Factor Copulas Research Articles

Structured factor copulas for modeling the systemic risk of European and United States banks

In this paper, we employ Credit Default Swaps (CDS) to model the joint and conditional distress probabilities of banks in Europe and the U.S. using factor copulas. We propose multi-factor, structured factor, and factor-vine models where the banks in the sample are clustered according to their geographic location. We find that within each region, the co-dependence between banks is best described using both, systematic and idiosyncratic, financial contagion channels. However, if we consider the banking system as a whole, then the systematic contagion channel prevails, meaning that the distress probabilities are driven by a latent global factor and region-specific factors. In all cases, the co-dependence structure of bank CDS spreads is highly correlated in the tail. The out-of-sample forecasts of several measures of systemic risk allow us to identify the periods of distress in the banking sector over the recent years including the COVID-19 pandemic, the interest rate hikes in 2022, and the banking crisis in 2023.

A study of one-factor copula models from a tail dependence perspective

AbstractModeling multivariate dependence in high dimensions is challenging, with popular solutions constructing multivariate copula as a composition of lower dimensional copulas. Pair-copula constructions do so by using bivariate linking copulas, but their parametrization, in size, being quadratic in the dimension, is not quite parsimonious. Besides, the number of regular vines grows super-exponentially with the dimension. One parsimonious solution is factor copulas, and in particular, the one-factor copula is touted for its simplicity – with the number of parameters linear in the dimension – while being able to cater to asymmetric non-linear dependence in the tails. In this paper, we add nuance to this claim from the point of view of a popular measure of multivariate tail dependence, the tail dependence matrix (TDM). We focus on the one-factor copula model with the linking copula belonging to the BB1 family, pointing out later the applicability of our results to a wider class of linking copulas. For this model, we derive tail dependence coefficients and study their basic properties as functions of the parameters of the linking copulas. Based on this, we study the representativeness of the class of TDMs supported by this model with respect to the class of all possible TDMs. We establish that since the parametrization is linear in the dimension, it is no surprise that the relative volume is zero for dimensions greater than three, and hence, by necessity, we present a novel manner of evaluating the representativeness that has a combinatorial flavor. We formulate the problem of finding the best representative one-factor BB1 model given a target TDM and suggest an implementation along with a simulation study of its performance across dimensions. Finally, we illustrate the results of the paper by modeling rainfall data, which is relevant in the context of weather-related insurance.

High-dimensional factor copula models with estimation of latent variables

Factor models are a parsimonious way to explain the dependence of variables using several latent variables. In Gaussian 1-factor and structural factor models (such as bi-factor and oblique factor) and their factor copula counterparts, factor scores or proxies are defined as conditional expectations of latent variables given the observed variables. With mild assumptions, the proxies are consistent for corresponding latent variables as the sample size and the number of observed variables linked to each latent variable go to infinity. When the bivariate copulas linking observed variables to latent variables are not assumed in advance, sequential procedures are used for latent variables estimation, copula family selection and parameter estimation. The use of proxy variables for factor copulas means that approximate log-likelihoods can be used to estimate copula parameters with less computational effort for numerical integration.

Fast inference methods for high-dimensional factor copulas

Abstract Gaussian factor models allow the statistician to capture multivariate dependence between variables. However, they are computationally cumbersome in high dimensions and are not able to capture multivariate skewness in the data. We propose a copula model that allows for arbitrary margins, and multivariate skewness in the data by including a non-Gaussian factor whose dependence structure is the result of a one-factor copula model. Estimation is carried out using a two-step procedure: margins are modelled separately and transformed to the normal scale, after which the dependence structure is estimated. We develop an estimation procedure that allows for fast estimation of the model parameters in a high-dimensional setting. We first prove the theoretical results of the model with up to three Gaussian factors. Then, simulation results confirm that the model works as the sample size and dimensionality grow larger. Finally, we apply the model to a selection of stocks of the S&P500, which demonstrates that our model is able to capture cross-sectional skewness in the stock market data.

Implicit Copulas: An Overview

Implicit copulas are the most common copula choice for modeling dependence in high dimensions. This broad class of copulas is introduced and surveyed, including elliptical copulas, skew t copulas, factor copulas, time series copulas and regression copulas. The common auxiliary representation of implicit copulas is outlined, and how this makes them both scalable and tractable for statistical modeling. Issues such as parameter identification, extended likelihoods for discrete or mixed data, parsimony in high dimensions, and simulation from the copula model are considered. Bayesian approaches to estimate the copula parameters, and predict from an implicit copula model, are outlined. Particular attention is given to implicit copula processes constructed from time series and regression models, which is at the forefront of current research. Two econometric applications—one from macroeconomic time series and the other from financial asset pricing—illustrate the advantages of implicit copula models.

Quantifying systemic risk with factor copulas

ABSTRACT We propose a tail dependence based network approach to study systemic risk in a network of systemically important financial institutions (SIFIs). We utilize a flexible factor copula based method which allows to measure the level of extreme risk in a portfolio when dependence is driven by one or several factors. We identify the most ‘connected’ SIFIs based on an eigenvector centrality approach applied to copula-implied dependence structures as ‘central’ SIFIs. We then demonstrate that the level of systemic risk implied by such SIFIs chosen as conditioning factors in the factor copula setup exceeds that which is implied by non-central SIFIs in terms of portfolio Value-at-Risk and the portfolio return under stress. This study contributes to quantification and ranking of the systemic importance of SIFIs which is important for setting adequate capital requirements in particular and stability of financial markets in general.

Variational inference for high dimensional structured factor copulas

Factor copula models have been recently proposed for describing the joint distribution of a large number of variables in terms of a few common latent factors. A Bayesian procedure is employed in order to make fast inferences for multi-factor and structured factor copulas. To deal with the high dimensional structure, a Variational Inference (VI) algorithm is applied to estimate different specifications of factor copula models. Compared to the Markov Chain Monte Carlo (MCMC) approach, the variational approximation is much faster and could handle a sizeable problem in limited time. Another issue of factor copula models is that the bivariate copula functions connecting the variables are unknown in high dimensions. An automatic procedure is derived to recover the hidden dependence structure. By taking advantage of the posterior modes of the latent variables, the bivariate copula functions are selected by minimizing the Bayesian Information Criterion (BIC). Simulation studies in different contexts show that the procedure of bivariate copula selection could be very accurate in comparison to the true generated copula model. The proposed procedure is illustrated with two high dimensional real data sets.

Mortality Risk Management Under the Factor Copula Framework—With Applications to Insurance Policy Pools

Mortality risk is one of the core risks that life insurers undertake. The uncertain future lifetime of each insured represents one risk factor, and the dependence structure among these risk factors determines the aggregate risk of an insurance policy pool. We propose using factor copulas to describe the dependence structure among the future lifetimes of numerous insureds. This differs from Chen, MacMinn, and Sun (2015) in that their focus is on pricing the securities linked to several mortality indexes. To mitigate the systematic mortality risk associated with an insurance pool, the insurer may purchase an asset exposed to similar systematic risk. We thus set up a two-factor copula framework and solve for the optimal investment amount in the asset. In numerical illustrations, we employ real-case data from a life insurer and a life settlement market maker involving hundreds of policies.

Quantifying systemic risk with factor copulas

We propose a tail dependence based network approach to study systemic risk in a network of systemically important financial institutions (SIFIs). We utilize a flexible factor copula-based method which allows us to measure the level of extreme risk in a portfolio when dependence is driven by one or several factors. We identify the most ``connected'' SIFIs based on an eigenvector centrality approach applied to copula-implied dependence structures as ``central'' SIFIs. We then demonstrate that the level of systemic risk implied by such SIFIs chosen as conditioning factors in the factor copula setup exceeds that which is implied by non-central SIFIs in terms of portfolio Value-at-Risk and the portfolio return under stress. This study contributes to the quantification and ranking of the systemic importance of SIFIs which is important for setting adequate capital requirements in particular and stability of financial markets in general.

Model and Moment Selection in Factor Copula Models

AbstractThis article develops a simultaneous model and moment selection procedure for factor copula models. Since the density of the factor copula is generally not known in closed form, widely used likelihood or moment-based model selection criteria cannot be directly applied on factor copulas. The new approach is inspired by the methods for generalized methods of moments proposed by Andrews (1999) and Andrews and Lu (2001). The consistency of the procedure is proved and Monte Carlo simulations show its good performance in finite samples in different scenarios of sample sizes and dimensions. The impact of the choice of moments in selected regions of the support on model selection and value-at-risk prediction is further examined by simulation and an application to a portfolio consisting of ten stocks in the Deutscher Aktienindex (DAX30) index.

Risk Measurement of Stock Markets in BRICS, G7, and G20: Vine Copulas versus Factor Copulas

Multivariate copulas have been widely used to handle risk in the financial market. This paper aimed to adopt two novel multivariate copulas, Vine copulas and Factor copulas, to measure and compare the financial risks of the emerging economy, developed economy, and global economy. In this paper, we used data from three groups (BRICS, which stands for emerging markets, specifically, those of Brazil, Russia, India, China, and South Africa; G7, which refers to developed countries; and G20, which represents the global market), separated into three periods (pre-crisis, crisis, and post-crisis) and weighed Value at Risk (VaR) and Expected Shortfall (ES) (based on their market capitalization) to compare among three copulas, C-Vine, D-Vine, and Factor copulas. Also, real financial data demonstrated that Factor copulas have stronger stability and perform better than the other two copulas in high-dimensional data. Moreover, we showed that BRICS has the highest risk and G20 has the lowest risk of the three groups.

Parallel Bayesian Inference for High-Dimensional Dynamic Factor Copulas*

To account for asymmetric dependence in extreme events, we propose a dynamic generalized hyperbolic skew Student-t factor copula where the factor loadings follow generalized autoregressive score pr ...

Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community.[...]

Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Husler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.

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

Factor modeling is a popular strategy to induce sparsity in multivariate models as they scale to higher dimensions. We develop Bayesian inference for a recently proposed latent factor copula model, which utilizes a pair copula construction to couple the variables with the latent factor. We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling for posterior simulation: Gibbs sampling enables application to Bayesian problems, while ARMS is an adaptive strategy that replaces traditional Metropolis-Hastings updates, which typically require careful tuning. Our simulation study shows favorable performance of our proposed approach both in terms of sampling efficiency and accuracy. We provide an extensive application example using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES).

Modeling Dependence in High Dimensions With Factor Copulas

This article presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high-dimensional applications, involving 50 or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider “scree” plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk. Supplementary materials for this article are available online.

Market risk forecasting for high dimensional portfolios via factor copulas with GAS dynamics

In this paper we propose forecasting market risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES), for large dimensional portfolios via copula modeling. For that we compare several high dimensional copula models, from naive ones to complex factor copulas, which are able to simultaneously tackle the curse of dimensionality and introduce a high level of complexity into the model. We explore both static and dynamic copula fitting. In the dynamic case we allow different levels of flexibility for the dependence parameters which are driven by a GAS (Generalized Autoregressive Scores) model, in the spirit of Oh and Patton (2015). Our empirical results, for assets negotiated at Brazilian BOVESPA stock market from January, 2008 to December, 2014, suggest that, compared to the other copula models, the GAS dynamic factor copula approach has a superior performance in terms of AIC (Akaike Information Criterion) and a non-inferior performance with respect to VaR and ES forecasting.

Modelling Dependence in High Dimensions with Factor Copulas

This paper presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high dimensional applications, involving fifty or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider "scree" plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk.

Modelling Dependence in High Dimensions with Factor Copulas

his paper presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high dimensional applications, involving fifty or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider scree plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk.

Modelling Dependence in High Dimensions with Factor Copulas

Modelling Dependence in High Dimensions with Factor Copulas