# Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Generalized Pareto Distribution Copula Approach

#### 3.2. Portfolio Optimization

## 4. Results and Interpretations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Elliptical Copula Parameters

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- The bivariate Gaussian copula (N)—it has no tail dependence, hence ${\tau}^{U}$ = ${\tau}^{L}$ = 0. Therefore, modeling the dependence structure of the series by a Gaussian (normal) copula is consistent with the estimation of this dependence by the linear correlation coefficient such that $-1<\rho <1$. The copula density is given by (see e.g., Cherubini et al. (2004))$${C}_{N}\left(u,v|\rho \right)=\underset{-\infty}{\overset{{\mathsf{\Phi}}^{-1}\left(u\right)}{{\displaystyle \int}}}{{\displaystyle \int}}_{-\infty}^{{\mathsf{\Phi}}^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{1-{\rho}^{2}}}\xb7exp\left\{\frac{-({r}^{2}-2\rho rs+{s}^{2})}{2\left(1-{\rho}^{2}\right)}\right\}\mathrm{drds},$$
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- The t-copula—it also has a correlation coefficient such that $-1<\rho <1$ however, it shows some tail dependence. Specifically, it has symmetric tail dependence. It may be expressed as follows (see e.g., Cherubini et al. (2004)):$${C}_{ST}\left(u,v|\rho ,\vartheta \right)={{\displaystyle \int}}_{-\infty}^{{t}_{\vartheta}^{-1}\left(u\right)}{{\displaystyle \int}}_{-\infty}^{{t}_{\vartheta}^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{1-{\rho}^{2}}}\xb7exp\left\{\frac{-({r}^{2}-2\rho rs+{s}^{2})}{2\left(1-{\rho}^{2}\right)}\right\}\mathrm{drds},$$

Family | Parameters | |
---|---|---|

Upper Tail Dependence | Lower Tail Dependence | |

Gaussian-copula | ${\tau}^{U}=0$ | ${\tau}^{L}$ = 0 |

t-copula | ${\tau}^{U}=2{t}_{\vartheta +1}^{}\left(-\sqrt{\vartheta +1}\sqrt{1-\rho}/\sqrt{1+\rho}\right)$ | ${\tau}^{L}=2{t}_{\vartheta +1}^{}\left(-\sqrt{\vartheta +1}\sqrt{1-\rho}/\sqrt{1+\rho}\right)$ |

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**Table 1.**Piecewise distribution (lower and upper tails) of G7, BRICS and emerging markets over the period 1997–2018.

Panel A. G7 Economies | ||||
---|---|---|---|---|

Lower Tail | Upper Tail | |||

US | −Inf < x < −0.05 | −0.04 (0.03) | 0.05 < x < Inf | −0.50 (0.02) |

Canada | −Inf < x < −0.06 | 0.36 (0.02) | 0.07 < x < Inf | 0.40 (0.01) |

UK | −Inf < x < −0.07 | 0.26 (0.04) | 0.08 < x < Inf | 0.08 (0.02) |

France | −Inf < x < −0.07 | −0.47 (0.09) | 0.07 < x < Inf | 0.00 (0.02) |

Italy | −Inf < x < −0.09 | −0.25 (0.06) | 0.09 < x < Inf | −0.09 (0.02) |

Germany | −Inf < x < −0.07 | −0.30 (0.09) | 0.08 < x < Inf | −0.01 (0.01) |

Japan | −Inf < x < −0.06 | 0.06 (0.02) | 0.06 < x < Inf | −0.57 (0.05) |

Panel B. BRICS Economies | ||||

Brazil | −Inf < x < −0.12 | −0.36 (0.14) | 0.14 < x < Inf | −0.68 (0.08) |

Russia | −Inf < x < −0.11 | −0.37 (0.09) | 0.14 < x < Inf | 0.16 (0.06) |

India | −Inf < x < −0.10 | −0.21 (0.08) | 0.10 < x < Inf | 0.10 (0.04) |

China | −Inf < x < −0.09 | −0.54 (0.11) | 0.10 < x < Inf | −0.24 (0.05) |

South Africa | −Inf < x < −0.15 | 0.16 (0.06) | 0.15 < x < Inf | −0.23 (0.07) |

Panel C. Emerging Economies | ||||

Brazil | −Inf < x < −0.12 | −0.36 (0.14) | 0.14 < x < Inf | −0.68 (0.08) |

Russia | −Inf < x < −0.11 | −0.37 (0.09) | 0.13 < x < Inf | 0.16 (0.03) |

India | −Inf < x < −0.09 | 0.21 (0.08) | 0.10 < x < Inf | 0.04 (0.04) |

China | −Inf < x < −0.09 | −0.54 (0.11) | 0.10 < x < Inf | 0.10 (0.03) |

South Africa | −Inf < x < −0.15 | 0.16 (0.05) | 0.15 < x < Inf | −0.23 (0.07) |

Chile | −Inf < x < −0.07 | 0.05 (0.05) | 0.08< x < Inf | −0.01 (0.03) |

Mexico | −Inf < x < −0.07 | 0.04 (0.06) | 0.10 < x < Inf | −0.33 (0.05) |

Peru | −Inf < x < −0.08 | 0.29 (0.03) | 0.12 < x < Inf | 0.26 (0.03) |

Czech Republic | −Inf < x < −0.07 | 0.03 (0.07) | 0.09 < x < Inf | 0.06 (0.03) |

Greece | −Inf < x < −0.14 | 0.13 (0.05) | 0.12 < x < Inf | −0.64 (0.07) |

Hungary | −Inf < x < −0.09 | 0.12 (0.07) | 0.11 < x < Inf | −0.11 (0.04) |

Poland | −Inf < x < −0.10 | 0.39 (0.03) | 0.11 < x < Inf | −0.24 (0.04) |

UAE | −Inf < x < −0.08 | −0.45 (0.11) | 0.09 < x < Inf | −0.46 (0.04) |

Indonesia | −Inf < x < −0.08 | 0.31 (0.04) | 0.10 < x < Inf | −0.13 (0.07) |

Korea | −Inf < x < −0.08 | 0.01 (0.05) | 0.10 < x < Inf | −0.34 (0.02) |

Malaysia | −Inf < x < −0.05 | −0.16 (0.04) | 0.06 < x < Inf | −0.20 (0.03) |

Philippines | −Inf < x < −0.07 | 0.10 (0.05) | 0.09 < x < Inf | −0.53 (0.04) |

Taiwan | −Inf < x < −0.09 | −0.00 (0.05) | 0.09 < x < Inf | −0.15 (0.04) |

Thailand | −Inf < x < −0.07 | 0.37 (0.02) | 0.10 < x < Inf | −0.23 (0.03) |

Multivariate Normal VaR | t-Copula VaR | Gaussian Copula VaR | |
---|---|---|---|

Panel A. G7 Stock Markets | |||

99% VaR | 11.81% | 14.67% | 14.65% |

99%CVaR | 13.48% | 18.35% | 18.01% |

Panel B. BRICS Markets | |||

99% VaR | 15.10% | 19.33% | 18.53% |

99%CVaR | 17.30% | 23.09% | 21.77% |

Panel C. Emerging Markets | |||

99% VaR | 12.77% | 15.21% | 14.90% |

99%CVaR | 14.59% | 19.71% | 19.06% |

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**MDPI and ACS Style**

Trabelsi, N.; Tiwari, A.K.
Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation. *Risks* **2019**, *7*, 78.
https://doi.org/10.3390/risks7030078

**AMA Style**

Trabelsi N, Tiwari AK.
Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation. *Risks*. 2019; 7(3):78.
https://doi.org/10.3390/risks7030078

**Chicago/Turabian Style**

Trabelsi, Nader, and Aviral Kumar Tiwari.
2019. "Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation" *Risks* 7, no. 3: 78.
https://doi.org/10.3390/risks7030078