# The Wasserstein Metric and Robustness in Risk Management

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Suitable Distance for Risk Management

**Definition 1.**

**Definition 2.**

- 1.
- $F\left(x\right)=G\left(x\right)\phantom{\rule{0.277778em}{0ex}}for\text{}all\text{}x\in \mathbb{R}\text{}iff\text{}d(F,G)=0$,
- 2.
- $d(F,G)=d(G,F)\phantom{\rule{0.277778em}{0ex}}for\text{}all\phantom{\rule{0.277778em}{0ex}}F,G\in \overline{\mathbb{F}}$,
- 3.
- $d(F,G)\le d(F,H)+d(G,H)\phantom{\rule{0.277778em}{0ex}}for\text{}all\phantom{\rule{0.277778em}{0ex}}F,G,H\in \overline{\mathbb{F}}$.

- Does the chosen metric fit well in the context of application, i.e., has the chosen metric a natural interpretation? Is it perhaps the canonical metric for a particular field of application?
- Is the metric not too strong? This means—do we have a sufficiently rich set of continuous objects?

**Example 1.**

**Value-at-Risk.**For a fixed $0<\alpha <1$ (usually $\alpha \le 0.05$), we define the Value-at-Risk at level α (VaR${}_{\alpha}$) as$$Va{R}_{\alpha}=-{F}^{-1}\left(\alpha \right).$$$$T\left({F}_{n}\right)=-{F}_{n}^{-1}\left(\alpha \right)=-{\int}_{0}^{1}{F}_{n}^{-1}\left(\tau \right)d{\delta}_{\alpha}\left(\tau \right),$$**Tail-VaR (TVaR).**1 The statistical functional for estimating TVaR is$$\begin{array}{cc}\hfill T\left({F}_{n}\right)& =-{(1-\alpha )}^{-1}{\int}_{\alpha}^{1}{F}_{n}^{-1}\left(x\right)dx.\hfill \end{array}$$

**Definition 3.**

**Example 2.**

**Definition 4.**

**Proposition 1.**

- 1.
- For $X,Y$ random variables we have the following scaling properties of ${W}_{p}$:$$\begin{array}{cc}\hfill {W}_{p}(aX,aY)& =|a|{W}_{p}(X,Y),\phantom{\rule{1.em}{0ex}}for\text{}any\text{}scale\text{}a.\hfill \end{array}$$
- 2.
- For ${F}_{1},{F}_{2},{G}_{1},{G}_{2}$ in ${\mathsf{\Gamma}}_{2}$ and $\u03f5\in (0,1)$$$\begin{array}{c}\hfill {W}_{2}(\u03f5{F}_{1}+(1-\u03f5){F}_{2},\u03f5{G}_{1}+(1-\u03f5){G}_{2})\le \u03f5{W}_{2}({F}_{1},{G}_{1})+(1-\u03f5){W}_{2}({F}_{2},{G}_{2}).\end{array}$$$$\begin{array}{c}\hfill {W}_{2}({F}_{1}+{F}_{2},{F}_{1}+{G}_{2})\le {W}_{2}({F}_{2},{G}_{2}).\end{array}$$
- 3.
- Let ${U}_{j},{V}_{i},i,j=1,\dots ,m$ be independent and assume that the laws are in ${\mathsf{\Gamma}}_{2}$. Then,$$\begin{array}{c}\hfill {W}_{2}(\sum _{i=1}^{m}{U}_{j},\sum _{j=1}^{m}{V}_{j})\le \sum _{j=1}^{m}{W}_{2}({U}_{j},{V}_{j}).\end{array}$$
- 4.
- The Wasserstein metric is a one-ideal metric (see the Appendix A for a definition of one- ideal).

**Example 3.**

**Proposition 2.**

**Theorem 1.**

**Proof.**

**Definition 5.**

**Proposition 3.**

**Example 4.**

## 3. Application of the Wasserstein Metric in Risk Management

**Example 5.**

**Example 6.**

**Example 7.**

**Example 8.**

**Example 9.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A Ideal Metric

**Definition A1**(s-ideal metric).

- 1.
- for any rvs $X,Y,Z$ with ${F}_{X},{F}_{Y},{F}_{Z}\in \overline{\mathbb{F}}$, X and Y independent of Z,$$d(X+Z,Y+Z)\le d(X,Y),and$$
- 2.
- for any rvs $X,Y$ with ${F}_{X},{F}_{Y}\in \overline{\mathbb{F}}$ and $c\in [0,\infty )$, it holds that$$d(cX,cY)={c}^{s}d(X,Y).$$

## Appendix B Proof of Theorem 1

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**Sample Availability:**Samples of the compounds are available from the authors.

^{1.}TVaR is also referred to as Expected Shortfall (ES)^{2.}[15] calls them distribution-free^{3.}The space ${H}^{p}$ is called Orlicz heart, see [22]. Actually, as the so-called ${\Delta}_{2}$ condition is satisfied, the Orlicz heart ${H}^{p}$ coincides with ${L}^{p}$ in the current setting.^{4.}This is a consequence of the fact that the ${\Delta}_{2}$ condition is satisfied, see [16].^{5.}We restrict ourselves to measures on the real line.^{6.}of course this is due to the fact that we consider the loss in the VaR definition.

**Figure 1.**The sequence of box plots and related outliers represent distributions from eleven risk categories that are aggregated and displayed in the first box plot from above. The coloured points above the box plot for the aggregate depict the fifty worst case scenarios. These worst case scenarios are traced back to the risk categories by coloured dots. As the graphs show, these fifty worst cases for the aggregate are not necessarily located in the tail for the contributing risk categories. Hence, the risk management cannot only focus on the tails but on the whole distribution function. The Wasserstein distance fits to this need in a very particular way.

**Figure 2.**The graph depicts distribution functions that follow a stochastic dominance ordering. The distribution functions may be interpreted as those functions that describe the risk of particular portfolios or segments.

p | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${W}_{p}$ | 0.054 | 0.067 | 0.075 | 0.085 |

${\mathsf{\Theta}}_{p}$ | 0.054 | 0.039 | 0.038 | 0.038 |

Normal | Gamma | GEV | |
---|---|---|---|

${E}_{1}$ | 5.14 | 4.08 | 14.36 |

${E}_{2}$ | 4.74 | 1.55 | 1.18 |

${E}_{3}$ | 5.17 | 1.78 | 1.27 |

${E}_{4}$ | 4.95 | 1.59 | 1.60 |

${E}_{5}$ | 6.41 | 5.45 | 9.22 |

${E}_{6}$ | 9.90 | 5.92 | 1.15 |

${E}_{7}$ | 7.55 | 4.53 | 1.39 |

© 2016 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**

Kiesel, R.; Rühlicke, R.; Stahl, G.; Zheng, J. The Wasserstein Metric and Robustness in Risk Management. *Risks* **2016**, *4*, 32.
https://doi.org/10.3390/risks4030032

**AMA Style**

Kiesel R, Rühlicke R, Stahl G, Zheng J. The Wasserstein Metric and Robustness in Risk Management. *Risks*. 2016; 4(3):32.
https://doi.org/10.3390/risks4030032

**Chicago/Turabian Style**

Kiesel, Rüdiger, Robin Rühlicke, Gerhard Stahl, and Jinsong Zheng. 2016. "The Wasserstein Metric and Robustness in Risk Management" *Risks* 4, no. 3: 32.
https://doi.org/10.3390/risks4030032