A Neyman-Pearson Perspective on Optimal Reinsurance With Constraints

Abstract

The formulation of optimal reinsurance policies that take various practical constraints into account is a problem commonly encountered by practitioners. In the context of a distortion-risk-measure-based optimal reinsurance model without moral hazard, this article introduces and employs a variation of the Neyman–Pearson Lemma in statistical hypothesis testing theory to solve a wide class of constrained optimal reinsurance problems analytically and expeditiously. Such a Neyman–Pearson approach identifies the unit-valued derivative of each ceded loss function as the test function of an appropriate hypothesis test and transforms the problem of designing optimal reinsurance contracts to one that resembles the search of optimal test functions achieved by the classical Neyman–Pearson Lemma. As an illustration of the versatility and superiority of the proposed Neyman–Pearson formulation, we provide complete and transparent solutions of several specific constrained optimal reinsurance problems, many of which were only partially solved in the literature by substantially more difficult means and under extraneous technical assumptions. Examples of such problems include the construction of the optimal reinsurance treaties in the presence of premium budget constraints, counterparty risk constraints and the optimal insurer–reinsurer symbiotic reinsurance treaty considered recently in Cai et al. (2016).

On randomized reinsurance contracts

“…We would like to point out that Gajek and Zagrodny [19] also discovered randomized reinsurance treaties as 'curious' possible solutions in the presence of discrete loss variables when the goal is to minimize the ruin probability of an insurer and there is a constraint on the available reinsurance premium, a problem which they nicely linked to the Neyman-Pearson lemma in statistical hypothesis testing (and in that case the performance of these randomized treaties can not be matched by a deterministic treaty). This connection between optimal reinsurance and the design of most powerful tests in statistics was recently studied in more detail in Lo [24].…”

Optimal Reinsurance From the Viewpoints of Both an Insurer and a Reinsurer Under the Cvar Risk Measure and Vajda Condition

“…Cai and Weng (2016) studied optimal reinsurance with the expectile. For more studies about optimal reinsurance under risk measures, we refer to Zheng and Cui (2014), Assa (2015), Lo (2017aLo ( , 2017b, etc. Most of the results obtained are from the insurer's point of view or from the reinsurer's point of view.…”

Universally Marketable Insurance Under Multivariate Mixtures

“…Among the wide spectrum of feasible insurance indemnity schedules, those with the non-decreasing and 1-Lipschitz condition (i.e., those I such that 0 ≤ I(x) − I(y) ≤ x − y for all y ≤ x), also informally known as the slowly growing condition, have unquestionably gained in popularity in the recent insurance literature. Ensuring that the indemnified loss never increases faster than the policyholder's ground-up loss, the non-decreasing and 1-Lipschitz condition has been used in an abundance of papers as a starting point for formulating optimal (re)insurance policies, which are typically in the form of insurance layers, see, for example, Chi and Tan (2011), Cai et al (2017), Cheung and Lo (2017), and Lo (2017). For all the mathematical benefits it brings, theoretical or empirical justifications of the economic suitability and practical relevance of the non-decreasing and 1-Lipschitz condition to the insurance business have long been lacking, undermining the conceptual foundation of many existing results it underlies.…”