diversification. Moreover, the contagion between risks, albeit weak, should not be overlooked, as shown in Figure 1, where we visualize the empirical estimation for systemic risk measures, $\mathrm{CoVaR}_{p,q}\left(X\big|Y\right)$ , $\mathrm{CoES}_{p,q}\left(X\big|Y\right)$ , $\Delta\mathrm{CoVaR}_{p,q}\left(X\big|Y\right)$ , and $\Delta\mathrm{CoES}_{p,q}\left(X\big|Y\right)$ under strong and weak contagion scenarios. Choose $10^7$ samples of X and Y both following the Pareto distribution (4.10) with $\alpha=3$ and $\kappa=1$ . A rotated Clayton copula defined as $\widehat{C}(u,v)=\left(u^{-r_1}+v^{-r_1}-1 \right)^{-1/r_1}$ is assumed to represent the strong contagion, while an FGM copula (2.9) and an independent copula (2.11) are assumed to represent the weak contagion caused by the dependence structure (2.7). Although there is a gap between strong and weak contagion cases, the weak contagion arising from the dependence structure (2.7) still exists, which motivates us to quantitatively capture their tail behaviors based on the following asymptotic study.

Figure 1. Empirical estimation for $\mathrm{CoVaR}_{p,q}\left(X\big|Y\right)$ , $\mathrm{CoES}_{p,q}\left(X\big|Y\right)$ , $\Delta\mathrm{CoVaR}_{p,q}\left(X\big|Y\right)$ , and $\Delta\mathrm{CoES}_{p,q}\left(X\big|Y\right)$ with $p=0.85$ based on $10^7$ samples of $\left(X, Y \right)$ , where X and Y both follow Pareto (3,1) and are dependent via a rotated Clayton copula with parameter $r_1=1$ , an FGM copula with parameter $r_2=0.5$ , or an independent copula.

3. Main results

This section establishes asymptotics for the CoHM defined in (2.1) by considering the aforementioned assumptions, where the primary risk X belongs to the Fréchet, Gumbel, or Weibull MDA.

We first focus on the Fréchet case. Denote by $\mathrm{B}(\cdot,\cdot)$ the beta function, namely, $\mathrm{B}(a,b)=\int_0^1x^{a-1}{}(1-x)^{b-1}\,\mathrm{d}x$ with $a,b \gt 0$ .

Theorem 3.1. Consider the risk measure CoHM defined in (2.1) for some $k\geq1$ with a bivariate risk vector $(X,Y).$ If Assumption 2.1 holds and $F \in \mathrm{MDA}\left(\Phi_{\gamma}\right)$ for some $\gamma \gt k$ , then as $q \uparrow 1,$ we have

(3.1) \begin{align}\mathrm{CoHM}_{p, q}\left(X\big|Y\right) \sim C_p^{1 / \gamma} \frac{\gamma(\gamma-k)^{(k / \gamma)-1}}{k^{(k-1) / \gamma}} \,\mathrm{B}(\gamma-k, k)^{1 / \gamma} F^{\leftarrow}(1-(1-q)^k).\end{align}

Next, we turn to the Gumbel case, in which $E\left[X_{+}^{k}\right] \lt \infty$ for $k \geq 1$ holds automatically because either $\hat{x} \lt \infty$ or $\overline{F} \in \mathcal{R}_{-\infty}(\infty)$ . Denote by $\Gamma({\cdot})$ the gamma function, namely, $\Gamma(a)=\int_0^{\infty} x^{a-1}\mathrm{e}^{-x}\,\mathrm{d}x$ with $a \gt 0$ .

Theorem 3.2. Consider the risk measure CoHM defined in (2.1) for some $k\geq1$ with a bivariate risk vector $(X,Y).$ If Assumption 2.1 holds and $F \in \mathrm{MDA}(\Lambda)$ with $0 \lt \hat{x}\leq \infty$ , then as $q \uparrow 1,$ we have

1. (i) for $\hat{x}=\infty$ we have

   (3.2) \begin{align}\mathrm{CoHM}_{p, q}\left(X\big|Y\right) \sim F^{\leftarrow}\left(1- C_p^{-1} \frac{k^k}{\Gamma(k+1)}(1-q)^{k}\right);\end{align}

2. (ii) for $\hat{x} \lt \infty$ we have

   (3.3) \begin{align}\hat{x}-\mathrm{CoHM}_{p, q}\left(X\big|Y\right) \sim \hat{x}-F^{\leftarrow}\left(1- C_p^{-1} \frac{k^k}{\Gamma(k+1)}(1-q)^{k}\right).\end{align}

We point out that the asymptotic formula in (3.2) can be replaced by $F^{\leftarrow}\left(1-w_2(1-q)^k\right)$ for any $w_2 \gt 0$ due to $F^{\leftarrow}\left(1-\cdot\right) \in \mathcal{R}_{0}(0{+})$ and the asymptotic formula in (3.3) can be replaced by $\hat{x}-F^{\leftarrow}\left(1-w_2(1-q)^k\right)$ for any $w_2 \gt 0$ due to $\hat{x}-F^{\leftarrow}\left(1-\cdot\right) \in \mathcal{R}_{0}(0{+})$ .

Finally, we consider the Weibull case. Notice that $E\left[X_{+}^{k}\right] \lt \infty$ for $k \geq 1$ holds automatically due to $\hat{x} \lt \infty$ .

Theorem 3.3. Consider the risk measure CoHM defined in (2.1) for some $k\geq1$ with a bivariate risk vector $(X,Y).$ If Assumption 2.1 holds and $F \in \mathrm{MDA}\left(\Psi_{\gamma}\right)$ for $\gamma \gt 0$ and $0 \lt \hat{x} \lt \infty$ , then as $q \uparrow 1,$ we have

(3.4) \begin{align}\hat{x}-\mathrm{CoHM}_{p, q}\left(X\big|Y\right) \sim C_p^{-1/ \gamma}\frac{\gamma k^{(k-1)/\gamma}}{(\gamma+k)^{(k/\gamma)+1}}\mathrm{B}(\gamma+1, k)^{-1/ \gamma} (\hat{x}-F^{\leftarrow}(1-(1-q)^k).\end{align}

4. Numerical illustrations

In this section, we extend our main results of the CoHM risk measure to a special case of the CoHG risk measure proposed by Xun et al. (2021), which generalizes the CoES. Following the proofs of our main results, asymptotic results for the CoHG are easily derived when the primary risk X belongs to the Fréchet, Gumbel, or Weibull MDA. We then examine the accuracy of the asymptotic estimates and conduct various sensitivity tests against a certain risk parameter.

4.1. The CoHG risk measure

Recall that a normalized Young function $\varphi({\cdot})$ is nonnegative and convex on $[0,\infty)$ and satisfies $\varphi(0)=0$ , $\varphi(1)=1$ , $\varphi(\infty)=\infty$ . $L_{0}^{\varphi}$ denotes the Orlicz heart associated with the Young function $\varphi({\cdot})$ , defined to be $\{X\,:\, E\left[\varphi(c X_+)\right] \lt \infty \text { for any } c \gt 0\}$ . For a real-valued random variable $X \in L_{0}^{\varphi}$ , distributed by $F=1-\overline{F}$ on $\mathbb{R}$ and a parameter $q\in(0,1)$ , its HG risk measure is defined as:

\begin{equation*}\mathrm{HG}_{q}(X)=\inf _{-\infty \lt x \lt \hat{x}} \left\{x+\mathrm{HG}_{q}\left((X-x)_{+}\right)\right\},\end{equation*}

where $\mathrm{HG}_{q}\left((X-x)_{+}\right)$ is the unique solution h to the equation:

\begin{equation*}E\left[\varphi\left(\frac{(X-x)_{+}}{h}\right)\right]=1-q,\end{equation*}

in case $\overline{F}(x) \gt 0$ and $\mathrm{HG}_{q}\left((X-x)_{+}\right)=0$ otherwise. In this risk measure, the convexity of the Young function $\varphi({\cdot})$ represents the degree of risk aversion and the parameter q represents the confidence level. A straightforward extension from the HG to the CoHG is proposed by Xun et al. (2021) as follows:

Definition 4.1 Given $X\in L_{0}^{\varphi}$ and Y, two parameters p, $q \in(0,1)$ , and a Young function $\varphi({\cdot})$ , the CoHG risk measure of X given $Y \gt \mathrm{VaR}_p(Y)$ is defined as:

\begin{equation*}\mathrm{CoHG}_{p, q}\left(X\big|Y\right)=\inf _{-\infty \lt x \lt \hat{x}} \left\{x+\mathrm{HG}_{p, q}\left((X-x)_{+}\big| Y\right)\right\},\end{equation*}

where $\mathrm{HG}_{p, q}\left((X-x)_{+}\big| Y\right)$ is the unique solution h to the equation:

\begin{equation*}E\left[\left. \varphi\left(\frac{(X-x)_{+}}{h}\right) \right|Y \gt y_p\right]=1-q,\end{equation*}

in case $P(X \gt x\big|Y \gt y_p) \gt 0$ and $\mathrm{HG}_{p, q}\left((X-x)_{+}\big| Y\right)=0$ otherwise, with $y_p=G^{\leftarrow}(p)$ .

We consider a special case that a Young function $\varphi(t)$ is restricted to $t^{k}$ for $k \geq 1$ . Note that, under this assumption, $L_{0}^{\varphi}$ and $L^{\varphi}$ coincide, given by $\{X\,:\, E[X_+^{k}] \lt \infty\}$ . Similar to the CoHM risk measure, for $k=1$ , the CoHG reduces to the CoES. For $k \gt 1$ , the CoHG risk measure is given by:

(4.1) \begin{align}\mathrm{CoHG}_{p, q}\left(X\big|Y\right)=x+\left(\frac{E\left[\left(X-x\right)_{+}^{k} \big| Y \gt y_p\right]}{1-q}\right)^{\frac{1}{k}},\end{align}

where $x \in\left({-}\infty, \hat{x}\right)$ is the unique solution of the equation:

(4.2) \begin{align}\frac{\left(E\left[(X-x)_{+}^{k-1}\big|Y \gt y_p\right]\right)^{k}}{\left(E\left[(X-x)_{+}^{k}\big|Y \gt y_p\right]\right)^{k-1}}=1-q.\end{align}

A numerical method has to be involved since solving Equation (4.2) for (4.1) analytically might be impossible. In that case, asymptotics for the CoHG risk measure can reduce computational complexity while ensuring accuracy. Following similar arguments from Theorems 3.1–3.3 with corresponding modifications mentioned in Section 2.2, we derive the following:

Corollary 4.1. Consider the risk measure CoHG in Definition 4.1 with a power young function $\varphi(t)=t^{k}$ for $k \geq 1$ with a bivariate risk vector (X, Y) satisfying Assumption 2.1.

1. (a) If $F \in \mathrm{MDA}\left(\Phi_{\gamma}\right)$ for some $\gamma \gt k$ , then as $q \uparrow 1$

   (4.3) \begin{align}\mathrm{CoHG}_{p, q}\left(X\big|Y\right)\sim C_p^{1 / \gamma} \frac{\gamma(\gamma-k)^{(k / \gamma)-1}}{k^{(k-1) / \gamma}} \,\mathrm{B}(\gamma-k, k)^{1 / \gamma} F^{\leftarrow}(q),\end{align}

2. (b) If $F \in \mathrm{MDA}(\Lambda)$ with $0 \lt \hat{x}\leq \infty$ , then as $q \uparrow 1$

   1. (i) for $\hat{x}=\infty$ we have

      (4.4) \begin{align}\mathrm{CoHG}_{p, q}\left(X\big|Y\right)\sim F^{\leftarrow}\left(1-C_p^{-1}\frac{k^{k}}{\Gamma(k+1)}(1-q)\right);\end{align}

   2. (ii) for $\hat{x} \lt \infty$ we have

      (4.5) \begin{align}\hat{x}-\mathrm{CoHG}_{p, q}\left(X\big|Y\right)\sim \hat{x}-F^{\leftarrow}\left(1-C_p^{-1} \frac{k^{k}}{\Gamma(k+1)}(1-q)\right).\end{align}

3. (c) If $F \in \mathrm{MDA}\left(\Psi_{\gamma}\right)$ for $\gamma \gt 0$ and $0 \lt \hat{x} \lt \infty$ , then as $q \uparrow 1$

   (4.6) \begin{align}\hat{x}-\mathrm{CoHG}_{p, q}\left(X\big|Y\right) \sim C_p^{-1/ \gamma}\frac{\gamma k^{(k-1)/\gamma}}{(\gamma+k)^{(k/\gamma)+1}\mathrm{B}(\gamma+1, k)^{-1/ \gamma} } (\hat{x}-F^{\leftarrow}(q)).\end{align}

In the remaining parts of this section, we first examine the accuracy of the asymptotic estimates obtained from (4.3)–(4.6) and then conduct sensitivity tests. Under the specification of bivariate risks (X, Y) satisfying Assumption 2.1 via an FGM copula in (2.9) with $r\in ({-}1,1)$ , the following relations are useful for calculating exact and asymptotic values:

(4.7) \begin{align}&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! P(X \gt x\big|Y \gt y_p)=\overline{F}(x)\cdot (1+rpF(x)), \\[-25pt] \nonumber \end{align}

(4.8) \begin{align}&\!\!\!P(X=x\big|Y \gt y_p)= f(x)\cdot (rp-1-2rpF(x)),\\[-25pt]\nonumber\end{align}

(4.9) \begin{align}&P(Y_h \gt y_p) = 1-p+rp-rp^{2} =(1-p)(rp+1).\end{align}

4.2. Examination of accuracy

For the first task, we compare the asymptotic estimates with their corresponding exact values. For this purpose, denote by $\mathrm{\widetilde{CoHG}}_{p,q}\left(X\big|Y\right)$ the asymptotic results for $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ obtained in Corollary 4.1. The relative error between $\mathrm{\widetilde{CoHG}}_{p,q}\left(X\big|Y\right)$ and $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ can be expressed by:

\begin{align*}\frac{\left| \mathrm{CoHG}_{p,q}\left(X\big|Y\right)-\mathrm{\widetilde{CoHG}}_{p,q}\left(X\big|Y\right) \right|}{\mathrm{CoHG}_{p,q}\left(X\big|Y\right)}\,=\!:\,r(q). \end{align*}

Example 4.1. (The Fréchet case) Assume X follows a Pareto distribution:

(4.10) \begin{align}F(x) = 1-\left(\frac{\kappa}{\kappa+x}\right)^{\alpha}, \quad x, \alpha,\kappa \gt 0,\end{align}

indicating that $F \in \mathrm{MDA}\left(\Phi_{\gamma}\right)$ with $\gamma = \alpha$ . On the one hand, we start the computation of the exact value from the left-hand side of Equation (4.2). By applying (4.7), we see that

\begin{align*}\begin{aligned}\frac{\left(E\left[(X-x)_{+}^{k-1}\big|Y \gt y_p\right]\right)^{k}}{\left(E\left[(X-x)_{+}^{k}\big|Y \gt y_p\right]\right)^{k-1}} &= \frac{\left( \int_{x}^{\infty} (k-1)(z-x)^{k-2} \cdot P(X \gt z\big|Y \gt y_p) \,\mathrm{d}z \right)^{k}}{\left(\int_{x}^{\infty} k(z-x)^{k-1} \cdot P(X \gt z\big|Y \gt y_p) \,\mathrm{d}z \right)^{k-1}} \\&= \frac{\left( \int_{x}^{\infty} (k-1)(z-x)^{k-2} \cdot \overline{F}(z)(1+rpF(z)) \,\mathrm{d}z \right)^{k}}{\left(\int_{x}^{\infty} k(z-x)^{k-1} \cdot \overline{F}(z)(1+rpF(z)) \,\mathrm{d}z \right)^{k-1}}.\end{aligned} \end{align*}

We then solve (4.2) by vpasolve and plug the solution into (4.1) to have the exact value of $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ . On the other hand, we use Corollary 4.1 (a) together with (4.9) to approximate it asymptotically.

Figure 2 shows that the relative error r(q) converges to 0 as $q \uparrow 1$ , indicating the asymptotic results are more accurate at a high confidence level.

Figure 2. Comparison of the asymptotic values with the exact values of $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ and the relative errors r(q) in Example 4.1.

Example 4.2 (The Gumbel case) Assume X follows a lognormal distribution:

\begin{align*}f(x)=\frac{1}{\sqrt{2 \pi} \sigma x} \exp \left\{-\frac{(\ln x-\mu)^{2}}{2 \sigma^{2}}\right\},\quad x \gt 0, \mu\in\mathbb{R}, \sigma \gt 0, \end{align*}

or it can be alternatively written as $F(x)=N\left((\mathrm{ln}\,x-\mu)/\sigma\right)$ . Notice that the auxiliary function is given by:

\begin{align*}a(x)=\frac{\overline{N}\left((\mathrm{ln}x-\mu)/\sigma \right)\sigma x}{N^{\prime}\left((\mathrm{ln}x-\mu)/\sigma \right)}. \end{align*}

$N({\cdot})$ is the standard normal distribution function and $N^{\prime}({\cdot})$ is its corresponding density function; see details in Page 150 of Embrechts et al. (1997). On the one hand, we can compute the solution of (4.2) by vpasolve in the same way as in the Fréchet case and substitute it into (4.1) to get the exact value of $ \mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ . On the other hand, similar to (A.9) and (A.10), we obtain

(4.11) \begin{align}\mathrm{CoHG}_{p, q}\left(X\big|Y\right)-x =\left(\frac{E\left[\left(X-x\right)_{+}^{k} \big| Y \gt y_p\right]}{1-q}\right)^{1/k} \sim ka(x),\end{align}

and

(4.12) \begin{eqnarray}x & \sim & F^{\leftarrow}\left(1-C_p^{-1} \frac{k^{k}}{\Gamma(k+1)}(1-q)\right).\end{eqnarray}

We then use (4.9), (4.11), and (4.12) to asymptotically approximate $x+ka(x)$ .

Figure 3 presents that the relative error r(q) also converges to 0 as $q \uparrow 1$ , which implies that the asymptotic results are more accurate at a high confidence level.

Figure 3. Comparison of the asymptotic values with the exact values of $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ and the relative errors r(q) in Example 4.2.

Example 4.3 (The Weibull case) Assume X follows a beta distribution:

\begin{align*}f(x)=\frac{x^{a-1}(1-x)^{b-1}}{\mathrm{B}(a,b)}, \quad 0 \lt x \lt 1, a,b \gt 0, \end{align*}

indicating that $F \in \mathrm{MDA}(\Lambda)$ with $\gamma = b$ . On the one hand, we start the computation of the exact value from the left-hand side of (4.2). By applying (4.8), we see that

\begin{align*}\begin{aligned}\frac{\left(E\left[(X-x)_{+}^{k-1}\big|Y \gt y_p\right]\right)^{k}}{\left(E\left[(X-x)_{+}^{k}\big|Y \gt y_p\right]\right)^{k-1}} &= \frac{\left(\int_{x}^{1} (z-x)^{k-1} \cdot P(X=z\big|Y \gt y_p) \,\mathrm{d}z \right)^{k}}{\left(\int_{x}^{1} (z-x)^{k} \cdot P(X=z\big|Y \gt y_p) \,\mathrm{d}z \right)^{k-1}} \\&= \frac{\left(\int_{x}^{1} (z-x)^{k-1} \cdot f(z)(rp-1-2rpF(z)) \,\mathrm{d}z \right)^{k}}{\left(\int_{x}^{1} (z-x)^{k} \cdot f(z)(rp-1-2rpF(z)) \,\mathrm{d}z \right)^{k-1}}.\end{aligned} \end{align*}

We then solve (4.2) by vpasolve and plug the solution into (4.1) to have the exact value of $\hat{x}-\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ . On the other hand, we use Corollary 4.1 (c) together with (4.9) to approximate it asymptotically.

In Figure 4 , the relative error r(q) still converges to 0 as $q \uparrow 1$ , which reveals that the asymptotic results are more accurate at a high confidence level.

Figure 4. Comparison of the asymptotic values with the exact values of $\mathrm{CoHG}_{p,q}\left(X\big|Y\right)$ and the relative errors r(q) in Example 4.3.

4.3. Sensitivity tests

In the spirit of some risk contribution measures such as the $\Delta \mathrm{CoVaR}$ in Adrian and Brunnermeier (2016) and the $\Delta \mathrm{CoES}$ in Fang and Li (2018) and Sordo et al. (2018), Xun et al. (2021) define the risk contribution of a given CoHG risk measure as:

\begin{align*}\Delta \mathrm{CoHG}_{p,q}\left(X\big|Y\right) = \mathrm{CoHG}_{p,q}\left(X\big|Y\right) - \mathrm{HG}_{q}\left(X\right), \qquad p,q \in (0,1). \end{align*}

For the second task, we conduct sensitivity tests of asymptotic estimate for the $\Delta \mathrm{CoHG}$ , denoted by $\widetilde{\Delta \mathrm{CoHG}}_{p,q}\left(X\big|Y\right)=\mathrm{\widetilde{CoHG}}_{p,q}\left(X\big|Y\right)-\mathrm{\widetilde{HG}}_{q}\left(X\right)$ , which is based on the asymptotic estimates for the CoHG in Corollary 4.1 and that of the HG in Theorems 4.1–4.3 of Tang and Yang (2012). Table 1 summarizes percentage changes in the risk contribution $\widetilde{\Delta \mathrm{CoHG}}$ with respect to percentage changes in the parameters $\alpha$ , $\kappa$ in a Pareto distribution given by (4.10), and r in an FGM copula of form (2.9) for $q = 99.5\%, 99.6\%$ and $99.7\%$ , respectively. We set $\alpha=2.6$ , $ \kappa=1.6$ , and $r=0.48$ as the benchmark. Other parameters are consistent with Example 4.1. The table shows that the risk contribution $\widetilde{\Delta \mathrm{CoHG}}$ increases as $\alpha$ decreases or as $\kappa$ increases, which is expected due to the heavier tail of the primary risk X. Additionally, it can be seen that the $\widetilde{\Delta \mathrm{CoHG}}$ increases as r increases due to a stronger positive dependency between X and Y. On the other hand, it shows that the sensitivity of the $\widetilde{\Delta \mathrm{CoHG}}$ to $\kappa$ and r remains unchanged for different levels of q, but the $\widetilde{\Delta \mathrm{CoHG}}$ is increasingly sensitive to changes in $\alpha$ as q increases, since in this example, changes in $\alpha$ affect the $(1-q)^{-1/\alpha}$ component in the $\widetilde{\Delta \mathrm{CoHG}}$ , which is predictable from the asymptotic results.

Table 1. Sensitivity tests on $\widetilde{\Delta \mathrm{CoHG}}_{p,q}\left(X\big|Y\right)$ with $k=1.5$ and $p=0.97$ against $\alpha $ , $\kappa$ , and r (with the benchmark $\alpha=2.6$ , $\kappa=1.6$ , and $r=0.48$ ), where the asymptotic estimate is based on Corollary 4.1 and Theorems 4.1–4.3 of Tang and Yang (2012).

5. Refinements and simulation study

In this section, we perform simulation to briefly examine the finite sample performance of two different estimators for the CoHM risk measure when F is in the Fréchet MDA. In doing so, we can calculate the exact value of $\mathrm{CoHM}_{p,q}\left(X\big|Y\right)$ similarly in Example 4.1.

Based on a random sample $(X_1,Y_1)$ , $\ldots$ , $(X_n,Y_n)$ , we now construct the estimation of $\mathrm{CoHM}_{p,q}\left(X\big|Y\right)$ , where $q = q(n) \rightarrow 1$ as $n \rightarrow \infty$ in two approaches. A natural nonparametric estimator based on (2.3) and (2.4) is

\begin{align*}\mathrm{\widehat{CoHM}}^{emp}_{p,q}\left(X\big|Y\right) = \hat{x}^{\ast}+\frac{1}{1-q} \left( \frac{1}{n-\lfloor np \rfloor} \sum_{i=1}^n \left(X_i-\hat{x}^{\ast}\right)_{+}^k 1_{\left(Y_i \gt Y_{\lfloor np \rfloor, n}\right)} \right)^{1/k}, \end{align*}

where $\hat{x}^{\ast}$ , the empirical estimator for $x^{\ast}$ , is the unique solution to the empirical equation:

\begin{align*}\frac{\left(\frac{1}{n-\lfloor np \rfloor} \sum_{i=1}^n \left(X_i-\hat{x}^{\ast}\right)_{+}^{k-1} 1_{\left(Y_i \gt Y_{\lfloor np \rfloor, n}\right)}\right)^{\frac{1}{k-1}}}{\left(\frac{1}{n-\lfloor np \rfloor} \sum_{i=1}^n \left(X_i-\hat{x}^{\ast}\right)_{+}^k 1_{\left(Y_i \gt Y_{\lfloor np \rfloor, n}\right)}\right)^{\frac{1}{k}}}=(1-q)^{\frac{1}{k-1}}. \end{align*}

Here, $Y_{1, n} \leq Y_{2, n} \leq \cdots \leq Y_{n, n}$ are the order statistics of $\left\{Y_1, Y_2, \ldots, Y_n\right\}$ . Alternatively, exploiting the asymptotics in Theorem 3.1, we propose another estimator of $\mathrm{CoHM}_{p,q}\left(X\big|Y\right)$ via an extrapolation. This is an analog of the estimation of extreme quantiles in Chapter 4 of de Haan and Ferreira (2006). More precisely, with $m =m(n)$ a sequence such that $m \rightarrow \infty$ and $\frac{m}{n} \rightarrow 0$ as $n \rightarrow \infty$ , by (3.1) we have

\begin{align*}\mathrm{CoHM}_{p,q}\left(X\big|Y\right) \sim \left( \frac{m}{n(1-q)} \right)^\frac{k}{\gamma} \mathrm{CoHM}_{p,1-\frac{m}{n}}\left(X\big|Y\right). \end{align*}

Thus, we estimate $\mathrm{CoHM}_{p,q}\left(X\big|Y\right)$ by extrapolating the nonparametric estimator at level $(1-\frac{m}{n})$ to level q as $n \rightarrow \infty$ ,

\begin{align*}\mathrm{\widehat{CoHM}}_{p,q}\left(X\big|Y\right) =\left( \frac{m}{n(1-q)} \right)^\frac{k}{\hat{\gamma}} \mathrm{\widehat{CoHM}}^{emp}_{p,1-\frac{m}{n}}\left(X\big|Y\right). \end{align*}

where $\hat{\gamma}$ is the Hill estimator for $\gamma$ , namely

\begin{align*}\hat{\gamma}=\frac{1}{m_1} \sum_{i=1}^{m_1} \log X_{n-i+1, n}-\log X_{n-m_1, n} \end{align*}

with an intermediate sequence $m_1=m_1(n) \rightarrow \infty$ and $\frac{m_1}{n} \rightarrow 0$ as $n \rightarrow \infty$ .

Similar to Example 4.1, we assume both X and Y follow the Pareto distribution given by (4.10) and they are dependent via the FGM copula of form (2.9) with parameter $r_3$ or the Clayton copula of form (2.10) with parameter $r_4$ . We conduct 1000 replications with the sample size $n=500,000$ . By adjusting $k=1.35$ , $1.5$ , $1.65$ , and $1.8$ in both examples, we compute the estimators $\mathrm{\widehat{CoHM}}^{emp}_{p,q}\left(X\big|Y\right)$ for $q=99.9\%$ and $\mathrm{\widehat{CoHM}}_{p,q}\left(X\big|Y\right)$ for different $q=99.9\%$ , $99.98\%$ , and $99.99\%$ correspondingly. The comparison of the estimators is shown in Figure 5 for the FGM copula and in Figure 6 for the Clayton copula, with boxplots of the ratios of the estimates and the exact values. In all cases of different values of k, our estimator $\mathrm{\widehat{CoHM}}_{p,q_1}\left(X\big|Y\right)$ using extrapolation outperforms the empirical estimator $\mathrm{\widehat{CoHM}}^{emp}_{p,q_1}\left(X\big|Y\right)$ at level $q_1= 99.9\%$ as it has a smaller variance and similar or even smaller bias. When the confidence level q becomes extreme, our estimator $\mathrm{\widehat{CoHM}}_{p,q_1}\left(X\big|Y\right)$ still provides good performance with relatively stable behavior.

Figure 5. (X,Y) follows an FGM copula with identical Pareto marginals ( $r_3=0.48$ , $\alpha=2.2$ , $\kappa=1.2$ , $p=0.01$ ). Boxplot of ratios of estimates and exact values with $k=1.35$ , $1.5$ , $1.65$ , and $1.8$ at level $q_1=99.9\%$ , $q_2=99.98\%$ , and $q_3=99.99\%$ .

Figure 6. (X,Y) follows a Clayton copula with identical Pareto marginals ( $r_4=1$ , $\alpha=2.5$ , $\kappa=1$ , $p=0.01$ ). Boxplot of ratios of estimates and exact values with $k=1.35$ , $1.5$ , $1.65$ , and $1.8$ at level $q_1=99.9\%$ , $q_2=99.98\%$ , and $q_3=99.99\%$ .

6. Concluding remarks

In this paper, we conduct an asymptotic study of the CoHM coherent risk measure in the presence of extreme risks under the weak contagion. As our main results, we derive the following asymptotic relations as the underlying confidence level becomes high:

• • If F is in the Fréchet MDA, $\mathrm{CoHM}_{p, q}(X|Y) \sim w_{1} F^{\leftarrow}\left(1- (1-q)^{k}\right)$ ;

• • If F is in the Gumbel MDA, $\mathrm{CoHM}_{p, q}(X|Y)\sim F^{\leftarrow}\left(1- w_{2} (1-q)^{k}\right)$ provided $\hat{x}=\infty$ or $\hat{x}-\mathrm{CoHM}_{p, q}(X|Y) \sim \hat{x}-F^{\leftarrow}\left(1- w_{2}(1-q)^{k}\right)$ provided $\hat{x} \lt \infty$ ;

• • If F is in the Weibull MDA, $\hat{x}-\mathrm{CoHM}_{p, q}(X|Y) \sim w_{3}\left(\hat{x}-F^{\leftarrow}\left(1-(1-q)^{k}\right)\right).$

The coefficients $w_{1}, w_{2}$ , and $w_{3}$ are explicitly given in Theorems 3.1–3.3. We conclude from above that the heavier the tail of the primary risk X, the faster $\mathrm{CoHM}_{p, q}(X|Y)$ will approach $\infty$ (for unbounded risk) or $\widehat{x}$ (for bounded risk): When F is unbounded, $\mathrm{CoHM}_{p, q}(X|Y)$ goes to $\infty$ as $q \uparrow 1$ and is asymptotically proportional to $F^{\leftarrow}\left(1-(1-q)^{k}\right)$ , which is regularly varying with index $-1/\gamma$ in the Fréchet case and is slowly varying in the Gumbel case. When F is bounded, $\hat{x}-\mathrm{CoHM}_{p, q}(X|Y)$ decays to 0 as $q \uparrow 1$ and is asymptotically proportional to $\hat{x}-F^{\leftarrow}\left(1- (1-q)^{k}\right)$ , which is slowly varying in the Gumbel case and is regularly varying with index $1/\gamma$ in the Weibull case. Moreover, the asymptotic proportionality constants $w_1$ and $w_3$ depend not only on the marginal distribution of X but also on the dependence structure of two risks. The above results can be further extended to the approximation of the CoHG risk measure under the high confidence level, with accuracy examined by numerical examples. As an application, we also demonstrate through the numerical examples that the asymptotic approximation can be easily applied to the sensitivity test of the risk contribution based on the HG risk measure. A simulation and comparison study is implemented to investigate the finite sample performance of our estimator using an extrapolation based on the asymptotic relation when the primary risk X belongs to the Fréchet MDA.

The following three extensions of our work are desired. The first is the case of the strong contagion, where an ideal modeling framework would be the multivariate regular variation that integrates heavy-tailedness and tail dependence. The second extension is to study the convolution-based risk measure, for example, see, Theorem 1 of Krokhmal (2007), which is a general case of the CoHM risk measure. The convolution-based risk measure promises an important role in both stochastic optimization and risk management. An asymptotic study of such convolution-based risk measures would be very interesting by applying a unified approach covering the Fréchet, Gumbel, and Weibull MDA. The work of Tang and Yang (2014) sheds some light on research in this direction. In addition, for the estimator of the CoHM risk measure via an extrapolation we proposed in Section 5, we remark that its statistical properties including consistency and asymptotic normality could be further studied in a future project.