Budget-constrained optimal reinsurance design under coherent risk measures

ABSTRACT

Reinsurance is a versatile risk management strategy commonly employed by insurers to optimize their risk profile. In this paper, we study an optimal reinsurance design problem minimizing a general law-invariant coherent risk measure of the net risk exposure of a generic insurer, in conjunction with a general law-invariant comonotonic additive convex reinsurance premium principle and a premium budget constraint. Due to its intrinsic generality, this contract design problem encompasses a wide body of optimal reinsurance models commonly encountered in practice. A three-step solution scheme is presented. Firstly, the objective and constraint functions are exhibited in the so-called Kusuoka's integral representations. Secondly, the mini-max theorem for infinite dimensional spaces is applied to interchange the infimum on the space of indemnities and the supremum on the space of probability measures. Thirdly, the recently developed Neyman–Pearson methodology due to Lo (2017a) is adopted to solve the resulting infimum problem. Analytic and transparent expressions for the optimal reinsurance policy are provided, followed by illustrative examples.

KEYWORDS:

• Budget constraint
• distortion
• TVaR
• mini-max theorem
• Neyman–Pearson

JEL CODE:

• G22
• C61

Acknowledgments

The authors are grateful to an anonymous reviewer for his/her careful reading and insightful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that $c^{\ast }=inf\{c\in [-\mathrm{\infty },0]\mid \mathcal{G}(c)>\pi \}$ always exists as a real negative number in Case (ii) because the set $\{c\in [-\mathrm{\infty },0]\mid \mathcal{G}(c)>\pi \}$ is non-empty.

2 For one of the possible candidates, see the appendix.

3 Although ${\mu }_2$ is not a probability measure, this does not affect the validity of the results in this paper.

Additional information

Funding

Ka Chun Cheung acknowledges the financial support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 17324516). Wing Fung Chong is supported by start-up funds provided by the Department of Mathematics and Department of Statistics, University of Illinois at Urbana-Champaign (Project Codes 257406 and 583115). Ambrose Lo is supported by a start-up research fund provided by the Department of Statistics and Actuarial Science, The University of Iowa, and a Centers of Actuarial Excellence (CAE) Research Grant (2018–2021) from the Society of Actuaries (SOA). Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA.