# Generalized Hyperbolic Distribution and Portfolio Efficiency in Energy and Stock Markets of BRIC Countries

^{*}

## Abstract

**:**

## 1. Introduction

#### Multivariate Normal Inverse Gaussian (MNIG) Distribution

_{1}, …, X

_{d})

^{t}if X = μ + AZ, where Z = (Z

_{1}, …, Z

_{k})

^{t}is an independent identically distributed vector of random variables with univariate Standard Normal distribution, A∈R

^{d×k}and μ∈R

^{d}. With this specification, we can express the multivariate distribution with the next density (Mardia et al. 1979):

_{λ}is the Bessel function of third kind defined as:

## 2. Methodology

_{0}: the empirical data follow a MNIG distribution. We used R software to obtain these results, as well as the later ones.

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 1: Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY.

$\overline{\mathsf{\alpha}}$ (Shape Parameter) |
---|

0.9539552 |

**Table A2.**MNIG Location parameter $\mathsf{\mu}$ for Vector 1: Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY.

μ (Location Parameter) | |||||
---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | IBOV | NIFTY |

0.001879721 | 0.001595145 | 0.001626657 | 0.001476737 | 0.001679586 | 0.001929382 |

**Table A3.**MNIG Dispersion parameter $\mathsf{\Sigma}$ for Vector 1: Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY.

Σ (Dispersion Parameter) | ||||||
---|---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | IBOV | NIFTY | |

Daqing | 5.871779 × 10^{−4} | 2.712689 × 10^{−4} | 1.309442 × 10^{−4} | 5.422485 × 10^{−5} | 6.821899 × 10^{−5} | 4.617644 × 10^{−5} |

Urals | 2.712689 × 10^{−4} | 5.949218 × 10^{−4} | 4.288536 × 10^{−4} | 4.788936 × 10^{−5} | 1.563676 × 10^{−4} | 7.103873 × 10^{−5} |

Marlim | 1.309442 × 10^{−4} | 4.288536 × 10^{−4} | 7.116214 × 10^{−4} | 5.093432 × 10^{−5} | 2.259380 × 10^{−4} | 7.561874 × 10^{−5} |

SHCOMP | 5.422485 × 10^{−5} | 4.788936 × 10^{−5} | 5.093432 × 10^{−5} | 3.316850 × 10^{−4} | 7.773826 × 10^{−5} | 7.598589 × 10^{−5} |

IBOV | 6.821899 × 10^{−5} | 1.563676 × 10^{−4} | 2.259380 × 10^{−4} | 7.773826 × 10^{−5} | 7.406710 × 10^{−4} | 1.347001 × 10^{−4} |

NIFTY | 4.617644 × 10^{−5} | 7.103873 × 10^{−5} | 7.561874 × 10^{−5} | 7.598589 × 10^{−5} | 1.347001 × 10^{−4} | 3.245957 × 10^{−4} |

$\gamma $ (Skewness Parameter) | |||||
---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | IBOV | NIFTY |

−0.001627277 | −0.001301048 | −0.001322349 | −0.001166164 | −0.001325997 | −0.001475250 |

**Table A5.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 2: Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV.

$\overline{\mathsf{\alpha}}$(Shape Parameter) |
---|

0.9329531 |

**Table A6.**MNIG Location parameter $\mathsf{\mu}$ for Vector 2: Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV.

$\mu $ (Location Parameter) | |||||
---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | RTSI | IBOV |

0.002233566 | 0.001820986 | 0.001726932 | 0.001666436 | 0.002339706 | 0.001424282 |

**Table A7.**MNIG Dispersion parameter $\mathsf{\Sigma}$ for Vector 2: Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV.

$\mathbf{\Sigma}$ (Dispersion Parameter) | ||||||
---|---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | RTSI | IBOV | |

Daqing | 5.873262 × 10^{−4} | 2.711392 × 10^{−4} | 1.324914 × 10^{−4} | 5.300234 × 10^{−5} | 1.496152 × 10^{−4} | 6.690959 × 10^{−5} |

Urals | 2.711392 × 10^{−4} | 5.945670 × 10^{−4} | 4.274482 × 10^{−4} | 4.606771 × 10^{−5} | 2.085086 × 10^{−4} | 1.575886 × 10^{−4} |

Marlim | 1.324914 × 10^{−4} | 4.274482 × 10^{−4} | 7.133162 × 10^{−4} | 4.995115 × 10^{−5} | 1.913549 × 10^{−4} | 2.280843 × 10^{−4} |

SHCOMP | 5.300234 × 10^{−5} | 4.606771 × 10^{−5} | 4.995115 × 10^{−5} | 3.350986 × 10^{−4} | 8.076749 × 10^{−5} | 7.668110 × 10^{−5} |

RTSI | 1.496152 × 10^{−4} | 2.085086 × 10^{−4} | 1.913549 × 10^{−4} | 8.076749 × 10^{−5} | 4.841467 × 10^{−4} | 2.336435 × 10^{−4} |

IBOV | 6.690959 × 10^{−5} | 1.575886 × 10^{−4} | 2.280843 × 10^{−4} | 7.668110 × 10^{−5} | 2.336435 × 10^{−4} | 7.442306 × 10^{−4} |

**Table A8.**MNIG Skewness parameter $\mathsf{\gamma}$ for Vector 2: Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV.

$\mathsf{\gamma}$ (Skewness Parameter) | |||||
---|---|---|---|---|---|

Daqing | Urals | Marlim | SHCOMP | RTSI | IBOV |

−0.001980172 | −0.001526153 | −0.001421928 | −0.001355208 | −0.002084975 | −0.001070141 |

**Table A9.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 3: Daqing, Marlim, SHCOMP, IBOV, NIFTY.

$\overline{\mathsf{\alpha}}$ (Shape Parameter) |
---|

0.878861 |

**Table A10.**MNIG Location parameter $\mathsf{\mu}$ for Vector 3: Daqing, Marlim, SHCOMP, IBOV, NIFTY.

$\mathsf{\mu}$ (Location Parameter) | ||||
---|---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV | NIFTY |

0.001674036 | 0.001738409 | 0.001599045 | 0.001764533 | 0.002011464 |

**Table A11.**MNIG Dispersion parameter $\mathsf{\Sigma}$ for Vector 3: Daqing, Marlim, SHCOMP, IBOV, NIFTY.

$\mathbf{\Sigma}$ (Dispersion Parameter) | |||||
---|---|---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV | NIFTY | |

Daqing | 6.073313 × 10^{−4} | 1.305301 × 10^{−4} | 5.362729 × 10^{−5} | 6.796856 × 10^{−5} | 4.580358 × 10^{−5} |

Marlim | 1.305301 × 10^{−4} | 7.325409 × 10^{−4} | 5.049618 × 10^{−5} | 2.278790 × 10^{−4} | 7.484912 × 10^{−5} |

SHCOMP | 5.362729 × 10^{−5} | 5.049618 × 10^{−5} | 3.300815 × 10^{−4} | 7.684276 × 10^{−5} | 7.592939 × 10^{−5} |

IBOV | 6.796856 × 10^{−5} | 2.278790 × 10^{−4} | 7.684276 × 10^{−5} | 7.428642 × 10^{−4} | 1.334200 × 10^{−4} |

NIFTY | 4.580358 × 10^{−5} | 7.484912 × 10^{−5} | 7.592939 × 10^{−5} | 1.334200 × 10^{−4} | 3.233551 × 10^{−4} |

**Table A12.**MNIG Skewness parameter $\mathsf{\gamma}$ for Vector 3: Daqing, Marlim, SHCOMP, IBOV, NIFTY.

$\mathsf{\gamma}$ (Skewness Parameter) | ||||
---|---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV | NIFTY |

−0.001420577 | −0.001433104 | −0.001287578 | −0.001409962 | −0.001556247 |

**Table A13.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 4: Daqing, Marlim, SHCOMP, IBOV.

$\overline{\mathsf{\alpha}}$ (Shape Parameter) |
---|

0.821983 |

$\mathsf{\mu}$ (Location Parameter) | |||
---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV |

0.001724782 | 0.001827421 | 0.001652227 | 0.001572860 |

$\mathbf{\Sigma}$ (Dispersion Parameter) | ||||
---|---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV | |

Daqing | 6.037915 × 10^{−4} | 1.330179 × 10^{−4} | 5.284570 × 10^{−5} | 6.921679 × 10^{−5} |

Marlim | 1.330179 × 10^{−4} | 7.310414 × 10^{−4} | 5.031627 × 10^{−5} | 2.346953 × 10^{−4} |

SHCOMP | 5.284570 × 10^{−5} | 5.031627 × 10^{−5} | 3.279021 × 10^{−4} | 7.639908 × 10^{−5} |

IBOV | 6.921679 × 10^{−5} | 2.346953 × 10^{−4} | 7.639908 × 10^{−5} | 7.536017 × 10^{−4} |

$\mathsf{\gamma}$ (Skewness Parameter) | |||
---|---|---|---|

Daqing | Marlim | SHCOMP | IBOV |

−0.001471361 | −0.001522132 | −0.001340790 | −0.001218463 |

**Table A17.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 5: Daqing, Urals, SHCOMP, RTSI.

$\overline{\mathsf{\alpha}}$ (Shape Parameter) |
---|

0.7936773 |

$\mu $ (Location Parameter) | |||
---|---|---|---|

Daqing | Urals | SHCOMP | RTSI |

0.001836324 | 0.001395869 | 0.001723940 | 0.001924799 |

$\mathbf{\Sigma}$ (Dispersion Parameter) | ||||
---|---|---|---|---|

Daqing | Urals | SHCOMP | RTSI | |

Daqing | 6.035950 × 10^{−4} | 2.811235 × 10^{−4} | 5.510528 × 10^{−5} | 0.0001540243 |

Urals | 2.811235 × 10^{−4} | 6.138225 × 10^{−4} | 4.647012 × 10^{−5} | 0.0002135782 |

SHCOMP | 5.510528 × 10^{−5} | 4.647012 × 10^{−5} | 3.343305 × 10^{−4} | 0.0000807863 |

RTSI | 1.540243 × 10^{−4} | 2.135782 × 10^{−4} | 8.078630 × 10^{−5} | 0.0004928031 |

$\mathsf{\gamma}$ (Skewness Parameter) | |||
---|---|---|---|

Daqing | Urals | SHCOMP | RTSI |

−0.001582824 | −0.001101023 | −0.001412450 | −0.001669955 |

**Table A21.**MNIG Shape parameter $\overline{\mathsf{\alpha}}$ for Vector 6: Urals, Marlim, RTSI, IBOV.

$\overline{\mathsf{\alpha}}$ (Shape Parameter) |
---|

0.8879731 |

$\mathsf{\mu}$ (Location Parameter) | |||
---|---|---|---|

Urals | Marlim | RTSI | IBOV |

0.001780400 | 0.002149520 | 0.002453884 | 0.001978662 |

$\mathbf{\Sigma}$ (Dispersion Parameter) | ||||
---|---|---|---|---|

Urals | Marlim | RTSI | IBOV | |

Urals | 0.0006007258 | 0.0004371611 | 0.0002083842 | 0.0001569735 |

Marlim | 0.0004371611 | 0.0007264886 | 0.0001957445 | 0.0002273840 |

RTSI | 0.0002083842 | 0.0001957445 | 0.0004714375 | 0.0002300495 |

IBOV | 0.0001569735 | 0.0002273840 | 0.0002300495 | 0.0007288311 |

$\mathsf{\gamma}$ (Skewness Parameter) | |||
---|---|---|---|

Urals | Marlim | RTSI | IBOV |

−0.001485447 | −0.00184417 | −0.002198906 | −0.001624147 |

## Appendix B

**Table A25.**Maximum Likelihood Variance-Covariance Matrices for portfolio 1: Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY. Elaborated by authors with data from Bloomberg.

Variance-Covariance Matrix Portfolio 1 | Daqing | Urals | Marlim | SHCOMP | IBOV | NIFTY |
---|---|---|---|---|---|---|

Daqing | 5.90 × 10^{−4} | 2.73 × 10^{−4} | 1.33 × 10^{−4} | 5.62 × 10^{−5} | 7.05 × 10^{−5} | 4.87 × 10^{−5} |

Urals | 2.73 × 10^{−4} | 5.97 × 10^{−4} | 4.31 × 10^{−4} | 4.95 × 10^{−5} | 1.58 × 10^{−4} | 7.31 × 10^{−5} |

Marlim | 1.33 × 10^{−4} | 4.31 × 10^{−4} | 7.13 × 10^{−4} | 5.26 × 10^{−5} | 2.28 × 10^{−4} | 7.77 × 10^{−5} |

SHCOMP | 5.62 × 10^{−5} | 4.95 × 10^{−5} | 5.26 × 10^{−5} | 3.33 × 10^{−4} | 7.94 × 10^{−5} | 7.78 × 10^{−5} |

IBOV | 7.05 × 10^{−5} | 1.58 × 10^{−4} | 2.28 × 10^{−4} | 7.94 × 10^{−5} | 7.43 × 10^{−4} | 1.37 × 10^{−4} |

NIFTY | 4.87 × 10^{−5} | 7.31 × 10^{−5} | 7.77 × 10^{−5} | 7.78 × 10^{−5} | 1.37 × 10^{−4} | 3.27 × 10^{−4} |

**Table A26.**Maximum Likelihood Variance–Covariance Matrices for portfolio 2: Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV. Elaborated by authors with data from Bloomberg.

Variance−Covariance Matrix Portfolio 2 | Daqing | Urals | Marlim | SHCOMP | RTSI | IBOV |
---|---|---|---|---|---|---|

Daqing | 5.92 × 10^{−4} | 2.74 × 10^{−4} | 1.36 × 10^{−4} | 5.59 × 10^{−5} | 1.54 × 10^{−4} | 6.92 × 10^{−5} |

Urals | 2.74 × 10^{−4} | 5.97 × 10^{−4} | 4.30 × 10^{−4} | 4.83 × 10^{−5} | 2.12 × 10^{−4} | 1.59 × 10^{−4} |

Marlim | 1.36 × 10^{−4} | 4.30 × 10^{−4} | 7.15 × 10^{−4} | 5.20 × 10^{−5} | 1.95 × 10^{−4} | 2.30 × 10^{−4} |

SHCOMP | 5.59 × 10^{−5} | 4.83 × 10^{−5} | 5.20 × 10^{−5} | 3.37 × 10^{−4} | 8.38 × 10^{−5} | 7.82 × 10^{−5} |

RTSI | 1.54 × 10^{−4} | 2.12 × 10^{−4} | 1.95 × 10^{−4} | 8.38 × 10^{−5} | 4.89 × 10^{−4} | 2.36 × 10^{−4} |

IBOV | 6.92 × 10^{−5} | 1.59 × 10^{−4} | 2.30 × 10^{−4} | 7.82 × 10^{−5} | 2.36 × 10^{−4} | 7.45 × 10^{−4} |

**Table A27.**Maximum Likelihood Variance–Covariance Matrices for portfolio 3: Daqing, Marlim, SHCOMP, IBOV, NIFTY. Elaborated by authors with data from Bloomberg.

Variance−Covariance Matrix Portfolio 3 | Daqing | Marlim | SHCOMP | IBOV | NIFTY |
---|---|---|---|---|---|

Daqing | 6.10 × 10^{−4} | 1.33 × 10^{−4} | 5.57 × 10^{−5} | 7.02 × 10^{−5} | 4.83 × 10^{−5} |

Marlim | 1.33 × 10^{−4} | 7.35 × 10^{−4} | 5.26 × 10^{−5} | 2.30 × 10^{−4} | 7.74 × 10^{−5} |

SHCOMP | 5.57 × 10^{−5} | 5.26 × 10^{−5} | 3.32 × 10^{−4} | 7.89 × 10^{−5} | 7.82 × 10^{−5} |

IBOV | 7.02 × 10^{−5} | 2.30 × 10^{−4} | 7.89 × 10^{−5} | 7.45 × 10^{−4} | 1.36 × 10^{−4} |

NIFTY | 4.83 × 10^{−5} | 7.74 × 10^{−5} | 7.82 × 10^{−5} | 1.36 × 10^{−4} | 3.26 × 10^{−4} |

**Table A28.**Maximum Likelihood Variance–Covariance Matrices for portfolio 4: Daqing, Marlim, IBOV, SHCOMP. Elaborated by authors with data from Bloomberg.

Variance−Covariance Matrix Portfolio 4 | Daqing | Marlim | IBOV | SHCOMP |
---|---|---|---|---|

Daqing | 6.06 × 10^{−4} | 1.36 × 10^{−4} | 7.14 × 10^{−5} | 5.52 × 10^{−5} |

Marlim | 1.36 × 10^{−4} | 7.34 × 10^{−4} | 2.37 × 10^{−4} | 5.28 × 10^{−5} |

IBOV | 7.14 × 10^{−5} | 2.37 × 10^{−4} | 7.55 × 10^{−4} | 7.84 × 10^{−5} |

SHCOMP | 5.52 × 10^{−5} | 5.28 × 10^{−5} | 7.84 × 10^{−5} | 3.30 × 10^{−4} |

**Table A29.**Maximum Likelihood Variance–Covariance Matrices for portfolio 5: Daqing, Urals, SHCOMP, RTSI. Elaborated by authors with data from Bloomberg.

Variance−Covariance Matrix Portfolio 5 | Daqing | Urals | SHCOMP | RTSI |
---|---|---|---|---|

Daqing | 6.07 × 10^{−4} | 2.83 × 10^{−4} | 5.79 × 10^{−5} | 1.57 × 10^{−4} |

Urals | 2.83 × 10^{−4} | 6.15 × 10^{−4} | 4.84 × 10^{−5} | 2.16 × 10^{−4} |

SHCOMP | 5.79 × 10^{−5} | 4.84 × 10^{−5} | 3.37 × 10^{−4} | 8.38 × 10^{−5} |

RTSI | 1.57 × 10^{−4} | 2.16 × 10^{−4} | 8.38 × 10^{−5} | 4.96 × 10^{−4} |

**Table A30.**Maximum Likelihood Variance–Covariance Matrices for portfolio 6: Urals, Marlim, RTSI, IBOV. Elaborated by authors with data from Bloomberg.

Variance−Covariance Matrix Portfolio 6 | Urals | Marlim | RTSI | IBOV |
---|---|---|---|---|

Urals | 6.03 × 10^{−4} | 4.40 × 10^{−4} | 2.12 × 10^{−4} | 1.60 × 10^{−4} |

Marlim | 4.40 × 10^{−4} | 7.30 × 10^{−4} | 2.00 × 10^{−4} | 2.31 × 10^{−4} |

RTSI | 2.12 × 10^{−4} | 2.00 × 10^{−4} | 4.77 × 10^{−4} | 2.34 × 10^{−4} |

IBOV | 1.60 × 10^{−4} | 2.31 × 10^{−4} | 2.34 × 10^{−4} | 7.32 × 10^{−4} |

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**Figure 1.**Efficient frontier curves for portfolios (described in Table 3) considering the MNIG and Empirical sample (MNormal) variance–covariance matrix. Elaborated by authors with data from Bloomberg.

**Table 1.**Descriptive statistics for logarithmic returns of Brazil, Russia, India and China (BRIC) indexes and oil mixes. High kurtosis values appeared in every data series distribution. Source: own elaboration with data from Bloomberg.

n | Mean | Variance | Skewness | Kurtosis | |
---|---|---|---|---|---|

IBOV | 3022 | 0.000353703 | 0.000773655 | −0.6705952 | 13.157355 |

RTSI | 3022 | 0.000253878 | 0.00058478 | −1.38471987 | 31.182532 |

NIFTY | 3022 | 0.000454259 | 0.000359029 | −0.02677346 | 13.064744 |

SHCOMP | 3022 | 0.000310674 | 0.000342458 | −0.40594055 | 4.142914 |

MARLIM | 3022 | 0.000304422 | 0.000780304 | −0.0322947 | 7.423756 |

URALS | 3022 | 0.000294209 | 0.000588745 | −0.02802895 | 4.068729 |

DAQING | 3022 | 0.000252584 | 0.000624716 | −0.0513526 | 5.094316 |

**Table 2.**p-Values for every proposed vector using the Cramer test considering a Multivariate Normal Inverse Gaussian (MNIG) distribution. In all cases the p-values are not statistically significant and indicate strong evidence for the null hypothesis. Elaborated by authors with data from Bloomberg.

p-Values | |||
---|---|---|---|

(Adjustment to a MNIG Distribution) | |||

Vector | Dimension | Cramer Test Significance 0.95 | Cramer Test Significance 0.99 |

Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY | 6 | 0.7052947 | 0.7112887 |

Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV | 6 | 0.1138861 | 0.1038961 |

Daqing, Marlim, IBOV, NIFTY, SHCOMP | 5 | 0.7102897 | 0.692307 |

Daqing, Marlim, IBOV, SHCOMP | 4 | 0.4215784 | 0.4155844 |

Daqing, Urals, RTSI, SHCOMP | 4 | 0.4035964 | 0.4115884 |

Marlim, Urals, IBOV, RTSI | 4 | 0.3416583 | 0.3606394 |

**Table 3.**Asset weights obtained for suggested portfolios 1 to 6 considering the 1st and 2nd moments of the adjusted MNIG distribution. From these weights, the expected returns and the standard deviation of each portfolio were calculated. Elaborated by authors with data from Bloomberg.

No. | PORTFOLIO | ω_{Daqing} | ω_{Urals} | ω_{Marlim} | ω_{SHCOMP} | ω_{RTSI} | ω_{IBOV} | ω_{NIFTY} |
---|---|---|---|---|---|---|---|---|

1 | Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY | 0.08657 | 0.04096 | 0.05774 | 0.25627 | - | 0.05879 | 0.49968 |

2 | Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV | 0.13106 | 0.08274 | 0.08798 | 0.47673 | 0.04694 | 0.17455 | - |

3 | Daqing, Marlim, SHCOMP, IBOV, NIFTY | 0.09874 | - | 0.07767 | 0.25804 | - | 0.06038 | 0.50517 |

4 | Daqing, Marlim, SHCOMP, IBOV | 0.16757 | - | 0.13729 | 0.50737 | - | 0.18777 | - |

5 | Daqing, Urals, SHCOMP, RTSI | 0.11166 | 0.18530 | - | 0.55792 | 0.14512 | - | - |

6 | Urals, Marlim, RTSI, IBOV | - | 0.27662 | 0.11611 | - | 0.24402 | 0.36325 | - |

**Table 4.**Return–standard deviation relationship for portfolios 1 to 6 using the particular variance-covariance matrix of the adjusted MNIG and MNormal distribution. Elaborated by authors with data from Bloomberg.

No. | PORTFOLIO | μ⁄σ | |
---|---|---|---|

MNIG | MNormal | ||

1 | Daqing, Urals, Marlim, SHCOMP, IBOV, NIFTY | 0.02912 | 0.02815 |

2 | Daqing, Urals, Marlim, SHCOMP, RTSI, IBOV | 0.02212 | 0.02134 |

3 | Daqing, Marlim, SHCOMP, IBOV, NIFTY | 0.02908 | 0.02780 |

4 | Daqing, Marlim, SHCOMP, IBOV | 0.02195 | 0.02104 |

5 | Daqing, Urals, SHCOMP, RTSI | 0.02056 | 0.02026 |

6 | Urals, Marlim, RTSI, IBOV | 0.01660 | 0.01581 |

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## Share and Cite

**MDPI and ACS Style**

Núñez-Mora, J.A.; Sánchez-Ruenes, E.
Generalized Hyperbolic Distribution and Portfolio Efficiency in Energy and Stock Markets of BRIC Countries. *Int. J. Financial Stud.* **2020**, *8*, 66.
https://doi.org/10.3390/ijfs8040066

**AMA Style**

Núñez-Mora JA, Sánchez-Ruenes E.
Generalized Hyperbolic Distribution and Portfolio Efficiency in Energy and Stock Markets of BRIC Countries. *International Journal of Financial Studies*. 2020; 8(4):66.
https://doi.org/10.3390/ijfs8040066

**Chicago/Turabian Style**

Núñez-Mora, José Antonio, and Eduardo Sánchez-Ruenes.
2020. "Generalized Hyperbolic Distribution and Portfolio Efficiency in Energy and Stock Markets of BRIC Countries" *International Journal of Financial Studies* 8, no. 4: 66.
https://doi.org/10.3390/ijfs8040066