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.     

How does the choice of Value-at-Risk estimator influence asset allocation decisions?

Considering the growing need for managing financial risk, Value-at-Risk (VaR) prediction and portfolio optimisation with a focus on VaR have taken up an important role in banking and finance. Motivated by recent results showing that the choice of VaR estimator does not crucially influence decision-making in certain practical applications (e.g. in investment rankings), this study analyses the important question of how asset allocation decisions are affected when alternative VaR estimation methodologies are used. Focusing on the most popular, successful and conceptually different conditional VaR estimation techniques (i.e. historical simulation, peak over threshold method and quantile regression) and the flexible portfolio model of Campbell et al. [J. Banking Finance. 2001, 25(9), 1789–1804], we show in an empirical example and in a simulation study that these methods tend to deliver similar asset weights. In other words, optimal portfolio allocations appear to be not very sensitive to the choice of VaR estimator. This finding, which is robust in a variety of distributional environments and pre-whitening settings, supports the notion that, depending on the specific application, simple standard methods (i.e. historical simulation) used by many commercial banks do not necessarily have to be replaced by more complex approaches (based on, e.g. extreme value theory).     

Gibbs posterior inference on value-at-risk

ABSTRACT

Accurate estimation of value-at-risk (VaR) and assessment of associated uncertainty is crucial for both insurers and regulators, particularly in Europe. Existing approaches link data and VaR indirectly by first linking data to the parameter of a probability model, and then expressing VaR as a function of that parameter. This indirect approach exposes the insurer to model misspecification bias or estimation inefficiency, depending on whether the parameter is finite- or infinite-dimensional. In this paper, we link data and VaR directly via what we call a discrepancy function, and this leads naturally to a Gibbs posterior distribution for VaR that does not suffer from the aforementioned biases and inefficiencies. Asymptotic consistency and root-n concentration rate of the Gibbs posterior are established, and simulations highlight its superior finite-sample performance compared to other approaches.     

The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management

The t copula is often used in risk management as it allows for modeling the tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, the grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a generalized grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on the finite sample properties of the maximum likelihood estimators and Kendall's tau approximation. This new copula is significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall.     

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.     

Modeling systemic risk and dependence structure between oil and stock markets using a variational mode decomposition-based copula method

This study combines the variational mode decomposition (VMD) method and static and time-varying symmetric and asymmetric copula functions to examine the dependence structure between crude oil prices and major regional developed stock markets (S&P500, stoxx600, DJPI and TSX indexes) during bear, normal and bull markets under different investment horizons. Furthermore, it analyzes the upside and downside short- and long-run risk spillovers between oil and stock markets by quantifying three market risk measures, namely the value at risk (VaR), conditional VaR (CoVaR) and the delta CoVaR (∆CoVaR). The results show that there is a tail dependence between oil and all stock markets for the raw return series. By considering time horizons, we show that there is an average dependence between the considered markets for the short-run horizons. However, the tail dependence is also found for the long-run horizons between the oil and stock markets, with the exception of the S&P500 index which exhibits average dependence with the oil market. Moreover, we find strong evidence of up and down risk asymmetric spillovers from oil to stock markets and vice versa in the short-and long run horizons. Finally, the market risk spillovers are asymmetric over the time and investment horizons.     

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.     

Model uncertainty and VaR aggregation

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.     

Bivariate Survival Models for Coupled Lives

This article presents alternative models for modeling the dependence of the time of deaths of coupled lives and applies these to a data set from a life annuity portfolio. The data indicates a very high positive correlation in the time of deaths between coupled lives. The analysis concludes that a mixed frailty copula model with Gompertz marginals fits the data very well. Another model that fits the data well is based on a generalized Frank copula.     

Value-at-risk in a market subject to regime switching

Many empirical researches report that value-at-risk (VaR) measures understate the actual 1% quantile, while for Inui, K., Kijima, M. and Kitano, A., VaR is subject to a significant positive bias. Stat. Probab. Lett., 2005, 72, 299–311. proved that VaR measures overstate significantly when historical simulation VaR is applied to fat-tail distributions. This paper resolves the puzzle by developing a regime switching model to estimate portfolio VaR. It is shown that our model is able to correct the underestimation problem of risk.     

The shifting dependence dynamics between the G7 stock markets

The growing interdependence between financial markets has attracted special attention from academic researchers and finance practitioners for the purpose of optimal portfolio design and contagion analysis. This article develops a tractable regime-switching version of the copula functions to model the intermarkets linkages during turmoil and normal periods, while taking into account structural changes. More precisely, Markov regime-switching C-vine and D-vine decompositions of the Student’s t copula are proposed and applied to returns on diversified portfolios of stocks, represented by the G7 stock market indices. The empirical results show evidence of regime shifts in the dependence structure with high contagion risk during crisis periods. Moreover, both the C- and D-vines highly outperform the multivariate Student’s t copula, which suggests that the shock transmission path is as important as the dependence itself, and is better detected with a vine copula decomposition.     

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.     

Modeling the dependence structures of financial assets through the Copula Quantile-on-Quantile approach

This paper considers a new approach of analyzing asset dependence by estimating how the distributions (in particular, quantiles) of assets are related. Combining the techniques of quantile regression and copula modeling, I propose the Copula Quantile-on-Quantile Regression approach to estimate the correlation that is associated with the quantiles of asset returns, which is able to uncover obscure nonlinear characteristics in asset dependence. The estimation procedure proposed here can also be used for analyzing dependence structures in other settings, such as for studying how macroeconomic covariates are nonlinearly related by looking at the relationship between their quantiles.     

Optimal insurance contract under a value-at-risk constraint

This study develops an optimal insurance contract endogenously under a value-at-risk (VaR) constraint. Although Wang et al. [2005] had examined this problem, their assumption implied that the insured is risk neutral. Consequently, this study extends Wang et al. [2005] and further considers a more realistic situation where the insured is risk averse. The study derives the optimal insurance contract as a single deductible insurance when the VaR constraint is redundant or as a double deductible insurance when the VaR constraint is binding. Finally, this study discusses the optimal coverage level from common forms of insurances, including deductible insurance, upper-limit insurance, and proportional coinsurance. JEL Classification G22     

Nonparametric Estimation of Expected Shortfall

The expected shortfall is an increasingly popular risk measurein financial risk management and it possesses the desired sub-additivityproperty, which is lacking for the value at risk (VaR). We considertwo nonparametric expected shortfall estimators for dependentfinancial losses. One is a sample average of excessive losseslarger than a VaR. The other is a kernel smoothed version ofthe first estimator (Scaillet, 2004 Mathematical Finance), hopingthat more accurate estimation can be achieved by smoothing.Our analysis reveals that the extra kernel smoothing does notproduce more accurate estimation of the shortfall. This is differentfrom the estimation of the VaR where smoothing has been shownto produce reduction in both the variance and the mean squareerror of estimation. Therefore, the simpler ES estimator basedon the sample average of excessive losses is attractive forthe shortfall estimation.     

Change analysis of a dynamic copula for measuring dependence in multivariate financial data

This paper proposes a new approach to measure dependencies in multivariate financial data. Data in finance and insurance often cover a long time period. Therefore, the economic factors may induce some changes within the dependence structure. Recently, two methods have been proposed using copulas to analyse such changes. The first approach investigates changes within the parameters of the copula. The second determines the sequence of copulas using moving windows. In this paper we take into account the non-stationarity of the data and analyse the impact of (1) time-varying parameters for a copula family, and (2) the sequence of copulas, on the computations of the VaR and ES measures. We propose tests based on conditional copulas and the goodness-of-fit to decide the type of change, and further give the corresponding change analysis. We illustrate our approach using the Standard & Poor 500 and Nasdaq indices in order to compute risk measures using the two previous methods.     

Empirical estimation of tail dependence using copulas: application to Asian markets

This paper introduces non-parametric estimators for upper and lower tail dependence whose confidence intervals are obtained with a bootstrap method. We call these estimators ‘naïve estimators’ as they represent a discretization of Joe's formulae linking copulas to tail dependence. We apply the methodology to an empirical data set composed of three composite indexes for the three Tigers (Thailand, Malaysia and Indonesia). The extremes show a dependence structure which is symmetric for the Thai and Malaysian markets and asymmetric for the Thai and Indonesian markets and for the Malaysian and the Indonesian markets. Using these results we estimate the copula (which belongs to the Student or Archimedean copula families) for each pair of markets by two methods. Finally, we provide risk measurements using the best copula associated with each pair of markets.     

Risk Measures for Hedge Funds: a Cross‐sectional Approach

This paper analyses the risk‐return trade‐off in the hedge fund industry. We compare semi‐deviation, value‐at‐risk (VaR), Expected Shortfall (ES) and Tail Risk (TR) with standard deviation at the individual fund level as well as the portfolio level. Using the Fama and French (1992) methodology and the combined live and defunct hedge fund data from TASS, we find that the left‐tail risk captured by Expected Shortfall (ES) and Tail Risk (TR) explains the cross‐sectional variation in hedge fund returns very well, while the other risk measures provide statistically insignificant or marginally significant results. During the period between January 1995 and December 2004, hedge funds with high ES outperform those with low ES by an annual return difference of 7%. We provide empirical evidence on the theoretical argument by Artzner et al. (1999) that ES is superior to VaR as a downside risk measure. We also find the Cornish‐Fisher (1937) expansion is superior to the nonparametric method in estimating ES and TR.     

Statistical analysis of financial time series under the assumption of local stationarity

The aim of this paper is to apply a nonparametric methodology developed by Donoho et al(2003 IEEE Trans. Signal Processing 53614–27) for estimating an autocovariance sequence to the statistical analysis of the return of securities and discuss the advantages offered by this approach over other existing methods such as fixed-window-length segmentation procedures. Theoretical properties of adaptivity of this estimation method have been proved for a specific class of time series, namely the class of locally stationary processes, with an autocovariance structure which varies slowly over time in most cases but might exhibit abrupt changes of regime. This method is based on an algorithm that selects empirically from the data the tiling of the time–frequency plane which exposes best in the least-squares sense the underlying second-order time-varying structure of the time series, and so may properly describe the time-inhomogeneous variations of speculative prices. The applications we consider here mainly concern the analysis of structural changes occurring in stock market returns, VaR estimation and the comparison between the variation structure of stock index returns in developed markets and in developing markets.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.