The Impact of Reinsurance Strategies on Capital Requirements for Premium Risk in Insurance

Abstract

New risk-based solvency requirements for insurance companies across European markets have been introduced by Solvency II and will come in force from 1 January 2016. These requirements, derived by a Standard Formula or an Internal Model, will be by far more risk-sensitive than the required solvency margin provided by the current legislation. In this regard, a Partial Internal Model for Premium Risk is developed here for a multi-line NonLife insurer. We follow a classical approach based on a Collective Risk Model properly extended in order to consider not only the volatility of aggregate claim amounts but also expense volatility. To measure the effect of risk mitigation, suitable reinsurance strategies are pursued. We analyze how naï ve coverage as conventional Quota Share and Excess of Loss reinsurance may modify the exact moments of the distribution of technical results. Furthermore, we investigate how alternative choices of commission rates in proportional treaties may affect the variability of distribution. Numerical results are also figured out in the last part of the paper with evidence of different effects for small and large companies. The main reasons for these differences are pointed out.

A Solvency II Partial Internal Model Considering Reinsurance and Counterparty Default Risk

“…In Table 1, the parameters of the LoBs in which the insurance company carries out its activity are reported. These parameters are derived from ANIA (2022), a report of the Italian association of insurance companies (ANIA) assuming an average insurance company operating in Italy and following the assumptions used in other works on the same subject (see, for instance, Clemente et al (2015) and Zanotto and Clemente ( 2022)) for the elements not directly available in the ANIA report (i.e., the coefficient of variation of the severity random variable CoV Zl and the policy limit pl l ). As described in the specific section, the results are general enough to be applied to the case of an insurer pursuing its activities in all the 12 LoBs.…”

“…Relatively recently, also pushed by the development of the European framework of Solvency II European Parliament and Council (2009), the actuarial literature shifted its focus towards the analysis of the determination of the required capital for ensuring the solvency of the firm. In this new emerging framework, Clemente et al (2015) develop a Partial Internal Model based on the Solvency II framework, which extends the classical collective risk model to also include expense as a stochastic variable. Moreover, they also analyse the presence of reinsurance treaties, both quota share (QS) and excess-of-loss (XL) treaties, on the exact moments of the distribution of technical result and investigate the effect of QS commission rates on the variability of distribution.…”

“…Following the idea of a partial internal model, based on the Solvency II framework, we extend the classic capital modelling of Clemente et al (2015) also considering the counterparty default risk, connected to the presence of reinsurance companies, as in Cai et al (2014). Hence, we propose a framework for the analysis of the insurance capital from the existing literature considering a realistic setting with the presence of reinsurance, potentially offered by multiple reinsurers and taking into account counterparty default risk.…”

Premium Risk Net of Reinsurance: From Short-Term to Medium-Term Assessment

“…Focusing on reinsurance recognition in the new regulation, the solution provided by Solvency II SF is based on a lognormal distribution assumption. This approach can lead to an overestimation of the risk mitigating effect of this kind of reinsurance (see (Clemente 2018)), because it neglects the systematic component that usually affects the number of claims (see (Daykin et al 1994); (Gisler 2009); and (Savelli and Clemente 2009)), while proportional reinsurance does not affect net aggregate claims' cost distribution, but insurers' risk profile can change because of reinsurance pricing (see (Clemente et al 2015)).…”

Capital requirement modeling for market and non-life premium risk in a dynamic insurance portfolio

“…The latter can be represented by many other distributions, as explained above, and clearly, the most appropriate one should be revealed using fitting procedures on appropriate empirical data. In addition, mixture, composite, or spliced distribution approaches can be used to have a proper distinction between the behavior of small-size amounts and medium-large amounts (see for instanceClemente et al, 2014). It is worth pointing out that we follow a total approach, without any distinction between small or medium-large claims.…”