Optimal consumption-leisure, portfolio and retirement selection based on α-maxmin expected CES utility with ambiguity

This article studies optimal consumption-leisure, portfolio and retirement selection of an infinitely lived investor whose preference is formulated by ??-maxmin expected CES utility which is to differentiate ambiguity and ambiguity attitude. Adopting the recursive multiplepriors utility and the technique of backward stochastic differential equations (BSDEs), we transform the ??-maxmin expected CES utility into a classical expected CES utility under a new probability measure related to the degree of an investor??s uncertainty. Our model investigates the optimal consumption-leisure-work selection, the optimal portfolio selection, and the optimal stopping problem. In this model, the investor is able to adjust her supply of labor flexibly above a certain minimum work-hour along with a retirement option. The problem can be analytically solved by using a variational inequality. And the optimal retirement time is given as the first time when her wealth exceeds a certain critical level. The optimal consumption-leisure and portfolio strategies before and after retirement are provided in closed forms. Finally, the distinctions of optimal consumption-leisure, portfolio and critical wealth level under ambiguity from those with no vagueness are discussed.     

Using genetic algorithm to solve a new multi-period stochastic optimization model

This paper presents a new asset allocation model based on the CVaR risk measure and transaction costs. Institutional investors manage their strategic asset mix over time to achieve favorable returns subject to various uncertainties, policy and legal constraints, and other requirements. One may use a multi-period portfolio optimization model in order to determine an optimal asset mix. Recently, an alternative stochastic programming model with simulated paths was proposed by Hibiki [N. Hibiki, A hybrid simulation/tree multi-period stochastic programming model for optimal asset allocation, in: H. Takahashi, (Ed.) The Japanese Association of Financial Econometrics and Engineering, JAFFE Journal (2001) 89-119 (in Japanese); N. Hibiki A hybrid simulation/tree stochastic optimization model for dynamic asset allocation, in: B. Scherer (Ed.), Asset and Liability Management Tools: A Handbook for Best Practice, Risk Books, 2003, pp. 269-294], which was called a hybrid model. However, the transaction costs weren’t considered in that paper. In this paper, we improve Hibiki’s model in the following aspects: (1) The risk measure CVaR is introduced to control the wealth loss risk while maximizing the expected utility; (2) Typical market imperfections such as short sale constraints, proportional transaction costs are considered simultaneously. (3) Applying a genetic algorithm to solve the resulting model is discussed in detail. Numerical results show the suitability and feasibility of our methodology.     

带息力的Erlang(2)风险过程下的一类积分方程

本文考虑了带息力的Erlang(2)风险模型，利用Sundt和Teugels(1995)，Yang和Zhang(2001a，2001b和2001c)文中的技巧，得到了生存概率所满足的积分方程和指数型的积分方程，然后研究了生存概率的Laplace-Stieltjes变换所满足的二阶微分方程．     

Efficient portfolios in financial markets with proportional transaction costs

In this article, we characterize efficient portfolios, i.e. portfolios which are optimal for at least one rational agent, in a very general multi-currency financial market model with proportional transaction costs. In our setting, transaction costs may be random, time-dependent, have jumps and the preferences of the agents are modeled by multivariate expected utility functions. We provide a complete characterization of efficient portfolios, generalizing earlier results of Dybvig (Rev Financ Stud 1:67–88, 1988) and Jouini and Kallal (J Econ Theory 66: 178–197, 1995). We basically show that a portfolio is efficient if and only if it is cyclically anticomonotonic with respect to at least one consistent price system that prices it. Finally, we introduce the notion of utility price of a given contingent claim as the minimal amount of a given initial portfolio allowing any agent to reach the claim by trading, and give a dual representation of it as the largest proportion of the market price necessary for all agents to reach the same expected utility level.     

Optimal Control for Utility Portfolio Selection with Liability

In this paper we use stochastic optimal control theory to investigate a dynamic portfolio selection problem with liability process, in which the liability process is assumed to be a geometric Brownian motion and completely correlated with stock prices. We apply dynamic programming principle to obtain Hamilton-Jacobi-Bellman (HJB) equations for the value function and systematically study the optimal investment strategies for power utility, exponential utility and logarithm utility. Firstly, the explicit expressions of the optimal portfolios for power utility and exponential utility are obtained by applying variable change technique to solve corresponding HJB equations. Secondly, we apply Legendre transform and dual approach to derive the optimal portfolio for logarithm utility. Finally, numerical examples are given to illustrate the results obtained and analyze the effects of the market parameters on the optimal portfolios.     

Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility.     

Mean-variance versus expected utility in dynamic investment analysis

Given the existence of a Markovian state price density process, this paper extends Merton??s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the risky fund can be chosen to be the growth optimal portfolio. The CAPM obtains without the assumption of log-normality for prices. The optimal investment policies for the case of a hyperbolic absolute risk aversion (HARA) utility function are derived analytically. It is proved that only the quadratic utility exhibits the global mean-variance efficiency among the family of HARA utility functions. A numerical comparison is made between the growth optimal portfolio and the mean-variance analysis for the case of log-normal prices. The optimal choice of target return which maximizes the probability that the mean-variance analysis outperforms the expected utility portfolio is discussed. Mean variance analysis is better near the mean and the expected utility maximization is better in the tails.     

具有任意Hurst参数的分数次Black-Scholes模型的最优资产组合

在由具有任意Hurst参数H ∈(0,1)的分数次布朗运动驱动的Black-Scholes型市场数学模型的基础上, 运用拟条件数学期望和随机-梯度等工具,解决了其在能量型效应函数时的最优资产组合问题.     

比例保本约束下的最优投资组合选择

本文研究了幂效用函数下带有比例保本约束的最优投资组合选择问题.利用拉格朗日乘子和投资组合复制方法,得到最优财富过程和最优投资组合,推广了带有限制的投资组合的相关结果.     

基于随机基准的最优投资组合选择问题研究

本文研究基于随机基准的最优投资组合选择问题.假设投资者可以投资于一种无风险资产和一种风险股票,并且选择某一基准作为目标.基准是随机的,并且与风险股票相关.投资者选择最优的投资组合策略使得终端期望绝对财富和基于基准的相对财富效用最大.首先,利用动态规划原理建立相应的HJB方程,并在幂效用函数下,得到最优投资组合策略和值函数的显示表达式.然后,分析相对业绩对投资者最优投资组合策略和值函数的影响.最后,通过数值计算给出了最优投资组合策略和效用损益与模型主要参数之间的关系.     

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.     

Consumption and portfolio rules for time-inconsistent investors

This paper extends the classical consumption and portfolio rules model in continuous time [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247–257, Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413] to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton–Jacobi–Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and power utilities) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.     

Building an Optimal Portfolio in Discrete Time in the Presence of Transaction Costs

Abstract

Portfolio theory covers different approaches to the construction of a portfolio offering maximum expected returns for a given level of risk tolerance where the goal is to find the optimal investment rule. Each investor has a certain utility for money which is reflected by the choice of a utility function. In this article, a risk averse power utility function is studied in discrete time for a large class of underlying probability distribution of the returns of the asset prices. Each investor chooses, at the beginning of an investment period, the feasible portfolio allocation which maximizes the expected value of the utility function for terminal wealth. Effects of both large and small proportional transaction costs on the choice of an optimal portfolio are taken into account. The transaction regions are approximated by using asymptotic methods when the proportional transaction costs are small and by using expansions about critical points for large transaction costs.     

Optimal portfolio strategies benchmarking the stock market

The paper investigates the impact of adding a shortfall risk constraint to the problem of a portfolio manager who wishes to maximize his utility from the portfolios terminal wealth. Since portfolio managers are often evaluated relative to benchmarks which depend on the stock market we capture risk management considerations by allowing a prespecified risk of falling short such a benchmark. This risk is measured by the expected loss in utility. Using the Black–Scholes model of a complete financial market and applying martingale methods, explicit analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.     

Optimal impulse control of a portfolio with a fixed transaction cost

The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent’s behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.     

Continuous-time mean-variance portfolios: a comparison

In this article, we present and compare three mean-variance optimal portfolio approaches in a continuous-time market setting. These methods are the L ²-projection as presented in Schweizer [M. Schweizer, Approximation of random variables by stochastic integrals, Ann. Prob. 22 (1995), pp. 1536–1575], the Lagrangian function approach of Korn and Trautmann [R. Korn and S. Trautmann, Continuous-time portfolio optimization under terminal wealth constraints, ZOR-Math. Methods Oper. Res. 42 (1995), pp. 69–92] and the direct deterministic approach of Lindberg [C. Lindberg, Portfolio optimization when expected stock returns are determined by exposure to risk, Bernoulli 15 (2009), pp. 464–474]. As the underlying model, we choose the recent innovative market parameterization introduced by Lindberg (2009) that has the particular aim to overcome the estimation problems of the stock price drift parameters. We derive some new results for the Lagrangian function approach, in particular explicit representations for the optimal portfolio process. Further, we compare the different optimization frameworks in detail and highlight their attractive and not so attractive features by numerical examples.     

变折现率下带含糊厌恶与预期的最优投资研究

本文采用折现率为时间的函数下的递推多先验效用,研究Merton模型在带预期条件下的最优消费和投资组合决策问题,其中含糊与风险是有区别的.在幂效用函数情形下,刻画了投资者最优投资决策,表明了含糊厌恶和预期对最优投资的影响.最优投资组合决策由倒向随机微分方程和Malliavin导数导出.     

Existence and uniqueness of an optimum in the infinite-horizon portfolio-cum-saving model with semimartingale investments

The model considered here is essentially that formulated in the author's previous paper Conditions for Optimality in the Infinite-Horizon Portfolio-cum-Saving Problem with Semimartingale Investments, Stochastics and Stochastics Reports 29 (1990), 133-171. In this model, the vector process representing returns to investments is a general semimartingale. Processes defining portfolio plans arc here required only to be predictable and non-negative. Existence of an optimal portfolio-cum-saving plan is proved under slight conditions of integrability imposed on the welfare functional; the proofs rely on properties of weak precompactness of portfolio and utility sequences in suitable L _p spaces together with dominated and monotone convergence arguments. Conditions are also obtained for the uniqueness of the portfolio plan generating a given returns process (i.e. for the uniqueness of the integrands generating a given sum of semimartingale integrals) and for the uniqueness of an optimal plan; here use is made of random measures associated with the jumps of a semimartingale     

幂效用函数的最优投资组合研究

本文对投资组合中较常用的风险厌恶型的幂效用函数进行研究。应用无差异曲线法求解出这种效用函数的最优投资比例,并对本文所得出的结论进行了实例应用分析。