Equilibrium mean–variance reinsurance and investment strategies for a general insurance company under smooth ambiguity

This work investigates the equilibrium investment and reinsurance strategies for a general insurance company under smooth ambiguity. The general insurance company holds shares of an insurance company and a reinsurance company. The claims of the insurer follow a compound Poisson process. The insurer can divide part of the insurance risk to the reinsurer. Besides, the insurer and reinsurer both participate in the financial market and invest in cash and stock. However, the general insurance company is ambiguous about the insurance and financial risks and is an ambiguity-averse manager (AAM). The uncertainties over the insurance and financial risks are described by second-order distributions. The AAM aims to maximize the average performance of the weighted sum surplus process of the insurer and reinsurer under the mean–variance criterion and smooth ambiguity. We present the extended Hamilton–Jacobi–Bellman (HJB) system for the optimization problem combining the mean–variance criterion and smooth ambiguity. In the case that the second-order distributions are Gaussian, we obtain the closed-forms of the equilibrium reinsurance and investment strategies. At the end of this work, sensitivity analyses are presented to show the economic behaviors of the AAM.     

Alpha-robust mean-variance reinsurance-investment strategy

Inspired by the α-maxmin expected utility, we propose a new class of mean-variance criterion, called α-maxmin mean-variance criterion, and apply it to the reinsurance-investment problem. Our model allows the insurer to have different levels of ambiguity aversion (rather than only consider the extremely ambiguity-averse attitude as in the literature). The insurer can purchase proportional reinsurance and also invest the surplus in a financial market consisting of a risk-free asset and a risky asset, whose dynamics is correlated with the insurance surplus. Closed-form equilibrium reinsurance-investment strategy is derived by solving the extended Hamilton–Jacobi–Bellman equation. Our results show that the equilibrium reinsurance strategy is always more conservative if the insurer is more ambiguity-averse. When the dependence between insurance and financial risks are weak, the equilibrium investment strategy is also more conservative if the insurer is more ambiguity-averse. However, in order to diversify the portfolio, a more ambiguity-averse insurer may adopt a more aggressive investment strategy if the insurance market is very ambiguous. For an ambiguity-neutral insurer, the investment strategy is identical to the non-robust investment strategy.     

Optimal time-consistent reinsurance and investment strategies for a jump–diffusion financial market without cash

This paper considers an optimal reinsurance and investment strategies for an insurer under mean–variance criterion within a game theoretic framework. Specially, it is assumed that the surplus process is governed by a Cramér–Lundberg model, and apart from purchasing reinsurance, the insurer is allowed to invest in a financial market with multiple assets that all can be risky, whose price processes are modeled by the jump–diffusion process. Due to the market without cash, the method of separating the variables is not viable any more. We turn to an alternative approach to solve the extended Hamilton–Jacobi–Bellman equation, and closed-form expressions of the optimal strategies and value function are not only derived but also proved to be uniqueness. Moreover, some special cases of our model are provided and several numerical analyses for our results are presented as well. Under this criterion, different from existing literature, we find that (i) the value function is not linear but quadratic with respect to the current wealth; (ii) the optimal reinsurance and investment strategies depend on the wealth process; (iii) the parameters of risky assets(insurance market) have impacts on the optimal reinsurance(investment) policy; (iv) the safety loading of the insurer affects the optimal strategies.     

Ambiguity on the insurer’s side: The demand for insurance

Empirical evidence suggests that ambiguity is prevalent in insurance pricing and underwriting, and that often insurers tend to exhibit more ambiguity than the insured individuals (e.g., Hogarth and Kunreuther, 1989). Motivated by these findings, we consider a problem of demand for insurance indemnity schedules, where the insurer has ambiguous beliefs about the realizations of the insurable loss, whereas the insured is an expected-utility maximizer. We show that if the ambiguous beliefs of the insurer satisfy a property of compatibility with the non-ambiguous beliefs of the insured, then optimal indemnity schedules exist and are monotonic. By virtue of monotonicity, no ex-post moral hazard issues arise at our solutions (e.g., Huberman et al., 1983). In addition, in the case where the insurer is either ambiguity-seeking or ambiguity-averse, we show that the problem of determining the optimal indemnity schedule reduces to that of solving an auxiliary problem that is simpler than the original one in that it does not involve ambiguity. Finally, under additional assumptions, we give an explicit characterization of the optimal indemnity schedule for the insured, and we show how our results naturally extend the classical result of Arrow (1971) on the optimality of the deductible indemnity schedule.     

An analytical solution for the robust investment-reinsurance strategy with general utilities

In financial markets, different investors have different attitudes or preferences on the investment policies and reinsurance problems. For investors with different investment utilities, how to provide an optimal investment strategy is not only a very hard problem, but also an urgent problem to be solved. In this paper, we derive an analytical solution for the optimal allocation problem of investment-reinsurance with general-form utility function. The general utility function allows for varying relative risk aversion coefficient, which is an important feature in finance theory. However, obtaining analytical solutions for general utility function has been difficult or impossible. The solution presented in this paper is constructed through the homotopy analysis method (HAM) and written in the form of a Taylor series expansion. The fully nonlinear Hamilton–Jacobi–Bellman (HJB) equation is decomposed into an infinite series of linear PDEs, which can be solved analytically. In the end, three examples are presented to illustrate the convergence and accuracy of the method, it also demonstrates that different risk reference investors have different investment-reinsurance strategies.     

Pricing a contract of linking home reversion plan and long-term care insurance via the principle of equivalent utility

Home reversion plans allow homeowners to tap into the value of their house and live in it until their death. The article considers a contract linking home reversion plan and long-term care insurance, which could better prepare seniors for their retirement and long-term care needs. Here, we assume the product exposes an insurer to two risks: the uncertainty of nursing care cost from disable, and the home value decreasing in real estate markets at the time of sale. Because the market is incomplete, we apply the principle of equivalent utility to price the contract under exponential utility.     

指数风险偏好下的最优保险策略

本文讨论了当投保个体和保险公司为指数风险偏好时，在保费约束下投保个体的最优保险策略问题。本文采用求解对偶优化问题的方法求解这个问题，并给出当损失服从指数分布时最优保险策略解的解析式。本文最后讨论了投保个体和保险公司风险厌恶程度以及保费预算变化对个体最优保险策略的影响。     

具有奖励机制与再保险安排的最优保险合约设计

在保险合约中引入奖励机制可以使投保人动态参与到保险合约中,赋予了投保人在面对索赔事件时是否执行索赔的可选择权,改变了传统保险合约中投保人执行索赔的单一权利,但却增加了保险人潜在的流动性风险。保险合约中再保险的安排则可以对冲由于奖励机制产生的潜在流动性风险,进一步分散保险人的风险,有助于保险人稳健经营。基于此,通过建立具有红利奖励机制与再保险安排的最优保险合约设计模型,最终求解得到最优保险合约是具有最优免赔额形式的保险合约。利用算例研究方法进行建模,研究结果显示,最优保险合约中的最优免赔额与奖励机制中的红利奖励之间具有正向关系,保费、自留额与最优免赔额之间则存在着显著的负向关系。     

Dynamic portfolio choice under ambiguity and regime switching mean returns

I examine a continuous-time intertemporal consumption and portfolio choice problem under ambiguity, where expected returns of a risky asset follow a hidden Markov chain. Investors with Chen and Epstein's (2002) recursive multiple priors utility possess a set of priors for unobservable investment opportunities. The optimal consumption and portfolio policies are explicitly characterized in terms of the Malliavin derivatives and stochastic integrals. When the model is calibrated to U.S. stock market data, I find that continuous Bayesian revisions under incomplete information generate ambiguity-driven hedging demands that mitigate intertemporal hedging demands. In addition, ambiguity aversion magnifies the importance of hedging demands in the optimal portfolio policies. Out-of-sample experiments demonstrate the economic importance of accounting for ambiguity.     

Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns

The existing literature on investment and reinsurance is limited to the study of continuous-time problems, while discrete-time problems are always ignored by researchers. In this study, we first discuss a multi-period investment and reinsurance optimization problem under the classical mean-variance framework. When the asset returns with a serially correlated structure, the time-consistent investment and reinsurance strategies are acquired via backward induction. In addition, we propose an alternative time-consistent mean-variance optimization model that contrasts with the classical mean-variance model, and the corresponding optimal strategy and value function are also derived. We find that the investment and reinsurance strategies are both independent of the current wealth for the above two optimization problems, which coincides with the conclusion presented in the continuous-time problems. Most importantly, the above investment strategies with serially correlated structures are both conditional mean-based strategies, rather than unconditional ones. Finally, we compare the investment and reinsurance strategies suggested above based on the simulation approach, to shed light on which investment-reinsurance strategies are more suitable for insurers.     

Risk, uncertainty, and option exercise

Many economic decisions can be described as an option exercise or optimal stopping problem under uncertainty. Motivated by experimental evidence such as the Ellsberg Paradox, we follow Knight (1921) and distinguish risk from uncertainty. To capture this distinction, we adopt the multiple-priors utility model. We show that the impact of ambiguity on the option exercise decision depends on the relative degrees of ambiguity about continuation payoffs and termination payoffs. Consequently, ambiguity may accelerate or delay option exercise. We apply our results to investment and exit problems, and show that the myopic NPV rule can be optimal for an agent having an extremely high degree of ambiguity aversion.     

Optimal growth under model uncertainty

The optimal growth of a wealth process toward a goal is studied under ambiguous markets with first- and second-order moment uncertainties relating to stock returns. Optimal strategies and value functions are solved explicitly. A verification theorem is proved to show that the results solve the original stochastic control problem. Quantitative analyses of the investment strategies indicate that a rational individual with ambiguity aversion reduces market participation when return and volatility are uncorrelated, while there is an exception for synchronous return and volatility. The welfare of shorting a discounted reward is computed, which demonstrates that in an ambiguous pricing economy, investors can generate a positive premium via appropriate asset allocations.     

寿险公司盈余管理研究——基于业务特性驱动视角

寿险公司业务经营具有跨期均衡性、资金融通性、偿付能力充足性三大特性。受业务特性驱动,寿险公司盈余管理策略具有独特性,这些策略主要包括再保险交易、保险准备金提取、金融资产分类和计量。合理适度的盈余管理有利于维护寿险公司和保险市场的稳定发展。     

Optimal insurance contract under VaR and CVaR constraints

This study endogenously develops an optimal insurance contractual form for maximizing insured expected utility under VaR and CVaR constraints. We find that CVaR constraint does not affect the contractual form, but may increase minimum insurance premium requirement. Additionally, when the VaR constraint is binding, the optimal contract is a double deductible insurance. However, if the contract is restricted to a regular form (both indemnity schedule and retained loss schedule are continuously nondecreasing) for avoiding moral hazard problem, the optimal contract is a piecewise linear deductible insurance. Finally, we provide intuitive comparison between this study result and relevant studies.     

Growth option,debt maturity and cash reserves with bank-tax-interaction

In this paper, we assume a small and micro enterprise(SME, henceforth) invests in a project, of which the investment cost is funded by the private lending and the bank-tax-interaction (BTI, henceforth). We build a tractable model of optimal investment, liquidity and default decisions based on cash flows with liquidity shocks and profitability uncertainty. In contrast to the case with pure private lending, we discover that BTI delays investment and increases the firm value. Furthermore, BTI causes the SME to retain more cash reserves. We also find that the SME prefers to select the BTI as the main financing policy under the higher liquidity risk and small profitability uncertainty. Besides, the impact of debt maturity on financial policies with BTI depends on liquidity shock.     

Are generalized call-spreads efficient?

In order to explain coexistence of a deductible for low values of the loss and an upper limit for high values of the loss in insurance contracts, we consider the exchange of risk between two rank dependent expected utility maximizers. It is shown that if the insurer (insured) takes more into account the lowest outcomes – hence maximal losses – than the insured (insurer), then the optimal contract has an upper limit (includes a deductible for high values of the loss). If furthermore, the insured (insurer) neglects the highest outcomes while the insurer (insured) does not, the optimal contract includes a deductible (full insurance) for low values of the loss.     

Incomplete markets, ambiguity, and irreversible investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of ambiguity over the appropriate no-arbitrage discount factor. The maxmin utility over multiple priors framework is used to model and solve the irreversible investment problem. Multiple priors are modeled using the notion of κ‐ignorance. This set-up is used to analyze finitely lived options. For infinitely lived options the notion of constant κ‐ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motion. It is argued that an increase in the set of priors delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect on investment.     

Optimal investment with taxes: an existence result

We study the deterministic control problem of maximizing utility from consumption of an agent who seeks to optimally allocate his wealth between consumption and investment in a financial asset subject to taxes on benefits with first-in–first-out priority rule on sales. Short sales are prohibited and consumption is restricted to be non-negative. Such a problem has been introduced in a previous paper by the same authors where the first-order conditions have been derived. In this paper, we establish an existence result for this non-classical optimal control problem.     

Discrete-time behavioral portfolio selection under cumulative prospect theory

We formulate and study three multi-period behavioral portfolio selection models under cumulative prospect theory: (i) S-shaped utility maximization without probability weighting in a market with one risky asset; (ii) S-shaped utility maximization without probability weighting in a market with multiple risky assets which follow a joint elliptical distribution; and (iii) S-shaped utility maximization with inverse-S-shaped probability weighting in a market with one risky asset. For the first two time consistent models, we identify the well-posedness conditions and derive the semi-analytical optimal policies. For the third time inconsistent model, we assume that the investor is aware of the time inconsistency but is unable to commit to his initial plan of action. Then, we reformulate the model into an intrapersonal game model and derive the semi-analytical subgame perfect Nash equilibrium (time consistent) policy under well-posedness condition. All the three policies take a piecewise linear feedback form. Our analysis of the three models not only partially explains the well documented phenomena of non-participation puzzle and horizon effect, but also extends the two fund separation theorem into multi-period S-shaped utility setting and pushes forward the study on time inconsistency issue incurred by probability weighting.