Optimal Reinsurance From the Viewpoints of Both an Insurer and a Reinsurer Under the Cvar Risk Measure and Vajda Condition

Abstract

In this paper, we study the optimal reinsurance contracts that minimize the convex combination of the Conditional Value-at-Risk (CVaR) of the insurer’s loss and the reinsurer’s loss over the class of ceded loss functions such that the retained loss function is increasing and the ceded loss function satisfies Vajda condition. Among a general class of reinsurance premium principles that satisfy the properties of risk loading and convex order preserving, the optimal solutions are obtained. Our results show that the optimal ceded loss functions are in the form of five interconnected segments for general reinsurance premium principles, and they can be further simplified to four interconnected segments if more properties are added to reinsurance premium principles. Finally, we derive optimal parameters for the expected value premium principle and give a numerical study to analyze the impact of the weighting factor on the optimal reinsurance.

The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer

“…For further research, the reinsurance optimization may be studied under general model settings by taking several constraints into account as in the works of Cai et al (2017) and Chen (2021). Furthermore, we may also use a more general risk measure, such as a distortion risk measure, to quantify the total loss covered by each party; see, e.g., Lo (2017), Jiang et al (2018), Lo andTang (2019), andJiang et al (2021).…”

Integrated fire/flight control coupler design for armed helicopters based on improved non-monotone adaptive trust region algorithm with distributionally robust optimization

“…In the CVaR method, the expectation calculation can be generally approximated by the normal distribution, even if the distribution of uncertain parameters is unknown. 35 Hence, we consider the values of uncertain parameters in the combat obey the truncated normal distribution (its value obeys the normal distribution in the range [Λmin,Λmax]), which can be describe as followswhere Φ(⋅) is the cumulative distribution function of standard normal distribution, and the truncated normal distribution function of Λ is 1/ΓΦ(Λ−u/Γ)Φ(normalΛmax−u/Γ)−Φ(normalΛmin−u/Γ).…”

“…In the CVaR method, the expectation calculation can be generally approximated by the normal distribution, even if the distribution of uncertain parameters is unknown. 35 Hence, we consider the values of uncertain parameters in the combat obey the truncated normal distribution (its value obeys the normal distribution in the range ½Λ min , Λ max ), which can be describe as follows…”

Optimal insurance with counterparty and additive background risk

“…In optimal insurance with one source of risk, it is assumed that the insured only faces a kind of risk which is total insurable. For instance, see Cai et al (2008), Tan et al (2020), Chen (2021), Meng et al (2022), and so on. However, in the insurance practice, the insured may face some multiplicative background risk like counterparty risk.…”