# Equivalent Risk Indicators: VaR, TCE, and Beyond

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## Abstract

**:**

## 1. Introduction

## 2. VaR and TCE

#### 2.1. Definitions

#### 2.2. Relation between VaR and TCE Quantiles

**Theorem**

**1.**

**Proof.**

#### 2.3. Illustration

## 3. A New High-Order TCE Indicator

#### 3.1. Definition

#### 3.2. Pareto Distributed Losses

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.3. GPD Losses

**Theorem**

**4.**

**Proof.**

- $\mu \xi -\sigma \ge 0$, to avoid the appearance of complex numbers in Equation (15).
- $\left|\frac{{\mathrm{VaR}}_{c}\phantom{\rule{4pt}{0ex}}\xi}{\mu \xi -\sigma}\right|<1$, which is a necessary property of the fourth parameter of the hypergeometric function.

## 4. Equivalence between High-Order Indicators

#### 4.1. Pareto Distributed Losses

**Proposition**

**1.**

#### 4.2. GPD Losses

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.2. Proof of Theorem 2

#### Appendix A.3. Proof of Theorem 3

#### Appendix A.4. Proof of Theorem 4

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**Figure 2.**Extended TCE quantile as a function of VaR quantile. (

**Left panel**): $m=2$ and $\theta =1$. (

**Right panel**): $m=2$ and $\theta =5$.

**Figure 3.**Extended TCE quantile as a function of VaR quantile. (

**Left panel**): $m=3$ and $\theta =1$. (

**Right panel**): $m=3$ and $\theta =5$.

**Figure 4.**Extended TCE quantile as a function of VaR quantile. (

**Left panel**): $m=2$, $\sigma =0.1$, and $\mu =-0.05$. (

**Right panel**): $m=2$, $\sigma =0.1$, and $\mu =+0.05$.

**Figure 5.**Extended TCE quantile as a function of VaR quantile. (

**Left panel**): $m=2$, $\sigma =0.1$, and $\mu =0$. (

**Right panel**): $m=2$, $\sigma =0.4$, and $\mu =0$.

**Figure 6.**Extended TCE quantile as a function of VaR quantile. (

**Left panel**): $m=3$, $\sigma =0.1$, and $\mu =0$. (

**Right panel**): $m=3$, $\sigma =0.4$, and $\mu =0$.

**Figure 7.**Extended TCE quantile at order $n=2$ as a function of extended TCE quantile at order $m=5$. (

**Left panel**): $\theta =1$. (

**Right panel**): $\theta =2$.

**Figure 8.**TCE quantile as a function of extended TCE quantile at order 2. (

**Left panel**): $\theta =1$. (

**Right panel**): $\theta =2$.

**Figure 9.**Extended TCE quantile at order $n=2$ as a function of extended TCE quantile at order $m=5$ with $\sigma =0.1$. (

**Left panel**): $\mu =-0.05$. (

**Right panel**): $\mu =+0.05$.

**Figure 10.**Extended TCE quantile at order $n=2$ as a function of extended TCE quantile at order $m=5$ with $\mu =0$. (

**Left panel**): $\sigma =0.1$. (

**Right panel**): $\sigma =0.4$.

**Figure 11.**TCE quantile as a function of extended TCE quantile at order $m=2$ with $\sigma =0.1$. (

**Left panel**): $\mu =-0.05$. (

**Right panel**): $\mu =+0.05$.

**Figure 12.**TCE quantile as a function of extended TCE quantile at order $m=2$ with $\mu =0$. (

**Left panel**): $\sigma =0.1$. (

**Right panel**): $\sigma =0.4$.

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**MDPI and ACS Style**

Faroni, S.; Le Courtois, O.; Ostaszewski, K.
Equivalent Risk Indicators: VaR, TCE, and Beyond. *Risks* **2022**, *10*, 142.
https://doi.org/10.3390/risks10080142

**AMA Style**

Faroni S, Le Courtois O, Ostaszewski K.
Equivalent Risk Indicators: VaR, TCE, and Beyond. *Risks*. 2022; 10(8):142.
https://doi.org/10.3390/risks10080142

**Chicago/Turabian Style**

Faroni, Silvia, Olivier Le Courtois, and Krzysztof Ostaszewski.
2022. "Equivalent Risk Indicators: VaR, TCE, and Beyond" *Risks* 10, no. 8: 142.
https://doi.org/10.3390/risks10080142