One-factor Copula Model Research Articles

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.

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.

Bayesian Inference for the One-Factor Copula Model

ABSTRACTWe develop efficient Bayesian inference for the one-factor copula model with two significant contributions over existing methodologies. First, our approach leads to straightforward inference on dependence parameters and the latent factor; only inference on the former is available under frequentist alternatives. Second, we develop a reversible jump Markov chain Monte Carlo algorithm that averages over models constructed from different bivariate copula building blocks. Our approach accommodates any combination of discrete and continuous margins. Through extensive simulations, we compare the computational and Monte Carlo efficiency of alternative proposed sampling schemes. The preferred algorithm provides reliable inference on parameters, the latent factor, and model space. The potential of the methodology is highlighted in an empirical study of 10 binary measures of socio-economic deprivation collected for 11,463 East Timorese households. The importance of conducting inference on the latent factor is motivated by constructing a poverty index using estimates of the factor. Compared to a linear Gaussian factor model, our model average improves out-of-sample fit. The relationships between the poverty index and observed variables uncovered by our approach are diverse and allow for a richer and more precise understanding of the dependence between overall deprivation and individual measures of well-being.

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.

Multi-Population Mortality Models: A Factor Copula Approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA-GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.

The Crash-NIG factor model

It is well known that the one-factor copula models are very useful for risk management and measurement applications involving the generation of scenarios for the complete universe of risk factors and the inclusion of CDO structures in a portfolio context. For this objective, it is necessary to have a simple and fast model that is also consistent with the scenario simulation framework. In this paper we present three extensions of the NIG one-factor copula model which jointly have not been considered so far: (1) tranches with different maturities modeled in a consistent way, (2) a portfolio with different rating buckets, relaxing the assumption of a large homogeneous portfolio, and (3) different correlation regimes. The regime-switching component of the proposed Crash-NIG factor model is especially important in view of the current credit crisis. We also introduce liquidity premiums into the Crash-NIG factor model and show that the actual credit crisis is substantially driven by liquidity effects.

One-Factor Copula-Based Model for Credit Portfolio

Aim of this research paper it to develop an alternative model for portfolio credit risk to those widely used - CreditRisk+ and CreditMetrics. The model aspires to patch the usual weak points of portfolio credit models and also is easy to implement. General engine relies on one-factor copula model and a concept of generating functions. Proposed model also uses only elementary inputs, which make it quick to deploy in the banking environment. The output is the usual credit VaR. We also illustrate various methods related to the implementation and incorporation of credit wrong-way-risk. As a particular case of the implementation, double t–copula is chosen in combination with a stochastic LGD feature. Finally, we show the incidence on capital requirement of various models by testing them on a sample portfolio.

The Crash-NIG copula model: Risk measurement and management of credit portfolios

The one-factor copula models became very popular for modelling dependence in credit portfolios and collateralised debt obligation (CDO) valuation owing to their simplicity. Still, it is also well known that they are too simple for an exact pricing. Nevertheless, it is possible to extend the model in various ways so that it is possible to describe historical correlation behaviour realistically. Such an extension of the one-factor copula model, called the Crash-NIG copula model, is proposed by the authors with the following characteristics: (i) more tail dependence than in the Gaussian case, (ii) consistent term structure dimension, (iii) different rating buckets, relaxing the assumption of a large homogeneous portfolio, and (iv) different correlation regimes. Here the authors demonstrate how to apply this model for generating rating transition and default scenarios of a credit portfolio together with the other relevant risk factors. Repricing of instruments on the simulated scenario paths, that is the most difficult problem for such complex instruments as CDO tranches, can also be done efficiently fast using the same model. Finally, portfolio optimisation can be performed on the derived profit and loss distributions that are shown to be very different from normal.

Correlation smile matching for collateralized debt obligation tranches with α-stable distributions and fitted Archimedean copula models

As an extension of the standard Gaussian copula model to price collateralized debt obligation (CDO) tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded in a framework of stochastic correlations. We furthermore generalize the linear dependence in the usual factor approach to a more general Archimedean copula dependence between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even throughout the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective, all these models are simple, since calculations can be reduced to one-dimensional numerical integrals.