Exponential Utility Of Terminal Wealth Research Articles

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Optimal reinsurance via BSDEs in a partially observable model with jump clusters

We investigate an optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximise expected exponential utility of terminal wealth and show that an optimal strategy exists. By exploiting both the Kushner–Stratonovich and Zakai approaches, we provide the equation governing the dynamics of the (infinite-dimensional) filter and characterise the solution of the stochastic optimisation problem in terms of a BSDE, for which we prove existence and uniqueness of a solution. After discussing the optimal strategy for a general reinsurance premium, we provide more explicit results in some relevant cases.

Optimal pairs trading of mean-reverting processes over multiple assets

<p style='text-indent:20px;'>This paper studies multi-asset pairs trading strategies of maximizing the expected exponential utility of terminal wealth. We model the log-relationship between each pair of stock prices as an Ornstein-Uhlenbeck (O-U) process, and formulate a portfolio optimization problem. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (HJB) equation, we characterize the optimal strategies and provide a verification result for the value function. Finally, we give some numerical results to show the characteristics of pairs trading.</p>

Optimal portfolio and reinsurance with two differential risky assets

This article considers the optimal reinsurance-portfolio problem that the insurer invests in two related risky assets described by different types: constant elasticity of variance model and jump-diffusion process model, besides a risk-free asset. There is a correlation between the diffusion processes of the two models. Meanwhile, the company purchases proportional reinsurance. Specially, assume the claim process follows a Lévy process and the reinsurance’s premium principle has not to be certain. Then based on stochastic control theory, a novel form of the optimal value function for solving the Hamilton-Jacobi-Bellman equation is constructed. Finally, the expressions of the optimal results are obtained under maximizing the expected exponential utility of terminal wealth. In addition, we listed several examples of the common premium principles. Numerical simulations are supplied for sensitivity analysis of parameters.

Bayesian optimal investment and reinsurance with dependent financial and insurance risks

Abstract Major events like the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined, it is important to consider models which reflect this dependence. In this paper, we make a proposal on how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the optimal investment strategy.

Asymptotic solution of optimal reinsurance and investment problem with correlation risk for an insurer under the CEV model

In this paper, we consider an optimal proportional reinsurance and investment problem for an insurer whose objective is to maximise the expected exponential utility of terminal wealth. Suppose that the insurer's surplus process follows a Brownian motion with drift. The insurer is allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. The correlation between risk model and the risky asset's price is considered. By applying dynamic programming approach, we derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation. The asymptotic expansions of the solution to the partial differential equation (PDE) derived from the HJB equation are presented as the parameter appearing in the exponent of the diffusion coefficient tends to 0. We use perturbation theory for partial differential equations (PDEs) to obtain the asymptotic solutions of the optimal reinsurance and investment strategies. Finally, we provide numerical examples and sensitivity analyses to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.

Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition

<p style='text-indent:20px;'>In this study, under the criterion of maximizing the expected exponential utility of terminal wealth, the optimal proportional reinsurance and investment strategy for an insurer is examined with the compound Poisson claim process. To make the model more realistic, the price process of the risky asset is modelled by the Brownian motion risk model with dividends and transaction costs, where the instantaneous of investment return follows as a mean-reverting Ornstein-Uhlenbeck process. At the same time, the net profit condition and variance reinsurance premium principle are also considered. Using stochastic control theory, explicit expressions for the optimal policy and value function are derived, and various numerical examples are given to further demonstrate the effectiveness of the model.</p>

Robust optimal investment and reinsurance problems with learning

In this paper, we consider an optimal investment and reinsurance problem with partially unknown model parameters which are allowed to be learned. The model includes multiple business lines and dependency between them. The aim is to maximize the expected exponential utility of terminal wealth which is shown to imply a robust approach. We can solve this problem using a generalized HJB equation where derivatives are replaced by Clarke's generalized gradient. The optimal investment strategy can be determined explicitly and the optimal reinsurance strategy is given in terms of the solution of an equation. Since this equation is hard to solve, we derive bounds for the optimal reinsurance strategy via comparison arguments.

Optimal investment and reinsurance problem with jump-diffusion model

In this paper, the optimal investment and reinsurance problem is investigated for a class of the jump-diffusion risk model. Here, the insurer can purchase excess-of-loss reinsurance and invest his or her surplus into a financial market consisting of one risk-free asset and one risk asset whose price is modeled by constant elasticity of variance (CEV) model, the net profit condition and the criterion of maximizing the expected exponential utility of terminal wealth are considered in the CEV financial market. By using stochastic control theory to solve the Hamilton-Jacobi-Bellman (HJB) equation, and the explicit form of the optimal policies and value functions can be obtained. Finally, numerical examples are presented to show the impacts of model parameters on the optimal strategies.

Robust optimal insurance and investment strategies for the government and the insurance company under mispricing phenomenon

This paper considers the robust optimal insurance-investment problem for the policyholder who is specified as a government and the insurance company with considering mispricing phenomenon. The government is allowed to purchase proportional insurance from the insurance company whose surplus process is assumed to follow the classical Cramér-Lundberg (C-L) model. Suppose the insurance company can invest in different financial markets and the government is limited to invest in one financial market. Thus, the insurance company is allowed to invest in a pair of mispriced stocks which offer statistical arbitrage opportunities, a risk-free asset, and a market index while the government is assumed to invest in a risk-free asset and a risky asset whose price process satisfies the geometric Brownian motion. Furthermore, we take the joint interest of the government and the insurance company into account. The government and the insurance company are both ambiguity-averse, aiming to maximize their expected joint product exponential utility of terminal wealth. By applying stochastic optimal control approach, we derive the robust equilibrium insurance-investment strategies and the corresponding equilibrium value function explicitly. In addition, we present some special cases of the model as extended results. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal strategies.

Optimal investment and reinsurance with premium control

This paper studies the optimal investment and reinsurance problem for a risk model with premium control. It is assumed that the insurance safety loading and the time-varying claim arrival rate are connected through a monotone decreasing function, and that the insurance and reinsurance safety loadings have a linear relationship. Applying stochastic control theory, we are able to derive the optimal strategy that maximizes the expected exponential utility of terminal wealth. We also provide a few numerical examples to illustrate the impact of the model parameters on the optimal strategy.

Optimal reinsurance and investment for an insurer with the jump diffusion risk model in A-C case

ABSTRACTIn this paper, we study the optimal reinsurance and investment problem for a class of jump-diffusion model, where the diffusion term represents the additional claims (i.e. A-C case). The insurer can purchase proportional reinsurance while allowing she/he to invest in a risk-free asset and a risky asset. The price process of the risky asset is driven by geometric Lévy process with dividend payouts. Applying stochastic control theory, the corresponding Hamilton–Jacobi–Bellman equation is established and the optimal reinsurance-investment strategies to maximize the expected exponential utility of terminal wealth are also established. Finally, the optimal strategies are analysed by the numerical simulation.

Optimal reinsurance-investment problem with dependent risks based on Legendre transform

<p style='text-indent:20px;'>This paper investigates an optimal reinsurance-investment problem in relation to thinning dependent risks. The insurer's wealth process is described by a risk model with two dependent classes of insurance business. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price follows CEV model. Our aim is to maximize the expected exponential utility of terminal wealth. Applying Legendre transform-dual technique along with stochastic control theory, we obtain the closed-form expression of optimal strategy. In addition, our wealth process will reduce to the classical Cramér-Lundberg (C-L) model when <inline-formula><tex-math id="M1">\begin{document}$ p = 0 $\end{document}</tex-math></inline-formula>, in this case, we achieve the explicit expression of the optimal strategy for Hyperbolic Absolute Risk Aversion (HARA) utility by using Legendre transform. Finally, some numerical examples are presented to illustrate the impact of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategy.

Robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with default risks and jumps for an insurer whose objective is to maximize the expected exponential utility of terminal wealth. The surplus process of the insurer is assumed to follow a Brownian motion with drift (which is an approximation of the classical compound Poisson risk model). Assume that the insurer is allowed to purchase proportional reinsurance and invest in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset whose price process is governed by a jump–diffusion model. Using stochastic control approach, we establish the robust Hamilton–Jacobi–Bellman equations for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed-form expressions of the optimal strategies and the corresponding value functions in both cases. Finally, we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies.

Optimal investment policy for an insurer with ambiguity aversion: maximizing exponential utility of terminal wealth

In this paper, we consider the optimal investment problem for an insurer who worries about the possibility of model misspecification. Without loss of generality, we assume there are only one risky asset and one risk-free asset in the financial market. With consideration of the existence of ambiguity, we aim to obtain the optimal investment policy with maximizing the expected utility of terminal wealth. By the dynamic optimal principle, we obtain closed-form solutions of the optimal investment policy and value function. Furthermore, we find that our study contains some results which are investigated in Browne (1995). We can see that if the insurer is ambiguity aversion, he will appear more conservative about the risky asset.

Optimal Insurance-Package and Investment Problem for an Insurer

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 reinsurance and investment problem in a defaultable market

ABSTRACTThis article investigates the optimal reinsurance and investment problem involving a defaultable security. The insurer can purchase reinsurance and allocate his wealth among three financial securities: a money account, a stock, and a defaultable corporate bond. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. Using techniques of stochastic control theory, we derive the corresponding Hamilton–Jacobi–Bellman equation and decompose the original optimization problem into a predefault case and a postdefault case. Explicit expressions for optimal strategies and the corresponding value functions are derived, and the verification theorem is given. Finally, we present numerical examples to illustrate our results.

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.

Robust reinsurance contracts in continuous time

We investigate reinsurance contract problems in a continuous-time principal-agent framework, where the reinsurer (principal) is concerned about potential model ambiguity in the claims process, but the insurer (agent) trusts the claims process, or vice versa. The reinsurer designs a robust reinsurance contract that maximizes his exponential utility of terminal wealth under the worst-case distribution, subject to the insurer’s incentive constraint. Optimal reinsurance contracts are explicitly derived in different ambiguity situations. We first show that the reinsurer’s robustness preference makes him become more conservative, which induces him to raise the reinsurance price, which then decreases the demand for reinsurance. However, the insurer’s robustness preference increases both the reinsurance price and the demand. Furthermore, the reinsurer continuously adjusts the reinsurance price, leading the insurer to always purchase a constant proportion of reinsurance, no matter who faces ambiguity, or whether ambiguity exists. Finally, the economic implications of model ambiguity are illustrated using numerical examples.

Optimal reinsurance and investment policies with the CEV stock market

In this paper, under the criterion of maximizing the expected exponential utility of terminal wealth, we study the optimal proportional reinsurance and investment policy for an insurer with the compound Poisson claim process. We model the price process of the risky asset to the constant elasticity of variance (for short, CEV) model, and consider net profit condition and variance reinsurance premium principle in our work. Using stochastic control theory, we derive explicit expressions for the optimal policy and value function. And some numerical examples are given.