Mean-variance problem for an insurer with default risk under a jump-diffusion risk model

Abstract

This paper considers an optimal investment-reinsurance problem with default risk under the mean-variance criterion. We assume that the insurer is allowed to purchase proportional reinsurance and invest his/her surplus in a risk-free asset, a stock and a defaultable bond. The goal is to maximize the expectation and minimize the variance of the terminal wealth. We first formulate the problem to stochastic linear-quadratic (LQ) control problem with constraints. Then the optimal investment-reinsurance strategies and the corresponding value functions are obtained via the viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations for the post-default case and pre-default case, respectively. Finally, we provide numerical examples to illustrate the effects of model parameters on the optimal strategies and value functions.     

Equilibrium excess-of-loss reinsurance–investment strategy for a mean–variance insurer under stochastic volatility model

This article considers an optimal excess-of-loss reinsurance–investment problem for a mean–variance insurer, and aims to develop an equilibrium reinsurance–investment strategy. The surplus process is assumed to follow the classical Cramér–Lundberg model, and the insurer is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Under the mean–variance criterion, equilibrium reinsurance–investment strategy and the corresponding equilibrium value function are derived by applying a game theoretic framework. Finally, numerical examples are presented to illustrate our results.     

Robust optimal investment and proportional reinsurance toward joint interests of the insurer and the reinsurer

In this article, we study a robust optimal investment and reinsurance problem for a general insurance company which holds shares of an insurance company and a reinsurance company. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance, and both the insurer and the reinsurer can invest in a risk-free asset and a risky asset. Besides, the general insurance company’s manager is an ambiguity-averse manager (AAM) who worries about model uncertainty in model parameters. The AAM’s objective is to maximize the minimal expected exponential utility of the weighted sum surplus process of the insurer and the reinsurer. By using techniques of stochastic control theory, we first derive the closed-form expressions of the optimal strategies and the corresponding value function, and then the verification theorem is given. Finally, we present numerical examples to illustrate the effects of model parameters on the optimal investment and reinsurance strategies, and analyze utility losses from ignoring model uncertainty.     

Risk minimization for an insurer with investment and reinsurance via g-expectation

Abstract

This paper is devoted to the study of a risk-based optimal investment and proportional reinsurance problem. The surplus process of the insurer and the risky asset process in the financial market are assumed to be general jump-diffusion processes. We use a convex risk measure generated by g-expectation to describe the risk of the terminal wealth with investment and reinsurance. Under the aim of minimizing the risk, the problem is solved by using techniques of stochastic maximum principles. Two interesting special cases are studied and the explicit expressions for optimal strategies and corresponding minimal risks are derived.     

Time-consistent investment and reinsurance under relative performance concerns

This article investigates the optimal time-consistent investment and reinsurance for two mean–variance insurance managers who take into account the relative performance by comparison to their peers. The unique time-consistent Nash equilibrium policies and the corresponding value functions are derived for asset concentration and diversification. No matter which case is chosen, when the two insurance managers are sensitive to each other’s wealth, they chase each other’s trading behaviors leading to under-reinsurance and overinvestment and lower utility relative to the standard case without relative concerns. The cost–benefit from asset diversification to asset concentration and economic implications of parameters are illustrated by numerical examples.     

Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles

Abstract

In this paper, we study Pareto-optimal reinsurance policies from the perspectives of an insurer and a reinsurer, assuming reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. By geometric approach, we determine the forms of the optimal policies among two classes of ceded loss functions, the class of increasing convex ceded loss functions and the class that the constraints on both ceded and retained loss functions are relaxed to increasing functions. Then we demonstrate the applicability of our results by giving the parameters of the optimal ceded loss functions under Dutch premium principle and Wang’s premium principle.     

A comparison of risk transfer strategies for a portfolio of life annuities based on RORAC

This paper aims to compare different reinsurance arrangements in order to reduce the longevity and financial risk originated by a life insurer while managing a portfolio of annuities policies. Linear and nonlinear reinsurance strategies as well as swap like agreements are evaluated via a discrete-time actuarial risk model. Specifically, longevity dynamics are represented by Lee–Carter type models, while interest rate is modeled by Cox–Ingersoll–Ross model. The reinsurance strategies effectiveness is evaluated according to the Return on Risk Adjusted Capital under a ruin probability constrain.     

Optimal impulse control for dividend and capital injection with proportional reinsurance and exponential premium principle

This article supposes that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals. The exponential premium principle and proportional reinsurance are adopted in business activities. We investigate the general situation that the company needs to pay both proportional and fixed costs for dividends and capital injections. The object of the company is to determine an optimal joint reinsurance–dividend–capital injection strategy for maximizing the expected present value of dividends less capital injections until the time of bankruptcy. In both cases of non cheap and cheap reinsurance, we obtain the explicit solutions for value function and optimal strategy.     

Optimal investment and premium control for insurers with ambiguity

Abstract

In this paper, we consider the optimal investment and premium control problem for insurers who worry about model ambiguity. Different from previous works, we assume that the insurer’s surplus process is described by a non-homogeneous compound Poisson model and the insurer has ambiguity on both the financial market and the insurance market. Our purpose is to find the impacts of model ambiguity on optimal policies. With the objective of maximizing the expected utility of terminal wealth, the closed-form solutions of the optimal investment and premium policies are obtained by solving HJB equations. Finally, numerical examples are also given to illustrate the results.     

The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims

This article studies a continuous-time bidimensional risk model, in which an insurer simultaneously confronts two kinds of claim sharing a common renewal claim-number process. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution with extended regularly varying margins, we derive an explicit asymptotic formula for the corresponding infinite-time ruin probability.     

Optimal asset allocation for participating contracts with mortality risk under minimum guarantee

Abstract

We investigate an optimal investment problem of participating insurance contracts with mortality risk under minimum guarantee. The insurer aims to maximize the expected utility of the terminal payoff. Due to its piecewise payoff structure, this optimization problem is a non-concave utility maximization problem. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the portfolio insurance constraint impacts the optimal terminal wealth.     

Optimal financing and dividend policy with Markovian switching regimes

This work investigates an optimal financing and dividend problem for an insurer whose surplus process is modulated by an observable continuous-time and finite-state Markov chain. We assume that the insurer should never go bankrupt by issuing new equity. The goal of the insurer is to maximize the expected present value of the dividends payout minus the discounted cost of equity issuance. We obtain the optimal policies and explicit expressions for the value functions when the risk reserve process is modeled by both upward jump model and its diffusion approximation. Numerical illustrations of the sensitivities of the model parameters are provided.     

Unrestricted consumption under a deterministic wealth and an Ornstein–Uhlenbeck process as a discount rate

We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein–Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.     

基于Copula-EVT模型的巨灾再保险定价

基于巨灾损失具有厚尾分布的特征,采用POT极值模型分别估计两个保险标的的边缘分布,并用二元Copula函数刻画这两个标的的关联性,同时应用Monte Carlo模拟方法估算巨灾再保险的纯保费。通过对洪水损失数据的实证分析表明:Clayton Copula函数能较好地反映两标的间的相关结构;起赔点的设定是影响纯保费的重要因素,且起赔点按条件分位点取值更优更合理。研究结果对保险人开发多元保险标的的巨灾再保险具有重要的参考价值。     

Optimal generalized case–cohort sampling design under the additive hazard model

Generalized case–cohort designs have been proved to be a cost-effective way to enhance effectiveness in large epidemiological cohort. In generalized case–cohort design, we first select a subcohort from the underlying cohort by simple random sampling, and then sample a subset of the failures in the remaining subjects. In this article, we propose the inference procedure for the unknown regression parameters in the additive hazards model and develop an optimal sample size allocations to achieve maximum power at a given budget in generalized case–cohort design. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm's Tumor Study Group.     

The entropic measure transform

We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk-measure and to obtain a convenient formula for the conditional expectation of a process that admits an affine representation under a related measure. The EMT is then used to provide a new characterization of defaultable bond prices, forward prices and futures prices when a jump-diffusion drives the asset. The characterization of these pricing problems in terms of the EMT provides economic interpretations as maximizing the returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default-free bond prices of Gombani & Runggaldier (2013). These methods are illustrated numerically with an example in the defaultable bond setting. The Canadian Journal of Statistics 48: 97–129; 2020 © 2020 Statistical Society of Canada     

The pricing of defaultable bonds under a regime-switching jump-diffusion model with stochastic default barrier

In this paper, we investigate the price for the zero-coupon defaultable bond under a structural form credit risk with regime switching. We model the value of a firm and the default threshold by two dependent regime-switching jump-diffusion processes, in which the Markov chain represents the states of an economy. The price is associated with the Laplace transform of the first passage time and the expected discounted ratio of the firm value to the default threshold at default. Closed-form results used for calculating the price are derived when the jump sizes follow a regime-switching double exponential distribution. We present some numerical results for the price of the zero-coupon defaultable bond via Gaver-Stehfest algorithm.     

A density function connected with a non-negative self-decomposable random variable

The innovation random variable for a non-negative self-decomposable random variable can have a compound Poisson distribution. In this case, we provide the density function for the compounded variable. When it does not have a compound Poisson representation, there is a straightforward and easily available compound Poisson approximation for which the density function of the compounded variable is also available. These results can be used in the simulation of Ornstein–Uhlenbeck type processes with given marginal distributions. Previously, simulation of such processes used the inverse of the corresponding tail Lévy measure. We show this approach corresponds to the use of an inverse cdf method of a certain distribution. With knowledge of this distribution and hence density function, the sampling procedure is open to direct sampling methods.     

Asymptotic ruin probabilities for a bidimensional risk model with heavy-tailed claims and non-stationary arrivals

Abstract

This article studies a bidimensional risk model, in which an insurer simultaneously confronts two kinds of claims sharing a common non-stationary arrival process. Assuming that the arrival process satisfies a large deviation principle and the claim-size distributions are heavy tailed, an asymptotic formula for the corresponding ruin probability of this bidimensional risk model is obtained.