Modeling and Management of Nonlinear Dependencies-Copulas in Dynamic Financial Analysis

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ABSTRACT

We study the influence of nonlinear dependencies on a non-life insurer's risk and return profile. To achieve this, we integrate several copula models in a dynamic financial analysis framework and conduct numerical tests. We also test risk management strategies in response to adverse outcomes. Nonlinear dependencies have a crucial influence on the insurer's risk profile that can hardly be affected by the analyzed management strategies. We find large differences in risk assessment for the ruin probability and for the expected policyholder deficit. This has important implications for insurers, regulators, and rating agencies that use these measures as a foundation for internal risk models, capital standards, and ratings.

INTRODUCTION

Dynamic financial analysis (DFA) is a financial modeling approach that projects financial results under a variety of possible scenarios, showing how outcomes might be affected by changing internal and external conditions (see Casualty Actuarial Society, 2006). DFA has become an important tool for decision making and an essential part of enterprise risk management (ERM), particularly within the field of non-life insurance. The DFA results and the quality of decisions derived from them depend on an appropriate modeling of the stochastic behavior of assets and liabilities. In this context, the correct mapping of nonlinear dependencies is of central concern. Although many DFA models and most practitioners still focus on linear correlation, the literature suggests that solely considering linear correlation is not appropriate in modeling dependence structures between heavy-tailed and skewed risks, which are frequent in the insurance context (see, e.g., Embrechts, McNeil, and Straumann, 2002). These risks are especially relevant in case of extreme events, e.g., the September 11, 2001, terrorist attacks that resulted in insurance companies experiencing large losses both from their underwriting business and the related capital markets plunge (see, e.g., Achleitner, Biebel, and Wichels, 2002). A more recent example is the subprime credit crisis, in which insurers have sustained large losses from their investments, e.g., in mortgage-backed securities, as well as from insuring structured credit products such as collateralized debt obligations (American International Group (AIG) is the most prominent example).

In this article, we evaluate the influence of such extreme events on a nonlife insurer's risk and return profile. We integrate nonlinear dependencies in a DFA framework using the copulas concept and evaluate their effects on the insurer's risk and return distribution within a simulation study. As one cannot generally say which copula describes reality best, we compare different forms of copulas (i.e., the Gauss, t, Gumbel, Clayton, and Frank copulas) and evaluate the possible impact in a stress-testing sense.

In our simulation study, we find that nonlinear dependencies have a strong influence on the insurer's default risk and performance. We also find different impacts of nonlinear dependencies on ruin probability and expected policyholder deficit, a result that is of special relevance for policyholders, regulators, and rating agencies. For example, for some kinds of nonlinear dependencies, the expected policyholder deficit cannot be reduced by increasing equity capital. It thus seems that these tail dependencies are relevant not only for low-capitalized companies but also for well-capitalized ones. Furthermore, we test several risk management strategies implemented in response to adverse outcomes generated by nonlinear dependencies. Our simulation results show that simple risk-reduction strategies are of little use. For example, a reinsurance strategy can delimit the high ruin probability generated by nonlinear dependencies, but not necessarily the expected policyholder deficit.

Our article builds upon two branches of literature--DFA and the copulas concept. …