Factor Copula Model Research Articles

Unveiling Mispricing Risks: Nonlarge Homogeneous Portfolio Factor Copula Models for Enhanced Valuation of Subordinated Loan Securitization

AbstractThis paper presents an innovative factor copula model for collateralized loan obligation (CLO) tranche valuation, incorporating non‐Gaussian distributions and dynamic correlations without relying on the large homogeneous portfolio (LHP) assumption. Through numerical analyses and comparisons with LHP models, I find that non‐LHP models produce higher tranche spreads, especially for lower‐rated tranches. Sensitivity analysis reveals varying sensitivities to changes in the number of collaterals, risk‐free rate, average collateral ratings, recovery rates, and time to maturity. The non‐LHP one‐factor copula models, including stochastic correlation and random factor loading models, outperform LHP models in root mean squared errors when calibrated to market data. The results underscore the importance of considering model limitations in CLO tranche pricing and highlight potential mispricing of spread risk in higher‐rated tranches using LHP models. The proposed models contribute to a more comprehensive understanding of CLO tranche pricing by accounting for various factors and assumptions influencing fair premiums.

Likelihood Inference for Factor Copula Models with Asymmetric Tail Dependence.

For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.

Measuring systemic risk in Asian foreign exchange markets

This paper measures systemic risk in eight Asian foreign exchange markets between 2015 and 2021. We define systemic risk as the risk of significant devaluation in a large number of currencies. Our measures, derived using a time-varying factor copula model, can take into account heterogeneous and dynamic dependencies among markets. Our empirical findings reveal that (1) systemic risk spiked during the US-China trade conflict and the COVID-19 pandemic; (2) Among the currencies studied, the Japanese yen contributes most to systemic risk, while it is the least vulnerable to systemic shocks; (3) Higher levels of regional trade and financial integration increase a currency's vulnerability to systemic risk in the Asian foreign exchange markets.

Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.

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.

Dynamic factor copula-based modeling for market risk optimization with an application to the real industry in China

As finance returns to its fundamental purpose of serving the real economy, its connections with various industries are strengthening. Accurately depicting the interdependence among these industries and mitigating financial risks has become increasingly critical. The dependence among China's real industries is dynamic rather than static, which is particularly pronounced during the COVID-19 pandemic. In this paper, we propose a dynamic factor model to optimize the risk of high-dimensional portfolios. To describe the dependence structure, we employ the factor copula model, driven by a GAS (Generalized Autoregressive Score) model. By combining the dynamic factor model with a mean-ES (Expected Shortfall) model, we construct a dynamic factor copula-mean-ES model. Our empirical findings, based on an analysis of 24 industries in China, suggest that the dynamic heterogeneous factor copula model is the most suitable for describing portfolio risk. Furthermore, the mean-ES model ensures the lowest portfolio risk for a given expected return. Accurate return predictions enable leveraging market information to develop a "good knowledge" of dynamic copula and risk optimization. This "good knowledge" of dynamic copulas facilitates precise return prediction and effective risk optimization of portfolios, thereby addressing the relationship between risk prevention and sustainability. Moreover, it reveals the internal connection between China's real industry and the risk landscape of the financial market.

Factor Tree Copula Models for Item Response Data

Factor copula models for item response data are more interpretable and fit better than (truncated) vine copula models when dependence can be explained through latent variables, but are not robust to violations of conditional independence. To circumvent these issues, truncated vines and factor copula models for item response data are joined to define a combined model, the so-called factor tree copula model, with individual benefits from each of the two approaches. Rather than adding factors and causing computational problems and difficulties in interpretation and identification, a truncated vine structure is assumed on the residuals conditional on one or two latent variables. This structure can be better explained as a conditional dependence given a few interpretable latent variables. On the one hand, the parsimonious feature of factor models remains intact and any residual dependencies are being taken into account on the other. We discuss estimation along with model selection. In particular, we propose model selection algorithms to choose a plausible factor tree copula model to capture the (residual) dependencies among the item responses. Our general methodology is demonstrated with an extensive simulation study and illustrated by analyzing Post-Traumatic Stress Disorder.

Transform MCMC Schemes for Sampling Intractable Factor Copula Models

In financial risk management, modelling dependency within a random vector $$\mathcal{X}$$ is crucial, a standard approach is the use of a copula model. Say the copula model can be sampled through realizations of $$\mathcal{Y}$$ having copula function C: had the marginals of $$\mathcal{Y}$$ been known, sampling $$\mathcal{X}^{(i)}$$ , the i-th component of $$\mathcal{X}$$ , would directly follow by composing $$\mathcal{Y}^{(i)}$$ with its cumulative distribution function (c.d.f.) and the inverse c.d.f. of $$\mathcal{X}^{(i)}$$ . In this work, the marginals of $$\mathcal{Y}$$ are not explicit, as in a factor copula model. We design an algorithm which samples $$\mathcal{X}$$ through an empirical approximation of the c.d.f. of the $$\mathcal{Y}$$ -marginals. To be able to handle complex distributions for $$\mathcal{Y}$$ or rare-event computations, we allow Markov Chain Monte Carlo (MCMC) samplers. We establish convergence results whose rates depend on the tails of $$\mathcal{X}$$ , $$\mathcal{Y}$$ and the Lyapunov function of the MCMC sampler. We present numerical experiments confirming the convergence rates and also revisit a real data analysis from financial risk management.

Estimation and inference in factor copula models with exogenous covariates

A factor copula model is proposed in which factors are either simulable or estimable from exogenous information. Point estimation and inference are based on a simulated methods of moments (SMM) approach with non-overlapping simulation draws. Consistency and limiting normality of the estimator is established and the validity of bootstrap standard errors is shown. Doing so, previous results from the literature are verified under low-level conditions imposed on the individual components of the factor structure. Monte Carlo evidence confirms the accuracy of the asymptotic theory in finite samples and an empirical application illustrates the usefulness of the model to explain the cross-sectional dependence between stock returns.

Dynamic factor copula models with estimated cluster assignments

This paper proposes a dynamic multi-factor copula for use in high-dimensional time series applications. A novel feature of our model is that the assignment of individual variables to groups is estimated from the data, rather than being pre-assigned using SIC industry codes, market capitalization ranks, or other ad hoc methods. We adapt the k-means clustering algorithm for use in our application and show that it has excellent finite-sample properties. Applying the new model to returns on 110 US equities, we find around 20 clusters to be optimal. In out-of-sample forecasts, we find that a model with as few as five estimated clusters significantly outperforms an otherwise identical model with 21 clusters formed using two-digit SIC codes.

Conditional Dynamic Dependence and Risk Spillover between Crude Oil Prices and Foreign Exchange Rates: New Evidence from a Dynamic Factor Copula Model

This paper proposes a dynamic factor model to accurately characterize the dynamic dependence and risk spillovers between the crude oil and exchange rate returns of oil-trading countries from 2000–2020, conditional on the common factors. To this end, we first identify the common factors related to returns of crude oil prices and exchange rates from 14 typical oil-trading countries. Then, we use the AR-GARCH model to filter the respective idiosyncratic factors and conduct a comparative study of conditional dynamic dependence between the crude oil and exchange rate returns of oil importers and exporters. Finally, we combine the dynamic factor copula model with the CoVaR method to measure the conditional risk spillover effect between crude oil and exchange rate markets. The empirical study indicates that the classical factor analysis can be used to precisely identify the common factors related to both financial markets, with the similar trend of macro-economic indicators. Furthermore, the factor copula model can capture the dynamic structure between crude oil and exchange rate markets more accurately than the traditional Copula–GARCH model. Specifically, the idiosyncratic factors related to each return series still have a significant impact on the dependence between the crude oil and exchange rate returns of oil importers, while the common factors have played an important role in the relationship of oil exporters’ exchange rates with crude oil prices. Finally, the crude oil market has enjoyed a relative risk premium to the exchange rate markets of oil-trading countries. However, there is almost no conditional risk spillover from the corresponding exchange rates to the crude oil prices. Finally, we discuss the implications for investors, policymakers and respective exchange rate regulators from oil-trading countries with further insights from the macro-economic perspective.

Dynamic Factor Copula Models with Estimated Cluster Assignments

This paper proposes a dynamic multi-factor copula for use in high dimensional time series applications. A novel feature of our model is that the assignment of individual variables to groups is estimated from the data, rather than being pre-assigned using SIC industry codes, market capitalization ranks, or other ad hoc methods. We adapt the k-means clustering algorithm for use in our application and show that it has excellent finite-sample properties. Applying the new model to returns on 110 US equities, we find around 20 clusters to be optimal. In out-of-sample forecasts, we find that a model with as few as five estimated clusters significantly outperforms an otherwise identical model with 21 clusters formed using two-digit SIC codes.

EPU spillovers and stock return predictability: A cross-country study

Inspired by interconnectedness of economic policy uncertainty (EPU) across countries, this paper applies the high-dimensional factor-copula model to estimate EPU spillovers as expected probability in distress (EPD) and examines its effect on stock returns across 23 global markets. We find that EPD significantly and negatively predicts country-level returns in global markets. The predictability is significant for both countries in developed markets and countries in emerging markets by EPD that primarily originates from developed markets. Moreover, return predictability through the spillover channel is highly significant under various market conditions and dominates that through the domestic channel. The spillover effect remains strongly significant after the financial crisis, reflecting increased interconnectedness of the post-crisis global markets. Our results are robust to various model specifications and alternative measures of EPU spillovers.

A General Numerical Algorithm for CDO Pricing Based on Single Factor Copula Framework and Nonhomogeneous Assumptions

In view of the fact that different factor Copula models are only applicable to different practical problems in collateralized debt obligations (CDO) market and that there is no semianalytical solution under nonhomogeneous assumptions to CDO pricing model, we designed a general numerical algorithm which was based on the framework of single factor Copula model and randomized quasi-Monte Carlo (RQMC) simulation method. We took two single factor Copula models as examples to conduct empirical study, in which the simulation results of RQMC and Monte Carlo (MC) simulation method were compared and analyzed based on variance changes. The result showed that the algorithm in this paper was not only applicable to general single factor Copula model but also very stable. So, it was a general and efficient numerical method to solve the problem of CDO pricing under nonhomogeneous assumptions.

Application of RQMC for CDO Pricing with Stochastic Correlations under Nonhomogeneous Assumptions

In consideration of that the correlation between any two assets of the asset pool is always stochastic in the actual market and that collateralized debt obligation (CDO) pricing models under nonhomogeneous assumptions have no semianalytic solutions, we designed a numerical algorithm based on randomized quasi‐Monte Carlo (RQMC) simulation method for CDO pricing with stochastic correlations under nonhomogeneous assumptions and took Gaussian factor copula model as an example to conduct experiments. The simulation results of RQMC and Monte Carlo (MC) method were compared from the perspective of variance changes. The results showed that this numerical algorithm was feasible, efficient, and stable for CDO pricing with stochastic correlation under nonhomogeneous assumptions. This numerical algorithm is expected to be extended to other factor Copula models for CDO pricing with stochastic correlations under nonhomogeneous assumptions.

Safe haven assets for international stock markets: A regime-switching factor copula approach

This paper assesses the safe haven properties of a large number of assets during global-level crises in international stock markets. The dataset consists of returns on 36 possible safe haven assets from different asset classes (government bonds, currencies, and commodities) and returns on 46 developed and emerging stock markets over the period from 1999 to 2020. A novel approach based on a regime-switching factor copula model is proposed to examine safe haven properties with such a relatively high-dimensional dataset. We find that, on average over the full sample period, the Japanese yen is the safest asset and the US government bond is the second safest, followed by a particular set of currencies. Government bonds generally have a lower but moderate degree of safe haven property. The ranking among these assets, however, varies depending on major crisis episodes. There is no clear evidence of safe haven roles in the remaining assets.

A multi-year microlevel collective risk model

For a typical insurance portfolio, the claims process for a short period, typically one year, is characterized by observing frequency of claims together with the associated claims severities. The collective risk model describes this portfolio as a random sum of the aggregation of the claim amounts. In the classical framework, for simplicity, the claim frequency and claim severities are assumed to be mutually independent. However, there is a growing interest in relaxing this independence assumption which is more realistic and useful for the practical insurance ratemaking. While the common thread has been capturing the dependence between frequency and aggregate severity within a single period, the work of Oh et al. (2021) provides an interesting extension to the addition of capturing dependence among individual severities. In this paper, we extend these works within a framework where we have a portfolio of microlevel frequencies and severities for multiple years. This allows us to develop a factor copula model framework that captures various types of dependence between claim frequencies and claim severities over multiple years. It is therefore a clear extension of earlier works on one-year dependent frequency-severity models and on random effects model for capturing serial dependence of claims. We focus on the results using a family of elliptical copulas to model the dependence. The paper further describes how to calibrate the proposed model using illustrative claims data arising from a Singapore insurance company. The estimated results provide strong evidences of all forms of dependencies captured by our model.

Factor copula models for right-censored clustered survival data.

In this article we extend the factor copula model to deal with right-censored event time data grouped in clusters. The new methodology allows for clusters to have variable sizes ranging from small to large and intracluster dependence to be flexibly modeled by any parametric family of bivariate copulas, thus encompassing a wide range of dependence structures. Incorporation of covariates (possibly time dependent) in the margins is also supported. Three estimation procedures are proposed: both one- and two-stage parametric and a two-stage semiparametric method where marginal survival functions are estimated by using a Cox proportional hazards model. We prove that the estimators are consistent and asymptotically normally distributed, and assess their finite sample behavior with simulation studies. Furthermore, we illustrate the proposed methods on a data set containing the time to first insemination after calving in dairy cattle clustered in herds of different sizes.

FACTOR COPULA MODEL FOR PORTFOLIO CREDIT RISK

A critical aspect in the valuation and risk management of multi-name credit derivatives is the modeling of the dependence among sources of credit risk. The dependence modeling poses difficulties in the pricing of a multi-name credit derivatives, in the estimation of the value-at-risk of a portfolio, or in the pricing of some other basket credit derivative as the description not only on the default arrival in an individual reference entity but on the default dependence among entities in the portfolio should be considered. Although the elliptical models have been widely used due to their mathematical tractability, the dependence modeling using the multi-dimensional Lévy process has shown growing interest among researchers despite its complexity. In this paper, we introduce one factor copula model for portfolio credit risk based on Normal Tempered Stable (NTS) distribution and calibrate the model through 5-year synthetic Collateralized Debt Obligation (CDO) tranche spreads under a large homogeneous portfolio approximation. The calibration results show that the one factor copula model based on NTS distribution is more flexible and provides a dependence structure fitting market CDO tranche spreads. As one of the major applications of the dependence modeling in credit risk, this model shares the advantage of the Gaussian one factor model, and all extensions and implementation methods used for it can be utilized.

Dynamic Factor Copula Models with Estimated Cluster Assignments

<ns2:p>This paper proposes a dynamic multi-factor copula for use in high dimensional time series applications. A novel feature of our model is that the assignment of individual variables to groups is estimated from the data, rather than being pre-assigned using SIC industry codes, market capitalization ranks, or other ad hoc methods. We adapt the k-means clustering algorithm for use in our application and show that it has excellent finite-sample properties. Applying the new model to returns on 110 US equities, we find around 20 clusters to be optimal. In out-of-sample forecasts, we find that a model with as few as five estimated clusters significantly outperforms an otherwise identical model with 21 clusters formed using two-digit SIC codes.</ns2:p>