Score-driven copula models for portfolios of two risky assets

ABSTRACT

The precise measurement of the association between asset returns is important for financial investors and risk managers. In this paper, we focus on a recent class of association models: Dynamic Conditional Score (DCS) copula models. Our contributions are the following: (i) We compare the statistical performance of several DCS copulas for several portfolios. We study the Clayton, rotated Clayton, Frank, Gaussian, Gumbel, rotated Gumbel, Plackett and Student's t copulas. We find that the DCS model with the Student's t copula is the most parsimonious model. (ii) We demonstrate that the copula score function discounts extreme observations. (iii) We jointly estimate the marginal distributions and the copula, by using the Maximum Likelihood method. We use DCS models for mean, volatility and association of asset returns. (iv) We estimate robust DCS copula models, for which the probability of a zero return observation is not necessarily zero. (v) We compare different patterns of association in different regions of the distribution for different DCS copulas, by using density contour plots and Monte Carlo (MC) experiments. (vi) We undertake a portfolio performance study with the estimation and backtesting of MC Value-at-Risk for the DCS model with the Student's t copula.     

Estimating non-linear serial and cross-interdependence between financial assets

This paper proposes an approach based on copula families to determine shape and magnitude of non-linear serial and cross-interdependence between returns and volatilities of financial assets. It is evident the predominance of the student’s t copula in returns relationships. Association in tails is generally larger than the absolute. There is a fast decrease in association along time, but even after 5 days, there is still dependence between returns. For volatilities, Joe copula predominates in estimated bivariate relationships fit. Clayton copula rotated 180° (survival), Gumbel, BB6 and BB8 copulas also fit some relationships. The magnitude of lagged associations is larger for risks than returns. Persistence in the dependences is very high, and decreases very little after the first lag. The tail dependence has larger values than the absolute in most relationships. We present a practical application of the proposed approach, based on optimal investment allocation and risk prediction.     

Copula dynamics in CDOs

Values of tranche spreads of collateralized debt obligations (CDOs) are driven by the joint default performance of the assets in the collateral pool. The dependence between the entities in the portfolio mainly depends on current economic conditions. Therefore, a correlation implied from tranches can be seen as a measure of the general situation of the credit market. We analyse the European market of standardized CDOs using tranches of the iTraxx index in the periods before and during the global financial crisis. We investigate the evolution of the correlations using different copula models: the standard Gaussian, the NIG, the double-t, and the Gumbel copula model. After calibration of these models, one obtains a time varying vector of parameters. We analyse the dynamic pattern of these coefficients. That enables us to forecast future parameters and consequently calculate Value-at-Risk measures for iTraxx Europe tranches.     

Modelling Co-movements and Tail Dependency in the International Stock Market via Copulae

This paper examines international equity market co-movements using time-varying copulae. We examine distributions from the class of Symmetric Generalized Hyperbolic (SGH) distributions for modelling univariate marginals of equity index returns. We show based on the goodness-of-fit testing that the SGH class outperforms the normal distribution, and that the Student-t assumption on marginals leads to the best performance, and thus, can be used to fit multivariate copula for the joint distribution of equity index returns. We show in our study that the Student-t copula is not only superior to the Gaussian copula, where the dependence structure relates to the multivariate normal distribution, but also outperforms some alternative mixture copula models which allow to reflect asymmetric dependencies in the tails of the distribution. The Student-t copula with Student-t marginals allows to model realistically simultaneous co-movements and to capture tail dependency in the equity index returns. From the point of view of risk management, it is a good candidate for modelling the returns arising in an international equity index portfolio where the extreme losses are known to have a tendency to occur simultaneously. We apply copulae to the estimation of the Value-at-Risk and the Expected Shortfall, and show that the Student-t copula with Student-t marginals is superior to the alternative copula models investigated, as well the Riskmetics approach.     

Portfolio optimization in the presence of dependent financial returns with long memory: A copula based approach

In this paper, we seek to examine the effect of the presence of long memory on the dependence structure between financial returns and on portfolio optimization. First, we focus on the dependence structure using copulas. To select the best copula, in addition to the goodness of fit tests, we employ a graphical method based on visual comparison of the fitted copula density and the smoothed copula density estimated by wavelets. Moreover, we check the stability of the copula parameter. The empirical results show that the long memory affects the dependence structure. Second, we analyze the impact of this dependence structure on the optimal portfolio. We propose a new approach based on minimizing the Conditional Value at Risk and assuming that the dependence structure is modeled by the copula parameter. The empirical results show that our approach outperforms the traditional minimizing variance approach, where the dependence structure is represented by the linear correlation coefficient.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

Studying the properties of the Bitcoin as a diversifying and hedging asset through a copula analysis: Constant and time-varying

In this study, we analyze the properties of Bitcoin as a diversifier asset and hedge asset against the movement of international market stock indices: S&P500 (US), STOXX50 (EU), NIKKEI (Japan), CSI300 (Shanghai), and HSI (Hong Kong). For this, we use several copula models: Gaussian, Student-t, Clayton, Gumbel, and Frank. The analysis period runs from August 18, 2011 to June 31, 2019. We found that the Gaussian and Student-t copulas are best at fitting the structure dependence between markets. Also, these copulas suggest that under normal market conditions, Bitcoin might act as a hedge asset against the stock price movements of all international markets analyzed. However, the dependence on the Shanghai and Hong Kong markets was somewhat higher. Also, under extreme market conditions, the role of Bitcoin might change from hedge to diversifier. In a time-varying copula analysis, given by the Student-t copula, we found that even under normal market conditions, for some markets, the role of Bitcoin as a hedge asset might fail on a high number of days.     

Where is the distribution tail threshold? A tale on tail and copulas in financial risk measurement

Estimating the market risk is conditioned by the fat tail of the distribution of returns. But the tail index depends on the threshold of this distribution fat tail. We propose a methodology based on the decomposition of the series into positive outliers, Gaussian central part and negative outliers and uses the latter to estimate this cutoff point. Additionally, from this decomposition, we estimate extreme dependence correlation matrix which is used in the measurement of portfolio risk. For a sample consisting of six assets (Bitcoin, Gold, Brent, Standard&Poor-500, Nasdaq and Real Estate index), we find that our methodology presents better results, in terms of normality and volatility of the tail index, than the Kolmogorov–Smirnov distance, and its unnecessary capital consumption is lower. Also, in the measurement of the risk of a portfolio, the results of our proposal improve those of a t-Student copula and allow us to estimate the extreme dependence and the corresponding indexes avoiding the implicit restrictions of the elliptic and Archimedean copulas.     

Checking for asymmetric default dependence in a credit card portfolio: A copula approach

Traditional credit risk models adopt the linear correlation as a measure of dependence and assume that credit losses are normally-distributed. However some studies have shown that credit losses are seldom normal and the linear correlation does not give accurate assessment for asymmetric data. Therefore it is possible that many credit models tend to misestimate the probability of joint extreme defaults.This paper employs Copula Theory to model the dependence across default rates in a credit card portfolio of a large UK bank and to estimate the likelihood of joint high default rates. Ten copula families are used as candidates to represent the dependence structure. The empirical analysis shows that, when compared to traditional models, estimations based on asymmetric copulas usually yield results closer to the ratio of simultaneous extreme losses observed in the credit card portfolio.Copulas have been applied to evaluate the dependence among corporate debts but this research is the first paper to give evidence of the outperformance of copula estimations in portfolios of consumer loans. Moreover we test some families of copulas that are not typically considered in credit risk studies and find out that three of them are suitable for representing dependence across credit card defaults.     

Portfolio optimization with CVaR under VG process

Formal portfolio optimization methodologies describe the dynamics of financial instruments price with Gaussian Copula (GC). Without considering the skewness and kurtosis of assets return rate, optimization with GC underestimate the optimal CVaR of portfolio. In the present paper, we develop the approach for portfolio optimization by introducing Lévy processes. It focuses on describing the dynamics of assets’ log price with Variance Gamma copula (VGC) rather than GC. A case study for three Indexes of Chinese Stock Market is performed. On application purpose, we calculate the best hedge positions of Shanghai Index (SHI), Shenzhen Index (SZI) and Small Cap Index (SCI) with the performance function CVaR under VG model. It can be combined with Monte Carlo Simulation and nonlinear programming techniques. This framework is suitable for any investment companies.     

Testing the Gaussian copula hypothesis for financial assets dependences

Abstract

Using one of the key properties of copulas that they remain invariant under an arbitrary monotonic change of variable, we investigate the null hypothesis that the dependence between financial assets can be modelled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student copula, cannot be rejected if it has sufficiently many ‘degrees of freedom’. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the coefficient of correlation between the pairs of assets is too high, such that the tail dependence neglected by the Gaussian copula can became large, leading to the ignoring of extreme events which may occur in unison.     

Pricing of credit default index swap tranches with one-factor heavy-tailed copula models

In this paper, we provide two one-factor heavy-tailed copula models for pricing a collateralized debt obligation and credit default index swap tranches: (1) a one-factor double t distribution with fractional degrees of freedom copula model and (2) a one-factor double mixture distribution of t and Gaussian distribution copula model. A time-varying tail-fatness parameter is introduced in each model, allowing one to change the tail-fatness of the copula function continuously. Fitting our model to comprehensive market data, we find that a model with fixed tail-fatness cannot fit market data well over time. The two models that we propose are capable of fitting market data well over time when using a proper time-varying tail-fatness parameter. Moreover, we find that the time-varying tail-fatness parameters change dramatically over a one-year period.     

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.     

Dependence structure and extreme comovements in international equity and bond markets

Common negative extreme variations in returns are prevalent in international equity markets. This has been widely documented with statistical tools such as exceedance correlation, extreme value theory, and Gaussian bivariate GARCH or regime-switching models. We point to limits of these tools to characterize extreme dependence and propose an alternative regime-switching copula model that includes one normal regime in which dependence is symmetric and a second regime characterized by asymmetric dependence. We apply this model to international equity and bond markets, to allow for inter-market movements. Empirically, we find that dependence between international assets of the same type is strong in both regimes, especially in the asymmetric one, but weak between equities and bonds, even in the same country.     

Estimating portfolio risk for tail risk protection strategies

We forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, copula‐GARCH and dynamic generalized autoregressive score models. Utilizing a loss function that overcomes the lack of elicitability for expected shortfall, we propose a novel expected shortfall (and value‐at‐risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modeling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics.     

A paradigm shift from production function to production copula: statistical description of production activity of firms

The heterogeneity of economic agents is emphasized in a new trend in macroeconomics. Accordingly, the new emerging discipline requires one to replace the production function, one of the key ideas in conventional economics, by an alternative that can take explicit account of the distribution of firms' production activities. In this paper we propose a new idea referred to as a production copula; a copula is an analytic means for modeling the dependence among variables. Such a production copula predicts the value added by firms with given capital and labor in a probabilistic way. It is thereby in sharp contrast to the production function, where the output of firms is completely deterministic. We demonstrate the empirical construction of a production copula using financial data of listed Japanese firms. Analysis of the data shows that there are significant correlations among capital, labor and value added, and confirms that the values added are too widely scattered to be represented by a production function. We employ four models for the production copula, that is trivariate versions of Frank, Gumbel and survival Clayton and non-exchangeable trivariate Gumbel. The latter was found to be the best.     

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.