Optimal reinsurance in the presence of counterparty default risk

Abstract

The optimal reinsurance arrangement is identified whenever the reinsurer counterparty default risk is incorporated in a one-period model. Our default risk model allows the possibility for the reinsurer to fail paying in full the promised indemnity, whenever it exceeds the level of regulatory capital. We also investigate the change in the optimal solution if the reinsurance premium recognises or not the default in payment. Closed form solutions are elaborated when the insurer's objective function is set via some well-known risk measures. It is also discussed the effect of reinsurance over the policyholder welfare. If the insurer is Value-at-Risk regulated, then the reinsurance does not increase the policyholder's exposure for any possible reinsurance transfer, even if the reinsurer may default in paying the promised indemnity. Numerical examples are also provided in order to illustrate and conclude our findings. It is found that the optimal reinsurance contract does not usually change if the counterparty default risk is taken into account, but one should consider this effect in order to properly measure the policyholders's exposure. In addition, the counterparty default risk may change the insurer's ideal arrangement if the buyer and seller have very different views on the reinsurer's recovery rate.

How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk

“…Synthesizing the discussions in Asimit et al [13] and Zhuang et al [11], we begin with a precise description of a DRM-based insurance-reinsurance model in which three agents, namely a policyholder, an insurer and a default-prone reinsurer, coexist. In this model, the risk faced by an agent is evaluated by means of distortion risk measures (DRMs), whose definition calls for the notion of a distortion function (see Denneberg [15] and Wang [16]).…”

“…Given the recent implementation of the Solvency II regulatory framework in the countries of the European Union, we follow Asimit et al [13] and assume that the reinsurer operates in a VaR-regulated environment and thus prescribes its regulatory capital in accordance with the β-level VaR of the reinsured loss for some large probability level β (e.g., β = 99.5% in Solvency II). If f R : [0, f I (F −1 X (1))) → R + represents the reinsurance indemnity function and δ ∈ [0, 1] is the recovery rate of the loss given default, then the insurer is in effect compensated for…”

“…More remarkably, the striking symmetry between Formula (2) and Formula (4), when viewed in the correct light, will substantially facilitate understanding the marginal costs and benefits of insurance and reinsurance. Compared with the reinsurance premium principles in Asimit et al [13] and Zhuang et al [11], Formula (4) possesses two subtle differences. First, unlike Asimit et al [13], no allowance is made in Formula (4) for the possibility of default when the reinsurer charges the reinsurance premium.…”

Some mathematical aspects of price optimisation

“…Very recent contributions focus on the issues of price optimisation, mainly from the ethical and regulation points of view, see [5,6]. It is important to note that optimality issues in insurance and reinsurance business, not directly related to the problems treated in this contribution, have been discussed in various context, see [7,8,9,10,11,12,13] and the references therein.…”

Optimal insurance in the presence of reinsurance

“…These pioneering results are later extended to situation where there is a more sophisticated objective function and/or more realistic premium principles (see, e.g. Young 1999, Gajek & Zagrodny 2000, 2004, Kaluszka 2001, 2005, Cai & Tan 2007, Balbás et al 2009, 2015, Chi 2012, Asimit et al 2013, 2015, Cai et al 2013, Forthcoming, Chi & Tan 2013, Cui et al 2013, Cheung et al 2014, 2015, Bernard et al 2015, Cheung et al 2015, Boonen et al 2016. In the above-mentioned papers, the risk of the insurer is typically given and the objective boils down to determining an optimal strategy of transferring part of its risk to a reinsurer.…”