RICING INSURANCE IN ORDER TO MINIMIZING THE EXPECTED LOSS IN WEALTH VIA OPTIMAL CONTROL

Abstract. In this paper we are interested in Pricing insurance in order to minimizing the expected loss inwealth via optimal control. The objective is to find the policy which maximizes the total wealth in company insurances. For this purpose, First, a dynamic model is introduced to describe the process of receiving premium and paying claims. Then, we introduce the premium variable as the problem control variable. Next, we define an appropriate objective function for the control variable and state variables in order to reduce expected losses and increase the wealth. In the end, one of the main variables is estimated by statistical methods and we solve the optimal control problem by PMP method and finally, numerical example are presented.Keywords: Optimal control, Premium, Dynamical systems, expected loss, Optimization

POTENTIAL GAMES WITH AGGREGATION IN NON-COOPERATIVE GENERAL INSURANCE MARKETS

AbstractIn the global insurance market, the number of product-specific policies from different companies has increased significantly, and strong market competition has boosted the demand for a competitive premium. Thus, in the present paper, by considering the competition between each pair of insurers, an N-player game is formulated to investigate the optimal pricing strategy by calculating the Nash equilibrium in an insurance market. Under that framework, each insurer is assumed to maximise its utility of wealth over the unit time interval. With the purpose of solving a game of N-players, the best-response potential game with non-linear aggregation is implemented. The existence of a Nash equilibrium is proved by finding a potential function of all insurers' payoff functions. A 12-player insurance game illustrates the theoretical findings under the framework in which the best-response selection premium strategies always provide the global maximum value of the corresponding payoff function.</jats:p

Optimal strategies for a non-linear premium-reserve model in a competitive insurance market

AbstractThe calculation of a fair premium is always a challenging topic in the real-world insurance applications. In this paper, a non-linear premium-reserve (P-R) model is presented and the premium is derived by minimising a quadratic performance criterion. The reserve is a stochastic equation, which includes an additive random non-linear function of the state, premium and not necessarily Gaussian noise, which is, however, independently distributed in time, provided only that the mean value and the covariance of the random function is 0 and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further by the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company’s reputation. The quadratic utility function concerns the present value of the reserve. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also dependent on the company’s reserve among the other parameters. Finally, a numerical application illustrates the main findings of the paper.</jats:p

Моделювання динаміки розвитку страхового ринку в країнах OECD

Моделювання динаміки розвитку страхового ринку в країнах OECD. – Кваліфікаційна магістерська робота. Сумський державний університет, Суми, 2022 р. У роботі проаналізовано сутність процесів трансформації страхового ринку країн OECD під впливом соціально-економічних та геополітичних факторів. Cформовано вимоги до вимоги до моделі визначення інтегрального показника розвитку страхового ринку країн OECD та їх кластеризації. Отримані результати та перевірка адекватності побудованої моделі дозволили розробити рекомендації Страховим комітетом OECD та національними органами регулювання страхового ринку задля гармонізації розвитку страхового сектору.Modeling the dynamics of insurance market development in OECD countries. – Masters-level Qualification Thesis. Sumy State University, Sumy, 2022 The master’s thesis analyzes the essence of the processes of transformation of the insurance market of OECD countries under the influence of socio-economic and geopolitical factors. The requirements for the model for determining the integral indicator of the development of the insurance market of OECD countries and their clustering have been formulated. The obtained results and verification of the adequacy of the constructed model allowed the OECD Insurance Committee and national insurance market regulatory bodies to develop recommendations for the harmonization of the development of the insurance sector

Game theoretical approaches for pricing of Non-life insurance policies into a competitive market environment

Standard actuarial approaches for non-life insurance products suggest that the premium is divided into three main components: the actuarial price, the safety loading, and the loading for expenses. The number of product-specific policies from different companies has increased significantly, and strong market competition has boosted the demand for a competitive premium in global insurance market. Thus, the actuarial premium could eventually be altered by an insurer's marketing and management department regarding the competitive environment. Thus in this thesis, considering the competition in insurance market, game theoretical approaches are applied to investigate the influence of competition on general insurance pricing. Firstly, a two-period deterministic N-player game is formulated to investigate the optimal pricing strategy by calculating the Nash equilibrium in an insurance market. Under that framework, each insurer is assumed to maximise its utility of wealth over the unit time interval. By analyzing the competition between each pair of insurers, the whole markets' competition is characterized through an aggregation. With the purpose of solving a game of N-players, the best-response potential game with non-linear aggregation is implemented. The existence of a Nash equilibrium is proved by finding a potential function of all insurers' payoff functions. A 12-player insurance game illustrates the theoretical findings under the framework in which the best-response selection premium strategies always provide the global maximum value of the corresponding payoff function. Secondly, deterministic differential games are constructed with the purpose of studying the insurers' equilibrium premium in a competitive market. We apply an optimal control theory to determine the open-loop Nash equilibrium premium strategies. In this direction, two models are formulated and studied. The market power of each insurance company is characterized by a price sensitive parameter, and the business volume is affected by the solvency ratio. Considering the average market premiums, the first model studies an exponential relation between premium strategies and volume of business. The other model initially characterizes the competition between any selected pair of insurers, then aggregates all the paired competitions in the market. Numerical examples illustrate the premium dynamics, and show that premium cycles may exist in equilibrium. Thirdly, a multi-stage stochastic game will be constructed. Insurers are considered to be risk-averse, that is, insurers will to set risk-premiums on their products with the purpose of avoiding risk. Mean-variance Utility function will be adopted. The expenditures of insurance companies will be discussed separately as exposure related costs and non-exposure related costs. The expenditures of insurance companies will be discussed separately as exposure related costs and non-exposure-based costs. The exposure-based component is assumed to be stochastic. Finally, summary of the conclusions complete the thesis

Optimal premium pricing strategies for nonlife products in competitive insurance markets

Non-life insurance pricing depends on different costs including claim and business acquisition costs, management expenses and other parameters such as margin for fluctuations in claims experience, expected profits etc. Nevertheless, in a competitive insurance market environment, company's premium should respond to changes in the level of premiums being offered by competitors. In this thesis, two major issues are being investigated. Primarily, it is explored how a company's optimal strategy can be determined in a competitive market and secondly a connection between this strategy and market's competition is established. More specifically, two functional equations for the volume of business are proposed. In the first place, the volume of business function is related to the past year's experience, the average premium of the market, the company's premium and a stochastic disturbance. Thus, an optimal premium strategy which maximizes the total expected linear discounted utility of company's wealth over a finite time horizon is defined analytically and endogenously. In the second place, the volume of business function is enriched with company's reputation, for the first time according to the author's knowledge. Moreover, the premium elasticity and reputation elasticity of the volume of business are taking into consideration. Thus, an optimal premium strategy which maximizes the total expected linear discounted utility of company's wealth over a finite time horizon is calculated and for some special cases analytical solutions are presented. Furthermore, an upper bound or a minimum premium excess strategy is found for a company with positive reputation and positive premium elasticity of the volume of business. Thirdly, the calculation of a fair premium in a competitive market is discussed. A nonlinear premium-reserve (P-R) model is presented and the premium is derived by minimizing a quadratic performance criterion concerns the present value of the reserve. The reserve is a stochastic equation, which includes an additive random nonlinear function of the state, premium and not necessarily Gaussian noise which is independently distributed in time, provided only that the mean value and the covariance of the random function is zero and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company's reputation. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also depended on the company's reserve among the other parameters. In each chapter numerical applications show the applicability of the proposed models and their results are further explained and analyzed. Finally, suggestions for further research and summary of the conclusions complete the thesis

Desenvolvimento de um modelo idiossincrático não-paramétrico de apoio ao cálculo de prémios de risco na contratação de seguros de saúde

Os modelos de cálculo de risco em seguros de saúde são ferramentas essenciais no desenvolvimento da atividade seguradora, pois servem para que as seguradoras consigam aferir, com antecedência, a potencial sinistralidade do risco, atendendo ao princípio do ciclo inverso da produção dos seguros. Este processo de avaliação é efetuado pelas seguradoras de diferentes formas, limitando os princípios de transparência, equidade e justiça no apuramento do risco, sendo um processo tendencialmente pouco objetivo e ambíguo. Recorrendo à combinação de técnicas de mapeamento cognitivo com a abordagem Measuring Attractiveness by a Categorical Based Evaluation Technique (MACBETH), este estudo visa tornar os processos de cálculo dos prémios de risco em seguros de saúde mais informados e transparentes, sendo o seu principal objetivo a criação de um modelo idiossincrático de orientação processual que permita apoiar a tomada de decisão no cálculo do nível de risco da pessoa segurada e o respetivo prémio de seguro. Os resultados da aplicação do sistema desenvolvido em contexto real, assim com as suas vantagens e limitações, são também alvo de análise e discussão.Health insurance risk analysis is crucial to the development of the insurance activity, allowing insurance companies to anticipate potential losses in this activity’s inverted production lifecycle. This evaluation process is, however, carried out in different ways, limiting the principles of transparency, fairness and justice in the calculation of risk-rewards. By integrating cognitive mapping and the measuring attractiveness by categorical based evaluation technique (MACBETH), this study sought to create a non-parametric idiosyncratic decision support system for health insurance risk analysis, which allows for greater transparency in the calculation of health insurance risk-rewards. The results of a real-world application of the system developed in this study, as well as its advantages and limitations, are also analyzed and discussed

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market's response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance. (C) 2011 Elsevier B.V. All rights reserved