Optimal life-insurance selection and purchase within a market of several life-insurance providers

We consider the problem faced by a wage-earner with an uncertain lifetime having to reach decisions concerning consumption and life-insurance purchase, while investing his savings in a financial market comprised of one risk-free security and an arbitrary number of risky securities whose prices are determined by diffusive linear stochastic differential equations. We assume that life-insurance is continuously available for the wage-earner to buy from a market composed of a fixed number of life-insurance companies offering pairwise distinct life-insurance contracts. We characterize the optimal consumption, investment and life-insurance selection and purchase strategies for the wage-earner with an uncertain lifetime and whose goal is to maximize the expected utility obtained from his family consumption, from the size of the estate in the event of premature death, and from the size of the estate at the time of retirement. We use dynamic programming techniques to obtain an explicit solution in the case of discounted constant relative risk aversion (CRRA) utility functions.     

Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach

We discuss an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero-coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. Finally, we derive the explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case.     

A note on optimal investment–consumption–insurance in a Lévy market

In Shen and Wei (2014) an optimal investment, consumption and life insurance purchase problem for a wage earner with Brownian information has been investigated. This paper discusses the same problem but extend their results to a geometric Itô–Lévy jump process. Our modelling framework is very general as it allows random parameters which are unbounded and involves some jumps. It also covers parameters which are both Markovian and non-Markovian functionals. Unlike in Shen and Wei (2014) who considered a diffusion framework, ours solves the problem using a novel approach, which combines the Hamilton–Jacobi–Bellman (HJB) and a backward stochastic differential equation (BSDE) in a Lévy market setup. We illustrate our results by two examples.     

Explicit solutions of optimal consumption,investment and insurance problems with regime switching

We consider an investor who wants to select his optimal consumption, investment and insurance policies. Motivated by new insurance products, we allow not only the financial market but also the insurable loss to depend on the regime of the economy. The objective of the investor is to maximize his expected total discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absolute risk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimal consumption, investment, and insurance problems when there is regime switching. We determine that the optimal insurance contract is either no-insurance or deductible insurance, and calculate when it is optimal to buy insurance. The optimal policy depends strongly on the regime of the economy. Through an economic analysis, we calculate the advantage of buying insurance.     

Stochastic utilities with subsistence and satiation: Optimal life insurance purchase,consumption and investment

We introduce stochastic utilities such that utility of any fixed amount of interest is a stochastic process or random variable. Also, there exist stochastic (or random) subsistence and satiation levels associated with stochastic utilities. Then, we consider optimal consumption, life insurance purchase and investment strategies to maximize the expected utility of consumption, bequest and pension with respect to stochastic utilities. We use the martingale approach to solve the optimization problem in two steps. First, we solve the optimization problem with an equality constraint which requires that the present value of consumption, bequest and pension is equal to the present value of initial wealth and income stream. Second, if the optimization problem is feasible, we obtain the explicit representations of the replicating life insurance purchase and portfolio strategies. As an application of our general results, we consider a family of stochastic utilities which have hyperbolic absolute risk aversion (HARA).     

Household consumption, investment and life insurance

This paper develops a continuous-time Markov model for utility optimization of households. The household optimizes expected future utility from consumption by controlling consumption, investments and purchase of life insurance for each person in the household. The optimal controls are investigated in the special case of a two-person household, and we present graphics illustrating how differences between the two persons affect the controls.     

Household Lifetime Strategies under a Self-Contagious Market

In this paper, we consider the optimal strategies in asset allocation, consumption, and life insurance for a household with an exogenous stochastic income under a self-contagious market which is modeled by bivariate self-exciting Hawkes jump processes. By using the Hawkes process, jump intensities of the risky asset depend on the history path of that asset. In addition to the financial risk, the household is also subject to an uncertain lifetime and a fixed retirement date. A lump-sum payment will be paid as a heritage, if the wage earner dies before the retirement date. Under the dynamic programming principle, explicit solutions of the optimal controls are obtained when asset prices follow special jump distributions. For more general cases, we apply the Feynman–Kac formula and develop an iterative numerical scheme to derive the optimal strategies. We also prove the existence and uniqueness of the solution to the fixed point equation and the convergence of an iterative numerical algorithm. Numerical examples are presented to show the effect of jump intensities on the optimal controls.     

Optimal investment,consumption and timing of annuity purchase under a preference change

In this paper, we study the optimal investment and consumption strategies for a retired individual who has the opportunity of choosing a discretionary stopping time to purchase an annuity. We assume that the individual receives a fixed annuity income and changes his/her preference after paying a fixed cost for annuitization. By using the martingale method and the variational inequality method, we tackle this problem and obtain the optimal strategies and the value function explicitly for the case of constant force of mortality and constant relative risk aversion (CRRA) utility function.     

Optimal investment and consumption decision of a family with life insurance

We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.     

Health shock risk,critical illness insurance,and housing services

This paper studies a consumption–investment problem involving health shock risk, perishable consumption, and consumption of housing services. Additionally to a risk-free asset and a stock index, the agent can invest in real estate. I analyze the impact of health shocks on the optimal consumption and investment decisions in model specifications with and without the possibility to buy critical illness insurance. I discuss the influence of critical illness insurance on the optimal strategy and analyze the drivers of the optimal critical illness insurance demand. The results indicate that health shock risk has potentially devastating consequences, especially for young agents. It turns out that critical illness insurance is an excellent instrument for hedging health shock risk and for consumption smoothing across different health states. Optimal critical illness insurance demand is decreasing in financial wealth and increasing in human wealth. Real estate prices have a minor influence on optimal critical illness insurance demand.     

Optimal portfolio, consumption and retirement decision under a preference change

We investigate an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion (CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We give some numerical results of the optimal policies. We also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level.     

随机利率条件下最优投资消费与寿险购买策略

随着我国利率市场化的深入发展, 利率的随机波动对投资者的最优投资消费策略将产生重要影响. 与此同时, 随着我国寿险市场的渐趋完善, 寿险购买也越来越受到投资者的重视, 投资者的最优策略也将发生改变. 现研究由 Vasicek 模型来刻画的随机利率条件下最优投资消费与寿险购买策略. 投资者的目标在于选择最优投资消费与寿险购买策略使期望效用最大化. 通过运用 Legendre 转换方法求出最优投资消费与寿险购买的显性解. 通过数值分析的方法, 实证分析相关变量的变化对投资者最优投资与寿险购买策略的影响.     

Online retailers’ promotional pricing,free-shipping threshold,and inventory decisions: A simulation-based analysis

We consider a two-stage decision problem, in which an online retailer first makes optimal decisions on his profit margin and free-shipping threshold, and then determines his inventory level. We start by developing the retailer’s expected profit function. Then, we use publicly-available statistics to find the best-fitting distribution for consumers’ purchase amounts and the best-fitting function for conversion rate (i.e., probability that an arriving visitor places an online order with the retailer). We show that: (i) a reduction of the profit margin does not significantly affect the standard deviation of consumers’ order sizes (purchase amounts) but increases the average order size; whereas, (ii) variations in a positive finite free-shipping threshold affect both the average value and the standard deviation of the order sizes. We then use Arena to simulate the online retailing system and OptQuest to find the retailer’s optimal decisions and maximum profit. Next, we perform a sensitivity analysis to examine the impact of the ratio of the unit holding and salvage cost to the unit shipping cost on the retailer’s optimal decisions. We also draw some important managerial insights.     

Optimal term life insurance — A practical solution

This paper aims to offer a reasonable, practical solution to the life insurance decision facing families with a single wage earner upon whom the remaining family members are dependent. The solution depends upon the individual's age, probability of dying, planning horizon, earnings, wealth, degree of risk aversion, and the social insurance that would be received by the family. It is shown that, under various realistic assumptions, although the essence of the problem is its multiperiod nature, the solution reduces to a sequence of single-period decisions.     

An optimal job,consumption/leisure,and investment policy

In this paper we investigate an optimal job, consumption, and investment policy of an economic agent in a continuous and infinite time horizon. The agent’s preference is characterized by the Cobb–Douglas utility function whose arguments are consumption and leisure. We use the martingale method to obtain the closed-form solution for the optimal job, consumption, and portfolio policy. We compare the optimal consumption and investment policy with that in the absence of job choice opportunities.     

Stochastic differential reinsurance games with capital injections

This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results.     

Advance selling to strategic consumers

Advance selling of goods and services is a form of separating purchase from consumption. It is often employed when consumers are uncertain about their consumption utilities until a short time period before consumption. A book to be released, a concert to attend, or a cruise to take are some examples. Invariably, in consumers’ mind inventory availability (of copies, seats, or rooms) is a concern. In this paper we study a retailer’s inventory and pricing decisions in an advance selling scenario that involves consumers who are strategic. Some consumers not only consider advance and spot prices, but also the uncertainty in future availability of the product (during the spot period) and in their consumption utility from it. We characterize the optimal inventory management and pricing policies, and discuss several interesting aspects of the solution. For example, it can be optimal for the retailer to limit advance sales even if there is more demand for it, and it can be optimal for the retailer to limit its inventory even though there is more capacity to keep it, but not both.     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

一种电子商务环境下耐用品的动态定价模型与策略

针对电子商务环境下消费者对价格歧视的抗拒问题,以及耐用品生命周期长、产品需求依赖于时间、价格等特点,提出了一种动态定价模型与策略。该模型通过构造转移概率矩阵,推导出在线消费者浏览到耐用品的不同价格状态下的概率,接着根据消费者多阶段效用函数分析消费者的购买决策行为,进而给出零售商利润达到最大化时的最优定价策略集合。为了验证模型与策略的有效性,通过数值模拟实验,分析模型主要参数变化对最优定价策略的影响。研究发现当效用折扣因子越高,零售商应该降低促销频率和高价格并且提高低价格,从而诱导高端消费者在高价格购买产品。折扣效用因子大小还决定了网上零售商是否要隐藏自己的促销概率。     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.