How Much Is Optimal Reinsurance Degraded by Error?

Abstract

Estimation error reduces reinsurance optimality under a fitted model to suboptimality under the true one. A mathematical formulation of this issue of degradation is offered and examined through asymptotics as the sample size n of the historical observations becoming infinite. Assuming economic or distortion pricing of reinsurance it is shown that the rate of degradation is either $O(1/n)$ or $O\left(1\sqrt{n}\right)$ depending on smoothness properties of the risk measure employed. Examples are conditional Value at Risk criteria, which tend to be $O(1/n),$ and Value at Risk, which is $O(1/\sqrt{n}).$ A numerical study investigates the issue for smaller n and suggests a need for developing more robust optimal reinsurance techniques that can with stand model errors better.

ACKNOWLEDGMENTS

The authors thank the editor and reviewers for careful reading and constructive comments that helped improve the presentation of the article and a number of references. We also thank Arne Huseby and Ingrid Hobaek Haff for their valuable comments and fruitful discussions.

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