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.     

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.     

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.     

Optimal investment-consumption and life insurance with capital constraints

Abstract

The aim of this paper is to solve an optimal investment, consumption and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions.     

Optimal investment strategy for a DC pension plan with mispricing under the Heston model

Abstract

In this article, we consider the optimal investment problem for a defined contribution (DC) pension plan with mispricing. We assume that the pension funds are allowed to invest in a risk-free asset, a market index, and a risky asset with mispricing, i.e. the prices are inconsistent in different financial markets. Assuming that the price process of the risky asset follows the Heston model, the manager of the pension fund aims to maximize the expected utility for the power utility function of terminal wealth. By applying stochastic control theory, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation. And the optimal investment strategy is obtained for the power utility function explicitly. Finally, numerical examples are provided to analyze effects of parameters on the optimal strategy.     

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.     

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.     

Robust portfolio choice for a defined contribution pension plan with stochastic income and interest rate

This paper considers a robust portfolio choice problem for a defined contribution pension plan with stochastic income and stochastic interest rate. The investment objective of the pension plan is to maximize the expected utility of the wealth at the retirement time. We assume that the financial market consists of a stock, a zero-coupon bond and a risk-free asset. And the member of defined contribution pension plan is ambiguity-averse, which means that the member is uncertain about the expected return rate of the bond and stock. Meanwhile, the member's ambiguity-aversion level toward these two financial assets is quite different. The closed-form expressions of the robust optimal investment strategy and the corresponding value function are derived by adopting the stochastic dynamic programming approach. Furthermore, the sensitive analysis of model parameters on the optimal investment strategy are presented. We find that the member's aversion on model ambiguity increases her hedging demand and has remarkable impact on the optimal investment strategy. Moreover, we demonstrate that ignoring model uncertainty will lead to significant utility loss for the ambiguity-averse member, and the model uncertainty about the stock dynamics implies greater effect on the outcome of the investment than the bond.     

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.     

Optimal investment and life insurance strategies in a mixed jump-diffusion framework

Abstract

This article investigates an optimal investment and life insurance strategies in a mixed jump-diffusion framework. The individual life insurance policyholder who has CRRA preferences. The market consists of riskless asset, a zero-coupon bond, a stock and life insurance. The instantaneous interest rate is modeled as the O-U model, while a zero-coupon bond with credit risk follows a BSDE and a risky asset be driven by MJD-fBm model. The problem is solved by the mixed jump diffusion fractional HJB SDE which satisfied the admissible strategy, then the closed form solution and optimal strategies are derived and the simulation of the various parameters are also given.     

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.     

Sub-optimal investment for insurers

Abstract

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.     

Nonparametric Estimation of Search Costs for Differentiated Products: Evidence from Medigap

Abstract

This article develops a method to estimate search frictions as well as preference parameters in differentiated product markets. Search costs are nonparametrically identified, which means our method can be used to estimate search costs in differentiated product markets that lack a suitable search cost shifter. We apply our model to the U.S. Medigap insurance market. We find that search costs are substantial: the estimated median cost of searching for an insurer is $30. Using the estimated parameters we find that eliminating search costs could result in price decreases of as much as $71 (or 4.7%), along with increases in average consumer welfare of up to $374.     

Optimal reinsurance and investment problem in a defaultable market

This 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.     

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.     

Hedging of contingent claims written on non traded assets under Markov-modulated models

ABSTRACT

This paper studies the hedging problem of European contingent claims when the underlying asset is non traded. We assume that the share prices of the assets are governed by Markov-modulated processes; that is, the market parameters switch over the time according to a finite-state continuous time Markov chain. Due to the presence of Markov chain the non traded asset, the market which we consider is incomplete, we shall use the local risk minimization method to obtain an optimal hedging strategy in a closed-form for an investor. Finally, numerical illustrations of an optimal hedging strategy are given by the Monte Carlo simulation.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

The Gerber-Shiu function for the compound Poisson Omega model with a three-step premium rate

Abstract

The compound Poisson Omega model is considered in the presence of a three-step premium rate. Firstly, the integral equations and the integro-differential equations for the Gerber-Shiu expected discounted penalty function are derived. Secondly, the integro-differential equations for the Gerber-Shiu expected discounted penalty function are determined in three different initial conditions. The results are then used to find the bankruptcy probability. Finally, the special cases where the claim size distribution is exponential be discussed in some detail in order to illustrate the effect of the model with three-step premium rate.     

Stochastic Spanning

ABSTRACT

This study develops and implements methods for determining whether introducing new securities or relaxing investment constraints improves the investment opportunity set for all risk averse investors. We develop a test procedure for “stochastic spanning” for two nested portfolio sets based on subsampling and linear programming. The test is statistically consistent and asymptotically exact for a class of weakly dependent processes. A Monte Carlo simulation experiment shows good statistical size and power properties in finite samples of realistic dimensions. In an application to standard datasets of historical stock market returns, we accept market portfolio efficiency but reject two-fund separation, which suggests an important role for higher-order moment risk in portfolio theory and asset pricing. Supplementary materials for this article are available online.     

A Simple Method to Study Sensitivity of BMP's

In this article we measure the local or infinitesimal sensitivity of a kind of Bayes estimates which appear in bonus–malus systems. Bonus–malus premiums can be viewed as a functional depending on the prior distribution. To measure when small changes in the prior cause large changes in the premium we compute the norm of the Fréchet derivative and propose a simple procedure to decide if a bonus–malus premium is robust. As an application, an example where the risk has a Poisson distribution and its parameter follows a Gamma prior distribution is presented under the net and variance premium principles.