Dynamic risk-sharing game and reinsurance contract design

This paper studies the optimal risk-sharing between an insurer and a reinsurer. The insurer purchases reinsurance for risk-control and decides her retention level with an objective to minimize her ruin probability. The reinsurer has control over the reinsurance price and aims to maximize her expected discounted profits up to the time when the insurer goes bankrupt. In a stochastic differential game-theoretic framework, we determine the insurer’s optimal reinsurance strategy and specify the reinsurance contract by solving a system of coupled Hamilton–Jacobi–Bellman equations. We obtain explicit solutions for the game problem when both the insurance and the reinsurance premiums are calculated according to the standard-deviation principle or the expected value principle, respectively. Our results show that, depending on the model parameters, the reinsurance contract is either provided with a peak price when the insurer has sufficient cash reserve and with a minimum price when otherwise, or is always provided with a peak price. We also perform some numerical analyses and provide economic interpretations for the results.     

保险公司和再保险公司之间的停止损失再保险策略选择博弈

本文在扩散逼近风险模型下考虑保险公司和再保险公司之间的停止损失再保险策略选择博弈问题.假设保险公司和再保险公司都以期望终端盈余效用增加作为购买停止损失再保险和接受承保的条件.在保险公司和再保险公司都具有指数效用函数条件下,运用动态规划原理,通过求解其对应的Hamilton-Jacobi-Bellman方程,得到了三种博...     

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework.The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly.The claim process of the insurer is governed by a Brownian motion with a drift.A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle.Both the insurer and the reinsurer are assumed to invest in a risky asset,which is distinct for each other and driven by a constant elasticity of variance model.The optimal decision is formulated on a weighted sum of the insurer’s and the reinsurer’s surplus processes.Upon a verification theorem,which is established with a formal proof for a more general problem,explicit solutions are obtained for the proposed investment-reinsurance model.Moreover,numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.     

CEV模型下时滞最优投资与再保险问题

在常方差弹性（constant elasticity of variance,CEV）模型下考虑了时滞最优投资与比例再保险问题.假设保险公司通过购买比例再保险对保险索赔风险进行管理,并将其财富投资于一个无风险资产和一个风险资产组成的金融市场,其中风险资产的价格过程服从常方差弹性模型.考虑与历史业绩相关的现金流量,保险公司的财富过程由一个时滞随机微分方程刻画,在负指数效用最大化的目标下求解了时滞最优投资与再保险控制问题,分别在投资与再保险和纯投资两种情形下得到最优策略和值函数的解析表达式.最后通过数值算例进一步说明主要参数对最优策略和值函数的影响.     

A Bowley solution with limited ceded risk for a monopolistic reinsurer

Borch (1969) advocated that the study of optimal reinsurance design should take into consideration the conflicting interests of both an insurer and a reinsurer. Motivated by this and exploiting a Bowley solution (or Stackelberg equilibrium game), this paper proposes a two-step model that tackles an optimal risk transfer problem between the insurer and the reinsurer. From the insurer’s perspective, the first step of the model provisionally derives an optimal reinsurance policy for a given reinsurance premium while reflecting the reinsurer’s risk appetite. The reinsurer’s risk appetite is controlled by imposing upper limits on the first two moments of the coverage. Through a comparative analysis, the effect of the insurer’s initial wealth on the demand for reinsurance is then examined, when the insurer’s risk aversion and prudence are taken into account. Based on the insurer’s provisional strategy, the second step of the model determines the monopoly premium that maximizes the reinsurer’s expected profit while still satisfying the insurer’s incentive condition. Numerical examples are provided to illustrate our Bowley solution.     

N.E.W.S.: A model for the evaluation of non-life insurance companies

This paper describes a model which can be used by the Dutch insurance supervisor to determine the priority a non-life insurer should have for further examination. This model combines a traditional statistical technique (an ordered logit model) with artificial intelligence techniques (a neural network and an expert system). The output of the model consists of the priority for further examination (high, medium, or low), and a report which summarizes the main findings of the model. The model was able to adequately classify 93% of the companies in a 1993 data set.     

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.     

Defensive financial decisions support for retailers in Greek pharmaceutical industry

In this paper, we present a decisions support solution designed for Greek pharmacies comprising a cash flow management system for early warning of financial distress and a financial advisor based on a neural network. The cash flow monitoring system integrates accounting elements with real time transactions and a predictive linear regression model while the decision support module is developed with the help of a neural network. For any given business unit the system associates accounting entries with information about credit times to reflect the precise instants of cash flows and using inflows/outflows equations monthly, eventually build its liquidity curve and cash flow balance over time. Alongside, a linear regression module is introduced to estimate future cash reserves based on past profitability ratios. Lastly, combining the power of artificial neural networks with expertise in this sector of pharmaceutical business, the financial decision support tool focuses on the retailers that face financial difficulties and suggests alternative solutions for escaping from distress and insolvency. The model has an ambitious and useful purpose, to inform and consult the owners of the business units and other members of the pharmaceutical chain, thus reduce financial risk for the chain.     

Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion

This paper considers the robust equilibrium reinsurance and investment strategies for an ambiguity-averse insurer under a dynamic mean–variance criterion. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market consisting of a risk-free asset and a credit default swap (CDS). Following a game theoretic approach, robust equilibrium strategies and equilibrium value functions for the pre-default case and the post-default case are derived, respectively. For the ambiguity-averse insurer, in general the equilibrium strategies can be characterized by unique solutions to some algebraic equations. For the degenerate case with an ambiguity-neutral insurer, closed-form expressions of equilibrium strategies and equilibrium value functions are obtained. Numerical examples demonstrate that the consideration of model uncertainty and CDS investment improves the insurer’s utility. In this regard, our paper establishes theoretical and numerical support for the importance of ambiguity aversion, credit risk and their interplay in insurance business.     

Worst allocations of policy limits and deductibles

In the literature, orderings of optimal allocations of policy limits and deductibles were established with respect to a policyholder’s preference. However, from the viewpoint of an insurer, the orderings are not enough for the purpose of pricing. In this paper, by applying the equivalent utility premium principle, we study worst allocations of policy limits and deductibles for an insurer, which give rise to the maximum fair premiums. Closed-form solutions are derived. Then we present a result concerning the optimality in a general risk-sharing scheme, by which we obtain optimal allocations for policyholders directly from worst allocations for an insurer. Several results in Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance Math. Econom. 41, 382–391] are generalized here.     

Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer

In this paper, we consider the optimal investment and optimal reinsurance problems for an insurer under the criterion of mean-variance with bankruptcy prohibition, i.e., the wealth process of the insurer is not allowed to be below zero at any time. The risk process is a diffusion model and the insurer can invest in a risk-free asset and multiple risky assets. In view of the standard martingale approach in tackling continuous-time portfolio choice models, we consider two subproblems. After solving the two subproblems respectively, we can obtain the solution to the mean-variance optimal problem. We also consider the optimal problem when bankruptcy is allowed. In this situation, we obtain the efficient strategy and efficient frontier using the stochastic linear-quadratic control theory. Then we compare the results in the two cases and give a numerical example to illustrate our results.     

Optimal proportional reinsurance and investment under partial information

In this paper, we study the optimal proportional reinsurance and investment strategy for an insurer that only has partial information at its disposal, under the criterion of maximizing the expected utility of the terminal wealth. We assume that the surplus of the insurer is governed by a jump diffusion process, and that reinsurance is used by the insurer to reduce risk. In addition, the insurer can invest in financial markets. We give a characterization for the optimal strategy within a non-Markovian setting. Malliavin calculus for Lévy processes is used for the analysis.     

Reinsurance contract design when the insurer is ambiguity-averse

This paper investigates proportional and excess-loss reinsurance contracts in a continuous-time principal–agent framework, in which the insurer is the agent and the reinsurer is the principal. Insurance claims follow the classic Cramér–Lundberg process. The insurer believes that the claim intensity is uncertain and he chooses robust risk retention levels to maximize the penalty-dependent multiple-priors utility. The reinsurer designs reinsurance contracts subject to the insurer’s incentive compatibility constraints. The analytical expressions of the two robust reinsurance contracts are derived. Our results show that the robust reinsurance demand and price are greater than their respective standard values without model ambiguity, and increase as the insurer’s ambiguity aversion increases. Moreover, the reinsurer specifies a decreasing reinsurance price to induce increasing demand over time. Specifically, the price of excess-loss reinsurance is higher, relative to that of proportional reinsurance. Further, only if the insurer’s risk aversion is high or the reinsurer’s risk aversion is low, the insurer prefers the excess-loss reinsurance contract.     

Derivatives trading for insurers

We investigate optimal strategies for a constant absolute risk aversion (CARA) insurer to manage its business risk through not only equity investment and proportional reinsurance but also trading derivatives of the equity. We obtain the optimal strategies in closed-form and quantify the value of derivatives trading by means of certainty-equivalence. Some numerical examples and sensitivity analysis are presented to illustrate our theoretical results. Our numerical results show that, unlike standard CRRA investors, the gain from trading derivatives to a CARA insurer is small and the insurer needs to expose itself to a relatively large position to fully enjoy the gain.     

Equilibrium relationships between insurance and self-protection activity

When an insured understakes some costly self-protection activity that reduces the probability of loss, a competitive insurer will increase the insurance coverage, given a fixed premium per dollar of coverage, to reflect the lower insurance risk.However, an imperfectly informed insurer cannot correctly adjust the coverage; while he can observe the self-protection activity of the insured, the insurer cannot determine the cost to the insured of such activity, nor can the insurer determine the reduction in the loss probability of the insured due to the self-protection activity.This paper demonstrates in an equilibrium model that insurers may be able to use the amount of self-protection activity by an insured as a screen to indicate to the insurer what the loss probability of the insured is, thus allowing the insurer to provide correctly priced insurance to all individuals. The model points out that insurers operating in a market with moral hazard may be able to overcome the adverse incentives of insureds by selectively offering certain insurance contracts contingent upon the insured meeting certain screening requirements; in the model here, self-protection activity is the screen.     

Multivariate reinsurance designs for minimizing an insurer’s capital requirement

This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.     

Optimal investment and risk control policies for an insurer in an incomplete market

In this paper, we apply the martingale approach to investigate the optimal investment and risk control problem for an insurer in an incomplete market. The claim risk of per policy is characterized by a compound Poisson process with drift, and the insurer can be invested in multiple risky assets whose price processes are described by the geometric Brownian motions model. By ‘complete’ the incomplete market, closed-form solutions to the problems of mean–variance criterion and expected exponential utility maximization are obtained. Moreover, numerical simulations are presented to illustrate the results with the basic parameters.     

An object-oriented neural network approach to short-term traffic forecasting

This paper discusses an object-oriented neural network model that was developed for predicting short-term traffic conditions on a section of the Pacific Highway between Brisbane and the Gold Coast in Queensland, Australia. The feasibility of this approach is demonstrated through a time-lag recurrent network (TLRN) which was developed for predicting speed data up to 15 minutes into the future. The results obtained indicate that the TLRN is capable of predicting speed up to 5 minutes into the future with a high degree of accuracy (90–94%). Similar models, which were developed for predicting freeway travel times on the same facility, were successful in predicting travel times up to 15 minutes into the future with a similar degree of accuracy (93–95%). These results represent substantial improvements on conventional model performance and clearly demonstrate the feasibility of using the object-oriented approach for short-term traffic prediction.     

Applying simulation optimization to the asset allocation of a property–casualty insurer

Proper asset allocations are vital for property–casualty insurers to be competitive and solvent. Theories of finance offer little practical guidance in constructing such asset allocations however. This research integrates simulation models with a newly developed evolutionary algorithm for the multi-period asset allocation problem of a property–casualty insurer. We first construct a simulation model to simulate operations of a property–casualty insurer. Then we develop multi-phase evolution strategies (MPES) to be used with the simulation model to search for promising asset allocations for the insurer. A thorough experiment is conducted to evaluate the performance of our simulation optimization approach. Computational results show that MPES is an effective search algorithm. It dominates the grid search method by a significant margin. The re-allocation strategy resulting from MPES outperforms re-balancing strategies significantly. This research further demonstrates that the simulation optimization approach can be used to study economic issues related to multi-period asset allocation problems in practical settings.     

Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model

In this article, we consider the optimal reinsurance and dividend strategy for an insurer. We model the surplus process of the insurer by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted dividend payments until ruin. We give the definition of viscosity solution in the presence of regime switching. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and a verification theorem is also obtained.