# Risk Measurement and Risk Modelling Using Applications of Vine Copulas

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## Abstract

**:**

## 1. Introduction

## 2. Background and Models

**Definition**

**1.**

- $V=\left\{{T}_{1},\dots ,{T}_{n-1}\right\},$
- ${T}_{1}$ is a connected tree with nodes ${N}_{1}=\left\{1,\dots ,n\right\}$, plus edges ${E}_{1}$; for $i=2,\dots ,n-1,$ ${T}_{i}$ is a tree with nodes ${N}_{i}={E}_{i-1}$,
- (proximity) for $i=2,\dots ,n-1,\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}\left\{a,b\right\}\in {E}_{i},\phantom{\rule{0.166667em}{0ex}}with\phantom{\rule{0.166667em}{0ex}}a=\{{a}_{1},{a}_{2}\}\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}b=\{{b}_{1},{b}_{2}\}\phantom{\rule{0.166667em}{0ex}}it\phantom{\rule{0.166667em}{0ex}}must\phantom{\rule{0.166667em}{0ex}}hold\phantom{\rule{0.166667em}{0ex}}that$$\phantom{\rule{0.166667em}{0ex}}\{\#(a\cap b)=1$, where ∩ denotes the symmetric difference operator and # denotes the cardinality of a set.

**Definition**

**2.**

#### 2.1. Modelling Vines

#### 2.2. Regular Vines

#### 2.3. Prior Work with R-Vines

## 3. Sample

## 4. Results

#### 4.1. Dependence Modelling Using Vine Copula

#### 4.2. Pre-GFC

#### 4.3. GFC Period

#### 4.4. Post-GFC Period

## 5. R Vine Copulas

#### 5.1. The Pre-GFC Period

#### 5.2. R-Vines GFC

#### 5.3. Post-GFC R-Vines

## 6. An Empirical Application

#### Empirical Example

- Convert the data sample to log returns.
- Select a moving window of 250 returns.
- Fit GARCH(1,1) with Student-t innovations to convert the log returns into an i.i.d. series. We fit the same GARCH(1,1) with student-t in all the iterations to maintain uniformity in the method, and this approach also makes the method a little less computationally intensive.
- Extract the residuals from Step-3 and standardize them with the Standard deviations obtained from Step-3.
- Convert the standardized residuals to student-t marginals for Copula estimation. The steps above are repeated for all the 10 stocks to obtain a multivariate matrix of uniform marginals.
- Fit an R-Vine to the multivariate data with the same copulas as used in Section 1.
- Generate simulations using the fitted R-Vine model. We generate 1000 simulations per stock for forecasting a day ahead VaR.
- Convert the simulated uniform marginals to standardized residuals.
- Simulate returns from the simulated standardized residuals using GARCH simulations.
- Generate a series of simulated daily portfolio returns to forecast 1% and 5% VaR.
- Repeat step 1 to 10 for a moving window.

- Convert the data sample to log returns.
- Select a moving window of 250 returns.
- Fit GARCH(1,1) with Normal innovations to convert the log returns into an i.i.d. series.
- Extract the fit from step-3 and simulate 1000 returns per asset.
- Repeat step-3 and 4 for all the stocks and then calculate the portfolio return from the simulated series.
- Generate a series of simulated daily portfolio returns to forecast 1% and 5% VaR.
- Repeat step 1 to 10 for a moving window.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Notation and Properties of Bivariate Elliptical and Archimedean Copula Families included in CDVine.

Reuters RIC Code | Index |
---|---|

.FTSE | British FTSE Index |

.GDAXI | German DAX Index |

.FCHI | French CAC 40 Index |

.AEX | AEX Amsterdam Index |

.IBEX | Spanish Ibex 35 Index |

.OMXC20 | OMX Copenhagen 20 Index |

.OMXSPI | OMX Stockholm All Share Index |

.OMXHPI | OMX Helsinki All Share Index |

.BVLG | Portuguese PSI General |

.BFX | Belgian Bell 20 Index |

.STOXX50 | European STOXX 50 |

Index | PRE-GFC January 2005–June 2007 | GFC July 2007–August 2009 | Post GFC September 2009–December 2013 |
---|---|---|---|

.FTSE | |||

Mean | 0.000555640 | −0.000895714 | 0.000295718 |

Median | 0.000738054 | 0.000203498 | 0.000584525 |

St. Deviation | 0.00813613 | 0.0238888 | 0.0129783 |

Skewness | −0.149519 | 0.0528700 | −0.224389 |

Ex-Kurtosis | 1.19971 | 4.78545 | 2.23948 |

Bera-Jarque test | 41.467 (0.00) | 539.38 0.000 | 245.837 (0.00) |

Hurst Exponent | 41.467 | 0.57554 | 0.453993 |

.GDAXI | |||

Mean | 0.000965496 | −0.000573968 | 0.000457927 |

Median | 0.00117051 | 0.000537208 | 0.000326914 |

St. Deviation | 0.00984275 | 0.0235477 | 0.0166782 |

Skewness | −0.198918 | 0.188195 | −0.189233 |

Ex-Kurtosis | 1.06819 | 4.83638 | 2.14630 |

Bera-Jarque test | 35.2434 (0.00) | 553.989 (0.00) | 223.835 |

Hurst exponent | 0.513785 | 0.577037 | 0.503562 |

.FCHI | |||

Mean | 0.000704096 | −0.000794553 | 0.000107354 |

Median | 0.000603012 | 0.000142254 | 0.000167647 |

St. Deviation | 0.00919530 | 0.0244639 | 0.0178098 |

Skewness | −0.225337 | 0.187499 | −0.0293812 |

Ex-Kurtosis | 1.29375 | 4.94817 | 2.75582 |

Bera-Jarque test | 50.9105 (0.00) | 579.714 (0.00) | 2.75582 |

Hurst exponent | 0.480051 | 0.570031 | 0.4944 |

.AEX | |||

Mean | 0.000706398 | −0.00100306 | 0.000233513 |

Median | 0.000689552 | 0.000104563 | 0.000746154 |

St. Deviation | 0.00866571 | 0.0252091 | 0.0151689 |

Skewness | −0.168285 | 0.0194178 | −0.106574 |

Ex-Kurtosis | 1.79428 | 4.93931 | 2.18174 |

Bera-Jarque test | 90.3996 (0.00) | 574.377 (0.00) | 226.455 (0.00) |

Hurst exponent | 0.510652 | 0.606603 | 0.503613 |

.IBEX | |||

Mean | 0.000754932 | −0.000376594 | −0.000156397 |

Median | 0.000457241 | 0.00000 | 0.000000 |

St. Deviation | 0.00896277 | 0.0237746 | 0.0199591 |

Skewness | −0.179712 | 0.0160456 | 0.176078 |

Ex-Kurtosis | 1.19903 | 4.57761 | 3.89965 |

Bera-Jarque test | 42.5008 | 493.327 (0.00) | 722.488 (0.00) |

Hurst exponent | 0.51974 | 0.602261 | 0.552881 |

.OMXC20 | |||

Mean | 0.000812167 | −0.000564910 | 0.000499859 |

Median | 0.00132764 | 0.00000 | 0.000802048 |

St. Deviation | 0.00993412 | 0.0239638 | 0.0144641 |

Skewness | −0.765524 | −0.187144 | −0.116423 |

Ex-Kurtosis | 2.80674 | 4.22380 | 2.12358 |

Bera-Jarque test | 277.268 (0.00) | 423.095 (0.00) | 215.069 |

Hurst exponent | 0.498681 | 0.59553 | 0.520375 |

Index | PRE-GFC January 2005–June 2007 | GFC July 2007–August 2009 | Post GFC September 2009–December 2013 |
---|---|---|---|

.OMXSPI | |||

Mean | 0.000865551 | −0.000760510 | 0.000458071 |

Median | 0.000980873 | 0.00000 | 0.000724200 |

St. Deviation | 0.0111512 | 0.0276237 | 0.0177400 |

Skewness | −0.295747 | 0.230350 | −0.208441 |

Ex-Kurtosis | 3.58218 | 2.34544 | 2.67172 |

Bera-Jarque test | 357.559 | 134.501 (0.00) | 344.573 (0.00) |

Hurst exponent | 0.522449 | 0.574409 | 0.497336 |

.OMXHPI | |||

Mean | 0.000938877 | −0.000984965 | 0.000108658 |

Median | 0.000484177 | −0.000943853 | 0.000373821 |

St. Deviation | 0.0103049 | 0.0242644 | 0.0167601 |

Skewness | −0.115052 | 0.139482 | −0.115636 |

Ex-Kurtosis | 2.36500 | 2.40933 | 2.21433 |

Bera-Jarque test | 153.152 | 138.489 | 233.587 (0.00) |

Hurst exponent | 0.49008 | 0.61372 | 0.538587 |

.BVLG | |||

Mean | 0.000891032 | −0.000884323 | −0.000169933 |

Median | 0.000892271 | −8.71710e-005 | 5.77024e-005 |

St. Deviation | 0.00703327 | 0.0199733 | 0.0159551 |

Skewness | 0.0898224 | −0.160906 | −0.00138346 |

Ex-Kurtosis | 1.05671 | 6.43503 | 3.74001 |

Bera-Jarque test | 31.64 (0.01) | 977.29 (0.00) | 659.168 (0.00) |

Hurst exponent | 0.6217 | 0.614598 | 0.556314 |

.BFX | |||

Mean | 0.000692830 | −0.00108296 | 0.000151703 |

Median | 0.000817080 | 0.00000 | 0.000179149 |

St. Deviation | 0.00877648 | 0.0222434 | 0.0158848 |

Skewness | −0.225903 | −0.131889 | 0.00987544 |

Ex-Kurtosis | 1.59909 | 3.23142 | 2.90360 |

Bera-Jarque test | 74.898 | 247.462 (0.00) | 397.323 (0.00) |

Hurst exponent | 0.54556 | 0.641453 | 0.497547 |

.STOXX50 | |||

Mean | 0.000642054 | −0.000752410 | 6.45717e-005 |

Median | 0.000635064 | 0.000153018 | 0.00000 |

St. Deviation | 0.00922232 | 0.0241936 | 0.0179815 |

Skewness | −0.154465 | 0.0851528 | 0.00350895 |

Ex-Kurtosis | 1.33626 | 4.29802 | 2.88743 |

Bera-Jarque test | 51.0231 (0.00) | 435.568 (0.00) | 392.895 (0.00) |

Hurst exponent | 0.474577 | 0.587196 | 0.504895 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 9 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 3 | 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 10 | 10 | 10 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 2 | 2 | 2 | 10 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 4 | 4 | 4 | 4 | 10 | 5 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 5 | 5 | 5 | 5 | 5 | 10 | 7 | 0 | 0 | 0 | 0 |

OMXSPI | 7 | 7 | 7 | 7 | 7 | 7 | 10 | 8 | 0 | 0 | 0 |

OMXHPI | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 0 | 0 |

BVLG | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 0 |

BFX | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 2 | 3 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 5 | 2 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 5 | 2 | 1 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 3 | 2 | 3 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 5 | 1 | 5 | 5 | 3 | 4 | 5 | 0 | 0 | 0 | 0 |

OMXHPI | 1 | 3 | 2 | 3 | 6 | 5 | 1 | 5 | 0 | 0 | 0 |

BVLG | 5 | 1 | 5 | 5 | 5 | 2 | 1 | 3 | 1 | 0 | 0 |

BFX | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 0.152236 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 0.786802 | 0.437083 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 1.418514 | 0.053894 | 1.036334 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 1.033914 | 0.672986 | 0.162141 | 0.137522 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | −0.18660 | 1.068144 | 0.155770 | 0.076751 | 0.205075 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | −0.06799 | −0.011583 | −0.092367 | 0.201391 | 0.973316 | 0.083272 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | −0.16274 | 0.016648 | −0.261361 | 0.182827 | 0.714802 | 0.047323 | −0.025852 | 0 | 0 | 0 | 0 |

OMXHPI | 1.05367 | 0.087220 | 0.0779183 | 0.126660 | 0.071290 | −0.004308 | 0.031866 | 1.223109 | 0 | 0 | 0 |

BVLG | 1.56654 | 0.164849 | 1.2442506 | 0.408728 | −0.130783 | −0.129392 | 0.148569 | 0.226344 | 0.196334 | 0 | 0 |

BFX | 0.94972 | 0.871079 | 0.9334080 | 0.8309762 | 0.827286 | 0.973728 | 0.978593 | 0.693275 | 0.890633 | 0.718538 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 15.375671 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 9.544367 | 12.310803 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 9.401744 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 0 | 10.267206 | 10.233424 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 0 | 10.124756 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 0 | 8.646548 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXHPI | 0 | 0 | 8.390870 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

BVLG | 0 | 0 | 0 | 0 | 0 | 12.429237 | 0 | 0 | 0 | 0 | 0 |

BFX | 8.686229 | 5.378347 | 3.377834 | 8.575454 | 11.885624 | 7.882211 | 6.454538 | 9.626281 | 13.482133 | 8.332783 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.004067 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | 0.036096 | −0.029548 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.014419 | 0.030447 | 0.068843 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | 0.005771 | −0.091656 | −0.099620 | 0.025585 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | 0.155059 | −0.031242 | −0.053529 | −0.007041 | 0.075579 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | 0.036919 | −0.085254 | 0.030712 | −0.029132 | −0.109944 | 0.101804 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | 0.062762 | 0.067815 | 0.106788 | −0.007854 | 0.069387 | 0.037842 | 0.118612 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.084060 | 0.056764 | 0.258504 | 0.035254 | 0.039828 | 0.116247 | 0.152900 | 0.096636 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.121015 | 0.168311 | 0.082120 | −0.024858 | 0.162175 | 0.101571 | 0.173707 | 0.120451 | 0.224388 | 0.000000 | 0.000000 |

BFX | 0.673160 | 0.868040 | 0.620234 | 0.853751 | 0.797280 | 0.766361 | 0.478690 | 0.624435 | 0.472402 | 0.692177 | 0.000000 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 11 | 11 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 5 | 5 | 11 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 2 | 2 | 2 | 11 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 10 | 10 | 10 | 10 | 11 | 3 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 3 | 3 | 3 | 3 | 3 | 11 | 4 | 0 | 0 | 0 | 0 |

OMXSPI | 4 | 4 | 4 | 4 | 4 | 4 | 11 | 7 | 0 | 0 | 0 |

OMXHPI | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 11 | 8 | 0 | 0 |

BVLG | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | 11 | 0 |

BFX | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 3 | 1 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 5 | 3 | 4 | 5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 5 | 2 | 6 | 4 | 2 | 4 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 |

OMXHPI | 1 | 4 | 2 | 2 | 5 | 2 | 2 | 5 | 0 | 0 | 0 |

BVLG | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |

BFX | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.015375 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | −0.014081 | −0.039582 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.056810 | −0.066848 | −0.006416 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | −0.025651 | −0.042298 | −0.058784 | −0.083306 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | 0.036874 | 0.051407 | 0.139010 | −0.030941 | 0.108496 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | 0.153082 | 0.015462 | 0.035938 | 0.124012 | −0.006829 | 0.041608 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | 0.131300 | 0.032328 | −0.067132 | −0.064857 | 0.029947 | 0.132041 | 0.162546 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.124725 | 0.124603 | 0.080170 | 0.011683 | 0.194322 | 0.106381 | 0.103593 | 0.126699 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.143013 | 0.264481 | 0.064391 | 0.133197 | 0.195961 | 0.189751 | 0.154686 | 0.292464 | 0.153108 | 0.000000 | 0.000000 |

BFX | 0.735492 | 0.702116 | 0.790637 | 0.844666 | 0.625231 | 0.867456 | 0.801941 | 0.630802 | 0.718425 | 0.720695 | 0.000000 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 11 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 7 | 11 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 1 | 1 | 11 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 3 | 3 | 3 | 11 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 9 | 9 | 9 | 9 | 11 | 10 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 10 | 10 | 10 | 10 | 10 | 11 | 5 | 0 | 0 | 0 | 0 |

OMXSPI | 5 | 5 | 5 | 5 | 5 | 5 | 11 | 8 | 0 | 0 | 0 |

OMXHPI | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | 4 | 0 | 0 |

BVLG | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | 11 | 0 |

BFX | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 3 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 2 | 4 | 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 3 | 5 | 1 | 5 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 3 | 4 | 1 | 1 | 4 | 2 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 2 | 2 | 5 | 2 | 1 | 1 | 3 | 0 | 0 | 0 | 0 |

OMXHPI | 4 | 2 | 2 | 1 | 2 | 5 | 5 | 1 | 0 | 0 | 0 |

BVLG | 3 | 2 | 2 | 2 | 2 | 6 | 1 | 1 | 2 | 0 | 0 |

BFX | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.016107 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | 0.036144 | 0.035753 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.052784 | 0.024838 | 0.031775 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | −0.130688 | 0.044635 | 0.051311 | 0.046822 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | 0.047183 | 0.135747 | −0.023586 | −0.027756 | 0.086703 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | 0.017433 | 0.131760 | −0.035346 | 0.042679 | 0.088264 | 0.198073 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | −0.264456 | −0.021457 | −0.078220 | −0.228646 | 0.003425 | 0.202873 | 0.059906 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.118687 | 0.228880 | 0.072495 | 0.108799 | 0.324425 | 0.072847 | −0.019366 | 0.131302 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.142026 | 0.226714 | 0.302870 | 0.255696 | 0.209531 | 0.019685 | −0.207497 | 0.284246 | 0.239010 | 0.000000 | 0.000000 |

BFX | 0.856993 | 0.629740 | 0.718863 | 0.912498 | 0.736706 | 0.707156 | 0.819042 | 0.721526 | 0.847664 | 0.818948 | 0.000000 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 10 | 7 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 11 | 10 | 7 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 3 | 11 | 10 | 7 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 2 | 3 | 11 | 10 | 7 | 2 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 9 | 2 | 3 | 11 | 10 | 7 | 6 | 0 | 0 | 0 | 0 |

OMXSPI | 5 | 9 | 2 | 3 | 11 | 10 | 7 | 7 | 0 | 0 | 0 |

OMXHPI | 8 | 5 | 9 | 2 | 3 | 11 | 10 | 10 | 3 | 0 | 0 |

BVLG | 1 | 8 | 8 | 9 | 2 | 3 | 11 | 11 | 10 | 11 | 0 |

BFX | 6 | 6 | 6 | 6 | 6 | 6 | 3 | 3 | 11 | 10 | 10 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.070734 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | 0.086887 | 0.048472 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.154546 | 0.034327 | 0.035060 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | 0.019231 | 0.074440 | 0.103680 | 0.087828 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | −0.020726 | 0.063797 | 0.099572 | 0.036958 | 0.131488 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | −0.043319 | −0.007375 | −0.058887 | 0.091484 | 0.107138 | 0.009252 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | −0.104071 | 0.010599 | −0.029020 | 0.083757 | 0.079020 | 0.030138 | −0.016460 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.050943 | 0.055597 | 0.037498 | 0.059558 | 0.034418 | −0.002743 | 0.015683 | 0.133918 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.169959 | 0.105428 | 0.136164 | 0.268055 | −0.083498 | −0.082605 | 0.094934 | 0.145355 | 0.089392 | 0.000000 | 0.000000 |

BFX | 0.797280 | 0.673160 | 0.766361 | 0.624435 | 0.620234 | 0.853751 | 0.868040 | 0.487667 | 0.699477 | 0.510377 | 0.000000 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 4 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 5 | 5 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 3 | 2 | 1 | 6 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 2 | 1 | 2 | 5 | 2 | 2 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 0 | 0 | 0 | 0 |

OMXHPI | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 |

BVLG | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 0 | 0 |

BFX | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.020428 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | 0.040243 | −0.050241 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.090111 | −0.051601 | 0.034037 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | 0.174882 | −0.023305 | 0.201117 | 0.060031 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | 0.085350 | 0.133903 | 0.134654 | 0.037200 | 0.047610 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | 0.074012 | −0.026941 | 0.088942 | 0.067369 | 0.074824 | −0.048008 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | −0.020369 | -0.069500 | 0.087310 | 0.014613 | 0.038448 | 0.063628 | −0.004543 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.106049 | −0.070206 | 0.093532 | 0.133080 | 0.076088 | 0.052608 | −0.033877 | 0.031725 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.203436 | 0.147673 | 0.112357 | 0.125506 | 0.176601 | 0.208258 | 0.177847 | 0.135507 | 0.183817 | 0.000000 | 0.000000 |

BFX | 0.651592 | 0.844666 | 0.725372 | 0.749098 | 0.867456 | 0.801941 | 0.790637 | 0.702116 | 0.625231 | 0.720695 | 0.000000 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

GDAXI | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

FCHI | 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

AEX | 5 | 5 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

IBEX | 1 | 5 | 5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

STOXX50 | 1 | 6 | 5 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

OMXC20 | 3 | 5 | 3 | 6 | 3 | 1 | 0 | 0 | 0 | 0 | 0 |

OMXSPI | 6 | 2 | 1 | 4 | 5 | 3 | 1 | 0 | 0 | 0 | 0 |

OMXHPI | 5 | 1 | 5 | 2 | 2 | 1 | 2 | 2 | 0 | 0 | 0 |

BVLG | 2 | 2 | 2 | 5 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |

BFX | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 |

FTSE | GDAXI | FCHI | AEX | IBEX | STOXX50 | OMXC20 | OMXSPI | OMXHPI | BVLG | BFX | |
---|---|---|---|---|---|---|---|---|---|---|---|

FTSE | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

GDAXI | 0.021969 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

FCHI | −0.025819 | 0.051738 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

AEX | 0.109136 | −0.042114 | 0.091242 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

IBEX | 0.068208 | −0.055661 | 0.014128 | 0.092498 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

STOXX50 | −0.073813 | 0.023514 | 0.001263 | −0.036656 | 0.017084 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXC20 | 0.015055 | −0.035009 | 0.063244 | 0.067810 | 0.060644 | 0.183691 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXSPI | 0.013031 | 0.069506 | −0.091398 | 0.144184 | 0.117396 | 0.071469 | −0.043601 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

OMXHPI | 0.140392 | −0.019256 | 0.053089 | 0.065557 | −0.070606 | −0.119362 | 0.192945 | −0.047774 | 0.000000 | 0.000000 | 0.000000 |

BVLG | 0.211872 | 0.082607 | 0.329059 | 0.184564 | −0.294979 | −0.283996 | 0.101914 | 0.135948 | 0.179939 | 0.000000 | 0.000000 |

BFX | 0.657377 | 0.748438 | 0.750231 | 0.744893 | 0.856993 | 0.819042 | 0.855328 | 0.912498 | 0.818948 | 0.711242 | 0.000000 |

Expected Exceed | Actual Exceed | uc. H0 | uc LRstat | uc. Critical | u LRp | uc Decision | cc H0 | cc LRstat | cc Critical | cc LRp | cc Decision | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 1 | Correct Exceedances | 1.276879991 | 3.841458821 | 0.258479955 | Fail to Reject H0 | Correct Exceedances & Independent | 1.317699151 | 5.991464547 | 0.517446275 | Fail to Reject H0 |

2 | 0 | 0 | Correct Exceedances | 1.025134257 | 3.841458821 | 0.311304237 | Fail to Reject H0 | Correct Exceedances & Independent | 1.025134257 | 5.991464547 | 0.598956006 | Fail to Reject H0 |

Expected Exceed | Actual Exceed | uc. H0 | uc LRstat | uc. Critical | uc LRp | uc Decision | cc H0 | cc. LRstar | cc. Critical | cc. LRp | cc. Decision | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 29 | Correct Exceedances | 106.272188368284 | 3.84145882069412 | 0 | Reject H0 | Correct Exceedances & Independent | 106.283103143687 | 5.99146454710798 | 0 | Reject H0 |

2 | 0 | 21 | Correct Exceedances | 124.915738708942 | 3.84145882069412 | 0 | Reject H0 | Correct Exceedances & Independent | 124.926653484345 | 5.99146454710798 | 0 | Reject H0 |

© 2017 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/).

## Share and Cite

**MDPI and ACS Style**

Allen, D.E.; McAleer, M.; Singh, A.K.
Risk Measurement and Risk Modelling Using Applications of Vine Copulas. *Sustainability* **2017**, *9*, 1762.
https://doi.org/10.3390/su9101762

**AMA Style**

Allen DE, McAleer M, Singh AK.
Risk Measurement and Risk Modelling Using Applications of Vine Copulas. *Sustainability*. 2017; 9(10):1762.
https://doi.org/10.3390/su9101762

**Chicago/Turabian Style**

Allen, David E., Michael McAleer, and Abhay K. Singh.
2017. "Risk Measurement and Risk Modelling Using Applications of Vine Copulas" *Sustainability* 9, no. 10: 1762.
https://doi.org/10.3390/su9101762