Optimal life insurance purchase,consumption and investment on a financial market with multi-dimensional diffusive terms

We introduce an extension to Merton's famous continuous time model of optimal consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for a wage earner to have a random lifetime and to use a portion of the income to purchase life insurance in order to provide for his estate, while investing his savings in a financial market comprised of one risk-free security and an arbitrary number of risky securities driven by multi-dimensional Brownian motion. We then provide a detailed analysis of the optimal consumption, investment and insurance purchase strategies for the wage earner 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 methods to obtain explicit solutions for the case of discounted constant relative risk aversion utility functions and describe new analytical results which are presented together with the corresponding economic interpretations.     

Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces.Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim.Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.     

The role of longevity bonds in optimal portfolios

We study the optimal consumption and portfolio for an agent maximizing the expected utility of his intertemporal consumption in a financial market with: (i) a riskless asset, (ii) a stock, (iii) a bond as a derivative on the stochastic interest rate, and (iv) a longevity bond whose coupons are proportional to the population (stochastic) survival rate. With a force of mortality instantaneously uncorrelated with the interest rate (but not necessarily independent), we demonstrate that the wealth invested in the longevity bond must be taken from the ordinary bond and the riskless asset proportionally to the duration of the two bonds. This result is valid for both a complete and an incomplete financial market.     

Optimal consumption problems in discontinuous markets

We study an extension of Merton’s classical portfolio investment – consumption optimization problem (1969–1970) to a particular case of complete discontinuous market, with a single jump. The market consists of a non-risky asset, a ‘standard risky’ asset and a risky asset with discontinuous price dynamics (e.g. a defaultable bond or a mortality linked security). We consider three different problems of maximization of the expected utility from consumption: in the case when the investment horizon is fixed and finite, when it is finite, but possibly uncertain and when it is infinite. The innovative setting is the second one. In a general stochastic coefficients’ model, we solve the problems and we compare the three optimal consumption rates, finding quite interesting results. In the logarithmic and power utility cases, explicit solutions are provided. Furthermore, the benchmark – constant coefficients’ case is deeply investigated and a partial information setting is also studied in the uncertain time horizon case.     

Optimal Investment and Consumption with Proportional Transaction Costs in Regime-Switching Model

This paper is concerned with an infinite-horizon problem of optimal investment and consumption with proportional transaction costs in continuous-time regime-switching models. An investor distributes his/her wealth between a stock and a bond and consumes at a non-negative rate from the bond account. The market parameters (the interest rate, the appreciation rate, and the volatility rate of the stock) are assumed to depend on a continuous-time Markov chain with a finite number of states (also known as regimes). The objective of the optimization problem is to maximize the expected discounted total utility of consumption. We first show that for a class of hyperbolic absolute risk aversion utility functions, the value function is a viscosity solution of the Hamilton–Jacobi–Bellman equation associated with the optimization problem. We then treat a power utility function and generalize the existing results to the regime-switching case.     

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.     

Dynamic pricing and warranty policies for products with fixed lifetime

We consider a repairable product with known market entry and departure times. A warranty policy is offered with product purchase, under which a customer can have a failed item repaired free of charge in the warranty period. It is assumed that customers are heterogeneous in their risk attitudes toward uncertain repair costs incurred after the warranty expires. The objective is to determine a joint dynamic pricing and warranty policy for the lifetime of the product, which maximizes the manufacturer’s expected profit. In the first part of the analysis, we consider a linearly decreasing price function and a constant warranty length. We first study customers’ purchase patterns under several different pricing strategies by the manufacturer and then discuss the optimal pricing and warranty strategy. In the second part, we assume that the warranty length can be altered once during the product lifetime in developing a joint pricing and warranty policy. Numerical studies show that a dynamic warranty policy can significantly outperform a fixed-length warranty policy.     

Coping with the Renewable Portfolio Standards: A utility’s perspective

Renewable Portfolio Standards require utilities to supply a percentage of their energy from renewables. Utilities demonstrate their compliance by purchasing renewable energy or renewable energy certificates. Certificates can be traded in a secondary market. This flexibility is intended to foster an efficient environment. Utilities, however, have not been reacting strategically. We formulate the utility’s problem as a stochastic dynamic program and present optimal policies for the utility to trade in the REC market and to purchase from the energy market.     

Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection

This paper assesses optimal life cycle consumption and portfolio allocations when households have access to Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuities over their adult lifetimes. Our contribution is to evaluate demand for these products which provide access to equity investments with money-back guarantees, longevity risk hedging, and partially-refundable premiums, in a realistic world with uncertain labor and capital market income as well as mortality risk. Others have predicted that consumers will only purchase such annuities late in life, but we show that they will optimally purchase GMWBs prior to retirement, consistent with their recent rapid uptick in sales. Additionally, many individuals optimally adjust their portfolios and consumption streams along the way by taking cash withdrawals from the products. These products can substantially enhance consumption, by up to 10% for those who experience highly unfavorable experiences in the stock market.     

Dynamic Bertrand Oligopoly

We study continuous time Bertrand oligopolies in which a small number of firms producing similar goods compete with one another by setting prices. We first analyze a static version of this game in order to better understand the strategies played in the dynamic setting. Within the static game, we characterize the Nash equilibrium when there are N players with heterogeneous costs. In the dynamic game with uncertain market demand, firms of different sizes have different lifetime capacities which deplete over time according to the market demand for their good. We setup the nonzero-sum stochastic differential game and its associated system of HJB partial differential equations in the case of linear demand functions. We characterize certain qualitative features of the game using an asymptotic approximation in the limit of small competition. The equilibrium of the game is further studied using numerical solutions. We find that consumers benefit the most when a market is structured with many firms of the same relative size producing highly substitutable goods. However, a large degree of substitutability does not always lead to large drops in price, for example when two firms have a large difference in their size.     

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.     

Integrating optimal annuity planning with consumption-investment selections in retirement planning

Annuities can be effective tools in managing longevity risk in retirement planning. This paper develops a framework that merges annuity purchase decisions with consumption-investment selections in retirement planning. After introducing a pricing and a benefit payment model for an annuity, we construct a multi-period wealth evolution model. An optimization problem is formulated with an objective of maximizing lifetime utility of consumption and wealth. Optimal decisions are determined as a trade off between consumption and investment among an annuity, a risky and a risk-free asset. Computational results are provided to illustrate the practical implications of the framework.     

ͨ�ͷ����µ�³������Ͷ��������ٱ��������о�

??In this paper, we investigate a robust optimal portfolio and reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries about uncertainty in model parameters. We assume that the AAI is allowed to purchase proportional reinsurance and invest his/her wealth in a financial market which consists of a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal expected power utility of terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategies are obtained.     

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.     

Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for Markov regime-switching jump diffusion models

In this paper, we consider the optimal consumption and investment strategies for households throughout their lifetime. Risks such as the illiquidity of assets, abrupt changes of market states, and lifetime uncertainty are considered. Taking the effects of heritage into account, investors are willing to limit their current consumption in exchange for greater wealth at their death, because they can take advantage of the higher expected returns of illiquid assets. Further, we model the liquidity risks in an illiquid market state by introducing frozen periods with uncertain lengths, during which investors cannot continuously rebalance their portfolios between different types of assets. In liquid market, investors can continuously remix their investment portfolios. In addition, a Markov regime-switching process is introduced to describe the changes in the market’s states. Jumps, classified as either moderate or severe, are jointly investigated with liquidity risks. Explicit forms of the optimal consumption and investment strategies are developed using the dynamic programming principle. Markov chain approximation methods are adopted to obtain the value function. Numerical examples demonstrate that the liquidity of assets and market states have significant effects on optimal consumption and investment strategies in various scenarios.     

Altruistic motives,uncertain lifetime and urban public pension replacement rates

This article employs an overlapping generations model with altruistic motives and uncertain lifetime to investigate China's urban public pension system. We examine the effects of the individual account benefit replacement rate, social pool benefit replacement rate, life expectancy and population growth rate on the capital-labour ratio, pension benefits, consumption and utility. We also find the optimal social pool benefit replacement rate. Raising the individual account benefit replacement rate only increases the individual account benefits. Raising the social pool benefit replacement rate increases the social pool benefits and retirement-period consumption, whereas decreases the capital-labour ratio, individual account benefits, working-period consumption and utility. The fall in the population growth rate increases the capital-labour ratio, social pool benefits, individual account benefits, working-period consumption and utility, whereas decreases the retirement-period consumption. The rise in the life expectancy decreases the six variables. The optimal social pool benefit replacement rate falls in case of either risen life expectancy or fallen population growth rate. It further falls under the joint case of risen life expectancy and fallen population growth rate. It will do more good than harm to raise the individual account benefit replacement rate, reduce the social pool benefit replacement rate and strictly implement the population policy.     

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 commutable annuities to minimize the probability of lifetime ruin

We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and who can purchase a commutable life annuity. The surrender charge of a life annuity is a proportion of its value. Ruin occurs when the total of the value of the risky and riskless assets and the surrender value of the life annuity reaches zero. We find the optimal investment strategy and optimal annuity purchase and surrender strategies in two situations: (i) the value of the risky and riskless assets is allowed to be negative, with the imputed surrender value of the life annuity keeping the total positive; (ii) the value of the risky and riskless assets is required to be non-negative. In the first case, although the individual has the flexibility to buy or sell at any time, we find that the individual will not buy a life annuity unless she can cover all her consumption via the annuity and she will never sell her annuity. In the second case, the individual surrenders just enough annuity income to keep her total assets positive. However, in this second case, the individual’s annuity purchasing strategy depends on the size of the proportional surrender charge. When the charge is large enough, the individual will not buy a life annuity unless she can cover all her consumption, the so-called safe level. When the charge is small enough, the individual will buy a life annuity at a wealth lower than this safe level.     

Worst-Case Scenario Portfolio Optimization: a New Stochastic Control Approach

We consider the determination of portfolio processes yielding the highest worst-case bound for the expected utility from final wealth if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. A particular application of our setting is to model crash scenarios where both the number and the height of the crash are uncertain but bounded. Also the situation of changing market coefficients after a possible crash is analyzed.     

Market segmentation by maximum likelihood clustering using choice elasticities

We determine market segments by clustering households on the basis of their average choice elasticities across purchases and brands w.r.t. price, sales promotion and brand loyalty. The cluster analysis technique used is a maximum likelihood method which allows varying size and orientation and assumes constant volume. Elasticities originate from choice models with alternatively linear and nonlinear utility functions. Choice models are estimated on the basis of household scanner data. Segments are interpreted by means of multiple discriminant analysis and multinomial logit models whose predictors are elasticities of predictors and external variables (i.e. number of purchases, number of brands bought, income and household size), respectively.