Determination of the optimal level of risk deduction by an insurance company

We study the dependence of the insurance premium on the limit of liability of the insurance company with respect to individual risk. We determine the conditions under which the relative insurance surcharge will have a minimum. By choosing the optimal limit of liability corresponding to a minimum of insurance surcharge, the insurance company decreases the cost of insurance. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 151–159.     

Estimation of the ruin probability of an insurance company operating on a BS-market

We obtain an estimate for the ruin probability of an insurance company that invests a part of its capital in stocks and puts the rest of the capital in a bank account. An insurance premium is established depending on the capital of the insurance company. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1443–1453, November, 2007.     

A discrete model for the problem of optimizing the activity of an insurance company

This paper is devoted to the construction of a model for and an analysis of the behavior of an insurance company striving to extract maximum profit over some fixed (possibly infinite) time interval. The strategy of the company depends on the size chosen for its initial capital and the size of threshold payments. The purpose of the paper is to determine the optimal values of these quantities. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 165–170.     

On the regulator–insurer interaction in a structural model

In this paper, we provide a new insight to the previous work of Briys and de Varenne [E. Briys, F. de Varenne, Life insurance in a contingent claim framework: Pricing and regulatory implications, Geneva Papers on Risk and Insurance Theory 19 (1) (1994) 53–72], Grosen and Jørgensen [A. Grosen, P.L. Jørgensen, Life insurance liabilities at market value: An analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework, Journal of Risk and Insurance 69 (1) (2002) 63–91] and Chen and Suchanecki [A. Chen, M. Suchanecki, Default risk, bankruptcy procedures and the market value of life insurance liabilities, Insurance: Mathematics and Economics 40 (2007) 231–255]. We show that the particular risk management strategy followed by the insurance company can significantly change the risk exposure of the company, and that it should thus be taken into account by regulators. We first study how the regulator establishes regulation intervention levels in order to control for instance the default probability of the insurance company. This part of the analysis is based on a constant volatility. Given that the insurance company is informed of regulatory rules, we study how results can be significantly different when the insurance company follows a risk management strategy with non-constant volatilities. We thus highlight some limits of the prior literature and believe that the risk management strategy of the company should be taken into account in the estimation of the risk exposure as well as in that of the market value of liabilities.     

The survival probability in finite time period in fully discrete risk model

The probabilities of the following events are first discussed in this paper: the insurance company survives to any fixed time k and the surplus at time k equals x≥1. The formulas for calculating such probabilities are deduced through analytical and probabilistic arguments respectively. Finally, other probability laws relating to risk are determined based on the probabilities mentioned above.     

Optimal reinsurance with a rescuing procedure

We consider a large insurance company whose reserve is modeled by a diffusion process. The management of the insurance company makes a decision on reinsurance in order to reduce the insurance risk. An optimal decision is the one which minimizes the expected time to reach a goal before the reserve reaches a ruin level. We introduce a rescuing procedure to deal with the case that the company is “too big to fail”. We disclose that the optimal decision of the management heavily depends on how much time the company needs to wait for rescuing when it gets in trouble.     

On the solution of the basic integral equation of actuarial mathematics by the method of successive approximations

We study the basic integral equation of actuarial mathematics for the probability of (non)ruin of an insurance company regarded as a function of the initial capital. We establish necessary and sufficient conditions for the existence of a solution of this equation, general sufficient conditions for its existence and uniqueness, and conditions for the uniform convergence of the method of successive approximations for finding the solution. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1689–1698, December, 2007.     

The idle period of the finite G/M/1 queue with an interpretation in risk theory

We consider a G/M/1 queue with restricted accessibility in the sense that the maximal workload is bounded by 1. If the current workload V _t of the queue plus the service time of an arriving customer exceeds 1, only 1−V _t of the service requirement is accepted. We are interested in the distribution of the idle period, which can be interpreted as the deficit at ruin for a risk reserve process R _t in the compound Poisson risk model. For this risk process a special dividend strategy applies, where the insurance company pays out all the income whenever R _t reaches level 1. In the queueing context we further introduce a set-up time a∈[0,1]. At the end of every idle period, an arriving customer has to wait for a time units until the server is ready to serve it.     

Current problems in insurance economics

This paper presents a survey of the lectures held within the B.V.V.O.-chair for insurance economics, 1990–1991. Within this context, a ‘company mission’ for the Center for Insurance Studies is formulated. Future topics to be tackled are given and some results to questions raised during the lectures are presented. In this respect, the corporate strategist will be interested in Sections 2 (positioning of the industry vis-à-vis the competitive forces) and 4 (potential competition from 1992 and synergy with banks). The economist will find some reflections on the nature of insurance from both partial (industrial organisation) and general equilibrium analyses.     

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

The present paper investigates the net loss of a life insurance company issuing equity-linked pure endowments in the case of periodic premiums. Due to the untradability of the insurance risk which affects both the in- and outflow side of the company, the issued insurance claims cannot be hedged perfectly. Furthermore, we consider an additional source of incompleteness caused by trading restrictions, because in reality the hedging of the contingent claims is more likely to occur at discrete times. Based on Møller [Møller, T., 1998. Risk-minimizing hedging strategies for unit-linked life insurance contracts. Astin Bull. 28, 17–47], we particularly examine the situation, where the company applies a time-discretized risk-minimizing hedging strategy. Through an illustrative example, we observe numerically that only a relatively small reduction in ruin probabilities is achieved with the use of the discretized originally risk-minimizing strategy because of the accumulated extra duplication errors caused by discretizing. However, the simulated results are highly improved if the hedging model instead of the hedging strategy is discretized. For this purpose, Møller’s [Møller, T., 2001. Hedging equity-linked life insurance contracts. North Amer. Actuarial J. 5 (2), 79–95] discrete-time (binomial) risk-minimizing strategy is adopted.     

Fourier Inversion Formulas in Option Pricing and Insurance

Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max (S – K, 0) also arises in excess-of-loss or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for payoffs which occur in insurance. Numerical examples are provided.      

A doubly optimal stopping of a risk process (Poster Presentation)

Problems which lead to an optimal stopping of a risk process are considered. Let an insurance company be endowed with an initial capital a > 0, receive insurance premiums and pay out successive claims. The losses occur according to renewal process. At any moment the company may broaden or narrow down the offer, what entails the change of the parameters. These changes concern the rate of income, the intensity of renewal process and the distribution of claims. The model of the risk process with two types of claims stream is considered. After the change the management wants to know the moment of the maximal value of the capital assets. Our goal is to find two optimal stopping times: the best moment of change the parameters and the moment of maximal value of the capital assets. A dynamic programming method to calculate the expected capital at that times is used. Based on the model which combine two types of risk the model of reinsurance with two firms is formulated. In this case the aim is to find for the firms the equilibrium strategy. The equilibrium is constructed in class of strategies driven by their risks. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal investment and reinsurance of an insurer with model uncertainty

We introduce a novel approach to optimal investment–reinsurance problems of an insurance company facing model uncertainty via a game theoretic approach. The insurance company invests in a capital market index whose dynamics follow a geometric Brownian motion. The risk process of the company is governed by either a compound Poisson process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal investment–reinsurance problems with model uncertainty are formulated as two-player, zero-sum, stochastic differential games between the insurance company and the market. We provide verification theorems for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) solutions to the optimal investment–reinsurance problems and derive closed-form solutions to the problems.     

允许赤字的MAP风险过程首达时

保险公司作为负债经营的特殊企业,其偿付能力受到监管部门的约束,本文以公司负债经营为前提研究其各种首次时.考虑MAP风险过程,即存在一随机背景Markov过程,索赔到达与索赔大小同时受这一背景过程影响,索赔到达为Markov到达点过程(MAP),索赔大小对于不同的背景状态具有不同的分布.本文给出首达时满足的积分-微分方程,通过求解带边界条件的积分-微分方程,给出了盈余过程从初始盈余水平到达某一给定盈余水平的首达时的Laplace变换的矩阵表示式,并由此推得了盈余过程到达指定水平的若干首达事件概率.     

保险动态惩罚机制对网约车平台监督作用的演化博弈研究

采用演化博弈方法,研究保险公司与网约车平台之间的博弈演化过程,分析了在保险公司监督下,网约车平台策略选择的影响因素,并比较了静态惩罚机制和动态惩罚机制下网约车平台与保险公司博弈的均衡策略。研究发现,保险公司的惩罚性保费可以促使网约车平台的策略选择发生改变。在静态惩罚机制下,网约车平台和保险公司的策略选择呈周期波动模式,不能收敛;在动态惩罚机制下,网约车平台和保险公司的博弈呈现螺旋收敛的演化轨迹,且收敛的均衡点不随策略选择的初始概率不同而改变。研究结论明确了保险在网约车行业管理中的社会监督职能,并为保险公司保费的制定提供理论参考。     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

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??It is assumed that both an insurance company and a reinsurance company adopt the variance premium principle to collect premiums. Specifically, an insurance company is allowed to investment not only in a domestic risk-free asset and a risky asset, but also in a foreign risky asset. Firstly, we use a geometry Brownian motion to model the exchange rate risk, and assume that the insurance company could control the insurance risk by transferring the insurance business into the reinsurance company. Secondly, the stochastic dynamic programming principle is used to study the optimal investment and reinsurance problems in two situations. The first is a diffusion approximation risk model and the second is a classical risk model. The optimal investment and reinsurance strategies are obtained under these two situations. We also show that the exchange rate risk has a great impact on the insurance company's investment strategies, but has no effect on the reinsurance strategies. Finally, a sensitivity analysis of some parameters is provided.     

Optimal Investment-Reinsurance Strategy with Exchange Rate Risk under Variance Premium Principl

It is assumed that both an insurance company and a reinsurance company adopt the variance premium principle to collect premiums. Specifically, an insurance company is allowed to investment not only in a domestic risk-free asset and a risky asset, but also in a foreign risky asset. Firstly, we use a geometry Brownian motion to model the exchange rate risk, and assume that the insurance company could control the insurance risk by transferring the insurance business into the reinsurance company. Secondly, the stochastic dynamic programming principle is used to study the optimal investment and reinsurance problems in two situations. The first is a diffusion approximation risk model and the second is a classical risk model. The optimal investment and reinsurance strategies are obtained under these two situations. We also show that the exchange rate risk has a great impact on the insurance company's investment strategies, but has no effect on the reinsurance strategies. Finally, a sensitivity analysis of some parameters is provided.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if $b$ is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined $\epsilon$ within the time horizon $T$. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

Fitting mixed-effects models when data are left truncated

Damage sizes, i.e. all damages occurring to a policy and not only those that are reported to an insurance company, are modelled as a linear mixed model. Only those damages that are larger than their deductibles are reported to the company, and this fact should be taken into account when analyzing such data. In statistical terms, the problem is to make inference in a linear mixed model with left truncated data. Estimation methods based on a Monte Carlo simulation of the likelihood are proposed, and extensive simulations to evaluate the quality of the methods are reported. The proposed methods are then used to analyze claimsizes for some marine insurance data, where shipowners represent random effects and technical data about the ships represent fixed effects.