**Figure 1.**
Trace plots and density plots of a single run from each of our four sampling methods on Kendall’s $\tau $ scale. The output is based on $\mathrm{10,000}$ posterior samples of ${z}_{1}$ in the mixed $\tau $ scenario.

**Figure 1.**
Trace plots and density plots of a single run from each of our four sampling methods on Kendall’s $\tau $ scale. The output is based on $\mathrm{10,000}$ posterior samples of ${z}_{1}$ in the mixed $\tau $ scenario.

**Figure 2.**
Daily log-returns (grey) of ACA from 2004 through 2013 with DLM one-day ahead 90% confidence bands (blue) and forecast value (dark blue).

**Figure 2.**
Daily log-returns (grey) of ACA from 2004 through 2013 with DLM one-day ahead 90% confidence bands (blue) and forecast value (dark blue).

**Figure 3.**
Kendall’s $\tau $ posterior modes with corresponding 90% credible intervals of copula parameters of the latent factor copula model with Gumbel (red), Gaussian (blue) and survival Gumbel (green) linking copulas.

**Figure 3.**
Kendall’s $\tau $ posterior modes with corresponding 90% credible intervals of copula parameters of the latent factor copula model with Gumbel (red), Gaussian (blue) and survival Gumbel (green) linking copulas.

**Figure 4.**
Contour plots with standard normal margins for ACA and latent factor pairs $({u}_{\mathrm{ACA},t},{\widehat{v}}_{t})$ for 2005 (**Top**) and 1 July 2008 to 30 June 2009 (**Bottom**). Empirical densities are indicated by solid black lines, while dotted blue lines show the theoretical densities.

**Figure 4.**
Contour plots with standard normal margins for ACA and latent factor pairs $({u}_{\mathrm{ACA},t},{\widehat{v}}_{t})$ for 2005 (**Top**) and 1 July 2008 to 30 June 2009 (**Bottom**). Empirical densities are indicated by solid black lines, while dotted blue lines show the theoretical densities.

**Figure 5.**
Historical relative portfolio value of a constant mix strategy with equal weights in the 8 bank stocks ACA, BBVA, BNP, CBK, DBK, GLE, ISP and SAN for years 2006 to 2013. Portfolio weights are readjusted daily. The 90% empirical VaR is shown in blue and the 90% empirical ES is in green.

**Figure 5.**
Historical relative portfolio value of a constant mix strategy with equal weights in the 8 bank stocks ACA, BBVA, BNP, CBK, DBK, GLE, ISP and SAN for years 2006 to 2013. Portfolio weights are readjusted daily. The 90% empirical VaR is shown in blue and the 90% empirical ES is in green.

**Figure 6.**
Daily log-returns of the equally weighted constant mix portfolio with (negative) 90% VaR (blue line) and ES (green line) forecasts.

**Figure 6.**
Daily log-returns of the equally weighted constant mix portfolio with (negative) 90% VaR (blue line) and ES (green line) forecasts.

**Table 1.**
Density functions and Kendall’s $\tau $ as a function of the parameter of the Gaussian and Gumbel pair copulas.

**Table 1.**
Density functions and Kendall’s $\tau $ as a function of the parameter of the Gaussian and Gumbel pair copulas.

Copula | Density Function | $\mathit{\tau}=\mathit{h}\left(\mathit{\theta}\right)$ |
---|

Gaussian | ${c}_{\mathcal{N}}({u}_{1},{u}_{2};\theta )=\frac{1}{\sqrt{1-{\delta}^{2}}2\pi \varphi \left({\mathsf{\Phi}}^{-1}\left({u}_{1}\right)\right)\varphi \left({\mathsf{\Phi}}^{-1}\left({u}_{2}\right)\right)}\times \phantom{\rule{3.0pt}{0ex}}exp\left(-\frac{{\mathsf{\Phi}}^{-1}\left({u}_{1}\right)+{\mathsf{\Phi}}^{-1}\left({u}_{2}\right)-2\delta {\mathsf{\Phi}}^{-1}\left({u}_{1}\right){\mathsf{\Phi}}^{-1}\left({u}_{2}\right)}{2(1-{\delta}^{2})}\right)$ | ${h}_{\mathcal{N}}\left(\theta \right)=\frac{2}{\pi}arcsin\left(\theta \right)$ |

Gumbel | ${c}_{\mathcal{G}}({u}_{1},{u}_{2};\theta )=\frac{1}{{u}_{1}{u}_{2}}{\left({x}_{1}{x}_{2}\right)}^{\theta -1}\times exp\left(\frac{-{{x}_{1}}^{\theta}-{{x}_{2}}^{\theta}}{\theta}\right)\times \left((1-\theta ){\left(-{{x}_{1}}^{\theta}-{{x}_{2}}^{\theta}\right)}^{\frac{1}{\theta}-2}+{\left(-{{x}_{1}}^{\theta}-{x}_{2}^{\theta}\right)}^{\frac{2}{\theta}-2}\right)$, where ${x}_{1}=-ln{u}_{1}$ and ${x}_{2}=-ln{u}_{2}$ | ${h}_{\mathcal{G}}\left(\theta \right)=1-\frac{1}{\theta}$ |

Survival Gumbel | ${c}_{\mathcal{SG}}({u}_{1},{u}_{2};\theta )={c}_{\mathcal{G}}(1-{u}_{1},1-{u}_{2};\theta )$ | ${h}_{\mathcal{SG}}\left(\theta \right)=1-\frac{1}{\theta}$ |

**Table 2.**
Pairs plots for the Gaussian, Gumbel and survival Gumbel copulas for different parameter values. As the Gumbel and survival Gumbel copulas only exhibit positive dependence, the ${90}^{\circ}$ and ${270}^{\circ}$ rotations of the Gumbel copulas are shown for negative Fisher z parameters.

**Table 3.**
Copula parameters used in the simulation to model the dependence between each marginal series ${\mathit{u}}_{:,j}:=({u}_{1j},\dots ,{u}_{Tj})$ and the latent factor $\mathit{v}$.

**Table 3.**
Copula parameters used in the simulation to model the dependence between each marginal series ${\mathit{u}}_{:,j}:=({u}_{1j},\dots ,{u}_{Tj})$ and the latent factor $\mathit{v}$.

| ${\mathit{c}}_{\mathbf{1}}$ | ${\mathit{c}}_{\mathbf{2}}$ | ${\mathit{c}}_{\mathbf{3}}$ | ${\mathit{c}}_{\mathbf{4}}$ | ${\mathit{c}}_{\mathbf{5}}$ |
---|

| **Low** $\mathit{\tau}$ |

$\tau $ | 0.10 | 0.12 | 0.15 | 0.18 | 0.20 |

$\theta $ | 1.11 | 1.14 | 1.18 | 1.21 | 1.25 |

z | 0.10 | 0.13 | 0.15 | 0.18 | 0.20 |

| **High** $\mathit{\tau}$ |

$\tau $ | 0.50 | 0.57 | 0.65 | 0.73 | 0.80 |

$\theta $ | 2.00 | 2.35 | 2.86 | 3.64 | 5.00 |

z | 0.55 | 0.65 | 0.78 | 0.92 | 1.10 |

| **Mixed** $\mathit{\tau}$ |

$\tau $ | 0.10 | 0.28 | 0.45 | 0.62 | 0.80 |

$\theta $ | 1.11 | 1.38 | 1.82 | 2.67 | 5.00 |

z | 0.10 | 0.28 | 0.48 | 0.73 | 1.10 |

**Table 4.**
Mean absolute deviation (MAD), mean squared error (MSE), effective sample size (ESS) per minute, and realized coverage of 95% credible intervals (C.I.) averaged across all variables ${\tau}_{1},\dots ,{\tau}_{5}$ and $N=100$ replications. Runtime is for 10,000 MCMC iterations on a 16-core node.

**Table 4.**
Mean absolute deviation (MAD), mean squared error (MSE), effective sample size (ESS) per minute, and realized coverage of 95% credible intervals (C.I.) averaged across all variables ${\tau}_{1},\dots ,{\tau}_{5}$ and $N=100$ replications. Runtime is for 10,000 MCMC iterations on a 16-core node.

| ARMGS | MCM | EVM | IRW | MLE |
---|

| **Low** $\mathit{\tau}$ |

MAD | 0.1088 | **0.0900** | 0.1105 | 0.1032 | 0.0739 |

MSE | 0.0314 | **0.0129** | 0.0327 | 0.0262 | 0.0105 |

ESS/min | 6.1 | 0.5 | 0.2 | **9.2** | n/a |

95% C.I. | **0.96** | 0.61 | **0.94** | 0.93 | n/a |

| **High** $\mathit{\tau}$ |

MAD | **0.0292** | **0.0292** | 0.0293 | 0.0294 | 0.0212 |

MSE | **0.0014** | **0.0014** | **0.0014** | **0.0014** | 0.0007 |

ESS/min | **23.7** | 1.6 | 1.4 | 22.2 | n/a |

95% C.I. | **0.95** | 0.91 | **0.95** | **0.95** | n/a |

| **Mixed** $\mathit{\tau}$ |

MAD | 0.0509 | **0.0434** | 0.0517 | 0.0506 | 0.0339 |

MSE | 0.0043 | **0.0030** | 0.0045 | 0.0042 | 0.0017 |

ESS/min | 21.4 | 1.9 | 1.1 | **29.2** | n/a |

95% C.I. | **0.93** | 0.84 | **0.93** | 0.90 | n/a |

| **Average across All Scenarios** |

Runtime | 165s | 664s | 776s | **55s** |

**Table 5.**
Mean absolute deviation (MAD), mean squared error (MSE), effective sample size (ESS) per minute, and realized coverage of 95% credible intervals (C.I.) averaged across all latent variables ${v}_{1},\dots ,{v}_{T}$ and $N=100$ replications.

**Table 5.**
Mean absolute deviation (MAD), mean squared error (MSE), effective sample size (ESS) per minute, and realized coverage of 95% credible intervals (C.I.) averaged across all latent variables ${v}_{1},\dots ,{v}_{T}$ and $N=100$ replications.

| ARMGS | MCM | EVM | IRW |
---|

| **Low** $\mathit{\tau}$ |

MAD | 0.2808 | 0.2975 | **0.2805** | 0.2811 |

MSE | **0.1248** | 0.1397 | **0.1248** | 0.1251 |

ESS/min | 25.7 | 0.5 | 1.1 | **63.5** |

95% C.I. | **0.91** | 0.71 | 0.89 | 0.88 |

| **High** $\mathit{\tau}$ |

MAD | 0.0709 | **0.0678** | 0.0710 | 0.0709 |

MSE | 0.0095 | **0.0087** | 0.0095 | 0.0095 |

ESS/min | **43.7** | 1.6 | 2.6 | 38.8 |

95% C.I. | **0.95** | 0.80 | **0.95** | 0.94 |

| **Mixed** $\mathit{\tau}$ |

MAD | 0.0828 | 0.0898 | **0.0824** | 0.0826 |

MSE | 0.0132 | 0.0154 | **0.0131** | **0.0131** |

ESS/min | 25.8 | 1.5 | 2.0 | **34.3** |

95% C.I. | **0.88** | 0.78 | 0.87 | 0.83 |

**Table 6.**
Ticker symbol, company name and exchange of the selected bank stocks.

**Table 6.**
Ticker symbol, company name and exchange of the selected bank stocks.

Ticker | Company Name | Exchange |
---|

ACA.PA | Credit Agricole S.A. | Euronext - Paris |

BBVA.MC | Banco Bilbao Vizcaya Argentaria | Madrid Stock Exchange |

BNP.PA | BNP Paribas SA | Euronext - Paris |

CBK.DE | Commerzbank AG | XETRA |

DBK.DE | Deutsche Bank AG | XETRA |

GLE.PA | Societe Generale Group | Euronext - Paris |

ISP.MI | Intesa Sanpaolo S.p.A. | Borsa Italiana |

SAN.MC | Banco Santander | Madrid Stock Exchange |

**Table 7.**
Percentage of observed log-returns above and below sequentially out-of-sample 90% forecast interval.

**Table 7.**
Percentage of observed log-returns above and below sequentially out-of-sample 90% forecast interval.

| ACA | BBVA | BNP | CBK | DBK | GLE | ISP | SAN |
---|

Above 95% bound | 4.90% | 4.90% | 4.43% | 4.86% | 4.22% | 4.73% | 4.73% | 4.61% |

Below 5% bound | 4.35% | 5.07% | 4.65% | 4.43% | 5.29% | 4.78% | 5.33% | 5.16% |

**Table 8.**
Frequency of 90% VaR violations and 90% ES of an equals weights portfolio of all eight selected financial stocks, and p-values of conditional coverage test of VaR violations. $R{V}_{R}$ denotes the multivariate regular vine copula model; the other columns are for the dynamic factor copula model with the respective linking copula families. The best values are emphasized in bold.

**Table 8.**
Frequency of 90% VaR violations and 90% ES of an equals weights portfolio of all eight selected financial stocks, and p-values of conditional coverage test of VaR violations. $R{V}_{R}$ denotes the multivariate regular vine copula model; the other columns are for the dynamic factor copula model with the respective linking copula families. The best values are emphasized in bold.

| | Gumbel | Gaussian | Survival Gumbel | |
---|

| ${\mathit{RV}}_{\mathit{R}}$ | ARMGS | MLE | ARMGS | MLE | ARMGS | MLE |
---|

90% VaR viol. | 9.64% | **10.17%** | 10.60% | 9.59% | 9.36% | 9.35% | 9.17% |

90% ES | 4.07% | 4.00% | 3.88% | 4.10% | 4.08% | **4.13%** | 4.08% |

p-value, cond. coverage test | 0.24 | **0.44** | 0.07 | 0.30 | 0.01 | 0.10 | 0.03 |