Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments (original) (raw)
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2019
In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital r...
On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance
Journal of Applied Mathematics
We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.
Journal of Mathematics
In this paper, the intention was to reduce the possibility of ruin in the insurance company by maximizing its survival function. This paper uses a perturbed classical risk process as the basic model. The basic model was later compounded by refinancing and return on investment. The Hamilton–Jacobi–Bellman equation and integro-differential equation of Volterra type were obtained. The Volterra integro-differential equation for the survival function of the insurance company was converted to a third-order ordinary differential equation which was later converted into a system of first-order ordinary differential equations. This system was then solved numerically using the fourth-order Runge-Kutta method. The results show that the survival function increases with the increase in the intensity of the counting process but decreases with an increase in the instantaneous rate of stock return and return volatility. This is due to the fact that the insurance company faces more risk. Thus, this p...
Insurance: Mathematics and Economics, 2005
Let ψ(y) be the probability of ultimate ruin in the classical risk process compounded by a linear Brownian motion. Here y is the initial capital. We give sufficient conditions for the survival probability function φ = 1 − ψ to be four times continuously differentiable, which in particular implies that φ is 1 the solution of a second order integro-differential equation. Transforming this equation into an ordinary Volterra integral equation of the second kind, we analyze properties of its numerical solution when basically the block-by-block method in conjunction with Simpsons rule is used. Finally, several numerical examples show that the method works very well.
Optimal proportional reinsurance and investment for stochastic factor models
2018
In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramer-Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions of two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations ...
2014
Abstract—This work investigates the effect of stochastic capital reserve on actuarial risk analysis. The formulated mathematical problem is a risk-reserve process of an insurance company whose ruin and survival probabilities are analyzed via the solutions of a derived integro-differential equation (IDE). We further study the interplay between the parameters governing the ruin and the survival probabilities regarding the risk-reserve model; thereby establish a relationship between the probabilities and the initial risk reserve in terms of the other parameters.
Applied Stochastic Models in Business and Industry, 2012
In this paper, we consider the jump-diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment-reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies.
Optimal investment and reinsurance for mean-variance insurers under variance premium principle
International Journal of Business Marketing and Management, 2022
：This paper studies an optimal investment and reinsurance problem for a jump-diffusion risk model with short-selling constraint under the mean-variance criterion. Assume that the insurer is allowed to purchase proportional reinsurance from the reinsurer and invest in a risk-free asset and a risky asset whose price follows a geometric Brownian motion. In particular, both the insurance and reinsurance premium are assumed to be calculated via the variance principle with different parameters respectively. The diffusion term can explain the uncertainty associated with the surplus of the insurer (U-S case) or the additional small claims (A-C case), which are the uncertainty associated with the insurance market or the economic environment. By using techniques of stochastic control theory, with normal constraints on the control variables, closed-form expressions for the value functions and optimal investment-reinsurance strategies are derived in both A-C case and US case.