##
**Optimal reinsurance in relation to ordering of risks.**
*(English)*
Zbl 0683.62060

From the authors’ introduction: This paper examines the optimal choice of a reinsurance contract. Applying the stop-loss ordering of risks to the retained risks after reinsurance leads to short and elegant derivations.

Several types of reinsurance contracts and degrees of coverage are possible. The optimal form and amount, from the insurer’s point of view, depends on his optimization criterion. In case of maximizing the expected utility of the insurer using a concave utility function a classical result says that if the reinsurance premium is proportional to the expected value of the risk, then for every fixed reinsurance premium the optimal contract must be of stop-loss form.

This important result is extended to several other optimization criteria: a partial preference ordering of risks after reinsurance is shown to imply the same preferences for a number of criteria a risk-averse decision maker might use. So if there is a retained risk that is minimal in stop-loss order, then it will be optimal from the insurer’s point of view when optimizing any of these criteria. The results found for a single (total) risk can be extended to compound models for portfolios of risks, as well as to the case where the set of reinsurance contracts to choose from is restricted.

Several types of reinsurance contracts and degrees of coverage are possible. The optimal form and amount, from the insurer’s point of view, depends on his optimization criterion. In case of maximizing the expected utility of the insurer using a concave utility function a classical result says that if the reinsurance premium is proportional to the expected value of the risk, then for every fixed reinsurance premium the optimal contract must be of stop-loss form.

This important result is extended to several other optimization criteria: a partial preference ordering of risks after reinsurance is shown to imply the same preferences for a number of criteria a risk-averse decision maker might use. So if there is a retained risk that is minimal in stop-loss order, then it will be optimal from the insurer’s point of view when optimizing any of these criteria. The results found for a single (total) risk can be extended to compound models for portfolios of risks, as well as to the case where the set of reinsurance contracts to choose from is restricted.

Reviewer: W.R.Heilmann

### MSC:

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

### Keywords:

stop-loss ordering of risks; reinsurance contracts; degrees of coverage; expected utility; concave utility function; reinsurance premium; optimal contract; partial preference ordering of risks; compound models for portfolios of risks
PDF
BibTeX
XML
Cite

\textit{A. E. van Heerwaarden} et al., Insur. Math. Econ. 8, No. 1, 11--17 (1989; Zbl 0683.62060)

Full Text:
DOI

### References:

[1] | Borch, K., An attempt to determine the optimal amount of stop loss reinsurance, (), 597-610 |

[2] | Borch, K., The utility concept applied to the theory of insurance, ASTIN bulletin, 1, 245-255, (1961) |

[3] | Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1986), The Society of Actuaries Itasca, IL · Zbl 0634.62107 |

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

[5] | Goovaerts, M.J.; De Vylder, F.; Haezendonck, J., Insurance premiums, (1984), North-Holland Amsterdam · Zbl 0532.62082 |

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

[7] | Pesonen, M.I., Optimal reinsurances, Scandinavian actuarial journal, 67, 65-90, (1984) · Zbl 0544.62096 |

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.