Capital allocation for portfolios with non-linear risk aggregation |
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| Institution: | 1. Amsterdam School of Economics, University of Amsterdam, The Netherlands;2. Faculty of Actuarial Science and Insurance, Cass Business School, City University London, United Kingdom;3. RiskLab, Department of Mathematics, ETH Zurich, Switzerland;1. Cass Business School, City University, London EC1Y 8TZ, United Kingdom;2. College of Business and Public Administration, Drake University, 345 Aliber Hall, 2507 University Avenue, Des Moines, IA 50311, USA;1. Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005-Santander, Spain;2. Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, 35017-Las Palmas de G.C., Spain;1. University of Oslo, Norway;2. University of Barcelona, Spain;1. Amsterdam School of Economics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, The Netherlands;2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada |
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| Abstract: | Existing risk capital allocation methods, such as the Euler rule, work under the explicit assumption that portfolios are formed as linear combinations of random loss/profit variables, with the firm being able to choose the portfolio weights. This assumption is unrealistic in an insurance context, where arbitrary scaling of risks is generally not possible. Here, we model risks as being partially generated by Lévy processes, capturing the non-linear aggregation of risk. The model leads to non-homogeneous fuzzy games, for which the Euler rule is not applicable. For such games, we seek capital allocations that are in the core, that is, do not provide incentives for splitting portfolios. We show that the Euler rule of an auxiliary linearised fuzzy game (non-uniquely) satisfies the core property and, thus, provides a plausible and easily implemented capital allocation. In contrast, the Aumann–Shapley allocation does not generally belong to the core. For the non-homogeneous fuzzy games studied, Tasche’s (1999) criterion of suitability for performance measurement is adapted and it is shown that the proposed allocation method gives appropriate signals for improving the portfolio underwriting profit. |
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| Keywords: | Capital allocation Euler rule Fuzzy core Aumann–Shapley value Risk measures |
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