Time Value Of Ruin Research Articles

Cash Flow Optimization on Insurance: An Application of Fixed-Point Theory

The purpose of this paper is to explore a discrete-time cash flow optimization problem of the insurance company with time value of ruin under different interest rates. For the sake of considering the time value of ruin, we assume that the shareholders can get subsidies per unit time, as long as the insurance company is not bankrupt. The switching of different interest rates on the market is controlled by a stationary Markov chain. The dynamic programming principle is used to solve this optimization problem. By using the method of fixed-point theory, we show that the value function is the unique solution of the dynamic programming equation and a numerical algorithm is proposed to solve the value function as well as the optimal policy. Furthermore, two examples are revealed to illustrate the application of the main results obtained in the presented paper.

Tax optimization with a terminal value for the Lévy risk processes

In this paper, the surplus process (before taxes are deducted) for an insurance company evolves as a spectrally negative Lévy process with the usual exclusion of negative subordinator or deterministic drift. Tax payments are collected according to the very general loss-carry-forward tax system introduced in [20]. We consider an optimal tax problem taking into account both the expected discounted tax payments and the time value of ruin. The optimal tax value function and the optimal tax strategy are derived, some numerical examples are also provided.

Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

Equilibrium dividend strategies for spectrally negative Lévy processes with time value of ruin and random time horizon

In the spectrally negative Lévy risk model, we investigate the absolutely continuous dividend problem with a general discount function, which results in a time-inconsistent control problem. Under the assumptions of a time value of ruin and an exponential time horizon, we study the equilibrium dividend strategies within a game theoretic framework for the return function composed by the discount expected dividend before the ruin. Using the technique of extended Hamilton-Jacobi-Bellman system of equations, we show the verification theorem and give the property of return function. For a mixture of exponential discount function, we obtain closed-form equilibrium dividend strategies and the corresponding equilibrium value functions in both a Cramér–Lundberg model and its diffusion approximation. In addition, some numerical examples are presented to discuss the impacts of some parameters on the control problem.

Optimal Expected Utility of Dividend Payments with Proportional Reinsurance under VaR Constraints and Stochastic Interest Rate

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company taking into account the time value of ruin. We assume the preference of the insurer is of the CRRA form. The discounting factor is modeled as a geometric Brownian motion. We introduce the VaR control levels for the insurer to control its loss in reinsurance strategies. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategy. Finally, we provide some numerical examples to illustrate the results and analyze the VaR control levels on the optimal strategy.

On corrected phase-type approximations of the time value of ruin with heavy tails

Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.

The optimal policy for insurance company under consideration of internal competition and the time value of ruin

This paper considers the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside the company. The objective is to find a reinsurance policy and a dividend payment scheme so as to maximize the expected discounted value of the dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the corresponding constrained Hamilton-Jacobi-Bellman (HJB) equation, we obtain the value function and the optimal reinsurance policy and dividend payment.

Optimal dividend problem with a terminal value for spectrally positive Lévy processes

In this paper we consider a modified version of the classical optimal dividend problem taking into account both expected dividends and the time value of ruin. We assume that the risk process is modeled by a general spectrally positive Lévy process before dividends are deducted. Using the fluctuation theory of spectrally positive Lévy processes we give an explicit expression of the value function of a barrier strategy. Subsequently we show that a barrier strategy is the optimal strategy among all admissible ones. Our work is motivated by the recent work of Bayraktar, Kyprianou and Yamazaki (2013a).

The Ornstein-Uhlenbeck-Type Model with a Hybrid Dividend Strategy

We consider the Ornstein-Uhlenbeck-type model. We first introduce the model and then find the ordinary differential equations and boundary conditions satisfied by the dividend functions; closed-form solutions for the dividend value functions are given. We also study the distribution of the time value of ruin. Furthermore, the moments and moment-generating functions of total discounted dividends until ruin are discussed.

Optimal dividend strategies in the diffusion model with stochastic return on investments

This paper studies the optimal dividend problem in the diffusion model with stochastic return on investments. The insurance company invests its surplus in a financial market. More specially, the authors consider the case of investment in a Black-Scholes market with risky asset such as stock. The classical problem is to find the optimal dividend payment strategy that maximizes the expectation of discounted dividend payment until ruin. Motivated by the idea of Thonhauser and Albrecher (2007), we take the lifetime of the controlled risk process into account, that is, the value function considers both the expectation of discounted dividend payment and the time value of ruin. The authors conclude that the optimal dividend strategy is a barrier strategy for the unbounded dividend payment case and is of threshold type for the bounded dividend payment case.

A diffusion model for optimal dividend payment and risk control for a firm under consideration of the time value of ruin

In this paper, we investigate a model for an insurance company with constraint on risk control. The objective of the insurer is to find a business policy and a dividend payment scheme so as to maximize the expected discounted value of dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the constrained Hamilton-Jacobi-Bellman equation, we obtain the explicit expression for value function and the corresponding optimal strategies.

A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model

In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber–Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber–Shiu function can be expressed in terms of the original Gerber–Shiu function (see e.g. [Gerber, Hans U., Shiu, Elias, S.W., 1998. On the time value of ruin. North American Actuarial Journal 2(1), 48–72]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber–Shiu function is also shown to be closely related to the original Gerber–Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber–Shiu function. Numerical illustrations are then examined.

Pricing perpetual American catastrophe put options: A penalty function approach

The expected discounted penalty function proposed in the seminal paper by Gerber and Shiu [Gerber, H.U., Shiu, E.S.W., 1998. On the time value of ruin. North Amer. Actuarial J. 2 (1), 48–78] has been widely used to analyze the joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, and the related quantities in ruin theory. However, few of its applications can be found beyond except that Gerber and Landry [Gerber, H.U., Landry, B., 1998. On the discount penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263–276] explored its use for the pricing of perpetual American put options. In this paper, we further explore the use of the expected discounted penalty function and mathematical tools developed for the function to evaluate perpetual American catastrophe equity put options. We obtain the analytical expression for the price of perpetual American catastrophe equity put options and conduct a numerical implementation for a wide range of parameter values. We show that the use of the expected discounted penalty function enables us to evaluate the perpetual American catastrophe equity put option with minimal numerical work.

On the expected discounted penalty function for a perturbed risk process driven by a subordinator

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers [Gerber, H.U., Shiu, E.S.W., 1997. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Math. Econ. 21, 129–137; Gerber, H.U., Shiu, E.S.W., 1998a. On the time value of ruin. N. Am. Actuar. J. 2 (1), 48–78; Gerber, H.U., Shiu, E.S.W., 1998b. Pricing perpetual options for jump processes. N. Am. Actuar. J. 2 (3), 101–112]. Later, Gerber and Landry [Gerber, H.U., Landry, B., 1998. On a discounted penalty at ruin in a jump–diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263–276] extended the concept to the perturbed case. Recent papers have extended these results in more general settings [for instance Tsai and Willmot [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66], Li and Lu [ Li, S., Lu, Y., 2005. On the expected discounted penalty function for two classes of risk processes. Insurance: Math. Econ. 36, 179–193] and Li and Garrido [ Li, S., Garrido, J., 2005. On the Gerber–Shiu functions in a Sparre–Andersen risk model perturbed by diffusion. Scand. Actuar. J. 161–186]]. In this note we present yet another generalization that has not been considered before in the literature. We present a perturbed risk process with a subordinator as the model for the aggregate claims. We generalize existing results [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66] on the EDPF for the subordinator case.

Authors' Reply: The Time Value of Ruin in a Sparre Andersen Model, Hans U. Gerber and Elias S. W. Shiu, April 2005 - Discussion by Andrew C. Y. Ng, Chuancun Yin, and Sung Nok Chiu

Authors' Reply: The Time Value of Ruin in a Sparre Andersen Model, Hans U. Gerber and Elias S. W. Shiu, April 2005 - Discussion by Andrew C. Y. Ng, Chuancun Yin, and Sung Nok Chiu

“The Time Value of Ruin in a Sparre Andersen Model,’ Hans U. Gerber and Elias S. W. Shiu, April 2005

“The Time Value of Ruin in a Sparre Andersen Model,’ Hans U. Gerber and Elias S. W. Shiu, April 2005

“The Time Value of Ruin in a Sparre Andersen Model,’ Hans U. Gerber and Elias S. W. Shiu, April 2005

“The Time Value of Ruin in a Sparre Andersen Model,’ Hans U. Gerber and Elias S. W. Shiu, April 2005

“The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005

“The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005

“Authors’ Reply: The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005 - Discussions by Hanspeter Schmidli, Hansjörg Albrecher, Cary Chi-Liang Tsai, Shuanming Li

“Authors’ Reply: The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005 - Discussions by Hanspeter Schmidli, Hansjörg Albrecher, Cary Chi-Liang Tsai, Shuanming Li

“The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005

“The Time Value of Ruin in a Sparre Andersen Model,” Hans U. Gerber and Elias S. W. Shiu, July 2005