scholarly journals Equilibrium reinsurance-investment strategy with a common shock under two kinds of premium principles

2021 ◽  
Author(s):  
Junna Bi ◽  
Danping Li ◽  
Nan Zhang
Keyword(s):  
Investment Strategy ◽  
Expected Value ◽  
Value Premium ◽  
Equilibrium Strategy ◽  
Model Parameters ◽  
Optimal Reinsurance ◽  
Investment Problem ◽  
Common Shock ◽  
Game Theoretic ◽  
Hamilton Jacobi Bellman

This paper investigates the optimal mean-variance reinsurance-investment problem for an insurer with a common shock dependence under two kinds of popular premium principles: the variance premium principle and the expected value premium principle. We formulate the optimization problem within a game theoretic framework and derive the closed-form expressions of the equilibrium reinsurance-investment strategy and equilibrium value function under the two different premium principles by solving the extended Hamilton-Jacobi-Bellman system of equations. We find that under the variance premium principle, the proportional reinsurance is the optimal reinsurance strategy for the optimal reinsurance-investment problem with a common shock, while under the expected value premium principle, the excess-of-loss reinsurance is the optimal reinsurance strategy. In addition, we illustrate the equilibrium reinsurance-investment strategy by numerical examples and discuss the impacts of model parameters on the equilibrium strategy.

scholarly journals Optimal Reinsurance-Investment Problem under Mean-Variance Criterion with n Risky Assets

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Peng Yang
Keyword(s):  
Financial Market ◽  
Investment Strategy ◽  
Hjb Equation ◽  
Model Parameters ◽  
Optimal Reinsurance ◽  
Risky Assets ◽  
Terminal Wealth ◽  
Investment Problem ◽  
Mean Variance ◽  
Variance Criterion

Based on the mean-variance criterion, this paper investigates the continuous-time reinsurance and investment problem. The insurer’s surplus process is assumed to follow Cramér–Lundberg model. The insurer is allowed to purchase reinsurance for reducing claim risk. The reinsurance pattern that the insurer adopts is combining proportional and excess of loss reinsurance. In addition, the insurer can invest in financial market to increase his wealth. The financial market consists of one risk-free asset and n correlated risky assets. The objective is to minimize the variance of the terminal wealth under the given expected value of the terminal wealth. By applying the principle of dynamic programming, we establish a Hamilton–Jacobi–Bellman (HJB) equation. Furthermore, we derive the explicit solutions for the optimal reinsurance-investment strategy and the corresponding efficient frontier by solving the HJB equation. Finally, numerical examples are provided to illustrate how the optimal reinsurance-investment strategy changes with model parameters.

scholarly journals Optimal Reinsurance Strategy for an Insurer and a Reinsurer with Generalized Variance Premium Principle

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Danping Li ◽  
Chaohai Shen
Keyword(s):  
Regularization Parameter ◽  
Risk Process ◽  
Model Parameters ◽  
Value Functions ◽  
Optimal Reinsurance ◽  
Terminal Wealth ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
Optimal Value Functions

This paper focuses on the optimal reinsurance problem with consideration of joint interests of an insurer and a reinsurer. In our model, the risk process is assumed to follow a Brownian motion with drift. The insurer can transfer the risk to the reinsurer via proportional reinsurance, and the reinsurance premium is calculated according to the variance and standard deviation premium principles. The objective is to maximize the expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s terminal wealth, where the weight can be viewed as a regularization parameter to measure the importance of each party. By applying stochastic control theory, we establish the Hamilton–Jacobi–Bellman equation and obtain explicit expressions of optimal reinsurance strategies and optimal value functions. Furthermore, we provide some numerical simulations to illustrate the effects of model parameters on the optimal reinsurance strategies.

Optimal Insurance-Package and Investment Problem for an Insurer

2018 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
Delei Sheng ◽  
Linfang Xing
Keyword(s):  
Value Function ◽  
Investment Strategy ◽  
Optimal Combination ◽  
Hjb Equations ◽  
Terminal Wealth ◽  
Optimal Insurance ◽  
Yield Rate ◽  
Investment Problem ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

AbstractAn insurance-package is a combination being tie-in at least two different categories of insurances with different underwriting-yield-rate. In this paper, the optimal insurance-package and investment problem is investigated by maximizing the insurer’s exponential utility of terminal wealth to find the optimal combination-share and investment strategy. Using the methods of stochastic analysis and stochastic optimal control, the Hamilton-Jacobi-Bellman (HJB) equations are established, the optimal strategy and the value function are obtained in closed form. By comparing with classical results, it shows that the insurance-package can enhance the utility of terminal wealth, meanwhile, reduce the insurer’s claim risk.

Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles

2019 ◽  
Vol 53 (1) ◽  
pp. 179-206
Author(s):  
Junna Bi ◽  
Kailing Chen
Keyword(s):  
Risk Model ◽  
Optimal Strategies ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Model Parameters ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Common Shock ◽  
The Impact ◽  
Poisson Risk Model

This paper considers the optimal investment-reinsurance strategy in a risk model with two dependent classes of insurance business under two kinds of premium principles, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the expected value premium principle and the variance premium principle, we use the stochastic optimal control theory to derive the optimal strategy and the value function for the compound Poisson risk model as well as for the Brownian motion diffusion risk model. In particular, we find that the optimal investment strategy on the risky asset is independent to the reinsurance strategy and the reinsurance strategy for the compound Poisson risk model are very different from those for the diffusion model under both two kinds of premium principles, but the investment strategies are the same in this two risk models. Finally, numerical examples are presented to show the impact of model parameters in the optimal strategies.

scholarly journals Optimal per-loss reinsurance and investment to minimize the probability of drawdown

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xia Han ◽  
Zhibin Liang ◽  
Yu Yuan ◽  
Caibin Zhang
Keyword(s):  
Closed Form ◽  
Financial Market ◽  
Value Function ◽  
Risk Model ◽  
Investment Strategy ◽  
Insurance Company ◽  
Optimal Reinsurance ◽  
Claim Number ◽  
Common Shock ◽  
Insurance Business

<p style='text-indent:20px;'>In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (n\geq3) $\end{document}</tex-math></inline-formula> dependent classes of insurance business and the optimal results are derived explicitly as well.</p>

scholarly journals Pareto-Optimal Reinsurance Revisited: A Two-Stage Optimisation Procedure Approach

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ying Fang ◽  
Lu Wang ◽  
Zhongfeng Qu ◽  
Wenguang Yu
Keyword(s):  
Value At Risk ◽  
Integral Form ◽  
Expected Value ◽  
Value Premium ◽  
Pareto Optimal ◽  
Two Stage ◽  
Optimal Reinsurance ◽  
Incentive Compatible ◽  
Ex Post ◽  
Optimisation Procedure

In this paper, based on the Tail-Value-at-Risk (TVaR) measure, we revisit the Pareto-optimal reinsurance policies for the insurer and the reinsurer via a two-stage optimisation procedure. To reduce ex-post moral hazard, we assume that reinsurance contracts satisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al. (1983). We show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form. The proposed class of premium principles encompasses the net premium principle, expected value premium principle, TVaR premium principle, generalized percentile premium principle, and so on. We further use the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the Pareto-optimal reinsurance policies. We extend the results by Cai et al. (2017) when the expected value premium principle is replaced by the TVaR premium principle.

scholarly journals On the Stochastic Optimal Control Model of the Investments of Defined Contribution (DC) Pension Funds

2020 ◽  
Vol 24 (2) ◽  
pp. 63-269
Author(s):  
T. Latunde ◽  
O.O. Esan ◽  
J.O. Richard ◽  
D.D. Dare
Keyword(s):  
Sensitivity Analysis ◽  
Optimization Problems ◽  
Absolute Risk ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Pension Funds ◽  
Model Parameters ◽  
Defined Contribution ◽  
Risky Assets ◽  
Hamilton Jacobi Bellman

One of the major problems faced in the management of pension funds and plan is how to allocate and control the future flow of contribution likewise the proportion of portfolio value and investments in risky assets. In this work, optimal investment for a stochastic model of a Defined contribution (DC) is investigated such that the model design is analysed yielding an optimized expected utility of the members’ terminal wealth. An optimized solution is derived using the Hamilton Jacobi equation in solving the problem of investment strategy formulated by Constant absolute risk aversion (CARA). However, to consider the changes that occur in the dimension of optimal solutions in optimization problems, mostly, the optimal behaviour of parameters, the sensitivity analysis is considered. Thus, the analysis of the model is carried out herein by utilising the approach of the sensitivity analysis of parameters. This is carried out by using Maple software and varying the values of some model parameters such that the behaviour of each parameter relating to the pension funds invested in the risky assets is determined. The results are presented graphically and using tables thus discussed such that pension investors and stakeholders are advised. Keywords: Stochastic; DC Pension funds; Sensitivity analysis; Hamilton-Jacobi-Bellman equation; Optimal investment

Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown

2018 ◽  
Vol 13 (2) ◽  
pp. 268-294 ◽  
Author(s):  
Xia Han ◽  
Zhibin Liang ◽  
Caibin Zhang
Keyword(s):  
Risk Model ◽  
Model Parameters ◽  
Proportional Reinsurance ◽  
Surplus Process ◽  
Claim Number ◽  
Common Shock ◽  
Fixed Proportion ◽  
Minimum Probability ◽  
Hamilton Jacobi Bellman ◽  
The Impact

AbstractIn this paper, we study the optimal proportional reinsurance problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component, and the criterion is to minimise the probability of drawdown, namely, the probability that the value of the surplus process reaches some fixed proportion of its maximum value to date. By the method of maximising the ratio of drift of a diffusion divided to its volatility squared, and the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, we investigate the optimisation problem in two different cases. Furthermore, we constrain the reinsurance proportion in the interval [0,1] for each case, and derive the explicit expressions of the optimal proportional reinsurance strategy and the minimum probability of drawdown. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.

scholarly journals OPTIMAL MEAN–VARIANCE REINSURANCE WITH COMMON SHOCK DEPENDENCE

2016 ◽  
Vol 58 (2) ◽  
pp. 162-181 ◽  
Author(s):  
ZHIQIN MING ◽  
ZHIBIN LIANG ◽  
CAIBIN ZHANG
Keyword(s):  
Efficient Frontier ◽  
Model Parameters ◽  
Linear Quadratic ◽  
Claim Number ◽  
Common Shock ◽  
Hamilton Jacobi Bellman Equation ◽  
The Mean ◽  
Hamilton Jacobi Bellman ◽  
The Impact ◽  
Mean Variance

We consider the optimal proportional reinsurance problem for an insurer with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Using the technique of stochastic linear–quadratic control theory and the Hamilton–Jacobi–Bellman equation, we derive the explicit expressions for the optimal reinsurance strategies and value function, and present the verification theorem within the framework of the viscosity solution. Furthermore, we extend the results in the linear–quadratic setting to the mean–variance problem, and obtain an efficient strategy and frontier. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

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