# Some Results on Measures of Interaction between Paired Risks

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## Abstract

**:**

## 1. Introduction

## 2. Some Preliminaries

`Stochastic orders`

**Definition**

**1.**

- usual stochastic order, denoted as $X{\le}_{\mathrm{st}}Y$, if ${F}_{X}\left(t\right)\ge {F}_{Y}\left(t\right)$ for all $t\in \mathbb{R}$;
- increasing convex order, denoted as $X{\le}_{\mathrm{icx}}Y$, if $\mathrm{E}\left[\phi \right(X\left)\right]\le \mathrm{E}\left[\phi \right(Y\left)\right]$ for any increasing convex function φ, provided the expectations exit;
- dispersive order, denoted as $X{\le}_{\mathrm{disp}}Y$, if ${F}_{X}^{-1}\left(p\right)-{F}_{X}^{-1}\left(q\right)\le {F}_{Y}^{-1}\left(p\right)-{F}_{Y}^{-1}\left(q\right)$ for all $p,q$ such that $0<q<p<1$;
- excess wealth order, denoted as $X{\le}_{\mathrm{ew}}Y$, if $\mathrm{E}[max\{X-{F}_{X}^{-1}\left(p\right),0\}]\le \mathrm{E}[max\{Y-{F}_{Y}^{-1}\left(p\right),0\}]$ for all $p\in (0,1)$.

**Lemma**

**1.**

`Co-risk measures`

**Definition**

**2.**

- the CoVaR of Y at stress level β given that X is under stress at level α is$${\mathrm{CoVaR}}_{\alpha ,\beta}\left(Y\right|X)={\mathrm{VaR}}_{\beta}\left(Y\right|X>{\mathrm{VaR}}_{\alpha}\left(X\right));$$
- the CoES of Y at stress level β given that X is under stress at level α is$${\mathrm{CoES}}_{\alpha ,\beta}\left(Y\right|X)=\frac{1}{1-\beta}{\int}_{\beta}^{1}{\mathrm{CoVaR}}_{\alpha ,t}\left(Y\right|X)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t.$$

`Risk contribution measures`

**Definition**

**3.**

- the ΔCoVaR of Y at stress level β given that X is under stress at level α is$$\Delta {\mathrm{CoVaR}}_{\alpha ,\beta}\left(Y\right|X)={\mathrm{CoVaR}}_{\alpha ,\beta}\left(Y\right|X)-{\mathrm{VaR}}_{\beta}\left(Y\right);$$
- the ΔCoES of Y at stress level β given that X is under stress at level α is$$\Delta {\mathrm{CoES}}_{\alpha ,\beta}\left(Y\right|X)={\mathrm{CoES}}_{\alpha ,\beta}\left(Y\right|X)-{\mathrm{ES}}_{\beta}\left(Y\right).$$

`Statistical dependence`

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**2.**

`Arrangement monotonicity`

## 3. Co-Risk Measures

**Theorem**

**1.**

- if $C(u,v)$ is symmetric, then $X{\le}_{\mathrm{st}}Y$ is equivalent to (2) for $\alpha ,\beta \in (0,1)$,
- if $C(u,v)$ is AI, then $X{\le}_{\mathrm{st}}Y$ implies (2) for $\alpha ,\beta \in (0,1)$ such that $\beta \ge [\alpha -C(\alpha ,\alpha \left)\right]/(1-\alpha )$, and
- if $C(u,v)$ is AD, then $X{\le}_{\mathrm{st}}Y$ implies (2) for $\alpha ,\beta \in (0,1)$ such that $\beta \le [\alpha -C(\alpha ,\alpha \left)\right]/(1-\alpha )$.

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Risk Contribution Measures

**Theorem**

**3.**

- symmetric, then $X{\le}_{\mathrm{disp}}Y$ implies (7) for $\alpha ,\beta \in (0,1)$,
- AI, then $X{\le}_{\mathrm{disp}}Y$ implies (7) for $\alpha ,\beta \in (0,1)$ such that $\beta \ge [\alpha -C(\alpha ,\alpha \left)\right]/(1-\alpha )$, and
- AD, then $X{\le}_{\mathrm{disp}}Y$ implies (7) for $\alpha ,\beta \in (0,1)$ such that $\beta \le [\alpha -C(\alpha ,\alpha \left)\right]/(1-\alpha )$.

**Proof.**

**Theorem**

**4.**

**Proof.**

## 5. Numerical Examples Based on Simulation

- For each stress level $\alpha =\frac{1}{101},\cdots ,\frac{100}{101}$ and $i=1,\cdots ,{10}^{5}$, generate a sample of ${10}^{5}$ observations ${v}_{i,\alpha}$ of $\left(V\right|U>\alpha )$ and a sample of ${10}^{5}$ observations ${u}_{i,\alpha}$ of $\left(U\right|V>\alpha )$, respectively.
- For $\alpha =\frac{1}{101},\cdots ,\frac{100}{101}$, based on ${v}_{i,\alpha}$’s and ${u}_{i,\alpha}$’s calculate respectively the adjusted empirical distribution functions$$\widehat{F}\left(t\right)=\frac{1}{{10}^{5}+1}\sum _{i=1}^{{10}^{5}}\mathrm{I}({v}_{i,\alpha}\le t)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\widehat{G}\left(t\right)=\frac{1}{{10}^{5}+1}\sum _{i=1}^{{10}^{5}}\mathrm{I}({u}_{i,\alpha}\le t).$$
- At each stress level $\alpha =\frac{1}{101},\cdots ,\frac{100}{101}$, for each stress level $\beta =\frac{1}{101},\cdots ,\frac{100}{101}$, employ the sample $\beta $th-quantiles$${\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)=inf\left\{t:\widehat{F}\left(t\right)\ge \alpha \right\}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)=inf\left\{t:\widehat{G}\left(t\right)\ge \alpha \right\}$$$${\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(Y\right|X)=\frac{1}{(1+{10}^{5})(1-\beta )}\sum _{i=1}^{{10}^{5}}\mathrm{I}\left({v}_{i,\alpha}>{\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)\right),$$$${\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(X\right|Y)=\frac{1}{(1+{10}^{5})(1-\beta )}\sum _{i=1}^{{10}^{5}}\mathrm{I}\left({u}_{i,\alpha}>{\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)\right),$$
- As for the risk contribution measures $\Delta \mathrm{CoVaR}$ and $\Delta \mathrm{CoES}$, the following empirical estimators are used.$${\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)={\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)-{\mathrm{VaR}}_{\beta}\left(Y\right),$$$${\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)={\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)-{\mathrm{VaR}}_{\beta}\left(X\right),$$$${\widehat{\Delta \mathrm{CoES}}}_{\alpha ,\beta}\left(Y\right|X)={\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(Y\right|X)-{\mathrm{ES}}_{\beta}\left(Y\right),$$$${\widehat{\Delta \mathrm{CoES}}}_{\alpha ,\beta}\left(X\right|Y)={\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(X\right|Y)-{\mathrm{ES}}_{\beta}\left(X\right).$$It should be remarked here that to mitigate the approximation error we simply use the population version for the marginal VaR and ES when deriving these estimators.

- For $X\sim \mathcal{E}\left(2\right)$ and $Y\sim \mathcal{E}\left(1\right)$, exponentially distributions with parameters 2 and 1, respectively, it is plain that $X{\le}_{\mathrm{st}}Y$. By the second assertion of Theorem 1, we have$${\mathrm{CoVaR}}_{\alpha ,\beta}\left(Y\right|X)\ge {\mathrm{CoVaR}}_{\alpha ,\beta}\left(X\right|Y),\phantom{\rule{1.em}{0ex}}\mathrm{for}\mathrm{all}\beta \ge [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha ).$$Figure 3a plots ${\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)-{\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)$. The difference surface is seen to be always above the horizontal surface on the region $\{(\alpha ,\beta )\in {[0,1]}^{2}:\beta \ge [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )\}$, and this confirms the finding of Theorem 1. On the other hand, although the copula ${C}_{\mathbf{\theta}}(u,v)$ fails to satisfy condition in the third assertion of Theorem 1, the surface above the horizontal surface on the region $\{(\alpha ,\beta )\in {[0,1]}^{2}:\beta \le [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )\}$ in Figure 3a hints that the third assertion of Theorem 1 may still be true when the requirement on dependence structure is violated.
- For $X\sim \mathcal{N}(0,1)$ and $Y\sim \mathcal{N}(0,2)$, two normal distributions, according to Table 1.1 of (Müller and Stoyan 2002), $X{\le}_{\mathrm{icx}}Y$ is valid. By the second assertion of Theorem 2, ${\mathrm{CoES}}_{\alpha ,\beta}\left(Y\right|X)\ge {\mathrm{CoES}}_{\alpha ,\beta}\left(X\right|Y)$ for $\beta \ge [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )$. This is illustrated by the surface of Figure 3b. Actually, the corresponding difference is still nonnegative when $\beta \le [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )$.
- For $X\sim \mathcal{W}(1,1)$ and $Y\sim \mathcal{W}(1.5,1)$, two Weibull distributions, Example 16 of (Sordo et al. 2018) proves that $X{\le}_{\mathrm{disp}}Y$. Figure 3c plots the difference between ${\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)$ and ${\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)$, and this difference surface confirms the finding of the second assertion of Theorem 3.
- For $X\sim \mathcal{W}(1,2)$ and $Y\sim \mathcal{W}(1,1)$, two Weibull distributions, as per Example 24 of (Sordo et al. 2018), we have $X{\le}_{\mathrm{ew}}Y$. Under this setting, the second assertion of Theorem 4 proves that $\Delta {\mathrm{CoES}}_{\alpha ,\beta}\left(Y\right|X)\ge \Delta {\mathrm{CoES}}_{\alpha ,\beta}\left(X\right|Y)$ for $\beta \ge [\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )$. Also, this fact is justified by the difference surface of Figure 3d.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Surface of ratio ${C}_{\mathbf{\theta}}(u,v)/{C}_{\mathbf{\theta}}(v,u)$. (

**a**) $0\le u\le v\le 1$; (

**b**) $0\le v\le u\le 1$.

**Figure 2.**Surfaces of second-order derivatives. (

**a**) ${\partial}^{2}{C}_{\mathbf{\theta}}(u,v)/\partial {u}^{2}$; (

**b**) ${\partial}^{2}{C}_{\mathbf{\theta}}(u,v)/\partial {v}^{2}$.

**Figure 3.**Difference surfaces (red) and $\beta =[\alpha -{C}_{\mathbf{\theta}}(\alpha ,\alpha )]/(1-\alpha )$ (green). (

**a**) ${\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)-{\widehat{\mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)$; (

**b**) ${\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(Y\right|X)-{\widehat{\mathrm{CoES}}}_{\alpha ,\beta}\left(X\right|Y)$; (

**c**) ${\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(Y\right|X)-{\widehat{\Delta \mathrm{CoVaR}}}_{\alpha ,\beta}\left(X\right|Y)$; (

**d**) ${\widehat{\Delta \mathrm{CoES}}}_{\alpha ,\beta}\left(Y\right|X)-{\widehat{\Delta \mathrm{CoES}}}_{\alpha ,\beta}\left(X\right|Y)$.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Fang, R.; Li, X.
Some Results on Measures of Interaction between Paired Risks. *Risks* **2018**, *6*, 88.
https://doi.org/10.3390/risks6030088

**AMA Style**

Fang R, Li X.
Some Results on Measures of Interaction between Paired Risks. *Risks*. 2018; 6(3):88.
https://doi.org/10.3390/risks6030088

**Chicago/Turabian Style**

Fang, Rui, and Xiaohu Li.
2018. "Some Results on Measures of Interaction between Paired Risks" *Risks* 6, no. 3: 88.
https://doi.org/10.3390/risks6030088