# Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Latent Factor Copulas

#### Linking Copulas

#### Likelihood

#### Frequentist Inference

## 3. Bayesian Posterior Analysis

#### 3.1. Prior Distributions

#### Prior for Copula Parameters $\mathit{z}$

#### Prior for Latent Factor $\mathit{v}$

#### Joint Prior for $\mathit{z}$ and $\mathit{v}$

#### 3.2. Posterior Distribution and Full Conditionals

#### Posterior Density

#### Full Conditionals

#### 3.3. MCMC Methods for Posterior Simulation

- Metropolis-Hastings within Gibbs sampling (Metropolis et al. (1953); Hastings (1970); Gelfand and Smith (1990)):
- -
- MCM: Truncated normal proposals for $\mathit{z}$ and Beta proposals for $\mathit{v}$; proposal mode and curvature match those of the full conditionals;
- -
- EVM: Gamma proposals for $\mathit{z}$ and Beta proposals for $\mathit{v}$; proposal expectation and variance match those of the full conditionals;
- -
- IRW: Truncated normal random walk proposals for $\mathit{z}$ and uniform independence proposals $\mathit{v}$; and

- ARMGS: Adaptive rejection Metropolis sampling within Gibbs sampling (Gelfand and Smith (1990); Gilks et al. (1995)).

## 4. Simulation Study

#### 4.1. Simulation Setup

#### Scenarios

#### Simulation Data

#### Posterior Simulation

#### 4.2. Results

#### Kendall’s $\tau $ Pair Copula Parameters

#### Latent Factor States

## 5. Application: Portfolio Value at Risk of European Financial Stocks

#### 5.1. Marginal Analysis

#### Time Series DLMs

#### Results

#### 5.2. Bayesian Latent Factor Copula Analysis

#### 5.3. Value at Risk and Expected Shortfall Forecasts

#### Forecasting Method

#### Portfolio Composition

#### Value at Risk Forecasts

#### Expected Shortfall Forecasts

## 6. Summary Comments

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. MCMC Sampling Methods

#### Appendix A.1. Details on MCMC Sampling Schemes

#### Mode and Curvature Matching (MCM)

#### Expectation and Variance Matching (EVM)

#### Independence and Random Walk Samplers (IRW)

#### Adaptive Rejection Metropolis within Gibbs Sampling (ARMGS)

#### Appendix A.2. Strategy to Find Starting Values for MCMC

#### Starting Values for z:

#### Starting Values for v:

#### Formal Procedure:

- Calculate the empirical $d\times d$ Kendall’s $\tau $ matrix $\mathit{T}={\left({\tau}_{ij}\right)}_{i=1:d,j=1:d}$, where ${\tau}_{ij}$ denotes the empirical Kendall’s $\tau $ between series ${\mathit{u}}_{:,i}$ and ${\mathit{u}}_{:,j}$.
- Find the column ${j}^{*}$ of $\mathit{T}$ that maximizes the sum of absolute Kendall’s $\tau $’s,$${j}^{*}=\underset{j=1:d}{arg\; max}\left(\right)open="\{"\; close="\}">\sum _{i=1:d}\left|{\tau}_{ij}\right|$$
- Set the starting values ${\mathit{z}}^{0}=({z}_{1}^{0},\dots ,{z}_{d}^{0})$ for the Fisher z parameters $\mathit{z}$ of the linking copulas to$$\begin{array}{cc}\hfill {z}_{i}^{0}& =z\left(\right)open="("\; close=")">{\tau}_{i{j}^{*}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i\in \{1:d\}\backslash \left\{{j}^{*}\right\}\hfill \end{array}$$
- Set the starting values ${\mathit{v}}^{0}={\left({v}_{t}^{0}\right)}_{t=1:T}$ of the latent factor $\mathit{v}$ to$${v}_{t}^{0}=\underset{v\in (0,1)}{arg\; max}\left(\right)open="\{"\; close="\}">p\left(v\right|{\mathit{u}}_{t},{\mathit{z}}^{0})$$

## Appendix B. Detailed Simulation Results

**Table A1.**Mean absolute deviation (MAD), mean squared errors (MSE), effective sample size (ESS) per minute, and coverage of 90% and 95% credible intervals for Kendall’s $\tau $ posterior sample. Results are averages over 100 independent replications of the analysis.

Low $\mathit{\tau}$ | High $\mathit{\tau}$ | Mixed $\mathit{\tau}$ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

${\tau}_{1}$ | MAD | 0.0829 | 0.0762 | 0.0898 | 0.0829 | 0.0656 | 0.0359 | 0.0352 | 0.0359 | 0.0357 | 0.0260 | 0.0431 | 0.0427 | 0.0432 | 0.0431 | 0.0321 |

MSE | 0.0203 | 0.0104 | 0.0269 | 0.0203 | 0.0099 | 0.0020 | 0.0020 | 0.0020 | 0.0020 | 0.0011 | 0.0029 | 0.0028 | 0.0029 | 0.0029 | 0.0016 | |

ESS/min | 9.5 | 0.5 | 0.3 | 13.7 | n/a | 37.2 | 2.5 | 2.1 | 35.5 | n/a | 46.8 | 3.1 | 1.6 | 60.2 | n/a | |

90% C.I. | 0.92 | 0.65 | 0.93 | 0.92 | n/a | 0.89 | 0.88 | 0.89 | 0.88 | n/a | 0.97 | 0.96 | 0.97 | 0.97 | n/a | |

95% C.I. | 0.97 | 0.76 | 0.96 | 0.95 | n/a | 0.94 | 0.94 | 0.94 | 0.94 | n/a | 0.98 | 0.98 | 0.97 | 0.98 | n/a | |

${\tau}_{2}$ | MAD | 0.0936 | 0.0832 | 0.0831 | 0.0916 | 0.0652 | 0.0334 | 0.0338 | 0.0335 | 0.0336 | 0.0259 | 0.0487 | 0.0486 | 0.0483 | 0.0492 | 0.0388 |

MSE | 0.0246 | 0.0116 | 0.0156 | 0.0229 | 0.0076 | 0.0017 | 0.0018 | 0.0017 | 0.0018 | 0.0010 | 0.0037 | 0.0037 | 0.0036 | 0.0038 | 0.0022 | |

ESS/min | 7.9 | 0.6 | 0.2 | 11.6 | n/a | 33.2 | 2.2 | 1.8 | 31.5 | n/a | 37.6 | 2.9 | 1.8 | 46.7 | n/a | |

90% C.I. | 0.91 | 0.58 | 0.92 | 0.89 | n/a | 0.83 | 0.81 | 0.84 | 0.81 | n/a | 0.85 | 0.85 | 0.87 | 0.82 | n/a | |

95% C.I. | 0.96 | 0.67 | 0.96 | 0.93 | n/a | 0.92 | 0.88 | 0.92 | 0.92 | n/a | 0.92 | 0.91 | 0.92 | 0.92 | n/a | |

${\tau}_{3}$ | MAD | 0.1070 | 0.0913 | 0.1144 | 0.0930 | 0.0811 | 0.0276 | 0.0279 | 0.0276 | 0.0277 | 0.0198 | 0.0406 | 0.0399 | 0.0407 | 0.0406 | 0.0290 |

MSE | 0.0298 | 0.0131 | 0.0354 | 0.0163 | 0.0137 | 0.0012 | 0.0012 | 0.0012 | 0.0012 | 0.0006 | 0.0026 | 0.0025 | 0.0026 | 0.0026 | 0.0013 | |

ESS/min | 5.1 | 0.5 | 0.2 | 8.1 | n/a | 25.9 | 1.8 | 1.5 | 23.9 | n/a | 19.1 | 2.2 | 1.5 | 31.1 | n/a | |

90% C.I. | 0.91 | 0.50 | 0.86 | 0.88 | n/a | 0.90 | 0.87 | 0.90 | 0.90 | n/a | 0.87 | 0.87 | 0.86 | 0.84 | n/a | |

95% C.I. | 0.96 | 0.61 | 0.95 | 0.93 | n/a | 0.95 | 0.93 | 0.96 | 0.95 | n/a | 0.93 | 0.93 | 0.93 | 0.92 | n/a | |

${\tau}_{4}$ | MAD | 0.1142 | 0.0983 | 0.1144 | 0.1231 | 0.0787 | 0.0252 | 0.0239 | 0.0254 | 0.0253 | 0.0172 | 0.0404 | 0.0401 | 0.0407 | 0.0401 | 0.0288 |

MSE | 0.0302 | 0.0145 | 0.0301 | 0.0369 | 0.0120 | 0.0010 | 0.0009 | 0.0010 | 0.0010 | 0.0005 | 0.0026 | 0.0025 | 0.0026 | 0.0026 | 0.0013 | |

ESS/min | 4.7 | 0.6 | 0.2 | 6.8 | n/a | 15.7 | 1.2 | 1.0 | 14.4 | n/a | 2.8 | 0.8 | 0.4 | 6.8 | n/a | |

90% C.I. | 0.88 | 0.42 | 0.89 | 0.83 | n/a | 0.86 | 0.86 | 0.86 | 0.87 | n/a | 0.84 | 0.74 | 0.86 | 0.82 | n/a | |

95% C.I. | 0.94 | 0.49 | 0.91 | 0.89 | n/a | 0.93 | 0.90 | 0.92 | 0.93 | n/a | 0.89 | 0.79 | 0.89 | 0.86 | n/a | |

${\tau}_{5}$ | MAD | 0.1463 | 0.1008 | 0.1507 | 0.1251 | 0.0787 | 0.0241 | 0.0249 | 0.0244 | 0.0245 | 0.0170 | 0.0817 | 0.0459 | 0.0855 | 0.0801 | 0.0406 |

MSE | 0.0524 | 0.0150 | 0.0557 | 0.0346 | 0.0094 | 0.0009 | 0.0010 | 0.0010 | 0.0010 | 0.0004 | 0.0099 | 0.0034 | 0.0106 | 0.0094 | 0.0023 | |

ESS/min | 3.3 | 0.5 | 0.1 | 5.6 | n/a | 6.7 | 0.4 | 0.4 | 5.8 | n/a | 0.8 | 0.4 | 0.1 | 1.0 | n/a | |

90% C.I. | 0.94 | 0.49 | 0.91 | 0.92 | n/a | 0.97 | 0.80 | 0.98 | 0.97 | n/a | 0.81 | 0.48 | 0.82 | 0.71 | n/a | |

95% C.I. | 0.98 | 0.51 | 0.93 | 0.94 | n/a | 1.00 | 0.92 | 1.00 | 1.00 | n/a | 0.93 | 0.57 | 0.92 | 0.81 | n/a |

## Appendix C. Sequential Learning of DLMs

#### Priors at time t:

#### Forecasts at time t:

#### Posteriors at time t:

#### Evolution to time t + 1:

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**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 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 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.

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

Gumbel | ${c}_{\mathcal{G}}({u}_{1},{u}_{2};\theta )=\frac{1}{{u}_{1}{u}_{2}}{\left(\right)}^{{x}_{1}}\theta -1\times \left(\right)open="("\; close=")">(1-\theta ){\left(\right)}^{-}\frac{1}{\theta}-2$, 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.

Fisher’s z | $\mathit{z}=-1.2$ | $\mathit{z}=-0.5$ | $\mathit{z}=0.7$ | $\mathit{z}=1$ |
---|---|---|---|---|

Gaussian | ||||

Gumbel | ||||

Survival Gumbel |

**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.

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.

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 |

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.

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.

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 |

© 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**

Schamberger, B.; Gruber, L.F.; Czado, C.
Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting. *Econometrics* **2017**, *5*, 21.
https://doi.org/10.3390/econometrics5020021

**AMA Style**

Schamberger B, Gruber LF, Czado C.
Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting. *Econometrics*. 2017; 5(2):21.
https://doi.org/10.3390/econometrics5020021

**Chicago/Turabian Style**

Schamberger, Benedikt, Lutz F. Gruber, and Claudia Czado.
2017. "Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting" *Econometrics* 5, no. 2: 21.
https://doi.org/10.3390/econometrics5020021