##
**Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria.**
*(English)*
Zbl 1152.91583

Summary: This paper is concerned with the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk. We deal with the problem by exploring the relationship between maximizing the adjustment coefficient and maximizing the expected utility of wealth for the exponential utility function, both with respect to the retained risk of the insurer.

Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.

Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

### Keywords:

optimal reinsurance; risk; stop loss; ruin probability; adjustment coefficient; premium principles; exponential utility function
PDF
BibTeX
XML
Cite

\textit{M. Guerra} and \textit{M. L. Centeno}, Insur. Math. Econ. 42, No. 2, 529--539 (2008; Zbl 1152.91583)

Full Text:
DOI

### References:

[1] | Arrow, K.J., Uncertainty and the welfare of medical care, The American economic review, LIII, 941-973, (1963) |

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

[3] | Centeno, M.L., Excess of loss reinsurance and the probability of ruin in finite horizon, ASTIN bulletin, 27, 1, 59-70, (1997) |

[4] | Deprez, O.; Gerber, H.U., On convex principles of premium calculation, Insurance: mathematics and economics, 4, 179-189, (1985) · Zbl 0579.62090 |

[5] | Gamkrelidze, R.V., Principles of optimal control theory, (1978), Plenum Press · Zbl 0401.49001 |

[6] | Gerber, H.U., () |

[7] | Hesselager, O., Some results on optimal reinsurance in terms of the adjustment coefficient, Scandinavian actuarial journal, 1, 80-95, (1990) · Zbl 0728.62100 |

[8] | Kahn, P., Some remarks on a recent paper by borch, ASTIN bulletin, 1, 265-272, (1961) |

[9] | Lemaire, J., Sur la détermination d’un contrat optimal de ré assurance, ASTIN bulletin, 7, 165-180, (1973) |

[10] | Ohlin, J., On a class of measures of dispersion with application to optimal reinsurance, ASTIN bulletin, 5, 249-266, (1969) |

[11] | Rudin, W., Functional analysis, (1991), McGraw-Hill International Singapore · Zbl 0867.46001 |

[12] | Vajda, S., Minimum variance reinsurance, ASTIN bulletin, 2, 257-260, (1962) |

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.