The fundamental theorem of mutual insurance |
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| Institution: | 1. School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown PE, C1A 4P3 Canada;2. Dept. of Stats. and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou str., 185 34 Piraeus, Greece;3. Universität Stuttgart, Fachbereich Mathematik, Postfach 80 11 40, D-70511 Stuttgart, Germany;1. Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong;2. Department of Accountancy, Finance and Insurance, KU Leuven, Naamsestraat 69, B-3000 Leuven, Belgium;3. Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA;1. University of Connecticut, United States;2. Moscow State University, Russia;1. Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1;2. Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong;1. Actuarial Science Program, Smeal College of Business, Penn State University, University Park, PA 16802, USA;2. Actuarial Science Program, Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI, 54701, USA |
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| Abstract: | The essence of mutual insurance is the notion that re-distributing risk in a pool of risks is more beneficial than taking the risk alone. Interpreting ‘more beneficial’ as an increase in utility and considering sequences of exchangeable risks, we are able to formalize this notion from the policyholder’s perspective and demonstrate its validity for various alternative preference functionals (e.g., expected utility, Choquet expected utility, and distortion risk measures). To obtain this result, we exploit that for a sequence of exchangeable risks the corresponding sequence of arithmetical averages is a reversed martingale.We conclude that pooling risks is fundamental for understanding the mechanisms of insurance because it favourably affects the utility of policyholders, and we refer to this phenomenon as the ‘utility-improving effect of risk pooling’. Moreover, we demonstrate that the utility of the policyholder is (strictly) increasing with the size of the risk pool. |
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| Keywords: | Pooling risks Exchangeability Reversed martingales Choquet expected utility Distortion risk measures |
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