scholarly journals On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Andersen Model

2007 ◽  
Vol 37 (2) ◽  
pp. 203-233 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jürgen Hartinger ◽  
Stefan Thonhauser
Keyword(s):  
Risk Model ◽  
Infinite Horizon ◽  
Threshold Level ◽  
Threshold Strategy ◽  
Sparre Andersen Model ◽  
Dividend Payments ◽  
Andersen Model ◽  
Linear Barrier ◽  
Barrier Type ◽  
Premium Income

For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

scholarly journals On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Andersen Model

2007 ◽  
Vol 37 (02) ◽  
pp. 203-233 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jürgen Hartinger ◽  
Stefan Thonhauser
Keyword(s):  
Risk Model ◽  
Infinite Horizon ◽  
Threshold Level ◽  
Threshold Strategy ◽  
Sparre Andersen Model ◽  
Dividend Payments ◽  
Andersen Model ◽  
Linear Barrier ◽  
Barrier Type ◽  
Premium Income

For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

scholarly journals Moments of Discounted Dividend Payments in the Sparre Andersen Model with a Constant Dividend Barrier

2011 ◽  
Vol 02 (04) ◽  
pp. 444-451
Author(s):  
Jiyang Tan ◽  
Lin Xiao ◽  
Shaoyue Liu ◽  
Xiangqun Yang
Keyword(s):  
Sparre Andersen Model ◽  
Dividend Payments ◽  
Andersen Model ◽  
Constant Dividend Barrier

scholarly journals ON THE INTERFACE BETWEEN OPTIMAL PERIODIC AND CONTINUOUS DIVIDEND STRATEGIES IN THE PRESENCE OF TRANSACTION COSTS

2016 ◽  
Vol 46 (3) ◽  
pp. 709-746 ◽  
Author(s):  
Benjamin Avanzi ◽  
Vincent Tu ◽  
Bernard Wong
Keyword(s):  
Lower Cost ◽  
Risk Model ◽  
Optimal Strategies ◽  
Hybrid Strategy ◽  
Optimal Dividends ◽  
Surplus Process ◽  
Dividend Payments ◽  
Barrier Type ◽  
Continuous Strategies ◽  
First Time

AbstractIn the classical optimal dividends problem, dividend decisions are allowed to be made at any point in time — according to acontinuousstrategy. Depending on the surplus process that is considered and whether dividend payouts are bounded or not, optimal strategies are generally of a band, barrier or threshold type. In reality, while surpluses change continuously, dividends are generally paid on a periodic basis. Because of this, the actuarial literature has recently considered strategies where dividends are only allowed to be distributed at (random) discrete times — according to aperiodicstrategy.In this paper, we focus on the Brownian risk model. In this context, theoptimalcontinuous and periodic strategies have previously been shown (independently of one another) to be of barrier type. For the first time, we consider a model where both strategies are used. In such ahybridstrategy, decisions are allowed to be made either at any time (continuously), or periodically at a lower cost. This proves optimal in some cases. We also determine under which combination of parameters a pure continuous, pure periodic or hybrid (including both continuous and periodic dividend payments) barrier strategy is optimal. Interestingly, the hybrid strategy lies in-between periodic and continuous strategies, which provides some interesting insights. Results are illustrated.

scholarly journals Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Ying Fang ◽  
Zhongfeng Qu
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Insurance Company ◽  
Threshold Strategy ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
Optimal Injection ◽  
Force Of Interest ◽  
Capital Injections ◽  
Injection Strategies

As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.

Analysis of the Compound Poisson Surplus Model with Liquid Reserves, Interest and Dividends

2009 ◽  
Vol 39 (1) ◽  
pp. 225-247 ◽  
Author(s):  
Jun Cai ◽  
Runhuan Feng ◽  
Gordon E. Willmot
Keyword(s):  
Interest Rate ◽  
Risk Model ◽  
Threshold Level ◽  
Threshold Strategy ◽  
Compound Poisson ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
The Interest Rate ◽  
The Impact ◽  
Liquid Reserve

AbstractThe paper incorporates liquid reserves, interest and dividends in the compound Poisson surplus model. When an insurer's surplus is below a certain level, it is kept as liquid reserves. As the surplus attains the level, the excess of the surplus above the level will earn interest at a constant interest rate. If the surplus continues to surpass a higher level, the excess of the surplus above this higher level will be paid out as dividends to the insurer's shareholders at a constant dividend rate or by the threshold strategy. The lower and higher levels are called the liquid reserve level and the threshold level, respectively.This paper is to discuss the interactions of the liquid reserve level, the interest rate, the threshold level, and the dividend rate in the proposed risk model by studying the expected discounted penalty function and the expected present value of dividends paid up to the time of ruin. We derive expressions for the solutions to both quantities via the approach of integro-differential equation systems. We show that the dividend-penalty identity (Gerber et al. 2006, ASTIN Bulletin) still holds for the threshold strategy with liquid reserves and interest. We illustrate these results by deriving explicit solutions to the probability of ultimate ruin under the threshold strategy when claim sizes are exponentially distributed. In the end, we also discuss the impact of the liquid reserve level, the interest rate, the threshold level, and the dividend rate on the ruin probability by numerical examples.

Deficit distributions at ruin in a regime-switching Sparre Andersen model

2018 ◽  
Vol 24 (1) ◽  
pp. 99-107 ◽  
Author(s):  
Lesław Gajek ◽  
Marcin Rudź
Keyword(s):  
Fixed Point ◽  
Markov Chain ◽  
Regime Switching ◽  
Infinite Horizon ◽  
Wait Time ◽  
Risk Process ◽  
Upper Bounds ◽  
The State ◽  
Sparre Andersen Model ◽  
Andersen Model

Abstract In this paper, we investigate deficit distributions at ruin in a regime-switching Sparre Andersen model. A Markov chain is assumed to switch the amount and/or respective wait time distributions of claims while the insurer can adjust the premiums in response. Special attention is paid to an operator {\mathbf{L}} generated by the risk process. We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero, form a vector of functions, which is the n-th iteration of {\mathbf{L}} on the vector of functions being identically equal to zero. Moreover, in the case of infinite horizon, the deficit distributions at ruin are shown to be a fixed point of {\mathbf{L}} . Upper bounds for the vector of deficit distributions at ruin are also proven.

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