# 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

## References

- M. Morini. Understanding and Managing Model Risk: A Practical Guide for Quants, Traders and Validators. New York, NY, USA: Wiley, 2011. [Google Scholar]
- J. Fouque, and J. Langsam. Handbook on Systemic Risk. Cambridge, UK: Cambridge University Press, 2013. [Google Scholar]
- P. Jorion. “Risk Management Lessons from the Credit Crisis.” Eur. Financ. Manag. 15 (2009): 923–933. [Google Scholar] [CrossRef]
- J.C. Hull. “The Credit Crunch of 2007: What Went Wrong? Why? What Lessons Can Be Learned? ” J. Credit Risk 5 (2009): 3–18. [Google Scholar] [CrossRef]
- European Parliament and Council of the European Union. “Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance.” Off. J. Eur. Union 20 (2009): 7–25. [Google Scholar]
- Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS). CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II: Articles 120 to 126 Tests and Standards for Internal Model Approval. Frankfurt am Main, Germany: CEIOPS, 2009. [Google Scholar]
- P. Barrieu, and G. Scandolo. “Assessing financial model risk.” Eur. J. Oper. Res. 242 (2014): 546–556. [Google Scholar] [CrossRef]
- T. Gneiting. “Making and evaluating point forecasts.” J. Am. Stat. Assoc. 106 (2011): 746–762. [Google Scholar] [CrossRef]
- J. Ziegel. “Coherence and elicitability.” Math. Financ., 2013. [Google Scholar] [CrossRef]
- S. Weber. “Distribution-invariant risk measures, information and dynamic consistency.” Math. Financ. 16 (2006): 419–441. [Google Scholar] [CrossRef]
- P. Bickel, and D. Freedman. “Some asymptotic theory for the bootstrap.” Ann. Stat. 9 (1981): 1196–1217. [Google Scholar] [CrossRef]
- S. Rachev, S. Stoyanov, and F. Fabozzi. Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization. New York, NY, USA: John Wiley & Sons, 2008. [Google Scholar]
- V. Krätschmer, A. Schied, and H. Zähle. “Sensitivity of risk measures with respect to the normal approximation of total claim distributions.” Insur. Math. Econ. 49 (2011): 335–344. [Google Scholar] [CrossRef]
- V. Krätschmer, A. Schied, and H. Zähle. “Qualitative and infinitesimal robustness of tail- dependent statistical functionals.” J. Multivar. Anal. 103 (2012): 35–47. [Google Scholar] [CrossRef]
- R. Cont, R. Deguest, and G. Scandolo. “Robustness and sensitivity analysis of risk measurement procedures.” Quant. Financ. 10 (2010): 593–606. [Google Scholar] [CrossRef]
- V. Krätschmer, A. Schied, and H. Zähle. “Comparative and qualitative robustness for law-invariant risk measures.” Financ. Stoch. 18 (2014): 271–295. [Google Scholar] [CrossRef]
- P. Sibbertsen, G. Stahl, and C. Luedtke. “Measuring model risk.” J. Risk Model Valid. 2 (2008): 65–81. [Google Scholar]
- M. Busse, U. Müller, and M. Dacorogna. “Robust Estimation of Reserve Risk.” ASTIN Bull. 40 (2010): 453–489. [Google Scholar]
- H. Föllmer, and T. Knipsel. “Convex Risk Measures: Basic Facts, Law-invariance and beyond, Asymptotics for Large Portfolios.” In Handbook of the Fundamentals of Financial Decision Making. Edited by L. MacLean and W. Ziemba. Singapore: World Scientific, 2013, Volume II, pp. 507–554. [Google Scholar]
- R. Cont, R. Deguest, and X. He. “Loss-based risk measures.” Stat. Decis. 30 (2013): 133–167. [Google Scholar] [CrossRef]
- C. Heyde, S. Kou, and X. Peng. What Is a Good External Risk Measure: Bridging the Gaps between Robustness, Subadditivity, and Insurance Risk Measures. New York, NY, USA: Columbia University, 2007. [Google Scholar]
- P. Cheridito, and T. Li. “Risk measures on Orlicz hearts.” Math. Financ. 19 (2009): 189–214. [Google Scholar] [CrossRef]
- D. Filipovic, and G. Svindland. “The canonical model space for law-invariant risk measures is L
^{1}.” Math. Financ. 22 (2012): 585–589. [Google Scholar] [CrossRef] - P. Artzner, F. Delbaen, J. Eber, and D. Heath. “Coherent Measures of risk.” Math. Financ. 9 (1999): 203–228. [Google Scholar] [CrossRef]
- P.J. Huber, and E.M. Ronchetti. Robust Statistics, 2nd ed. New York, NY, USA: Wiley, 2009. [Google Scholar]
- G. Dall’Aglio. “Sugli estremi dei momenti delle funzioni di ripartizione doppia.” Ann. Sc. Norm. Super. Pisa 3 (1956): 35–74. [Google Scholar]
- C.L. Mallows. “A Note on Asymptotic Joint Normality.” Ann. Math. Stat. 43 (1972): 508–515. [Google Scholar] [CrossRef]
- G. Pflug, A. Pichler, and D. Wozabal. “The 1/N investment strategy is optimal under high model ambiguity.” J. Bank. Financ. 36 (2012): 410–417. [Google Scholar] [CrossRef]
- B.R. Clarke. “A Review of Differentiability in Relation to Robustness with an Application to Seismic Data Analysis.” Proc. Indian Natl. Sic. Acad. 66 (2000): 467–482. [Google Scholar]
- P. Davies. “Approximating Data.” J. Korean Stat. Soc. 37 (2008): 191–211. [Google Scholar] [CrossRef]

**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