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LINEAR VERSUS NONLINEAR ALLOCATION RULES IN RISK SHARING UNDER FINANCIAL FAIRNESS

Published online by Cambridge University Press:  06 August 2018

Johannes M. Schumacher
Johannes M. Schumacher*
Affiliation:
University of Amsterdam, Faculty of Economics and Business, Section Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
*
E-Mail: [email protected]
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Abstract

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In a risk exchange, participants trade a privately owned risk for a share in a pool. If participants agree on a valuation rule, it can be decided whether or not, according to the given rule, these trades take place at equal value. If equality of values holds for all participants, then the exchange is said to be “financially fair”. It has been shown by Bühlmann and Jewell (1979) that, under mild assumptions, the constraint of financial fairness singles out a unique solution among the set of all Pareto efficient risk exchanges. In this paper, we find that an analogous statement is true if we limit ourselves to linear exchanges. Conditions are provided for existence and uniqueness of linear sharing rules that are both financially fair and Pareto efficient among all linear sharing rules. The performance of the linear rule is compared to that of the general (nonlinear) rule in a number of specific cases.

Keywords

Risk sharingallocation rulesfinancial fairnessBorch theoremsyndicates
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 995 - 1024
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Astin Bulletin 2018

References

Bao, H., Ponds, E.H.M. and Schumacher, J.M. (2017) Multi-period risk sharing under financial fairness. Insurance: Mathematics and Economics, 72, 49–66.Google Scholar
Barrieu, P. and El Karoui, N. (2005) Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9, 269–298.10.1007/s00780-005-0152-0Google Scholar
Barrieu, P. and Scandolo, G. (2008) General Pareto optimal allocations and applications to multi-period risks. ASTIN Bulletin, 38, 105–136.10.1017/S0515036100015087Google Scholar
Biagini, S. and Guasoni, P. (2011) Relaxed utility maximization in complete markets. Mathematical Finance, 21, 703–722.Google Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424–444.10.2307/1909887Google Scholar
Brams, S.J. and Taylor, A.D. (1996) Fair Division: From Cake-Cutting to Dispute Resolution. New York: Cambridge University Press.10.1017/CBO9780511598975Google Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10, 243–262.10.1017/S0515036100005882Google Scholar
Carmona, R., ed. (2009) Indifference Pricing: Theory and Applications. Princeton, NJ: Princeton University Press.Google Scholar
Chateauneuf, A., Dana, R.-A. and Tallon, J.-M. (2000) Optimal risk-sharing rules and equilibria with Choquet-Expected-Utility. Journal of Mathematical Economics, 34, 191–214.10.1016/S0304-4068(00)00041-0Google Scholar
Chen, A., Pelsser, A. and Vellekoop, M. (2011) Modeling non-monotone risk aversion using SAHARA utility functions. Journal of Economic Theory, 146, 2075–2092.10.1016/j.jet.2011.06.011Google Scholar
Chen, D. and Beetsma, R. (2015) Mandatory participation in occupational pension schemes in the Netherlands and other countries: An update. Technical Report DP 10/2015-032, Netspar.10.2139/ssrn.2670476Google Scholar
Dai, R. and Schumacher, J.M. (2009) Welfare analysis of conditional indexation schemes from a two-reference-point perspective. Journal of Pension Economics and Finance, 8, 321–350.10.1017/S1474747208003867Google Scholar
Filipovic, D. and Svindland, G. (2008) Optimal capital and risk allocations for law- and cash-invariant convex functions. Finance and Stochastics, 12, 423–439.10.1007/s00780-008-0069-5Google Scholar
Gale, D. (1977) Fair division of a random harvest. Technical Report ORC 77-21, Operations Research Center, UC Berkeley.Google Scholar
Gale, D. and Sobel, J. (1979) Fair division of a random harvest: The finite case. In General Equilibrium, Growth, and Trade. Essays in Honor of Lionel McKenzie (eds. Green, J.R. and Scheinkman, J.), pp. 193–198. New York: Academic Press.10.1016/B978-0-12-298750-2.50016-2Google Scholar
Höffding, W. (1994) Scale-invariant correlation theory. In The Collected Works of Wassily Hoeffding (eds. Fisher, N.I. and Sen, P.K.), pp. 57–108. New York: Springer.10.1007/978-1-4612-0865-5_4Google Scholar
Homma, T. (1952) A theorem on continuous functions. Kodai Mathematical Seminar Reports, 4, 13–16.10.2996/kmj/1138843207Google Scholar
Huang, C.-F. and Litzenberger, R. (1985) On the necessary condition for linear sharing and separation: A note. Journal of Financial and Quantitative Analysis, 20, 381–384.10.2307/2331037Google Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18, 269–292.10.1111/j.1467-9965.2007.00332.xGoogle Scholar
Kamps, U. (1998) On a class of premium principles including the Esscher principle. Scandinavian Actuarial Journal, 1998 (1), 75–80.10.1080/03461238.1998.10413993Google Scholar
Kramkov, D. and Schachermayer, W. (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability, 9, 904–950.Google Scholar
Marić, V. and Tomić, M. (1990) A classification of solutions of second order linear differential equations by means of regularly varying functions. Publications de l'Institut Mathématique, 48, 199–207.Google Scholar
Molenaar, R.D.J. and Ponds, E.H.M. (2012/13) Risk sharing and individual lifecycle investing in funded collective pensions. Journal of Risk, 15 (2), 103–124.10.21314/JOR.2012.256Google Scholar
Pak, I. (2010) Lectures on Discrete and Polyhedral Geometry. http://www.math.ucla.edu/~pak/geompol8.pdf.Google Scholar
Pazdera, J., Schumacher, J.M. and Werker, B.J.M. (2017) The composite iteration algorithm for finding efficient and financially fair risk-sharing rules. Journal of Mathematical Economics, 72, 122–133.10.1016/j.jmateco.2017.07.008Google Scholar
Rotar, V.I. (2007) Actuarial Models. The Mathematics of Insurance. Boca Raton: Chapman & Hall/CRC.Google Scholar
Seal, H.L. (1969) Stochastic Theory of a Risk Business. New York: Wiley.Google Scholar
Tucker, A. (1995) The parallel climbers puzzle. Math Horizons, 3 (2), 22–24.10.1080/10724117.1995.11974954Google Scholar
van Heerwaarden, A.E., Kaas, R. and Goovaerts, M.J. (1989) Properties of the Esscher premium calculation principle. Insurance: Mathematics and Economics, 8, 261–267.Google Scholar
Whittaker, J.V. (1966) A mountain-climbing problem. Canadian Journal of Mathematics, 18, 873–882.10.4153/CJM-1966-087-xGoogle Scholar
Wilson, R. (1968) The theory of syndicates. Econometrica, 36, 119–132.10.2307/1909607Google Scholar
Wuerth, A.M. and Schumacher, J.M. (2011) Risk aversion for nonsmooth utility functions. Journal of Mathematical Economics, 47, 109–128.10.1016/j.jmateco.2010.10.003Google Scholar