Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACT

Empirical studies suggest that many insurance companies recontract with their clients on premiums by extrapolating past losses: a client is offered a decrease in premium if the monetary amounts of his claims do not exceed some prespecified quantities, otherwise, an increase in premium. In this paper, we formulate the empirical studies and investigate optimal reinsurance problems of a risk-averse insurer by introducing a loss-dependent premium principle, which uses a weighted average of history losses and the expectation of future losses to replace the expectation in the expected premium principle. This premium principle satisfies the bonus-malus and smoothes the insurer's wealth. Explicit expressions for the optimal reinsurance strategies and value functions are derived. If the reinsurer applies the loss-dependent premium principle to continuously adjust his premium, we show that the insurer always needs less reinsurance when he also adopts this premium principle than when he adopts the expected premium principle.

KEYWORDS:

• History loss
• loss-dependent premium principle
• proportional reinsurance
• utility loss

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The weighted premium principle is achieved via distorting the cumulative distribution function and giving great loss more weight (Furman & Zitikis 2008, Alary et al. 2013, Gollier 2014).

2. Klugman et al. (2004) and Zeng (2010) prove that when the insurance portfolio is large enough relative to an individual claim, or μ is large relative to σ, the diffusion model is a good approximation of the Cramér–Lundberg model.

3. The weighting method similar to ours, ${\mathrm{e}}^{-\beta t}\mu +\beta {\int }_0^t{\mathrm{e}}^{-\beta (t-s)}\mathrm{d}L(s-\mathrm{d}t)$ is already used in economic and financial research. Baltas et al. (2012) use this method to estimate the stock price's expectation. Constantinides (1990) and Kraft et al. (2017) adopt an exponential weighted average of past consumption and initial habit level to define the dynamic habit level.

4. Because the number of reinsurers is small, insurers who are tempted to terminate reinsurance contracts may find it difficult to get new coverage. This encourages long-term relationships between insurers and reinsurers, and makes the renew of contracts is common (Gilliam 1980, Jean-Bapitiste & Santomero 2000).

5. The exponential utility function (i.e. the CARA preference) plays an important role in insurance mathematics and actuarial practice, because it is the only function under which the principle of zero utility gives a fair premium that is independent of the level of reserves of an insurance company (Gerber 1979). Moreover, this utility function allows us to derive the explicit expressions of the optimal reinsurance strategies. Additionally, it is easily to verify that the main conclusions in this paper also hold for the mean-variance utility function.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant numbers 71521061, 71790593 71501065] and Humanity and Social Science Foundation of Ministry of Education of China [grant number 19YJC790037].