On the impossibility of fair risk allocation

Csóka, Péter and Pintér, Miklós (2010): On the impossibility of fair risk allocation.

Abstract

Measuring and allocating risk properly are crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using a coherent measure of risk it is impossible to allocate risk satisfying the natural requirements of (Solution) Core Compatibility, Equal Treatment Property and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games.

• Item Type: MPRA Paper
• Language: English
• Keywords: Coherent Measures of Risk, Risk Allocation Games, Totally Balanced Games, Exact Games, Shapley value, Solution core
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games; G - Financial Economics > G1 - General Financial Markets > G10 - General

References

Acerbi C., Scandolo G., 2008. Liquidity risk theory and coherent measures of risk. Quantitative Finance 8, 681-692.

Acerbi C., Tasche D., 2002. On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487-1504.

Artzner P.F., Delbaen F., Eber J.M., Heath D., 1999. Coherent measures of risk. Mathematical Finance 9, 203-228.

Branzei R., Tijs S., Zarzuelo J., 2009. Convex multi-choice games: Characterizations and monotonic allocation schemes. European Journal of Operational Research 198, 571-575.

Buch A., Dor eitner G., 2008. Coherent risk measures, coherent capital allocations and the gradient allocation principle. Insurance: Mathematics and Economics 42 (1), 235-242.

Calleja P., Borm P., Hendrickx R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730-747.

Casas-Mendez B., Garcia-Jurado I., van den Nouweland A., Vaazquez-Brage M., 2003. An extension of the -value to games with coalition structures. European Journal of Operational Research 148, 494-513.

Cont R., 2001. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1, 223-236.

Csoka P., Herings P.J.J., Koczy L.A., 2007. Coherent measures of risk from a general equilibrium perspective. Journal of Banking and Finance 31 (8), 2517-2534.

Csoka P., Herings P.J.J., Koczy L.A., 2009. Stable allocations of risk. Games and Economic Behavior 67 (1), 266-276.

Denault M., 2001. Coherent allocation of risk capital. Journal of Risk 4 (1),1-34.

Dubey P., Shapley L.S., 1984. Totally balanced games arising from controlled programming problems. Mathematical Programming 29, 245-267.

Gillies D.B., 1959. Solutions to general non-zero-sum games, vol. IV of Contributions to the Theory of Games. Princeton University Press.

Kalai E., Zemel E., 1982a. Generalized network problems yielding totally balanced games. Operations Research 30, 998-1008.

Kalai E., Zemel E., 1982b. Totally balanced games and games of ow. Mathematics of Operations Research 7, 476-478.

Kalkbrener M., 2005. An axiomatic approach to capital allocation. Mathematical Finance 15 (3), 425-437.

Kim J.H.T., Hardy M.R., 2009. A capital allocation based on a solvency exchange option. Insurance Mathematics and Economics 44, 357-366.

Legut J., 1990. On totally balanced games arising from cooperation in fair division. Games and Economic Behavior 2, 47-60.

Lintner J., 1965. The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13-37.

Owen G., 1975. On the core of linear production games. Mathematical Programming 9, 358-370.

Pinter M., 2009. Young's axiomatization of the Shapley value - a new proof. arXiv:0805.2797v2 .

Rabie M.A., 1981. A note on the exact games. International Journal of Game Theory 10 (3-4), 131-132.

Schmeidler D., 1972. Cores of exact games. Journal of Mathematical Analysis and Applications 40, 214-225.

Shapley L.S., 1953. A value for n-person games. In: H.W. Kuhn, A.W. Tucker (Eds.), Contributions to the Theory of Games II, vol. 28 of Annals of Mathematics Studies. Princeton University Press, Princeton, pages 307-317.

Shapley L.S., 1971. Cores of convex games. International Journal of Game Theory 1, 11-26.

Shapley L.S., Shubik M., 1969. On market games. Journal of Economic Theory 1, 9-25.

Sharpe W.F., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425-442.

Tijs S., Parthasarathy T., Potters J., Prassad V.R., 1984. Permutation games: Another class of totally balanced games. OR Spektrum 6, 119-123.

Valdez E.A., Chernih A., 2003. Wang's capital allocation formula for elliptically contoured distributions. Insurance: Mathematics and Economics 33, 517-532.

Young H.P., 1985. Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65-72.