原保险公司再保险效果的若干建议

文章将经典风险模型推广到具有超额损失—比例混合分保情形下的破产模型。在个体索赔额服从指数分布情形下,得到了原保险人破产概率的渐进表达式及调节系数。同时,分析了评价原保险公司通过再保险分散经营风险效果的度量。     

多重停止损失—比例混合再保险的破产问题研究

文章研究了具有多重停止损失—比例混合再保险合同的破产概率。在索赔额为指数分布情形下,针对直接保险公司、第一层和第二层的再保险公司分别得到了破产概率的表达式及调节系数,同时用蒙特卡罗模拟了所得的结果,该结果可以推广至多重次的分保情形,并验证了指数分布的无记忆性的性质。     

损失额分布的非参数估计方法研究

在财险实务中,理赔额的分布直接影响到了费率厘定,准备金评估,以及“偿二代”下风险资本计算等精算问题上。理赔额是基于损失额确定的,但是损失额的真实分布未知,如果对其进行假设,则分析结果的误差会比较大。本文提出了损失额分布的非参数估计方法,即最大惩罚似然估计法,该方法对损失额分布不作任何假设,损失额分布完全由记录在案的理赔数据决定。新方法引入一种迭代算法来解决约束最优化问题,得到损失额的危险函数和生存函数最大惩罚似然估计,估计曲线平滑,便于分析风险变化的趋势。同时本文也建立了计算估计渐进方差的数学公式,该公式可以用来建立预测值的置信区间。随机模拟结果显示保单数量越大,分布估计越接近于真实值,渐进方差计算公式越精确。在“偿二代”监管框架下,新方法可以被保险公司作为内部模型来进行风险分析。     

常利率下带投资和干扰的再保险风险模型研究

本文研究了在常数利息力和常数投资回报率的情况下,考虑投资、再保险因素以及干扰项的影响,设定保险公司的保单数和索赔次数服从负二项分布,建立了一类带投资和干扰项的再保险风险模型。利用概率统计和保险精算的方法研究了该模型的相关性质,给出了该模型破产概率的一个显式表达式,同时求出了破产概率的Lundberg上界值。由于模型的引入更加符合实际,该研究结果对于保险公司的经营和决策具有一定的指导意义。     

变额年金业务的风险度量与分析

变额年金产品具有给付投保人最低保证收益率的特点,使资本市场下滑风险从投保人转移到保险公司,但是风险的大小尚不明确。本文建立资产随机模型并使用破产概率和尾部期望损失两个指标度量保险公司销售最低生存利益保证保险(GMLB)和最低身故利益保证保险(GMDB)承担的风险。结果表明最低保证收益率对GMLB的风险有显著影响,而对GMDB没有,在最低保证收益率既定的条件下保险公司投资于股票的份额对风险有显著影响。     

随机利率离散时间风险模型的几个结果

本文研究了引入随机利率的离散时间风险模型．得到了保险公司在初始准备金为“时的生存概率,有限时间内的破产概率,破产后的赤字分布以及盈余首次低于某一水平的时间分布的递推公式     

美国破产金融机构索赔机制研究

2008年金融危机期间,美国大量银行机构破产倒闭,美国政府通过赋予联邦存款保险公司(FDIC)较大自主权力,以高度市场化的处置机制有效化解银行破产危机。在破产处置过程中,FDIC以破产接管方和处置方的双重身份开展了大量索赔,积累了丰富的经验。借鉴美国索赔追责机制经验,对建立健全我国破产金融机构索赔机制,明确索赔机构、对象及流程,具有一定现实意义。     

几类风险模型下保险公司的破产概率

在研究经典风险模型理论的基础上,结合保险公司实际运营情况,给出了几种风险模型的相关模型推广,分析了这些推广模型的破产模型,并根据概率论与随机控制理论推出破产概率的具体表达式.     

信用等级变动下企业破产概率的研究

本文建立了企业信用等级变动的模型。该模型包含了Yang(2003)考虑到的情形,也包含了违约情形。本文在假设信用等级转换遵循具有吸收状态的齐次马氏链前提下,推导了公司破产后赤字的分布的递推公式,并通过给出数值算例使得结果更加清晰。     

信用等级变动下企业破产概率的研究

本文建立了企业信用等级变动的模型。该模型包含了Yang(2003)考虑到的情形,也包含了违约情形。本文在假设信用等级转换遵循具有吸收状态的齐次马氏链前提下,推导了公司破产后赤字的分布的递推公式,并通过给出数值算例使得结果更加清晰。     

Authors’ Reply: Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by U.S. Insurers - Discussion by L. Lee Colquitt; Arlette C. Wilson; Gray G. Venter; Morton Lane; Joan Lamm-Tennant,January 1997

Abstract

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.     

Application of Risk Theory to Interpretation of Stochastic Cash-Flow-Testing Results

Abstract

In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.     

Oversigt over de udenlandske Aktuartidsskrifter

Abstract

In the classical compound Poisson risk model, Lundberg's inequality provides both an upper bound for, and an approximation to, the probability of ultimate ruin. The result can be applied only when the moment generating function of the individual claim amount distribution exists. In this paper we derive an upper bound for the probability of ultimate ruin when the moment generating function of the individual claim amount distribution does not exist.     

Ruin under stochastic dependence between premium and claim arrivals

We investigate, focusing on the ruin probability, an adaptation of the Cramér–Lundberg model for the surplus process of an insurance company, in which, conditionally on their intensities, the two mixed Poisson processes governing the arrival times of the premiums and of the claims respectively, are independent. Such a model exhibits a stochastic dependence between the aggregate premium and claim amount processes. An explicit expression for the ruin probability is obtained when the claim and premium sizes are exponentially distributed.     

Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

Abstract

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.     

Ruin time and aggregate claim amount up to ruin time for the perturbed risk process

We consider the classical Sparre-Andersen risk process perturbed by a Wiener process, and study the joint distribution of the ruin time and the aggregate claim amounts until ruin by determining its Laplace transform. This is first done when the claim amounts follow respectively an exponential/Phase-type distribution, in which case we also compute the distribution of recovery time and study the case of a barrier dividend. Then the general distribution is considered when ruin occurs by oscillation, in which case a renewal equation is derived.     

Impact of Underwriting Cycles on the Solvency of an Insurance Company

Abstract

This paper studies the solvency of an insurance firm in the presence of underwriting cycles. A small or medium-size insurance company with a price-taker position in the market is considered. Its premium income is assumed to obey an autoregressive process with cycles. Specifically, the premium income for a specific calendar year is influenced by the market experience for the last couple years. Under this classical AR(2) dynamics governing the premium income, an explicit expression for the ultimate ruin probability is derived, using a martingale approach, in the lighttailed claims case. Furthermore, the logarithmic asymptotic behavior of the ultimate ruin probability as well as the typical path to ruin are investigated. Then a comparison is made with the classical case where the same company operates on a market without such cycles. Asymptotically, the presence of market cycles is shown to increase the risk for the company. Numerical illustrations are performed on Canadian motor insurance market data and support the theoretical analysis.     

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.     

On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo is considered. We study a generalization of the Gerber–Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin; and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure, where the claim severities follow a combination of exponentials, is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.     

Parisian ruin probability with a lower ultimate bankrupt barrier

The paper deals with a ruin problem, where there is a Parisian delay and a lower ultimate bankrupt barrier. In this problem, we will say that a risk process get ruined when it stays below zero longer than a fixed amount of time ζ > 0 or goes below a fixed level ?a. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we identify the Laplace transform of the ruin probability in terms of so-called q-scale functions. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.