Stackelberg reinsurance chain under model ambiguity

Abstract

In this paper, we consider a continuous-time version of a reinsurance chain, which is sequentially formed by n+1 companies, with the first company being the primary insurer and the rest being reinsurers. Because of possible model misspecification, all companies are ambiguous about the original risk of the primary insurer. We model each reinsurance contracting problem as a Stackelberg game, in which the assuming reinsurer acts as the leader while the ceding company is the follower. Reinsurance is priced using the mean-variance premium principle and all companies are risk neutral under their own beliefs. We obtain equilibrium indemnities, premium loadings, and distortions in closed form, all of which are proportional to the original risk, with the corresponding proportions decreasing along the chain. We also show that the reinsurance chain with ambiguity aversions in increasing order is optimal from the perspectives of both selfish individual companies and an unselfish central planner.

Keywords:

• Knightian uncertainty
• ambiguity
• mean-variance premium principle
• Stackelberg differential game
• reinsurance

JEL Subject Classifications:

• C61
• C72
• C73
• D81
• G22

Acknowledgments

We are grateful to the anonymous reviewers for their insightful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See https://content.naic.org/cipr-topics/reinsurance.

2 Note that Stackelberg equilibria are reached in two steps: the follower optimizes first given an arbitrary but fixed strategy from the leader, and next the leader optimizes upon knowing the follower's optimal decisions. However, in a standard Cournot game, all players make their decisions simultaneously.

3 For the line of research on reinsurance tree, see Asimit et al. (2013), Chi & Meng (2014), Boonen et al. (2016), and Boonen & Ghossoub (2019). Cao et al. (2023b) compare both the reinsurance tree and chain in a market with two reinsurers.

4 Chen et al. (2020) argue that static models ‘cannot reflect the changes in the claim processes’ and, thus, ‘may not be sufficient for the managerial needs in the context of dynamic insurance risk management in the long run’ (see pp. 129-130 therein).

5 For a single, continuous-time Stackelberg reinsurance game with one insurer and one reinsurer, please see Chen & Shen (2018, 2019) and Li & Young (2022).

6 It is well documented in the literature that neither the insurer nor the reinsurer knows the precise distribution of the underlying loss in a reinsurance contract. This naturally motivates the incorporation of model ambiguity, also termed model uncertainty, in the study of reinsurance. For related literature, see, for instance, Zhang & Siu (2009), Hu et al. (2018), Gu et al. (2020), and Cao et al. (2022, 2023a).

7 This assumption is well-justified in the literature of insurance economics; see Rothschild & Stiglitz (1976).

8 This result lends support to Gerber (1984), who only considers proportional reinsurance, because we do not impose any assumption on the form of reinsurance.

9 If j=0, then $i=0,1,\dots ,j-1$ represents the empty set.

10 We fix the position of company 0, the primary insurer, in the reinsurance chain. It is straightforward, with only notational changes, to allow company 0 to switch places with company 1.

11 In our model, the only distinguishing characteristic of a reinsurer is its ambiguity parameter. Thus, without loss of generality, assume that the ambiguity parameters of the n reinsurers are n distinct positive numbers.

12 According to Insurance Information Institute,

‘The London Market is a distinct, separate part of the U.K. insurance and reinsurance industry centered in the City of London. Its main participants are insurance and reinsurance companies, Lloyd's of London syndicates, Marine Protection and Indemnity Clubs (P&I Clubs), and brokers who handle most of the business’.

Additional information

Funding

Jingyi Cao and Dongchen Li acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (grant numbers 05061 and 04958, respectively). V. R. Young thanks the Cecil J. and Ethel M. Nesbitt Professorship for financial support of her research.