On the Optimal Risk Sharing in Reinsurance with Random Recovery Rate

Abstract

This paper studies a Pareto-optimal reinsurance contract in the presence of negative statistical dependence between the insurance claim and the random recovery rate. In the context of symmetric information model and asymmetric information model, we investigate properties of the Pareto-optimal indemnity schedules. For risk neutral reinsurer with proportional cost and associated expense, we present possible forms of the Pareto-optimal indemnity schedule as well.Risks 2018, 6, 114 2 of 16 and uninsurable risks, Dana and Scarsini (2007) investigated qualitative properties of Pareto-optimal insurance contracts.Besides the additive one, the multiplicative background risk is quite common in the insurance practice. The counter-party risk in a reinsurance contract, which means the little chance that the reinsurer fails to pay the entire promised benefit to the insurer, is the default risk of the reinsurer. Readers may refer to Franke et al. (2006) for a comprehensive study on the multiplicative background risk. In the existing literature, researchers studied various characterizations of the counter-party risk together with its impact on the design of optimal reinsurance contract. For example, Tapiero et al. (1986) employed the perceived probability of ruin to characterize the probability of the insurer being affected by the reinsurer's default risk, and Schlesinger and Schulenburg (1987) considered a three-state model in which a loss that can not be indemnified occurs with a nonnegative probability, and discussed how insurance purchases are affected by the insurer's level of risk aversion. In the four-state model of Doherty and Schlesinger (1990), both the reinsurer and the insurer know the default probability, and the partial insurance is proved to be optimal when the reinsurer has a positive probability of total default. Later, such a result was extended to partial default by Mahul and Wright (2007). On the other hand, Cummins and Mahul (2003) derived the optimal insurance in the context that the reinsurer has a positive probability of total default and the reinsurer and insurer have divergent beliefs about this probability. Recently, Bernard and Ludkovski (2012) considered loss-dependent probability of default and partial recovery in the event of contract non-performance, and studied the Pareto-optimal reinsurance contracts with counter-party risk for risk neutral reinsurer and risk averse insurer. Thus, Bernard and Ludkovski (2012) extended the model of Dana and Scarsini (2007) to the case of multiplicative but not additive background risk.Usually, the background risk has significant impact on the Pareto-optimal reinsurance contract, and as is pointed out in Franke et al. (2006) and Bernard and Ludkovski (2012), the presence of multiplicative background risk can be more complex than the additive risk. On the other hand, in the presence of multiplicative background risk, Bernard and Ludkovski (2012) only dealt with binary recovery rate and risk neutral insurer with linear cost. In this study, we consider the reinsuranc...

Mean–variance Insurance Design With Counterparty Risk and Incentive Compatibility

“…In the literature, the recovery rate Y is generally assumed to be negatively correlated with X (see Bernard and Ludkovski 2012;Li and Li 2018). This is intuitive, as a larger loss would make the insurer more likely to default, which results in a smaller recovery rate.…”

“…Second, one assumes that the market with policyholders is "large", so that individual insurance transactions do not impact the likelihood of default. This assumption is imposed by Cummins and Mahul (2003), Bernard andLudkovski (2012), andLi andLi (2018), and in this paper we also impose this assumption. Cummins and Mahul (2003) study the optimal insurance problem when the insurer has a positive probability to default and the insured and insurer have divergent beliefs about this probability.…”

“…The optimal indemnity function is derived in an implicit way for the case where the insurer is risk neutral. Finally, Li and Li (2018) derive the optimal indemnity function in a Pareto-optimal insurance problem when the loss and recovery rate are negatively correlated for the cases where the information is symmetric and where the information is asymmetric. All these papers focus primarily on expected utility preferences of the insured, while our focus is on a mean-variance objective of the insured.…”

Optimal insurance with counterparty and additive background risk

“…Cai et al (2014) derive optimal insurance strategies by taking counterparty risk and the regulatory capital into insurance contracts. Li and Li (2018) study Pareto optimal insurance under counterparty risk. For more works on insurance under counterparty risk, see Asimit et al (2013), Filipovic et al (2015), Boonen and Jiang (2023), and references therein.…”

Pareto-optimal reinsurance with default risk and solvency regulation

“…In [7,10,24], default is modeled as an exogenous event, which is unaffected by the insurance contract. Li and Li [31] extend the work of Bernard and Ludkovski [7] to a bilateral setting in the presence of information asymmetry. Asimit et al [3] assume a constant default probability and recovery rate and study the optimal indemnity function in a risk minimization framework.…”