THE SIMULATION OF DEPENDENT INSURANCE COMPANY’S LOSSES EMPLOYING COPULAS Kristina Sutiene, Agne Paulauskaite-Taraseviciene, Henrikas Pranevicius Kaunas University of Technology, Department of Business Informatics, Studentu str. 56-442, Kaunas, Lithuania, kristina.sutiene@ktu.lt, agne@ifko.ktu.lt, henrikas.pranevicius@ktu.lt Abstract. The research is performed in the field of risk management of insurance company’s activity. The simulation model that allows to evaluate the dependency between two types of insurance company’s losses is presented in this paper. Since the company’s losses are dependent only in the right tails of distributions and are not normally distributed, the application of Pearson’s correlation coefficient would be inadequate. For their dependency structure, the alternative method – copula – is employed, which allows to construct the non-linear dependency structure between the dependent stochastic variables despite their type of distributions. The purpose of this work is to explore the copula effect on the liability portfolio of the insurance company. The developed simulation model is based on Piece Linear Aggregates (PLA) approach. Keywords: insurance company’s losses, liability scenario generator, copula, aggregate approach. 1 Introduction For an insurance company, the planning has to be carried out under uncertainty. Thus, the decision making model includes the parameters that are not completely known at the current point of time when the decision has to be taken [1], [9]. These random parameters can be named as risk factors. The modeling of randomness employees the set of available past data with the aim to build models for each individual stochastic parameter. These models are integrated in Scenario Generator (SG) and are used to generate a set of scenarios that represent the consistent depictions of pathways in the future based on assumptions about economic and technological developments [4], [19]. Thus, the factors driving risky events are approximated by a discrete set of scenarios. This process is known as scenario generation. Scenarios can be generated using various methods, based on different principles: conditional sampling, sampling from given marginals and correlations, moment matching, path based methods, optimal discretization, as in [7], [12], [14]. The activity of insurance company is affected by many risk factors. In this paper, the size and the frequency of both non-catastrophe losses and catastrophe losses are considered. These uncertain data are typical liabilities of insurance company. Usually these types of losses are simulated separately due to quite different statistical behaviour of catastrophe and non-catastrophe losses. But in the references [5], [11] it is argued that the dependency among the sizes of these losses is strengthened in the right side of the distribution if the catastrophe event occurs. The other notice is that the distributions of non-catastrophe losses and catastrophe losses are rarely of Gaussian type [1], [8], [10]. These restrictions of insurance company’s liabilities show that the application of Pearson’s correlation coefficient would be inadequate. Thus, in this paper the alternative method – copula – is applied to describe the dependency structure of non-linear type among non-normally distributed stochastic variables [6], [13]. Many copulas are available with differing characteristics that lead to different relationships among variables generated. Note that copulas differ not so much in the degree of association they provide, but rather in which part of the distributions the association is strongest: the behavior of copulas in the right and left tails can be used to distinguish among joint distributions that produce the same overall correlation. To describe the dependency for non-catastrophe losses and catastrophe losses, Gumbel copula [13] was employed since it has the feature to strengthen the dependency in the right tails of distributions of dependent variables. Since the scenarios are generated to represent the evolution of risk factors in future, the features of copula functions have to be reflected in the generated scenarios set. For this purpose, Liability Scenario Generator (LSG) with incorporated sub-model for copula based dependency measure was developed. The illustration of provided LSG is given in Figure 1, where 1 X , 2 X – the size of non-catastrophe and catastrophe losses respectively, f – the model for dependency structure based on Gumbel copula, Y – the output, which is the ultimate losses of insurance company. Thus, the bivariate distribution of losses has to be constructed. We named this dependency as event driven dependency since it is invoked by catastrophic event. The augmented scenario based risk management framework [17] was used to generate dependent scenarios for dependent losses incorporating a copula-based dependency measure. The simulation model of Liability Scenario Generator is based on Piece Linear Aggregates (PLA) formal notation [15]. The stages of model development from mathematical model to its realization are presented in Figure 2. - 219 -