# Portfolio Tail Risk: A Multivariate Extreme Value Theory Approach

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework and Estimation Methodology

#### 2.1. Theoretical Framework

**Theorem**

**1.**

#### 2.2. Estimation Methodology

#### 2.2.1. Orthogonalization

**Definition**

**1.**

**Λ**is a diagonal matrix of the eigenvalues of ${\mathbf{V}}_{\infty}$, ordered by descending values, ${\lambda}_{1}\ge {\lambda}_{2}\ge \dots \ge {\lambda}_{n}>0$. Further, let

#### 2.2.2. Filtering

#### 2.2.3. Estimating Independent Univariate Excess Distributions

#### 2.2.4. Tails of Univariate Distributions

#### 2.2.5. Estimating Univariate Var and Es

**Definition**

**2.**

#### 2.2.6. Portfolio-Level VaR and ES

#### 2.2.7. Summary of the Methodology

**Step 1.**Start by performing the orthogonalization on portfolio returns. In other words, use the eigenvalue decomposition of the unconditional covariance matrix ${\mathbf{V}}_{\infty}$ of return residuals ${\mathbf{\epsilon}}_{t}$ to obtain the matrix $\mathbf{L}$ following Definition 1. Then, use Equation (2) to obtain the set of orthogonal vectors ${\mathbf{z}}_{t}$. This step assures cross-sectional independence and allows working with univariate series.

**Step 2.**Filter out the heteroskedasticity on each component from step 1 to obtain serially i.i.d. random variables. In other words, use the set of orthogonal vectors ${\mathbf{z}}_{t}$ from step 1, calculate ${\tilde{\mathbf{E}}}_{t}={\mathbf{z}}_{t}{\mathbf{z}}_{t}^{\prime}$ and run a GARCH-like process such as one given by Equation (8) on ${\tilde{\mathbf{E}}}_{t}$. This results in a set of univariate estimates of the conditional variance ${\tilde{V}}_{t,i}$. This step is a prerequisite for using the univariate EVT.

**Step 3.**Use the estimator from Equation (11) to fit the tails of each of the independent series from step 2. This gives the parameters of GP distribution.

**Step 4.**Substitute the parameters from step 3 into Equation (12) or (13). This step allows us to calculate the quantile for each component, Equation (15).

**Step 5.**Substitute the quantiles from step 4 into the closed-form expressions from Equations (16) and (17) to obtain portfolio VaR and ES.

## 3. Illustration of the Method

## 4. Out-of-Sample Performance

## 5. Numerical Simulations

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Working with Fewer Than n Principal Components

## Appendix B. Illustration Using Exchange-Rate Data

**Table A1.**Actual and expected number of lower-tail VaR violations for an equally-weighted portfolio of 20 major currencies against USD.

Method | Confidence Level, $1-\mathit{\alpha}$ | ||||
---|---|---|---|---|---|

0.900 | 0.950 | 0.990 | 0.995 | 0.999 | |

mv GARCH | |||||

Normal | 264 | 169 | 53 | 34 | 20 |

Student | 248 | 147 | 32 | 21 | 9 |

GP | 217 | 115 | 25 | 13 | 2 |

mv GJR | |||||

Normal | 225 | 140 | 36 | 27 | 15 |

Student | 251 | 145 | 31 | 22 | 8 |

GP | 207 | 111 | 24 | 12 | 1 |

Expected | 200 | 100 | 20 | 10 | 2 |

**Table A2.**Pearson’s Q test statistics and corresponding p-values for an equally-weighted portfolio of 20 major currencies against USD.

Method | Q | p-Value |
---|---|---|

mv GARCH | ||

Normal | 193.33 | 0.000 |

Student | 43.20 | 0.000 |

GP | 2.98 | 0.704 |

mv GJR | ||

Normal | 96.40 | 0.000 |

Student | 38.85 | 0.000 |

GP | 2.82 | 0.727 |

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**Figure 2.**Distribution of exceedances in the lower tail of standardized residuals (log scale): (

**a**) first principal component; (

**b**) second principal component; (

**c**) third principal component; (

**d**) fourth principal component.

**Figure 3.**We show the ${log}_{10}{F}_{u}\left(x\right)$ of the first principal component for each of the six copula types obtained from the orthogonal-GJR(1,1) residuals. (

**a**) Gaussian copula (Pearson); (

**b**) Gaussian copula (Kendall); (

**c**) t copula; (

**d**) Clayton copula; (

**e**) Frank copula; (

**f**) Gumbel copula.

Confidence Level, $1-\mathit{\alpha}$ | |||||
---|---|---|---|---|---|

0.900 | 0.950 | 0.990 | 0.995 | 0.999 | |

mv GARCH – GP | |||||

VaR | $5.11$ | $6.77$ | $10.81$ | $12.63$ | $17.08$ |

ES | $7.57$ | $9.30$ | $13.52$ | $15.42$ | $20.05$ |

mv GJR – GP | |||||

VaR | $4.69$ | $6.33$ | $10.08$ | $11.67$ | $15.28$ |

ES | $7.04$ | $8.66$ | $12.35$ | $13.90$ | $17.46$ |

eig. | Cumulative ${R}^{2}$ | mv GARCH | mv GJR | ||||
---|---|---|---|---|---|---|---|

Normal | Student | GP | Normal | Student | GP | ||

1 | $0.4913$ | $0.0470$ *** | $0.0367$ *** | $0.0053$ | $0.0419$ *** | $0.0364$ *** | $0.0052$ |

2 | $0.5609$ | $0.0403$ *** | $0.0238$ * | $0.0060$ | $0.0410$ *** | $0.0247$ * | $0.0040$ |

3 | $0.6162$ | $0.0362$ *** | $0.0177$ | $0.0051$ | $0.0363$ *** | $0.0280$ ** | $0.0047$ |

4 | $0.6523$ | $0.0335$ *** | $0.0148$ | $0.0041$ | $0.0334$ *** | $0.0144$ | $0.0044$ |

5 | $0.6792$ | $0.0396$ *** | $0.0126$ | $0.0047$ | $0.0388$ *** | $0.0125$ | $0.0046$ |

6 | $0.7053$ | $0.0335$ *** | $0.0181$ | $0.0061$ | $0.0338$ *** | $0.0176$ | $0.0073$ |

7 | $0.7286$ | $0.0571$ *** | $0.0126$ | $0.0044$ | $0.0571$ *** | $0.0143$ | $0.0047$ |

8 | $0.7489$ | $0.0425$ *** | $0.0116$ | $0.0070$ | $0.0408$ *** | $0.0115$ | $0.0062$ |

9 | $0.7677$ | $0.0302$ ** | $0.0134$ | $0.0045$ | $0.0296$ ** | $0.0139$ | $0.0050$ |

10 | $0.7862$ | $0.0261$ ** | $0.0170$ | $0.0047$ | $0.0260$ ** | $0.0172$ | $0.0046$ |

11 | $0.8042$ | $0.0308$ ** | $0.0146$ | $0.0059$ | $0.0307$ ** | $0.0147$ | $0.0063$ |

12 | $0.8214$ | $0.0261$ ** | $0.0162$ | $0.0043$ | $0.0264$ ** | $0.0162$ | $0.0041$ |

13 | $0.8374$ | $0.0310$ *** | $0.0167$ | $0.0058$ | $0.0295$ ** | $0.0167$ | $0.0063$ |

14 | $0.8525$ | $0.0320$ *** | $0.0180$ | $0.0074$ | $0.0316$ *** | $0.0177$ | $0.0067$ |

15 | $0.8668$ | $0.0229$ | $0.0194$ | $0.0045$ | $0.0225$ | $0.0194$ | $0.0052$ |

16 | $0.8808$ | $0.0277$ ** | $0.0157$ | $0.0053$ | $0.0278$ ** | $0.0156$ | $0.0049$ |

17 | $0.8939$ | $0.0363$ *** | $0.0131$ | $0.0041$ | $0.0360$ *** | $0.0133$ | $0.0039$ |

18 | $0.9066$ | $0.0302$ ** | $0.0144$ | $0.0052$ | $0.0301$ ** | $0.0134$ | $0.0049$ |

19 | $0.9188$ | $0.0456$ *** | $0.0196$ | $0.0050$ | $0.0441$ *** | $0.0149$ | $0.0041$ |

20 | $0.9296$ | $0.0347$ *** | $0.0240$ * | $0.0053$ | $0.0365$ *** | $0.0149$ | $0.0042$ |

21 | $0.9396$ | $0.0574$ *** | $0.0110$ | $0.0058$ | $0.0572$ *** | $0.0114$ | $0.0056$ |

22 | $0.9488$ | $0.0405$ *** | $0.0168$ | $0.0044$ | $0.0405$ *** | $0.0164$ | $0.0042$ |

23 | $0.9576$ | $0.0356$ *** | $0.0148$ | $0.0035$ | $0.0357$ *** | $0.0150$ | $0.0040$ |

24 | $0.9657$ | $0.0266$ ** | $0.0177$ | $0.0076$ | $0.0264$ ** | $0.0173$ | $0.0072$ |

25 | $0.9736$ | $0.0272$ ** | $0.0140$ | $0.0040$ | $0.0280$ ** | $0.0143$ | $0.0040$ |

26 | $0.9804$ | $0.0519$ *** | $0.0097$ | $0.0036$ | $0.0519$ *** | $0.0107$ | $0.0036$ |

27 | $0.9863$ | $0.0318$ *** | $0.0139$ | $0.0052$ | $0.0321$ *** | $0.0140$ | $0.0043$ |

28 | $0.9919$ | $0.0362$ *** | $0.0196$ | $0.0046$ | $0.0364$ *** | $0.0196$ | $0.0047$ |

29 | $0.9972$ | $0.0400$ *** | $0.0152$ | $0.0062$ | $0.0400$ *** | $0.0160$ | $0.0065$ |

30 | $1.0000$ | $0.0326$ *** | $0.0136$ | $0.0040$ | $0.0327$ *** | $0.0140$ | $0.0048$ |

**Table 3.**Actual and expected number of lower-tail VaR violations for an equally-weighted portfolio of 30 stocks.

Method | Confidence Level, $1-\mathit{\alpha}$ | ||||
---|---|---|---|---|---|

0.900 | 0.950 | 0.990 | 0.995 | 0.999 | |

mv GARCH | |||||

Normal | 109 | 69 | 24 | 16 | 7 |

Student | 118 | 71 | 17 | 10 | 3 |

GP | 117 | 68 | 15 | 8 | 3 |

mv GJR | |||||

Normal | 107 | 64 | 25 | 15 | 8 |

Student | 117 | 65 | 19 | 10 | 1 |

GP | 115 | 62 | 14 | 8 | 0 |

Expected | 100 | 50 | 10 | 5 | 1 |

**Table 4.**Pearson’s Q test statistics and corresponding p-values for an equally-weighted portfolio of 30 stocks.

Method | Q | p-Value |
---|---|---|

mv GARCH | ||

Normal | $46.77$ | $0.000$ |

Student | $12.49$ | $0.029$ |

GP | $\phantom{\rule{3.33333pt}{0ex}}9.62$ | $0.087$ |

mv GJR | ||

Normal | $57.31$ | $0.000$ |

Student | $10.75$ | $0.057$ |

GP | $\phantom{\rule{3.33333pt}{0ex}}7.23$ | $0.204$ |

Confidence Level, $1-\mathit{\alpha}$ | |||||
---|---|---|---|---|---|

0.900 | 0.950 | 0.990 | 0.995 | 0.999 | |

Gaussian copula (Pearson) | |||||

VaR | $5.07$ | $7.19$ | $12.29$ | $14.58$ | $20.08$ |

ES | $8.19$ | $10.38$ | $15.66$ | $18.02$ | $23.71$ |

Gaussian copula (Kendall) | |||||

VaR | $5.26$ | $7.41$ | $12.34$ | $11.67$ | $19.25$ |

ES | $8.34$ | $10.47$ | $15.35$ | $17.42$ | $22.19$ |

t copula | |||||

VaR | $5.01$ | $7.17$ | $13.67$ | $17.28$ | $28.14$ |

ES | $8.72$ | $11.49$ | $19.86$ | $24.50$ | $38.47$ |

Clayton copula | |||||

VaR | $3.85$ | $5.50$ | $9.75$ | $11.76$ | $16.93$ |

ES | $6.38$ | $8.19$ | $12.83$ | $15.04$ | $20.68$ |

Frank copula | |||||

VaR | $4.24$ | $6.05$ | $10.63$ | $12.78$ | $18.21$ |

ES | $6.99$ | $8.94$ | $13.89$ | $16.21$ | $22.07$ |

Gumbel copula | |||||

VaR | $3.97$ | $6.02$ | $12.25$ | $15.73$ | $26.31$ |

ES | $7.51$ | $10.18$ | $18.97$ | $22.79$ | $36.52$ |

**Table 6.**Actual and expected number of lower-tail VaR violations for an equally-weighted portfolio (simulated data).

Method | Confidence Level, $1-\mathit{\alpha}$ | ||||
---|---|---|---|---|---|

0.900 | 0.950 | 0.990 | 0.995 | 0.999 | |

Gaussian copula (Pearson) | 110 | 62 | 13 | 5 | 2 |

Gaussian copula (Kendall) | 109 | 64 | 16 | 7 | 1 |

t copula | 104 | 58 | 8 | 4 | 0 |

Clayton copula | 110 | 52 | 12 | 6 | 0 |

Frank copula | 100 | 45 | 10 | 8 | 1 |

Gumbel copula | 101 | 51 | 12 | 8 | 1 |

Expected | 100 | 50 | 10 | 5 | 1 |

**Table 7.**Pearson’s Q test statistics and corresponding p-values for an equally-weighted portfolio (simulated data).

Method | Q | p-Value |
---|---|---|

Gaussian copula (Pearson) | 5.26 | 0.384 |

Gaussian copula (Kendall) | 6.39 | 0.270 |

t copula | 4.04 | 0.544 |

Clayton copula | 3.59 | 0.610 |

Frank copula | 5.18 | 0.394 |

Gumbel copula | 2.48 | 0.780 |

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Božović, M.
Portfolio Tail Risk: A Multivariate Extreme Value Theory Approach. *Entropy* **2020**, *22*, 1425.
https://doi.org/10.3390/e22121425

**AMA Style**

Božović M.
Portfolio Tail Risk: A Multivariate Extreme Value Theory Approach. *Entropy*. 2020; 22(12):1425.
https://doi.org/10.3390/e22121425

**Chicago/Turabian Style**

Božović, Miloš.
2020. "Portfolio Tail Risk: A Multivariate Extreme Value Theory Approach" *Entropy* 22, no. 12: 1425.
https://doi.org/10.3390/e22121425