A time-series risk model with constant interest for dependent classes of business

In this paper, we propose a discrete-time model with dependent classes of business using a time-series approach. Specifically, premiums and claims of all classes are supposed to satisfy a multivariate first-order autoregressive time-series model. A constant interest rate is also included in the model. A Lundberg-type inequality for the ruin probability is deduced. We also give an example with constant premiums and two classes of claims for which an expression as well as an exponential bound for the ruin probability is given. A simulation study is provided to help understanding the model.     

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.     

一类比例保险模型的破产概率问题

研究了比例再保险的破产概率问题,推广了Gramer-Lundberg经典模型的有关结果,并证明了基于比例模型合保问题最低保费的一个结果。     

On the discrete-time compound renewal risk model with dependence

In this paper, we study the discrete-time renewal risk model with dependence between the claim amount random variable and the interclaim time random variable. We consider several dependence structures between the claim amount random variable and the interclaim time random variable. Recursive formulas are derived for the probability mass function and the moments of the total claim amount over a fixed period of time. In the context of ruin theory, explicit expressions for the expected penalty (Gerber-Shiu) function are derived for special cases. We also discuss how the discrete-time compound renewal risk model with dependence can be used to approximate the corresponding continuous time compound renewal risk model with dependence. Numerical examples are provided to illustrate different topics discussed in the paper.     

一类带干扰且Cox相关的双险种风险模型

在带有随机扰动的环境中,考虑保单到达及索赔到达均为Cox点过程且两类索赔到达过程相关的一类双险种风险模型.利用鞅技巧,将破产概率的指数上界推广到了更一般的情形.     

双险种最优再保险策略

本文对双险种风险模型,在一险种采取比例再保险,另一险种采取超出损失再保险策略下,得到调节系数与再保险自留水平之间的函数关系式,在理赔额为指数分布和Erlang(2)分布的条件下,得到最优比例再保险和超出损失再保险的自留水平,以及调节系数最大值。     

An optimal investment strategy with maximal risk aversion and its ruin probability

In this paper we study an optimal investment problem of an insurer when the company has the opportunity to invest in a risky asset using stochastic control techniques. A closed form solution is given when the risk preferences are exponential as well as an estimate of the ruin probability when the optimal strategy is used. This work was partially supported by Grants IN103606 PAPIIT-UNAM, 37922E-CONACyT, and 61423-CONACYT Mexico.     

Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria

This paper is concerned with the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk. We deal with the problem by exploring the relationship between maximizing the adjustment coefficient and maximizing the expected utility of wealth for the exponential utility function, both with respect to the retained risk of the insurer.Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.     

Ruin theory in the linear model

The probability of ruin is examined in a model where the annual gains of an insurance company are dependent random variables. The model used is the linear model (well known in time-series analysis) which includes the autoregressive model and the moving average model as special cases. It is also shown that a certain credibility model can be interpreted as a first-order model of the mixed type.     

Computational methods in risk theory: A matrix-algorithmic approach

This paper is concerned with the numerical computation of the probability ψ(u) of ruin with initial reserve u. The basic assumption states that the claim size distribution is phase-type in the sense of Neuts. The models considered are: the classical compound Poisson risk process, the Sparre Anderse process and varying environments which are either governed by a Markov process or exhibit periodic fluctuations. The computational steps involve the iterative solution of a non-linear matrix equation Q = Ψ (Q) as well as the evaluation of matrix-exponentials e^Qu. A number of worked-out numerical examples are presented.     

Optimal consumption and investment with insurer default risk

We solve the optimal consumption and investment problem in an incomplete market, where borrowing constraints and insurer default risk are considered jointly. We derive in closed-form the optimal consumption and investment strategies. We find two main results by quantitative analysis. As insurer default risk increases, the proportion of wealth invested in stocks could increase when wealth is small, and decrease when wealth is large. As risk aversion increases, the voluntary annuity demand could increase when insurer default risk is low, and decrease when this risk is high.     

复合广义齐次Poisson过程的多险种破产概率

本文推广了经典的复合泊松风险模型，建立了两类复合广义齐次poisson过程的多险种破产模型．对于新模型，我们得到了初始资本为u的破产概率φ(u)的精确表达式以及特殊情况下φ(0)的表达式，并且导出了调节系数方程和调节系数R的上下界．     

On optimal periodic dividend strategies in the dual model with diffusion

The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, Bayraktar et al. (2013a) show that a dividend barrier strategy is optimal when dividend decisions are made continuously. In practice, however, companies that are capable of issuing dividends make dividend decisions on a periodic (rather than continuous) basis.In this paper, we consider a periodic dividend strategy with exponential inter-dividend-decision times and continuous monitoring of solvency. Assuming hyperexponential gains, we show that a periodic barrier dividend strategy is the periodic strategy that maximizes the expected present value of dividends paid until ruin. Interestingly, a ‘liquidation-at-first-opportunity’ strategy is optimal in some cases where the surplus process has a positive drift. Results are illustrated.     

Ruin probabilities of a bidimensional risk model with investment

We consider a classical risk model with the possibility of investment. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound for the infinite-time ruin probability with respect to the ruin time T_max(u₁,u₂). For each type of ruin, we derive an integral-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.     

Optimal investment to minimize the probability of drawdown

We determine the optimal investment strategy in a Black–Scholes financial market to minimize the so-called probability of drawdown, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund’s value.     

Asymptotics of discounted aggregate claims for renewal risk model with risky investment

Under the assumption that the claim size is subexponentially distributed and the insurance surplus is totally invested in risky asset, a simple asymptotic relation of tail probability of discounted aggregate claims for renewal risk model within finite horizon is obtained. The result extends the corresponding conclusions of related references.     

Optimal prevention strategies in the classical risk model

In this paper, we propose and study a first risk model in which the insurer may invest into a prevention plan which decreases claim intensity. We determine the optimal prevention investment for different risk indicators. In particular, we show that the prevention amount minimizing the ruin probability maximizes the adjustment coefficient in the classical ruin model with prevention, as well as the expected dividends until ruin in the model with dividends. We also show that the optimal prevention strategy is different if one aims at maximizing the average surplus at a fixed time horizon. A sensitivity analysis is carried out. We also prove that our results can be extended to the case where prevention starts to work only after a minimum prevention level threshold.     

Control of ruin probabilities by discrete-time investments

The control problem of controlling ruin probabilities by investments in a financial market is studied. The insurance business is described by the usual Cramer-Lundberg-type model and the risk driver of the financial market is a compound Poisson process. Conditions for investments to be profitable are derived by means of discrete-time dynamic programming. Moreover Lundberg bounds are established for the controlled model.     

An Analogue of the Cramér–Lundberg Approximation in the Optimal Investment Case

We consider ruin probabilities for an insurance company, which can also invest in the stock market. The risk process is modeled by a compound Poisson process and the stock price by geometric Brownian motion. We show that if the tails of the claims are light tailed, then the optimal strategy is asymptotically given by holding a constant $-value in the stock position. Furthermore, we show that a kind of Cramér–Lundberg approximation holds for the minimal ruin probability. Everything is shown under assumptions, which are analogous to the assumptions in the case of the classical Cramér–Lundberg approximation without investment.