Expected Premium Principle Research Articles

Research on the Estimation of The Scale of Catastrophe Insurance Funds in Various Provinces in China

The catastrophe insurance fund, as an important means for the steady and robust development of catastrophe risk management, has attracted significant attention from various sectors of society. However, natural disasters in different regions of China vary significantly in terms of frequency and severity, the centralized management of a uniform catastrophe insurance fund severely affects the efficiency and fairness of catastrophe risk management. This study selects the geological disaster loss data from various provinces of mainland China from 2004 to 2021 as the research sample. Based on the characteristics of relative losses caused by geological disasters, the geological disaster losses in various provinces of China are divided into four categories using cluster analysis, and the Generalized Pareto Distribution (GPD) is utilized to describe the loss distributions of each category; The scale of geological disaster insurance funds in each province is estimated through the optimal reinsurance strategy based on the Conditional Value-at-Risk (CVaR) - Expected Premium principle; Provinces with similar characteristics of relative losses from geological disasters have significantly different requirements for the scale of catastrophe insurance funds. Compared to traditional catastrophe models, this study introduces cluster analysis to consider the heterogeneity of geological disaster occurrences. In this paper, a new approach to enhance the management of catastrophe risks by fitting the distribution of geological disaster losses more reasonably is provided.

Multiple per-claim reinsurance based on maximizing the Lundberg exponent

In this paper, we consider the optimal per-claim reinsurance problem for an insurer who designs a reinsurance contract with multiple reinsurance participants. In contrast to using the value-at-risk as a short-term risk measure, we take the Lundberg exponent in risk theory as a risk measure for the insurer over a long-term horizon because the Lundberg upper bound performs better in measuring the infinite-time ruin probability. To reflect various risk preferences of the reinsurance participants, we adopt a type of combined premium principle in which the expected premium principle, variance premium principle, and exponential premium principle are all special cases. Based on maximization of the insurer's Lundberg exponent, the optimal reinsurance is formulated within a static setting, and we derive optimal multiple reinsurance strategies within a general admissible policies set. In general, these optimal strategies are shown to have non-piecewise linear structures, differing from conventional reinsurance strategies such as quota-share, excess-of-loss, or linear layer reinsurance arrangements. In some special cases, the optimal reinsurance strategies reduce to classical results.

Robust equilibrium reinsurance and investment strategy for the insurer and reinsurer under weighted mean-variance criterion

<abstract><p>This paper investigates the time-consistent robust optimal reinsurance problem for the insurer and reinsurer under weighted objective criteria. The joint objective criterion is obtained by weighting the mean-variance objectives of both the insurer and reinsurer. Specifically, we assume that the net claim process is approximated by a diffusion model, and the insurer can purchase proportional reinsurance from the reinsurer. The insurer adopts the loss-dependent premium principle considering historical claims, while the reinsurance contract still uses the expected premium principle due to information asymmetry. Both the insurer and reinsurer can invest in risk-free assets and risky assets, where the risky asset price is described by the constant elasticity of variance model. Additionally, the ambiguity-averse insurer and ambiguity-averse reinsurer worry about the uncertainty of parameter estimation in the model, therefore, we obtain a robust optimization objective through the robust control method. By solving the corresponding extended Hamilton-Jacobi-Bellman equation, we derive the time-consistent robust equilibrium reinsurance and investment strategy and corresponding value function. Finally, we examined the impact of various parameters on the robust equilibrium strategy through numerical examples.</p></abstract>

The Pareto-Optimal Stop-Loss Reinsurance

Reinsurance plays a role of a stabilizer of the insurance industry and can be an effective tool to reduce the risk for the insurer. This paper aims to provide the optimal reinsurance design associated with the stop-loss reinsurance under the criterion of value-at-risk (VaR) risk measure. In this paper, the probability levels in the VaRs used by the both reinsurance parties are assumed to be different and the optimality results of reinsurance are derived by minimizing linear combination of the VaRs of the cedent and the reinsurer. The optimal parameter values of the stop-loss reinsurance policy are formally derived under the expectation premium principle.

VaR and CTE Based Optimal Reinsurance from a Reinsurer’s Perspective

In this article, we study optimal reinsurance design. By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle, we study and obtain the optimal reinsurance treaty by minimizing the VaR (value at risk) of the reinsurer’s total risk exposure. When the distortion premium principle is specified to be the expectation premium principle, we also obtain the optimal reinsurance treaty by minimizing the CTE (conditional tail expectation) of the reinsurer’s total risk exposure. The present study can be considered as a complement of that of Cai et al. [5].

Optimal Time-Consistent Investment and Reinsurance Strategy Under Time Delay and Risk Dependent Model

In this paper, we consider the problem of investment and reinsurance with time delay under the compound Poisson model of two-dimensional dependent claims. Suppose an insurance company controls the claim risk of two kinds of dependent insurance businesses by purchasing proportional reinsurance and invests its wealth in a financial market composed of a risk-free asset and a risk asset. The risk asset price process obeys the geometric Brownian motion. By introducing the capital flow related to the historical performance of the insurer, the wealth process described by stochastic delay differential equation (SDDE) is obtained. The extended HJB equation is obtained by using the stochastic control theory under the framework of game theory. Under the reinsurance expected premium principle, optimal time-consistent investment and reinsurance strategy and the corresponding value function are obtained. Finally, the influence of model parameters on the optimal strategy is explained by numerical analysis.

Demand for non-life insurance under habit formation

This paper studies the optimal non-life insurance for an individual exhibiting internal habit formation in a life-cycle model. We show that the optimal indemnity is deductible under the expected premium principle. Under the additional assumption of exponential utility functions, we obtain the optimal strategies explicitly and find that habit formation reduces insurance coverage. Our model offers a potential explanation for global underinsurance phenomenon. Some numerical examples and sensitivity analysis are presented to highlight our theoretical results.

Optimal reinsurance for both an insurer and a reinsurer under general premium principles

A reinsurance contract should consider the conflicting interests of the insurer and the reinsurer. An optimal reinsurance contract for one party may not be optimal for another party and it might be unacceptable for another party. Therefore, in this paper, we study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of loss function which is defined by the joint value-at-risk, assuming that the reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. We derive the optimal reinsurance policies over three ceded loss function sets, the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the layer reinsurance is shown to be optimal; the quota-share reinsurance with a limit is always optimal when the ceded loss functions are in the class of increasing concave functions. We further use the expectation premium principle and Dutch premium principle to illustrate the application of our results by deriving the optimal parameters.

Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACTEmpirical 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.

Optimal Retention for a Quota-Share Reinsurance

The Indonesian Financial Services Authority (OJK) has instructed all insurance providers in Indonesia to apply a mandatory tariff for property insurance. The tariff has to be uniformly applied and the rule of set the maximum and minimum premium rates for protection against losses. Furthermore, the OJK issued the new rule regarding self-retention and domestic reinsurance. Insurance companies are obliged to have and implement self-retention for each risk in accordance with the self-retention limits. Fluctuations of total premium income and claims may lead the insurance company cannot fulfil the obligation to the insured, thus the company needs to conduct reinsurance. Reinsurance helps protect insurers against unforeseen or extraordinary losses by allowing them to spread their risks. Because reinsurer chargers premium to the insurance company, a properly calculated optimal retention would be nearly as high as the insurer financial ability. This paper is aimed at determining optimal retentions indicated by the risk measure Value at Risk (VaR), Expected Shortfall (ES) and Minimum Variance (MV). Here we use the expectation premium principle which minimizes individual risks based on their quota share reinsurance. Regarding to the data in an insurance property, we use a bivariate lognormal distribution to obtain VaR, ES and MV, and a bivariate exponential distribution to obtain MV. The bivariate distributions are required to derive the conditional probability of the amount of claim occurs given the benefit has occurred.

Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle

In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer’s risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., f * ( x ) = x ∧ c * + ( x - d * ) + with c * and d * being constants such that 0 ≤ c * ≤ d * .

The Optimal Strategy of Reinsurance-Investment Problem for an Insurer with Dynamic Income Under Stochastic Interest Rate

AbstractThis paper considers the reinsurance-investment problem for an insurer with dynamic income to balance the profit of insurance company and policy-holders. The insurer’s dynamic income is given by a net premium minus a dynamic reward budget item and the net premium is obtained according to the expected premium principle. Applying the stochastic control technique, a Hamilton-Jacobi-Bellman equation is established under stochastic interest rate model and the explicit solution is obtained by maximizing the insurer’s power utility of terminal wealth. In addition, the comparison with corresponding results under constant interest rate helps us to understand the role and influence of stochastic interest rates more in-depth.

Explicit Solution of Reinsurance-Investment Problem for an Insurer with Dynamic Income under Vasicek Model

Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous.

OPTIMAL DIVIDEND–REINSURANCE WITH TWO TYPES OF PREMIUM PRINCIPLES

A combined optimal dividend/reinsurance problem with two types of insurance claims, namely the expected premium principle and the variance premium principle, is discussed. Dividend payments are considered with both fixed and proportional transaction costs. The objective of an insurer is to determine an optimal dividend–reinsurance policy so as to maximize the expected total value of discounted dividend payments to shareholders up to ruin time. The problem is formulated as an optimal regular-impulse control problem. Closed-form solutions for the value function and optimal dividend–reinsurance strategy are obtained in some particular cases. Finally, some numerical analysis is given to illustrate the effects of safety loading on optimal reinsurance strategy.

Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles

Recently the optimal reinsurance strategy concerning the insurer’s risk attitude and the reinsurance premium principle has been an interesting topic. This paper discusses the optimal reinsurance problem with the insurer’s risk measured by distortion risk measure and the reinsurance premium calculated by a general principle including expected premium principle and Wang’s premium principle as its special cases. Explicit solutions of the optimal reinsurance strategy are obtained under the assumption that both the ceded loss and the retained loss are increasing with the initial loss. We present a new method for discussing the optimal problem. Based on our method, one can explain the optimal reinsurance treaty in the view of a balance between the insurer’s risk measure and the reinsurance premium principle.

Optimal reinsurance under the Haezendonck risk measure

In this work, we study the optimal reinsurance under the Haezendonck risk measure by minimizing the total risk of the insurer. Firstly, the optimal reinsurance model with the expectation premium principle is proposed. Then, on the basis of our model, the explicit solution is obtained, i.e. the stop-loss function. On the other hand, our result can be considered as a promotion of the optimal reinsurance under the CVaR risk measure since CVaR is only a specific case of the Haezendonck risk measure.

Optimal risk control and dividend distribution policies for a diffusion model with terminal value

In this paper we investigate the optimal risk control and dividend distribution problem for a diffusion model with a terminal value. Usually the insurer cedes risk by means of a reinsurance contract, and pays dividends out dynamically from the surplus. Consider that the insurer is trying to balance risk control and dividend payout in terms of reinsurance and dividend distribution policies. Then the objective is set to make a dynamic choice of reinsurance policy and dividend distribution policy, which maximizes the sum of the expected discounted dividends up to ruin time and the expected discounted terminal value.There are two novelties in this paper. Firstly, we formulate the optimal control problem in terms of general reinsurance control policies. Each of the proportional reinsurance, the excess-of-loss reinsurance and combination of the two can be treated as a special case. It is shown that, under an expected premium principle, the dynamic excess-of-loss reinsurance is of optimal type within the general reinsurance contracts. Secondly, considering the excess-of-loss reinsurance policy and terminal value, we obtain the explicit expressions for the value function and optimal control policies by solving the HJB equation method. At the end of this paper numerical calculations are done to illustrate the influence of the terminal value on the value function and optimal policies as well.

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

The quest for optimal reinsurance design has remained an interesting problem among insurers, reinsurers, and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an insurer and thereby enhance its value. This paper complements the existing research on optimal reinsurance by proposing another model for the determination of the optimal reinsurance design. The problem is formulated as a constrained optimization problem with the objective of minimizing the value-at-risk of the net risk of the insurer while subjecting to a profitability constraint. The proposed optimal reinsurance model, therefore, has the advantage of exploiting the classical tradeoff between risk and reward. Under the additional assumptions that the reinsurance premium is determined by the expectation premium principle and the ceded loss function is confined to a class of increasing and convex functions, explicit solutions are derived. Depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer, we establish conditions for the existence of reinsurance. When it is optimal to cede the insurer's risk, the optimal reinsurance design could be in the form of pure stop-loss reinsurance, quota-share reinsurance, or a combination of stop-loss and quota-share reinsurance.

Optimal Reinsurance Revisited – A Geometric Approach

AbstractIn this paper, we reexamine the two optimal reinsurance problems studied in Cai et al. (2008), in which the objectives are to find the optimal reinsurance contracts that minimize the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risk exposure under the expectation premium principle. We provide a simpler and more transparent approach to solve these problems by using intuitive geometric arguments. The usefulness of this approach is further demonstrated by solving the VaR-minimization problem when the expectation premium principle is replaced by Wang's premium principle.