The central idea of the paper is to present a general simple patchwork construction principle for multivariate copulas that create unfavourable VaR (i.e. Value at Risk) scenarios while maintaining given marginal distributions. This is of particular interest for the construction of Internal Models in the insurance industry under Solvency II in the European Union. The method is exemplified with a 19-dimensional real-life data set of insurance losses.
We propose a Monte Carlo simulation method to generate stress tests by VaR scenarios under Solvency II for dependent risks on the basis of observed data. This is of particular interest for the construction of Internal Models and requirements on evaluation processes formulated in the Commission Delegated Regulation. The approach is based on former work on partition-ofunity copulas, however with a direct scenario estimation of the joint density by product beta distributions after a suitable transformation of the original data.
An approach to the modelling of volatile time series using a class of uniformity-preserving transforms for uniform random variables is proposed. V-transforms describe the relationship between quantiles of the stationary distribution of the time series and quantiles of the distribution of a predictable volatility proxy variable. They can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with copula processes for the dynamics of the volatility proxy. The idea is illustrated using a Gaussian ARMA copula process and the resulting model is shown to replicate many of the stylized facts of financial return series and to facilitate the calculation of marginal and conditional characteristics of the model including quantile measures of risk. Estimation is carried out by adapting the exact maximum likelihood approach to the estimation of ARMA processes and the model is shown to be competitive with standard GARCH in an empirical application to Bitcoin return data.
We present a constructive approach to Bernstein copulas with an admissible discrete skeleton in arbitrary dimensions when the underlying marginal grid sizes are smaller than the number of observations. This prevents an overfitting of the estimated dependence model and reduces the simulation effort for Bernstein copulas a lot. In a case study, we compare different approaches of Bernstein and Gaussian copulas w.r.t. the estimation of risk measures in risk management.
We present a constructive and self-contained approach to data driven general partition-of-unity copulas that were recently introduced in the literature. In particular, we consider Bernstein-, negative binomial and Poisson copulas and present a solution to the problem of fitting such copulas to highly asymmetric data.
In this paper, we investigate the optimal management of defined contribution (abbr. DC) pension plan under relative performance ratio and Value-at-Risk (abbr. VaR) constraint. Inflation risk is introduced in this paper and the financial market consists of cash, inflation-indexed zero coupon bond and a stock. The goal of the pension manager is to maximize the performance ratio of the real terminal wealth under VaR constraint. An auxiliary process is introduced to transform the original problem into a self-financing problem first. Combining linearization method, Lagrange dual method, martingale method and concavification method, we obtain the optimal terminal wealth under different cases. For convex penalty function, there are fourteen cases while for concave penalty function, there are six cases. Besides, when the penalty function and reward function are both power functions, the explicit forms of the optimal investment strategies are obtained. Numerical examples are shown in the end of this paper to illustrate the impacts of the performance ratio and VaR constraint.