# Tail Dependence in Financial Markets: A Dynamic Copula Approach

## Abstract

**:**

## 1. Introduction

- Describe the evolution of the dependence in the tails via the computation of tail indices2.
- Forecast capital losses, computing and then forecasting one popular risk measure, like Value-at-Risk (VaR).

## 2. Model

#### 2.1. Data Description

#### 2.2. Estimation

#### 2.3. Marginal Distributions

#### 2.4. Joint Distribution

- If ${\theta}_{t}\in [1,+\infty )$, then we will set $\Lambda (x)={x}^{2}+1$.
- If ${\theta}_{t}\in (0,+\infty )$, we will set $\Lambda (x)=exp(x)$.

#### 2.5. Value-at-Risk

- First, generate a random sample $({u}_{i,t},{v}_{i,t})$ from the selected copula ${C}_{\theta}$.
- Second, create shocks from the copula probabilities using the marginal inverse cumulative distribution functions ${z}_{i,t}={T}_{\nu}^{-1}({u}_{i,t})$ on each asset.
- Third, create returns ${x}_{i,t}$, ${y}_{i,t}$ from shocks using the dynamic volatility models.

- Generate two independent uniform $(0,1)$ rvs u and s.
- Set $v={c}_{u}^{-1}(s)$, where ${c}_{u}^{-1}$ is the inverse of ${c}_{u}$.
- The desired pair is $(u,v)$.

## 3. Results

- Compute ${C}_{n}$, the empirical copula, from the uniform transforms $\widehat{{\mathrm{U}}_{\mathrm{i}}}$ and estimate the vector of copula parameters θ, say θ
_{n}, via maximum likelihood. - Compute the t.s. S
_{n}. - For some large N, repeat the following steps, $k=\{1,\cdots ,N\}$.
- (a)
- Generate a random sample ${\widehat{{\mathrm{Y}}_{1}}}^{(k)},\cdots ,{\widehat{{\mathrm{Y}}_{\mathrm{n}}}}^{(k)}$ from copula ${C}_{{\theta}_{n}}$, then compute their associated rank vectors ${\widehat{{\mathrm{R}}_{1}}}^{(k)},\cdots ,{\widehat{{\mathrm{R}}_{\mathrm{n}}}}^{(k)}$, $i=1,\cdots ,n$.
- (b)
- Compute ${\widehat{{\mathrm{U}}_{\mathrm{i}}}}^{(k)}={\widehat{{\mathrm{R}}_{\phantom{\rule{1.42262pt}{0ex}}\mathrm{i}}}}^{(k)}/(n+1)$ and let$${C}_{n}^{(k)}(\mathbf{u})=\frac{1}{n}\sum _{i=1}^{n}\mathbf{1}({\widehat{{\mathrm{U}}_{\mathrm{i}}}}^{(k)}\le \mathbf{u})\phantom{\rule{1.em}{0ex}}\mathbf{u}\in {[0,1]}^{2}$$
- (c)
- Compute an approximate realization of ${S}_{n}$ by$${S}_{n}^{(k)}=\sum _{i=1}^{n}{\{{C}_{n}^{(k)}({\widehat{{\mathrm{U}}_{\mathrm{i}}}}^{(k)})-{C}_{{\theta}_{n}^{(k)}}({\widehat{{\mathrm{U}}_{\mathrm{i}}}}^{(k)})\}}^{2}.$$

- An approximation for the p-value of the test is given by$${p}_{value}=\frac{1}{N}{\sum}_{i=1}^{N}\mathbf{1}({S}_{n}^{(k)}\ge {S}_{n}).$$$${p}_{value}=\frac{1}{N+1}\left(\sum _{i=1}^{N}\mathbf{1}({S}_{n}^{(k)}\ge {S}_{n})+\frac{1}{2}\right),$$

- Estimate the vector of parameters $\theta $, say ${\theta}_{n}$, via maximum likelihood.
- For some large N, say $N=1000$, repeat the following three steps, $k=\{1,\cdots ,N\}$.
- Generate a random sample ${\widehat{{\mathrm{U}}_{1}}}^{(k)},\cdots ,{\widehat{{\mathrm{U}}_{\mathrm{n}}}}^{(k)}$ from copula ${C}_{{\theta}_{n}}$.
- Estimate ${\theta}_{n}^{(k)}$ of $\theta $ from $\widehat{{\mathrm{U}}_{1}},\cdots ,\widehat{{\mathrm{U}}_{\mathrm{n}}}$ via maximum likelihood.
- Collect ${\theta}_{n}^{(k)}$.

## 4. Concluding Remarks

## Funding

## Conflicts of Interest

## Abbreviations

rv | random variable |

SJC | Symmetrized Joe-Clayton |

GAS | Generalized Autoregressive Score |

t.i. | tail index |

cdf | cumulative distribution function |

t.s. | test statistic |

DCC | Dynamic Conditional Correlation |

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1 | Uniform transforms can be defined in the following way: if x has cumulative distribution function ${F}_{1}$ and y has cumulative distribution function ${F}_{2}$, then $u={F}_{1}(x)$ and $v={F}_{2}(y)$ are standard uniform distributed. |

2 | We make use of the classical tail dependence indices for simplicity; as will be seen later, they can be easily obtained using a closed form formula, once a particular copula function is settled, but the reader should be aware of the shortcomings in the use of classical tail indices when moving apart from the "Gaussian world". For details, see Furman et al. (2016), where the authors urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. |

**Figure 1.**Upper and lower tail indices of FTSE MIB–DAX 30. (

**a**) BB1 with dynamic Equation (6); (

**b**) BB1 copula with dynamic Equation (7).

**Figure 2.**Upper and lower tail indices of FTSE MIB–IBEX 35; (

**a**) BB1 copula with dynamic Equation (6); (

**b**) BB1 copula with dynamic Equation (7).

**Figure 3.**Upper and lower tail indices of FTSE MIB–DAX 30; (

**a**) SJC copula with dynamic Equation (6); (

**b**) SJC copula with dynamic Equation (7).

**Figure 4.**Upper and lower tail indices of FTSE MIB–IBEX 35; (

**a**) SJC copula with dynamic Equation (6); (

**b**) SJC copula with dynamic Equation (7).

**Figure 5.**Value-at-Risk (VaR) forecasts time patterns considering an SJC copula with dynamic Equation (6) (

**a**,

**b**), an SJC–GAS copula (

**c**,

**d**), and a DCC model (

**e**,

**f**). (

**a**,

**c**,

**e**) refer to the FTSE MIB–DAX 30 portfolio VaR, while (

**b**,

**d**,

**f**) to the FTSE MIB–IBEX 35 portofolio.

**Table 1.**Descriptive statistics of the three stock indexes, FTSE (Financial Times Stock Exchange) MIB (Milano Indice di Borsa), IBEX 35 (Spanish Exchange Index), and DAX 30 (German Exchange Index).

FTSE MIB | IBEX 35 | DAX 30 | |
---|---|---|---|

Mean | −0.08599 | −0.00012 | 0.00020 |

Median | 0.00028 | 0.00062 | 0.00089 |

Maximum | 0.10874 | 0.13484 | 0.10797 |

Minimum | −0.08599 | −0.09586 | −0.07433 |

St. Deviation | 0.016844 | 0.01601 | 0.01463 |

Skewness | −0.06100 | 0.01222 | 0.01011 |

Kurtosis | 6.83680 | 8.4853 | 8.65864 |

FTSE MIB | IBEX 35 | DAX 30 | FTSE MIB | IBEX 35 | DAX 30 | ||
---|---|---|---|---|---|---|---|

${\theta}_{1}$ | −0.043168 | 0.044679 | −0.026672 | $\omega $ | −0.094826 | −0.166808 | −0.169652 |

(0.004125) | (0.041526) | (0.005380) | (0.001587) | (0.005469) | (0.002550) | ||

${\theta}_{2}$ | 0.944940 | 0.845535 | 0.955586 | $\alpha $ | −0.144421 | −0.152084 | −0.178421 |

(0.001412) | ( 0.074948) | (0.005338) | (0.013629) | (0.018368) | (0.018107) | ||

${\varphi}_{1}$ | 0.020101 | −0.009836 | 0.056801 | $\beta $ | 0.989424 | 0.980849 | 0.980906 |

(0.008237) | (0.044879) | (0.000259) | (0.000032) | (0.0008379) | (0.000058) | ||

${\varphi}_{2}$ | −0.921542 | −0.848789 | −0.939253 | $\gamma $ | 0.094497 | 0.111731 | 0.118315 |

(0.000008) | (0.083562) | (0.000007) | (0.014279) | (0.0173419) | (0.008174) | ||

${\varphi}_{3}$ | 0.053048 | - | - | $\nu $ | 9.392484 | 8.374261 | 7.671721 |

(0.008330) | - | - | (1.641448) | (0.901715) | (1.226093) |

FTSE MIB | IBEX 35 | DAX 30 | |
---|---|---|---|

KS statistic | 0.022156 | 0.015969 | 0.018821 |

p-value | 0.2089 | 0.6005 | 0.3891 |

**Table 4.**Fitted coefficients, Cramér–von Mises test statistic (t.s.), and log-likelihood for the BB1, and for the Symmetrized Joe-Clayton (SJC) copulas with dynamic as in Equation (6).

BB1 | SJC | ||||
---|---|---|---|---|---|

FTSE MIB–IBEX 35 | FTSE MIB–DAX 30 | FTSE MIB–IBEX 35 | FTSE MIB–DAX 30 | ||

${\omega}_{U}$ | −0.30000567 | 0.40884762 | ${\omega}_{U}$ | 0.24838033 | 0.09810473 |

(0.03716655) | (−0.28192092) | (0.24721904) | (0.2551627) | ||

${\alpha}_{U}$ | 0.03950425 | −0.94995981 | ${\alpha}_{U}$ | 0.85179066 | 0.96073183 |

(0.29152632) | (−0.19627552) ) | (0.85034764) | (0.8588131) | ||

${\beta}_{U}$ | −4.94732447 | −16.59315866 | ${\beta}_{U}$ | 0.02490790 | −0.31265168 |

(3.35765376) | (7.20800344) | (0.02766444) | (−0.4735605) | ||

${\omega}_{L}$ | 1.28118967 | 0.01167368 | ${\omega}_{L}$ | −0.04173219 | −0.07179274 |

(1.09177413) | (0.30021851) | (−0.04417824) | (−0.1359612) | ||

${\alpha}_{L}$ | 0.17951749 | 0.99249420 | ${\alpha}_{L}$ | 0.84924889 | 0.80251885 |

(0.46726049) | (0.83081313) | (0.84873280) | (0.5257496) | ||

${\beta}_{L}$ | −2.58514184 | −0.03026523 | ${\beta}_{L}$ | 0.03121317 | 0.14441369 |

(−2.39938572) | (−0.06393856) | (0.02994828) | (−0.1443658) | ||

${S}_{n}$ | 0.2059648 | 0.2754048 | ${S}_{n}$ | 0.01920454 | 0.3144344 |

p-value | 0.0004995 | 0.0004995 | p-value | 0.6728272 | 0.2442557 |

Log-Lik | 1576.834 | 1387.445 | Log-Lik | 1512.259 | 1343.717 |

**Table 5.**Fitted coefficients, Cramér–von Mises t.s., and log-likelihood for the BB1, and for the SJC copulas with generalized autoregressive score model (GAS) dynamic as in Equation (7).

BB1–GAS | SJC–GAS | ||||
---|---|---|---|---|---|

FTSE MIB–IBEX 35 | FTSE MIB–DAX 30 | FTSE MIB–IBEX 35 | FTSE MIB–DAX 30 | ||

${\omega}_{U}$ | −1.31443861 | −0.003463384 | ${\omega}_{U}$ | 0.25368733 | 0.28971776 |

(-0.89944562) | (−0.46987585) | (0.24816560) | (0.29316496) | ||

${\alpha}_{U}$ | −0.26458349 | 0.996577947 | ${\alpha}_{U}$ | 0.84285325 | 0.81958206 |

(0.05733857) | (0.26361708) | (0.84930233) | (0.80452288) | ||

${\beta}_{U}$ | 0.66099308 | 0.066533564 | ${\beta}_{U}$ | 0.10302827 | 0.04663823 |

(0.48235972) | (0.01623878) | (0.03045566) | (0.04545464) | ||

${\omega}_{L}$ | 0.06892752 | 0.080316329 | ${\omega}_{L}$ | −0.03158979 | −0.03777936 |

(0.08599858) | (0.08656701) | (−0.04297152) | (−0.05679195) | ||

${\alpha}_{L}$ | 0.94315204 | 0.926766139 | ${\alpha}_{L}$ | 0.85087467 | 0.79993059 |

(0.93582273) | (0.92903208) | (0.84992074) | (0.80080747) | ||

${\beta}_{L}$ | 0.05274755 | 0.068472429 | ${\beta}_{L}$ | 0.03762481 | 0.03008048 |

(0.05108619) | (0.06695990) | (0.01961471) | (0.04271727) | ||

${S}_{n}$ | 0.2372009 | 0.4471988 | ${S}_{n}$ | 0.08591551 | 0.03982476 |

p-value | 0.00075 | 0.000487 | p-value | 0.02147852 | 0.7787213 |

Log-Lik | 1584.072 | 1413.833 | Log-Lik | 1542.525 | 1319.508 |

**Table 6.**Test statistic (t.s.) and, in brackets, p-values for Kupiec test and Christoffersen test, considering $1\%$–VaR on an equally weighted portfolio.

FTSE MIB—IBEX 35 | ||
---|---|---|

Kupiec Test | Christoffersen Test | |

(Unconditional Coverage) | (Conditional Coverage) | |

SJC | 1.006591 | 3.670008 |

(0.3157209) | (0.1596129) | |

SJC - GAS | 0.3804048 | 3.774803 |

(0.5373867) | (0.1514649) | |

DCC | 72.16992 | 77.56573 |

(0) | (0) | |

FTSE MIB—DAX 30 | ||

Kupiec Test | Christoffersen Test | |

(Unconditional Coverage) | (Conditional Coverage) | |

SJC | 4.304982 | 5.927579 |

(0.03800091) | (0.05162293) | |

SJC—GAS | 5.80171 | 7.142801 |

(0.0160106) | (0.02811646) | |

DCC | 64.669 | 74.18366 |

(0) | (0) |

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**MDPI and ACS Style**

Cortese, F.P.
Tail Dependence in Financial Markets: A Dynamic Copula Approach. *Risks* **2019**, *7*, 116.
https://doi.org/10.3390/risks7040116

**AMA Style**

Cortese FP.
Tail Dependence in Financial Markets: A Dynamic Copula Approach. *Risks*. 2019; 7(4):116.
https://doi.org/10.3390/risks7040116

**Chicago/Turabian Style**

Cortese, Federico Pasquale.
2019. "Tail Dependence in Financial Markets: A Dynamic Copula Approach" *Risks* 7, no. 4: 116.
https://doi.org/10.3390/risks7040116