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.     

Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure

Abstract

In this paper, we consider the optimal proportional reinsurance problem in a risk model with the thinning-dependence structure, and the criterion is to minimize the probability that the value of the surplus process drops below some fixed proportion of its maximum value to date which is known as the probability of drawdown. The thinning dependence assumes that stochastic sources related to claim occurrence are classified into different groups, and that each group may cause a claim in each insurance class with a certain probability. By the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, the optimal reinsurance strategy and the corresponding minimum probability of drawdown are derived not only for the expected value principle but also for the variance premium principle. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.     

Optimal lifetime consumption and investment under a drawdown constraint

We consider the infinite-horizon optimal consumption-investment problem under a drawdown constraint, i.e., when the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the with constant coefficients. For a general class of utility functions, we provide the value function in explicit form and derive closed-form expressions for the optimal consumption and investment strategy.      

Portfolio sensitivity to changes in the maximum and the maximum drawdown

In this article, we define new ‘Greeks’ for financial derivatives: sensitivities to the running maximum and the running maximum drawdown of an underlying asset. Some types of portfolios, such as the net asset value of a hedge fund or performance fees, are sensitive to these parameters. In order to illustrate the concept of the new ‘Greeks’, we derive probabilistic representations of sensitivities for two classes of financial contracts: forwards on the maximum drawdown and lookback options. These results allow us to interpret the delta-hedge of the contracts in a novel way.     

A generalized penalty function for a class of discrete renewal processes

Analysis of a generalized Gerber–Shiu function is considered in a discrete-time (ordinary) Sparre Andersen renewal risk process with time-dependent claim sizes. The results are then applied to obtain ruin-related quantities under some renewal risk processes assuming specific interclaim distributions such as a discrete K _n distribution and a truncated geometric distribution (i.e. compound binomial process). Furthermore, the discrete delayed renewal risk process is considered and results related to the ordinary process are derived as well.     

Joint Insolvency Analysis of a Shared MAP Risk Process: A Capital Allocation Application

In recent years, multivariate insurance risk processes have received increasing attention in risk theory. First-passage-time problems in the context of these insurance risk processes are of primary interest for risk management purposes. In this article we study joint-ruin problems of two risk undertakers in a proportionally shared Markovian claim arrival process. Building on the existing work in the literature, joint-ruin–related quantities are thoroughly analyzed by capitalizing on existing results in certain univariate insurance surplus processes. Finally, an application is considered where the finite-time and infinite-time joint-ruin probabilities are used as risk measures to allocate risk capital among different business lines. The proposed joint-ruin allocation principle enables us to not only capture the risk dynamics over a given time horizon, but also overcome the “cross-subsidizing” effect of many existing allocation principles.     

保险资金介入保障房建设的退出机制研究

目前我国保障性住房建设的难题之一是资金的短缺,解决这一难题需要引入市场手段作为政府投入的必要补充。一方面是保障性住房建设的融资缺口,另一方面是保险公司的资金等待合理的配置,然而由于没有规范的退出机制,保险资金想要大规模地介入保障房建设还有待时日。文章从保障性住房和保险资金运用的基本现状入手,从保险资产负债管理和保险资金的良性循环两个方面说明建立保险资金介入保障房建设退出机制的必要性。并按其退出原因分为常态退出和非常态退出,具体分析了融资资金权益转让、资产证券化和政府回购三种退出方式。建立有效的保险资金退出机制应做好三方面工作：健全险资退出的法律体系及配套制度;建立和完善险资企业资产管理体制;加强危机状态下政府应急预案的建立。     

Risk model based on the first-order integer-valued moving average process with compound Poisson distributed innovations

This paper considers a discrete-time risk model by introducing a temporal dependence structure between the number of claims for each period. The risk model is based on the first-order integer-valued moving average (INMA(1)) process with compound Poisson distributed innovations. We derive the explicit expression for the moment generating function of the aggregate claim amount, which can be used for the calculation of some related quantities. We examine the properties of the adjustment coefficient for measuring the dangerousness of an insurance portfolio. Some special cases are included and numerical examples are provided to illustrate the results obtained in the paper.     

Asymptotics for large claims reinsurance in a time-dependent renewal risk model

Abstract

We study the asymptotic tail behaviour of reinsured amounts of the LCR and ECOMOR treaties under a time-dependent renewal risk model, in which a dependence structure is introduced between each claim size and the interarrival time before it. Assuming that the claim size distribution has a subexponential tail, we derive some precise asymptotic results for both treaties.     

Analytic value function for a pairs trading strategy with a Lévy-driven Ornstein–Uhlenbeck process

This paper studies the performance of pairs trading strategy under a specific spread model. Based on the empirical evidence of mean reversion and jumps in the spread between pairs of stocks, we assume that the spread follows a Lévy-driven Ornstein–Uhlenbeck process with two-sided jumps. To evaluate the performance of a pairs trading strategy, we propose the expected return per unit time as the value function of the strategy. Significantly different from the current related works, we incorporate an excess jump component into the calculation of return and time cost. Further, we obtain the analytic expression of strategy value function, where we solve out the probabilities of crossing thresholds via the Laplace transform of first passage time of the Lévy-driven Ornstein–Uhlenbeck process in one-sided and two-sided exit problems. Through numerical illustrations, we calculate the value function and optimal thresholds for a spread model with symmetric jumps, reveal the non-negligible contribution of incorporating the excess jumps into the value function, and analyze the impact of model parameters on the strategy performance.     

Optimal investment and risk control for an insurer with partial information in an anticipating environment

Abstract

This paper considers an optimal investment and risk control problem under the criterion of logarithm utility maximization. The risky asset process and the insurance risk process are described by stochastic differential equations with jumps and anticipating coefficients. The insurer invests in the financial assets and controls the number of policies based on some partial information about the financial market and the insurance claims. The forward integral and Malliavin calculus for Lévy processes are used to obtain a characterization of the optimal strategy. Some special cases are discussed and the closed-form expressions for the optimal strategies are derived.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

On a Class of Renewal Risk Processes

Abstract

In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.     

The Time of Recovery and the Maximum Severity of Ruin in a Sparre Andersen Model

Abstract

Phase-type distributions are one of the most general classes of distributions permitting a Markovian interpretation. Sparre Andersen risk models with phase-type claim interarrival times or phase-type claims can be analyzed using Markovian techniques, and results can be expressed in compact matrix forms. Computations involved are readily programmable in practice.

This paper studies some quantities associated with the first passage time and the time of ruin in a Sparre Andersen risk model with phase-type interclaim times. In an earlier discussion the present author obtained a matrix expression for the Laplace transform of the first time that the surplus process reaches a given target from the initial surplus. Using this result, we analyze (1) the Laplace transform of the recovery time after ruin, (2) the probability that the surplus attains a certain level before ruin, and (3) the distribution of the maximum severity of ruin. We also give a matrix expression for the expected discounted dividend payments prior to ruin for the Sparre Andersen model in the presence of a constant dividend barrier.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

On dividends in the phase–type dual risk model

The dual risk model assumes that the surplus of a company decreases at a constant rate over time, and grows by means of upward jumps which occur at random times with random sizes. In the present work, we study the dual risk renewal model when the waiting times are phase-type distributed. Using the roots of the fundamental and the generalized Lundberg’s equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Then, we address the calculation of expected discounted future dividends particularly when the individual common gains follow a phase-type distribution. We further show that the optimal dividend barrier does not depend on the initial reserve. As far as the roots of the Lundberg equations and the time of ruin are concerned, we address the existing formulae in the corresponding Sparre-Andersen insurance risk model for the first hitting time, and we generalize them to cover also the situations where we have multiple roots. We do that working a new approach and technique, approach we also use for working the dividends, unlike others, it can be also applied for every situation.     

基于交替更新过程的随机模糊破产模型

构建一个更能刻画保险公司现实运营状况的破产模型,对于保险公司来说具有重要意义。假设索赔额、盈余额和更新过程均是在随机模糊环境下,使得该模型不仅能反映事件的随机性,也能反映决策者的主观性;同时,假设保险公司赔偿时刻滞后与核赔事件发生时刻,建立了交替更新过程。基于以上两点假设,当索赔额和时间间隔均是服从指数分布时,建立了交替更新过程下的随机模糊破产模型,并给出了最终破产概率公式与最终破产机会均值公式。     

Guaranteed renewable life insurance under demand uncertainty

Guaranteed renewability (GR) is a prominent feature in many health and life insurance markets. We develop a model that includes unpredictable (and unobservable) fluctuations in demand for life insurance as well as changes in risk type (observable) over individuals' lifetimes. The presence of demand type heterogeneity leads to the possibility that optimal GR contracts may have a renewal price that is either above or below the actuarially fair price of the lowest risk type in the population. Individuals whose type turns out to be high risk but low demand renew more of their GR insurance than is efficient due to the attractive renewal price. This results in imperfect insurance against reclassification risk. Although a first‐best efficient contract is not possible in the presence of demand type heterogeneity, the presence of GR contracts nonetheless improves welfare relative to an environment with only spot markets.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.     

我国人身保险市场结构变化与预测分析

虽然保险业经受了本次金融危机的考验,但是保险业可能成为系统性风险传导链条上的一环,因此,保险监管机构需要转变审慎监管方式,建立适合保险业的宏观审慎监管框架。在这个背景下,对中国人身保险市场结构现状有一个较为清晰和准确的认识就显得十分必要。研究发现:从全国范围来说,我国寿险市场主体数量较少,人身险市场集中度过高,市场结构为寡头垄断,但各地区的人身险市场结构又各有特点,经济发达地区人身险保险市场结构已经过渡到垄断竞争阶段。根据成分数据预测结果分析,在"十二五"期间,我国人身险的市场集中度继续呈下降趋势。