Singular mixture copulas

We present a new family of copulas??the Singular Mixture Copulas. We begin with constructing singular copulas whose supports lie on the graphs of two given quantile functions. These copulas are then mixed with respect to a continuous distribution resulting in a nonsingular parametric copula. The Singular Mixture Copulas we construct have a Lebesgue density and a closed form representation. Moreover, they have positive lower and upper tail dependence. As an application we fit the copulas to flood level data. As the results show Singular Mixture Copulas provide an alternative to elliptical copulas, e.g., Gaussian and t-copulas, in modeling strongly dependent random variables.     

Evolution of multivariate copulas in continuous and discrete processes

There has been much interest in copulas, which are known to provide a flexible tool for analyzing the dependence structure among random variables. Dependence relations must be dynamic rather than static in nature. However, copulas are useful mainly for static matters. Thus we introduce evolving multivariate copulas, which transform through time autonomously governed by the multivariate heat equation. Our aims are to prove their existences and solutions to analyze their transitions. Moreover, we construct discrete type to apply empirical data analysis and investigate their properties, and prove that they converge to their original continuous type.     

Sampling from Archimedean copulas

We develop sampling algorithms for multivariate Archimedean copulas. For exchangeable copulas, where there is only one generating function, we first analyse the distribution of the copula itself, deriving a number of integral representations and a generating function representation. One of the integral representations is related, by a form of convolution, to the distribution whose Laplace transform yields the copula generating function. In the infinite-dimensional limit there is a direct connection between the distribution of the copula value and the inverse Laplace transform. Armed with these results, we present three sampling algorithms, all of which entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula. We implement and compare the various methods. For more general cases, in which an N-dimensional Archimedean copula is given by N?1 nested generating functions, we present algorithms in which each new variate is drawn conditional only on the value of the copula of the previously drawn variates. We also discuss the use of composite nested and exchangeable copulas for modelling random variates with a natural hierarchical structure, such as ratings and sectors for obligors in credit baskets.     

Remarks on a copula-based conditional value at risk for the portfolio problem

We deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula-based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula-based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results.     

A new time-varying optimal copula model identifying the dependence across markets

This paper proposes a new time-varying optimal copula (TVOC) model to identify and capture the optimal dependence structure of bivariate time series at every time point. In the TVOC model, half-rotated copulas are constructed to measure the nonlinear and asymmetric negative dependence, and the distribution-free test for independence is introduced to verify the dependent relationship and reduce the computational time. The TVOC model is then employed to research the dependence structure between security and commodity markets. We find evidence that the dependence structures across different markets vary over time and that emergencies are usually the major cause of sudden changes in the dependence structure. We also show that the TVOC model captures the dynamic characteristics of the direction and intensity of the dependence as well as the dynamic characteristics of the types of dependence structure. In particular, the half-rotated copulas can accurately describe the asymmetric negative extreme dependence across different markets.     

Measuring financial risks with copulas

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the concept of copulas. We select some special copulas and identify the type of dependency captured by each one. We fit copulas to daily returns and simulate from the fitted models. We compare the effect of the choice of copula on risk measures and assess the variability of one-step-ahead predictions of portfolio losses. We analyze extreme scenarios and fit extreme value copulas to the block maxima and minima from daily returns. The stress scenarios constructed are compared to those obtained using models from the extreme value theory. We illustrate the usefulness of the copula approach using two stock market indexes.     

Recovering copulas from limited information and an application to asset allocation

This paper proposes an entropy-based method to construct a new class of copulas - the most entropic canonical copulas (MECC). Our empirical study focuses on an investment problem for an investor with a constant relative risk aversion (CRRA) utility function allocating wealth between the Dow Jones Large-Cap and Small-Cap indices, of which the contemporaneous dependence can be modeled by the MECC or other commonly-used copulas. Both the theoretical analysis of the method and the empirical study indicate the potential for enormous statistical and economic gains as a result of using the MECC.     

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.     

Flexible dependence modeling of operational risk losses and its impact on total capital requirements

Operational risk data, when available, are usually scarce, heavy-tailed and possibly dependent. In this work, we introduce a model that captures such real-world characteristics and explicitly deals with heterogeneous pairwise and tail dependence of losses. By considering flexible families of copulas, we can easily move beyond modeling bivariate dependence among losses and estimate the total risk capital for the seven- and eight-dimensional distributions of event types and business lines. Using real-world data, we then evaluate the impact of realistic dependence modeling on estimating the total regulatory capital, which turns out to be up to 38% smaller than what the standard Basel approach would prescribe.     

An empirical analysis of multivariate copula models

Since the pioneering work of Embrechts and co-authors in 1999, copula models have enjoyed steadily increasing popularity in finance. Whereas copulas are well studied in the bivariate case, the higher-dimensional case still offers several open issues and it is far from clear how to construct copulas which sufficiently capture the characteristics of financial returns. For this reason, elliptical copulas (i.e. Gaussian and Student-t copula) still dominate both empirical and practical applications. On the other hand, several attractive construction schemes have appeared in the recent literature promising flexible but still manageable dependence models. The aim of this work is to empirically investigate whether these models are really capable of outperforming its benchmark, i.e. the Student-t copula and, in addition, to compare the fit of these different copula classes among themselves.     

Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets

A traditional Monte Carlo simulation using linear correlations induces estimation bias in measuring portfolio value-at-risk (VaR), due to the well-documented existence of fat-tail, skewness, truncations, and non-linear relations in return distributions. In this paper, we consider the above issues in modeling VaR and evaluate the effectiveness of using copula-extreme-value-based semiparametric approaches. To assess portfolio risk in six Asian markets, we incorporate a combination of extreme value theory (EVT) and various copulas to build joint distributions of returns. A backtesting analysis using a Monte Carlo VaR simulation suggests that the Clayton copula-EVT evinces the best performance regardless of the shapes of the return distributions, and that in general the copulas with the EVT provide better estimations of VaRs than the copulas with conventionally employed empirical distributions. These findings still hold in conditional-coverage-based backtesting. These findings indicate the economic significance of incorporating the down-side shock in risk management.     

Longitudinal modeling of insurance claim counts using jitters

Modeling insurance claim counts is a critical component in the ratemaking process for property and casualty insurance. This article explores the usefulness of copulas to model the number of insurance claims for an individual policyholder within a longitudinal context. To address the limitations of copulas commonly attributed to multivariate discrete data, we adopt a ‘jittering’ method to the claim counts which has the effect of continuitizing the data. Elliptical copulas are proposed to accommodate the intertemporal nature of the ‘jittered’ claim counts and the unobservable subject-specific heterogeneity on the frequency of claims. Observable subject-specific effects are accounted in the model by using available covariate information through a regression model. The predictive distribution together with the corresponding credibility of claim frequency can be derived from the model for ratemaking and risk classification purposes. For empirical illustration, we analyze an unbalanced longitudinal dataset of claim counts observed from a portfolio of automobile insurance policies of a general insurer in Singapore. We further establish the validity of the calibrated copula model, and demonstrate that the copula with ‘jittering’ method outperforms standard count regression models.     

Jackknife Empirical Likelihood Intervals for Spearman’s Rho

Abstract

In connection with copulas, rank correlation such as Kendall’s tau and Spearman’s rho has been employed in risk management for summarizing dependence between two variables and estimating parameters in bivariate copulas and elliptical models. In this paper a jackknife empirical likelihood method is proposed to construct confidence intervals for Spearman’s rho without estimating the asymptotic variance. A simulation study confirms the advantages of the proposed method.     

Dependence structure between the equity market and the foreign exchange market–A copula approach

This paper investigates the dependence structure between the equity market and the foreign exchange market by using copulas. In particular, several copulas with different dependence structure are compared and used to directly model the underlying dependence structure. We find that there exists significant symmetric upper and lower tail dependence between the two financial markets, and the dependence remains significant but weaker after the launch of the euro. Our findings have important implications for both global investment risk management and international asset pricing by taking into account joint tail risk.     

CreditRisk+ Model with Dependent Risk Factors

The CreditRisk⁺ model is widely used in industry for computing the loss of a credit portfolio. The standard CreditRisk⁺ model assumes independence among a set of common risk factors, a simplified assumption that leads to computational ease. In this article, we propose to model the common risk factors by a class of multivariate extreme copulas as a generalization of bivariate Fréchet copulas. Further we present a conditional compound Poisson model to approximate the credit portfolio and provide a cost-efficient recursive algorithm to calculate the loss distribution. The new model is more flexible than the standard model, with computational advantages compared to other dependence models of risk factors.     

Forecasting Surrender Rates Using Elliptical Copulas and Financial Variables

A multistage stochastic model to forecast surrender rates for life insurance and pension plans is proposed. Surrender rates are forecasted by means of Monte Carlo simulation after a sequence of GLM, ARMA-GARCH, and copula fitting is executed. The model is illustrated by applying it to age-specific time series of surrender rates derived from pension plans with annuity payments of a Brazilian insurer. In the GLM process, the only macroeconomic variable used as an explanatory variable is the Brazilian real short-term interest rate. The advantage of such a variable is that we can take future market expectation through the current term structure of interest rates. The GLM residuals of each age/gender group are then modeled by ARMA-GARCH processes to generate i.i.d. residuals. The dependence among these residuals is then modeled by multivariate Gaussian and Student's t copulas. To produce a conditional forecast on a stock market index, in our application we used the residuals of an ARMA-GARCH model fitted to the Brazilian stock market index (Ibovespa) returns, which generates one of the marginal distributions used in the dependence modeling through copulas. This strategy is adopted to explain the high and uncommon surrender rates observed during the recent economic crisis. After applying known simulation methods for elliptical copulas, we proceeded backwards to obtain the forecasted distributions of surrender rates by application, in the sequel, of ARMA-GARCH and GLM models. Additionally, our approach produced an algorithm able to simulate multivariate elliptical copulas conditioned on a marginal distribution. Using this algorithm, surrender rates can be simulated conditioned on stock index residuals (in our case, the residuals of the Ibovespa returns), which allows insurers and pension funds to simulate future surrender rates assuming a financial stress scenario with no need to predict the stock market index.     

Copula‐Based Formulas to Estimate Unexpected Credit Losses (The Future of Basel Accords?)

The model used to estimate the capital required to cover unexpected credit losses in financial institutions (Basel II) has some drawbacks that reduce its ability to capture potential joint extreme losses in downturns. This paper suggests an alternative approach based on Copula Theory to overcome such flaws. Similarly to Basel II, the suggested model assumes that defaults are driven by a latent variable which varies as a response to an unobserved factor. On the other hand, the use of copulas allows the identification of asymmetric dependence between defaults which has been registered in the literature. As an example, a specific copula family (Clayton) is adopted to represent the association between the latent variables and a formula to estimate potential unexpected losses at a certain level of confidence is derived. Simulations reveal that, in most of the cases, the alternative model outperforms Basel II for portfolios with right‐tail‐dependent probabilities of default (supposedly, a good representation for real loan portfolios).     

Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests

Lévy subordinated hierarchical Archimedean copulas (LSHAC) are flexible models in high dimensional modeling. However, there is limited literature discussing their applications, largely due to the challenges in estimating their structures and their parameters. In this paper, we propose a three-stage estimation procedure to determine the hierarchical structure and the parameters of a LSHAC. This is the first paper to empirically examine the modeling performances of LSHAC models using exchange traded funds. Simulation study demonstrates the reliability and robustness of the proposed estimation method in determining the optimal structure. Empirical analysis further shows that, compared to elliptical copulas, LSHACs have better fitting abilities as well as more accurate out-of-sample Value-at-Risk estimates with less parameters. In addition, from a financial risk management point of view, the LSHACs have the advantage of being very flexible in modeling the asymmetric tail dependence, providing more conservative estimations of the probabilities of extreme downward co-movements in the financial market.     

Hierarchies of Archimedean copulas

We present a flexible class of hierarchical copulas capable of modelling multidimensional joint distributions of asset returns with a richer rank correlation structure than existing models. We derive estimators and simulation techniques. The methods are applied to an illustrative portfolio consisting of a subset of DAX stocks.     

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.