Copula Simulation Research Articles

Trivariate Probabilistic Assessments of the Compound Flooding Events Using the 3-D Fully Nested Archimedean (FNA) Copula in the Semiparametric Distribution Setting

Severe flooding in coastal areas can result from the joint probability of oceanographic, hydrological, and meteorological factors, resulting in compound flooding (CF) events. Recently, copula functions have been used to enable a much more flexible environment in joint modelling. Incorporating a higher dimensional copula framework via symmetric 3-D Archimedean or elliptical copulas has statistical limits, and the preservation of all lower-level dependencies would be impossible. The heterogeneous dependency in CF events can be effectively modelled via a fully nested Archimedean (FNA) copula. Incorporating FNA under parametric settings is insufficiently flexible since it is restricted by the prior distributional assumption of the function type for both marginal density and copulas in parametric fittings. If the marginal density is of a specific parametric distribution, it could be problematic if underlying assumptions are violated. This study introduces a 3-D FNA copula simulation in a semiparametric setting by introducing nonparametric marginal distributions conjoined with parametric copula density. The derived semiparametric FNA copula is applied in the trivariate model in compounding the joint impact of rainfall, storm surge, and river discharge based on 46 years of observations on the west coast of Canada. The performance of the derived model has also been compared with the FNA copula constructed with parametric marginal density. It is concluded that the performance of FNA with nonparametric marginals outperforms the FNA copula built under parametric settings. The derived model is employed in multivariate analysis of flood risks in trivariate primary joint and conditional joint return periods. The trivariate hydrologic risk is analyzed using failure probability (FP) statistics. The investigation reveals that trivariate events produce a higher FP than bivariate (or univariate) events; thus, neglecting trivariate joint analysis results in FP being underestimated. Moreover, it indicates that trivariate hydrologic risk values increase with an increase in the service time of hydraulic facilities.

A v-transformed copula-based simulation model for lithological classification in an Indian copper deposit

Copula functions are widely used for modeling multivariate dependence. Since the multivariate data may not necessarily be linear and Gaussian, the copula model is very often brought into the picture for modeling such multivariate phenomena. The lithological classification in spatial domain is a class of problems dealing with categorical variables. A generalized class of copula model is effective for such classification tasks. In this paper, a non-Gaussian copula (v-transformed copula) model has been used for lithotype classification of an Indian copper deposit. Coupling of Markov chain Monte Carlo (MCMC) simulation and copula discriminant function is performed for this purpose. Specifically, four lithotypes, e.g., granite, quartz, basic, and aplite are simulated in the case study deposit. The efficacy of v-transformed copula discriminant function-based simulation is compared with those of Gaussian copula, t copula, and sequential indicator simulations. Finally, the classification accuracy of all the approaches is examined with ground-truth lithological classes obtained from blast hole information. The results show that the v-transformed copula simulation has a relatively higher classification accuracy (76%) than those of Gaussian copula (70%), t copula (69%), and sequential indicator (70%) simulations.

Copula-Based Multivariate Simulation Approach for Flood Risk Transfer of Multi-Reservoirs in the Weihe River, China

The interplay of multi-reservoirs is critical in reservoir joint disposal and water conservancy projects. As the flood risk of upstream hydrological stations could be transferred and unevenly distributed to downstream tributary stations, flood risk transfer through multi-reservoirs warrants further investigation. This study proposed a copula simulation approach to develop a joint flood risk distribution of multi-reservoirs (spanning Xianyang, Huaxian County, and Zhangjiashan) in a drainage tributary of the Weihe River. Pair-copulas of each reservoir pair were constructed to analyse the correlations between the reservoir sites. The approach was then used to create a joint flood risk distribution for the reservoirs. The flood risk and corresponding flood volume of Zhangjiashan were calculated based on the flood risk levels of Xianyang and Huaxian County. The results indicate that the flood risks of Huaxian County would be transferred to Xianyang and Zhangjiashan to some extent, and Xianyang could mitigate more flood risks from Huaxian County than from Zhangjiashan. The findings have significance for informed decision-making regarding the Zhangjiashan reservoir construction project.

Modeling the Interdependence Structure between Rain and Radar Variables Using Copulas: Applications to Heavy Rainfall Estimation by Weather Radar

In radar quantitative precipitation estimates (QPE), the progressive evolution of rainfall algorithms has been guided by attempts to reduce the uncertainties in rainfall retrieval. However, because most of the algorithms are based on the linear dependence between radar and rain variables and designed for rain rates ranging from light to moderate rainfall, they result in misleading estimations of intense or strong rainfall rates. In this paper, based on extensive data gathered during the AMMA and Megha-Tropiques data campaigns, we provided a way to improve the estimation of intense rainfall rates from radar measurements. To this end, we designed a formulation of the QPE algorithm that accounts for the co-dependency between radar observables and rainfall rate using copula simulation synthetic datasets and using the quantile regression features for a more complete picture of covariate effects. The results show a clear improvement in heavy rainfall retrieval from radar data using copula-based R(KDP) algorithms derived from a realistic simulated dataset. For a better performance, Gaussian copula-derived algorithms require a 0.8 percentile distribution to be considered. Conversely, lower percentiles are better for Student’s, Gumbel and HRT copula estimators when retrieving heavy rainfall rates (R > 30). This highlights the need to investigate the entire conditional distribution to determine the performance of radar rainfall estimators.

Ranking and optimizing the rain-gauge networks using the entropy–copula approach (Case study of the Siminehrood Basin, Iran)

The existence of reliable rain-gauge networks is a necessity in managing water sources and the relevant environmental issues in any basin. The present study aimed to investigate the rain-gauge network and rank the rain-gauge stations in the Tazehkand sub-basin in the southwest of Lake Urmia, Iran, using the entropy–copula approach using the data obtained from six rain-gauge stations in the Siminehrood basin in Northwestern Iran between 1981 and 2019. The interaction effects of the stations were investigated using the copula simulation approach instead of the multivariate regression analysis. The R-vine was selected as the most convenient copula after investigating various vines using the statistics related to error investigation. Then, the interaction effects of the stations were investigated using the R-vine. The rainfall in each station was estimated by considering the rainfall of other stations. Moreover, the more prevalent multivariate regression analysis was implemented, and its results were compared with the results obtained from the vine simulation approach. The results of the comparison according to the NSE statistic showed that the vine simulation approach was above 90% across all stations. The estimates for the ITI and N(i) indices of each station using the entropy theory showed the shortage of stations in the investigated area. The stations were in average, above-average, and surplus modes according to the ITI index. The rating concerning the N(i) index indicated that the Dashband station was the most appropriate one in terms of communicating with other stations and conveniently covering the plain.

Simulation of potential evapotranspiration values based on vine copula

AbstractVine copula had a great impact on the study and analysis of dependence structures in various sciences. In multivariate analyses with dimensions of more than two variables, it is associated with computational complexities that solve vine copulas and these problems. In this study, in order to provide an approach to simulate potential evapotranspiration based on meteorological parameters in Birjand meteorological station from different family copulas including R‐vine, independent R‐vine, Gaussian, independent Gaussian, C‐vine, C‐vine independent, D‐vine and D‐vine independent were used. In this regard, vine copula simulation and conditional density were used. In pair correlation analysis of the studied variables using Kendall's tau statistic, dependence structure confirmed the studied parameters. The results showed a minimum correlation of −0.32 and a maximum of 0.77. The results of Akaike's information criteria (AIC), Bayesian information criteria (BIC) and LogLike statistics in evaluating the performance of vine copula dependency structure introduced the C‐vine copula as the superior copula for analysing the pair dependence of the studied variables. By introducing the superior dependency structure and internal copulas, the tree sequence of the pair of values under study was obtained. Pair of simulated values was performed using vine copula. Comparison of Kendall's tau values in both simulation and observation modes showed that Kendall's tau values were close to each other in both modes and were approximately similar. The simulation results of vine copula potential evapotranspiration values and precipitation, temperature and relative humidity values showed 92% efficiency. The efficiency of C‐vine copula in dependence analysis and simulation of potential evapotranspiration (PET) values is very high, which shows the ability of vine family copulas in multivariate analysis.

Optimizing the market-risk of major cryptocurrencies using CVaR measure and copula simulation

ABSTRACT In this paper, we utilize the conditional value-at-risk to quantify the risk exposure and the generalized Pareto distribution copula technique to analyse extreme events which helps in finding out the efficient portfolio selection. The sample data covers nine cryptocurrencies covering the period from September 2016 to August 2018. Our results using the efficient frontier indicate that if a minimum variance portfolio is constructed using chosen cryptocurrencies, investment in Bitcoin is preferred being the least risky currency on the bottom of the efficient frontier. These results find prime importance for investors and risk managers.

Consistent Spread Dynamics for CVA Risk Charge and Historical Value-at-Risk by Means of Cross Sectional / Consolidated Bucket Link Copula Simulation

This paper describes the modelling of spread risk, in case of missing or illiquid market data, by using a subset of good quality liquid bond/credit default swap (CDS) spread time series. The proposed method links copula simulation to the actual historical spread dynamics. This is important when calculating credit valuation adjustment (CVA) risk charge and Value-at-Risk (VaR) with historical simulation. The methodology center around buckets of similar spreads. Buckets with good data, are straight forward, whereas buckets without data rely on a cross sectional model spread based on other buckets with good data. Residuals from regression of the bucket spread returns against market index returns are used to derive a link for each bucket. The link is subsequently used for simulating the spread dynamics in case of missing or illiquid spreads, using a modified one factor copula. A link between the actual and simulated residuals maintaining the risk dynamics is thus ensured. The result of the copula simulation is transformed into quantiles that are plugged into residual distributions from actual quality data, thereby maintaining the properties of actual market data such that the choice of copula only affects the risk dynamics, not the distributions of the risk factors.

Long-term stochastic risk models: the sixth generation of modern actuarial models?

Abstract This paper discusses the use of modelling techniques for the purpose of risk management within life insurers. The key theme of the paper is that life insurance is long-term business and carries with it long-term risks, yet much of modern actuarial risk management is focussed on short-term modelling approaches. These typically include the use of copula simulation models within a 1-year Value-at-Risk (VaR) framework. The paper discusses the limitations inherent within the techniques currently used in the UK and discusses how the focus of the next generation of actuarial models may be on long-term stochastic projections. The scope of the paper includes a discussion of how existing techniques, together with new approaches, may be used to develop such models and the benefits this can bring. The paper concludes with a practical example of how a long-term stochastic risk model may be implemented.

Non-Gaussian Copula Simulation for Estimation of Recoverable Reserve in an Indian Copper Deposit

The present study developed a geostatistical simulation technique based on non-Gaussian copula for recoverable reserve estimation considering support effect of a well-known open-pit mine of a copper deposit in India. The focus was to examine the efficacy of copula-based simulation model in recoverable reserve estimation. It was assessed by comparing three selectivity curves like grade–ore tonnage, grade–metal tonnage and ore–metal tonnage of reserve constructed using the copula-based simulation, disjunctive kriging and multi-Gaussian kriging with respect to the production data of blasting. The results informed that the copula-based simulation technique provided better accuracy than the nonlinear geostatistical techniques of disjunctive kriging and multi-Gaussian kriging. The root mean square error and standard error analysis indicated that the copula-based simulation model provided more accurate estimates compared with disjunctive kriging and multi-Gaussian kriging. All the three techniques yielded biased estimates, but the least bias was attributed to the copula-based simulation technique.

A nonparametric copula distribution framework for bivariate joint distribution analysis of flood characteristics for the Kelantan River basin in Malaysia

The joint distribution analysis of multidimensional flood characteristics i.e., flood peak flow, volume and duration, often facilitates a comprehensive understanding in the hydrologic risk assessments. Copula-based methodology are frequently incorporated via parametric approach to model dependence structure of parametric based univariate marginal distributions. But, if the targeted copulas and univariate marginal distributions belongs to some specific parametric families, it might be problematic, if the underlying assumption are violated. Also, no universal rules and literatures are imposed to model any hydrologic vectors and their joint dependence structure through any fixed or pre-defined distributions. In this literature, a nonparametric copula simulation are incorporated and applied as a case study for 50 years annual maximum flood samples of the Kelantan River basin at the Gulliemard bridge station in Malaysia. In this study, a combination of both parametric and nonparametric marginal distribution separately conjoined by a nonparametric copulas framework, which is based on the Beta kernel function. The Beta kernel copula function are incorporated to estimate bivariate copula density which further used to derived joint cumulative density of flood peak-volume, volume-duration and peak-duration pairs and their associated joint as well as conditional return periods.

Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation

In this paper, the generalized Pareto distribution (GPD) copula approach is utilized to solve the conditional value-at-risk (CVaR) portfolio problem. Particularly, this approach used (i) copula to model the complete linear and non-linear correlation dependence structure, (ii) Pareto tails to capture the estimates of the parametric Pareto lower tail, the non-parametric kernel-smoothed interior and the parametric Pareto upper tail and (iii) Value-at-Risk (VaR) to quantify risk measure. The simulated sample covers the G7, BRICS (association of Brazil, Russia, India, China and South Africa) and 14 popular emerging stock-market returns for the period between 1997 and 2018. Our results suggest that the efficient frontier with the minimizing CVaR measure and simulated copula returns combined outperforms the risk/return of domestic portfolios, such as the US stock market. This result improves international diversification at the global level. We also show that the Gaussian and t-copula simulated returns give very similar but not identical results. Furthermore, the copula simulation provides more accurate market-risk estimates than historical simulation. Finally, the results support the notion that G7 countries can provide an important opportunity for diversification. These results are important to investors and policymakers.

Goodness-of-fit tests for multi-dimensional copulas: Expanding application to historical drought data

The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt's transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.

Vine copulas and applications to the European Union sovereign debt analysis

European sovereign debt crisis has become a very popular topic since late 2009. In this paper, sovereign debt crisis is investigated by calculating the probabilities of the potential future crisis of 11 countries in the European Union. We use sovereign spreads of the European countries against Germany as targets and apply the GARCH based vine copula simulation technique. The methodology solves the difficulties of calculating the probabilities of rarely happening events and takes sovereign debt movement dependence, especially tail dependence, into consideration. Results indicate that Italy and Spain are the most likely next victims of the sovereign debt crisis, followed by Ireland, France and Belgium. The UK, Sweden and Denmark, which are outside the euro area, are the most financially stable countries in the sample.

Conditional copula simulation for systemic risk stress testing

Since the financial crisis of 2007–2009 there is an active debate by regulators and academic researchers on systemic risk, with the aim of preventing similar crises in the future or at least reducing their impact. A major determinant of systemic risk is the interconnectedness of the international financial market. We propose to analyze interdependencies in the financial market using copulas, in particular using flexible vine copulas, which overcome limitations of the popular elliptical and Archimedean copulas. To investigate contagion effects among financial institutions, we develop methods for stress testing by exploiting the underlying dependence structure. New approaches for Archimedean and, especially, for vine copulas are derived. In a case study of 38 major international institutions, 20 insurers and 18 banks, we then analyze interdependencies of CDS spreads and perform a systemic risk stress test. The specified dependence model and the results from the stress test provide new insights into the interconnectedness of banks and insurers. In particular, the failure of a bank seems to constitute a larger systemic risk than the failure of an insurer.

Decomposition of portfolio VaR and expected shortfall based on multivariate Copula simulation

Portfolio risk-adjusted performance measurement involves the calculation of the risk contribution for each asset it contains. This paper uses multivariate Copula functions to model the dependence structure among the assets in a portfolio, then, based on a simulation, decomposes the portfolio VaR and Expected Shortfall. The research shows this simulation approach provides a way to test if the risk contributions of various assets are significantly different, and also displays results insusceptible to confidence level and risk measures. Furthermore, with this approach, the risk contribution calculated using Expected Shortfall is more robust, and its estimation error can be reduced by increasing the simulation sample size. For the equally-weighted portfolios of five Shanghai industrial stock indices, empirical evidence shows that the Real Estate Index has the largest risk contribution of the whole portfolio.

Adaptive Importance Sampling for simulating copula-based distributions

In this paper, we propose a generalization of importance sampling, called Adaptive Importance Sampling, to approximate simulation of copula-based distributions. Unlike existing methods for copula simulation that have appeared in the literature, this algorithm is broad enough to be used for any absolutely continuous copula. We provide details of the algorithm including rules for stopping the iterative process and consequently assess its performance using extensive Monte Carlo experiments. To assist in its extension to several dimensions, we discuss procedures for identifying the crucial parameters in order to achieve desirable results especially as the size of the dimension increases. Finally, for practical illustration, we demonstrate the use of the algorithm to price First-to-Default credit swap, an important credit derivative instrument in the financial market. The method works exquisitely well even for large dimensions making it a valuable tool for simulating from many different classes of copulas including those which have been difficult to sample from using traditional techniques.

Validation of Economic Capital Models: State of the Practice, Supervisory Expectations and Results from a Bank Study

A challenge in economic capital modeling within financial institutions is developing a coherent approach to model validation. This has been motivated by rapid financial innovation, developments in supervisory standards (Pillar II of the Basel II framework) and the recent financial turmoil. We survey various practices in validating economic capital models, both quantitative and qualitative approaches, and discuss supervisory expectations and concerns regarding this process. We then illustrate several of these approaches (benchmarking, sensitivity analysis and testing for predictive accuracy) utilizing data from major banking institutions' loss experience (from supervisory call reports), estimate and compare alternative established frameworks for risk aggregation (including alternative copula models). Results suggests that practitioners may want to consider implementing a simple non-parametric methodology (empirical copula simulation - ECS) in order to quantify integrated risk, in that it is found to be more conservative, as wells as more stable than the other models, in a non-parametric bootstrap experiment.

Validation of economic capital models: State of the practice, supervisory expectations and results from a bank study

A challenge in economic capital modelling within financial institutions is developing a coherent approach to model validation. This has been motivated by rapid financial innovation, developments in supervisory standards (Pillar 2 of the Basel II framework) and the recent financial turmoil. Various practices are surveyed in validating economic capital models, both quantitative and qualitative approaches, and supervisory expectations and concerns regarding this process are discussed. The paper then illustrates several of these approaches (benchmarking, sensitivity analysis and testing for predictive accuracy) utilising data from major banking institutions’ loss experience (from supervisory call reports), and estimates and compares alternative established frameworks for risk aggregation (including alternative copula models). Results suggest that practitioners may want to consider implementing a simple non-parametric methodology (empirical copula simulation (ECS)) in order to quantify integrated risk, in that it is found to be more conservative, as well as more stable than the other models, in a nonparametric bootstrap experiment.

Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital

A challenge in enterprise risk measurement for diversified financial institutions is developing a coherent approach to aggregating different risk types. This has been motivated by rapid financial innovation, developments in supervisory standards (Basel 2) and recent financial turmoil. The main risks faced - market, credit and operational – have distinct distributional properties, and historically have been modeled in differing frameworks. We contribute to the modeling effort by providing tools and insights to practitioners and regulators. First, we extend the scope of the analysis to liquidity and interest rate risk, having Basel Pillar II of Basel implications. Second, we utilize data from major banking institutions’ loss experience from supervisory call reports, which allows us to explore the impact of business mix and inter-risk correlations on total risk. Third, we estimate and compare alternative established frameworks for risk aggregation (including copula models) on the same data-sets across banks, comparing absolute total risk measures (Value-at-Risk – VaR and proportional diversification benefits-PDB), goodness-of-fit (GOF) of the model as data as well as the variability of the VaR estimate with respect to sampling error in parameter. This benchmarking and sensitivity analysis suggests that practitioners consider implementing a simple non-parametric methodology (empirical copula simulation- ECS) in order to quantify integrated risk, in that it is found to be more conservatism and stable than the other models. We observe that ECS produces 20% to 30% higher VaR relative to the standard Gaussian copula simulation (GCS), while the variance-covariance approximation (VCA) is much lower. ECS yields the highest PDBs than other methodologies (127% to 243%), while Archimadean Gumbel copula simulation (AGCS) is the lowest (10-21%). Across the five largest banks we fail to find the effect of business mix to exert a directionally consistent impact on total integrated diversification benefits. In the GOF tests, we find mixed results, that in many cases most of the copula methods exhibit poor fit to the data relative to the ECS, with the Archimadean copulas fitting worse than the Gaussian or Student-T copulas. In a bootstrapping experiment, we find the variability of the VaR to be significantly lowest (highest) for the ECS (VCA), and that the contribution of the sampling error in the parameters of the marginal distributions to be an order or magnitude greater than that of the correlation matrices.