Optimal reinsurance under general law-invariant risk measures.

*(English)*Zbl 1401.91110Summary: In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under value at risk and conditional tail expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than value at risk and conditional tail expectation which are more commonly used. Several illustrative examples will be provided.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

##### Keywords:

optimal reinsurance; convex risk measures; stop-loss insurance; insurance layer; geometric approach
PDF
BibTeX
XML
Cite

\textit{K. C. Cheung} et al., Scand. Actuar. J. 2014, No. 1, 72--91 (2014; Zbl 1401.91110)

Full Text:
DOI

##### References:

[1] | Acerbi, C. and Tasche, D.2002. On the coherence of expected shortfall. Journal of Banking & Finance, 26(7): 1487-1503. |

[2] | Annaert, J., van Osselaer, S. and Verstraete, B.2009. Performance evaluation of portfolio insurance strategies using stochastic dominance criteria. Journal of Banking & Finance, 33(2): 272-280. |

[3] | Arrow, K. J.1963. Uncertainty and the welfare economics of medical care. American Economic Review, 53: 941-973. |

[4] | Artzner, P., Delbaen, F., Eber, J. M. and Heath, D.1999. Coherent measures of risk. Mathematical Finance, 9(3): 203-228. · Zbl 0980.91042 |

[5] | Balbás, A., Balbás, B. and Heras, A.2009. Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics, 44(3): 374-384. |

[6] | Borch , K.1960 . An attempt to determine the optimum amount of stop loss reinsurance . Transactions of the 16th International Congress of Actuaries , 1 , 597 - 610 . |

[7] | Cai, J. and Tan, K. S.2007. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. ASTIN Bulletin, 37(1): 93-112. · Zbl 1162.91402 |

[8] | Cai, J., Tan, K. S., Weng, C. and Zhang, Y.2008. Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 43(1): 185-196. · Zbl 1140.91417 |

[9] | Cheung, K. C.2010. Optimal reinsurance revisited-geometric approach. ASTIN Bulletin, 40: 221-239. · Zbl 1230.91070 |

[10] | Delbaen, F.2000. Coherent risk measures, Cattedra Galileiana: Scuola Normale Superiore di Pisa. · Zbl 1060.91077 |

[11] | Delbaen, F.2002. Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, 1-37. Berlin: Springer-Verlag. · Zbl 1020.91032 |

[12] | Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kass, R. and Vyncke, D.2005. Comonotonic approximations for optimal portfolio selection problems. Journal of Risk and Insurance, 72(2): 253-301. |

[13] | Denuit, M. and Vermandele, C.1998. Optimal reinsurance and stop-loss order. Insurance: Mathematics and Economics, 22: 229-233. · Zbl 0986.62085 |

[14] | Filipović , D. & Svindland , G.2008 . Convex risk measures beyond bounded risks, or the canonical model space for law-invariant convex risk measures is L\^{1} . Preprint . |

[15] | Föllmer, H. and Schied, A.2002. Convex measures of risk and trading constraints. Finance and Stochastics, 6(4): 429-447. · Zbl 1041.91039 |

[16] | Föllmer, H. and Schied, A.2004. Stochastic finance: an introduction in discrete time, Berlin: Walter de Gruyter. |

[17] | Frittelli, M. and Rosazza Gianin, E.2002. Putting order in risk measures. Journal of Banking & Finance, 26(7): 1473-1486. |

[18] | Frittelli, M. and Rossaza Gianin, E.2005. Law invariant convex risk measures. Advances in Mathematical Economics, 7: 33-46. · Zbl 1149.91320 |

[19] | Heath, D. and Ku, H.2004. Pareto Equilibria with coherent measures of risk. Mathematical Finance, 14(2): 163-172. · Zbl 1090.91033 |

[20] | Jouini, E., Schachermayer, W. and Touzi, N.2006. Law invariant risk measures have the Fatou property. Advances in Mathematical Economics, 9: 49-71. · Zbl 1198.46028 |

[21] | Kaluszka, M.2004a. Mean-variance optimal reinsurance arrangements. Scandinarian Actuarial Journal, 1: 28-41. · Zbl 1117.62115 |

[22] | Kaluszka, M.2004b. An extension of Arrow’s result on optimality of a stop loss contract. Insurance: Mathematics and Economics, 35(3): 527-536. · Zbl 1122.91343 |

[23] | Kaluszka, M.2005. Optimal reinsurance under convex principles of premium calculation. Insurance: Mathematics and Economics, 36(3): 375-398. · Zbl 1120.62092 |

[24] | Kusuoka, S.2001. On law invariant coherent risk measures. Advances in Mathematical Economics, 3: 83-95. · Zbl 1010.60030 |

[25] | Rockafellar , R. T. & Uryasev , S.2001 . Conditional value-at-risk for general loss distributions . Research report 2001-5 . ISE Dept., University of Florida . |

[26] | Schied, A.2007. Optimal investments for risk- and ambiguity-averse preferences: a duality approach. Finance and Stochastics, 11(1): 107-129. · Zbl 1143.91021 |

[27] | Song, Y. and Yan, J. A.2009a. Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders. Insurance: Mathematics and Economics, 45(3): 459-465. · Zbl 1231.91237 |

[28] | Song, Y. and Yan, J. A.2009b. An overview of representation theorems for static risk measures. Science in China Series A: Mathematics, 52(7): 1412-1422. · Zbl 1184.91114 |

[29] | Sung, K. C. J., Yam, S. C. P., Yung, S. P. and Zhou, J. H.2011. Behavioral optimal insurance. Insurance: Mathematics and Economics, 49(3): 418-428. · Zbl 1229.91167 |

[30] | Young, V. R.1999. Optimal insurance under Wang’s premium principle. Insurance: Mathematics and Economics, 25(2): 109-122. · Zbl 1156.62364 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.