随机模糊的等比例-最小方差投资组合优化研究

Markowitz的均值-方差模型在投资组合优化中得到了广泛的运用和拓展,其中多数拓展模型仅局限于对随机投资组合或模糊投资组合的研究,而忽略了实际问题同时包含了随机信息和模糊信息两个方面。本文首先定义随机模糊变量的方差用以度量投资组合的风险,提出具有阀值约束的最小方差随机模糊投资组合模型,基于随机模糊理论,将该模型转化为具有线性等式和不等式约束的凸二次规划问题。为了提高上述模型的有效性,本文以投资者期望效用最大化为压缩目标对投资组合权重进行压缩,构建等比例-最小方差混合的随机模糊投资组合模型,并求解该模型的最优解。最后,运用滚动实际数据的方法,比较上述两个模型的夏普比率以验证其有效性。     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

Mean-variance models for portfolio selection with fuzzy random returns

This paper develops two novel types of mean-variance models for portfolio selection problems, in which the security returns are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. In the proposed models, we take the expected return of a portfolio as the investment return and the variance of the expected return of a portfolio as the investment risk. We assume that the security returns are triangular fuzzy random variables. To solve the proposed portfolio problems, this paper first presents the variance formulas for triangular fuzzy random variables. Then this paper applies the variance formulas to the proposed models so that the original portfolio problems can be reduced to nonlinear programming ones. Due to the reduced programming problems include standard normal distribution in the objective functions, we cannot employ the conventional solution methods to solve them. To overcome this difficulty, this paper employs genetic algorithm (GA) to solve them, and verify the obtained optimal solutions via Kuhn-Tucker (K-T) conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed models and methods.     

Stochastic linear knapsack programming problem and its application to a portfolio selection problem

In this paper a probability maximization model of a stochastic linear knapsack problem is considered where the random variables consist of several groups with mutually correlated ones. We propose a solution algorithm to the equivalent nonlinear fractional programming problem with a simple ranking method. This approach will be effectively applied to one of the portfolio selection problems.     

Extended scenario analysis

Scenario analysis, as proposed by Rockafellar and Wets, is a stochastic programming technique employing discrete scenarios with known probabilities, usually covering several time periods. The requirement of nonaticipativity (not using future information to make present decisions) is enforced during the computational solution by using Spingarn's method of partial inverses. The scenario analysis method as proposed relies on separability (with respect to scenarios) of all problem elements except the nonanticipativity constraint.We show how, by making a little more use of the partial inverse technique, one can include nonseparable convex constraints in such a procedure. As an illustrative example, we show how to analyze a portfolio optimization problem of Markowitz type (minimize variance for a given return) using scenarios. This offers the prospect of extending classical portfolio analysis from models based on historical behavior to models incorporating future scenarios of any desired type.The research reported here was sponsored by the National Science Foundation under Grant CCR-8801489, and by the Air Force Systems Command, USAF, under Grant No. AFOSR-89-0058. The US Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

Robust-based interactive portfolio selection problems with an uncertainty set of returns

This paper considers a robust portfolio selection problem with an uncertainty set of future returns and satisfaction levels in terms of the total return and robustness parameter. Since the proposed model is formulated as an ill-defined problem due to uncertainty and is bi-objective, that is, to maximize both the abovementioned satisfaction levels, it is difficult to solve the model directly without introducing some criterion of optimality for the bi-objective functions. Therefore, by introducing fuzzy goals and an interactive fuzzy satisficing method, the proposed model is transformed into a deterministic equivalent problem. Furthermore, to obtain the exact optimal portfolio analytically, a solution method is developed by introducing the auxiliary problem and performing equivalent transformations. In order to compare the proposed model with previous useful models, numerical examples are provided, and the results show that it is important to maximize the robustness parameter and total return using the interactive process for adjusting investor’s satisfaction levels.     

Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization

This paper considers Stackelberg solutions for two-level linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random two-level linear programming problem, an α-stochastic two-level linear programming problem is defined through the introduction of α-level sets of fuzzy random variables. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through fractile criterion optimization in stochastic programming, the transformed stochastic two-level programming problem can be reduced to a deterministic two-level programming problem. An extended concept of Stackelberg solution is introduced and a numerical example is provided to illustrate the proposed method.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

Multi-objective stochastic programming for portfolio selection

Generally, in the portfolio selection problem the Decision Maker (DM) considers simultaneously conflicting objectives such as rate of return, liquidity and risk. Multi-objective programming techniques such as goal programming (GP) and compromise programming (CP) are used to choose the portfolio best satisfying the DM’s aspirations and preferences. In this article, we assume that the parameters associated with the objectives are random and normally distributed. We propose a chance constrained compromise programming model (CCCP) as a deterministic transformation to multi-objective stochastic programming portfolio model. CCCP is based on CP and chance constrained programming (CCP) models. The proposed program is illustrated by means of a portfolio selection problem from the Tunisian stock exchange market.     

基于模糊系数的投资组合优化模型及其实证分析

采用模糊数处理不确定性信息．以模糊期望收益率最大为目标函数,使总的风险不高于给定的模糊数,建立了一种新的模型．在给定的截集下,期望收益率转化为区间数,目标函数转化为对该区间数的下限求最大值．基于模糊数大小的概率比较,从而将模糊优化模型转化为不等式约束下的线性规划模型．利用Matlab编程可解得其最优解．最后通过实例分析...     

Efficient optimization of the reward-risk ratio with polyhedral risk measures

In problems of portfolio selection the reward-risk ratio criterion is optimized to search for a risky portfolio offering the maximum increase of the mean return, compared to the risk-free investment opportunities. In the classical model, following Markowitz, the risk is measured by the variance thus representing the Sharpe ratio optimization and leading to the quadratic optimization problems. Several polyhedral risk measures, being linear programming (LP) computable in the case of discrete random variables represented by their realizations under specified scenarios, have been introduced and applied in portfolio optimization. The reward-risk ratio optimization with polyhedral risk measures can be transformed into LP formulations. The LP models typically contain the number of constraints proportional to the number of scenarios while the number of variables (matrix columns) proportional to the total of the number of scenarios and the number of instruments. Real-life financial decisions are usually based on more advanced simulation models employed for scenario generation where one may get several thousands scenarios. This may lead to the LP models with huge number of variables and constraints thus decreasing their computational efficiency and making them hardly solvable by general LP tools. We show that the computational efficiency can be then dramatically improved by alternative models based on the inverse ratio minimization and taking advantages of the LP duality. In the introduced models the number of structural constraints (matrix rows) is proportional to the number of instruments thus not affecting seriously the simplex method efficiency by the number of scenarios and therefore guaranteeing easy solvability.     

Multi-stage portfolio selection problem with dynamic stochastic dominance constraints

We study the multi-stage portfolio selection problem where the utility function of an investor is ambiguous. The ambiguity is characterized by dynamic stochastic dominance constraints, which are able to capture the dynamics of the random return sequence during the investment process. We propose a multi-stage dynamic stochastic dominance constrained portfolio selection model, and use a mixed normal distribution with time-varying weights and the K-means clustering technique to generate a scenario tree for the transformation of the proposed model. Based on the scenario tree representation, we derive two linear programming approximation problems, using the sampling approach or the duality theory, which provide an upper bound approximation and a lower bound approximation for the original nonconvex problem. The upper bound is asymptotically tight with infinitely many samples. Numerical results illustrate the practicality and efficiency of the proposed new model and solution techniques.

     

Portfolio adjusting optimization under credibility measures

This paper discusses portfolio adjusting problems for an existing portfolio. The returns of risky assets are regarded as fuzzy variables and a class of credibilistic mean-variance adjusting models with transaction costs are proposed on the basis of credibility theory. Under the assumption that the returns of risky assets are triangular fuzzy variables, the optimization models are converted into crisp forms. Furthermore, we employ the sequential quadratic programming method to work out the optimal strategy. Numerical examples illustrate the effectiveness of the proposed models and the influence of the transaction costs in portfolio selection.     

Risk-control approach for bottleneck transportation problem with randomness and fuzziness

Solving transportation problems is essential in engineering and supply chain management, where profitability depends on optimal traffic flow. This study proposes risk-control approaches for two bottleneck transportation problems with random variables and preference levels to objective functions with risk parameters. Each proposed model is formulated as a multiobjective programming problem using robust-based optimization derived from stochastic chance constraints. Since it is impossible to obtain a transportation pattern that optimizes all objective functions, our proposed models are numerically solved by introducing an aggregation function for the multiobjective problem. An exact algorithm that performs deterministic equivalent transformations and introduces auxiliary problems is also developed.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.     

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.The author's work was supported in part by the National Science Foundation under Grant DDM-9215921.     

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

A cutting plane algorithm for MV portfolio selection model

This paper deals with a portfolio selection problem with fuzzy return rates. A possibilistic mean variance (FMVC) portfolio selection model was proposed. The possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC). The nonlinear programming problem can be solved by sequence linear programming problem. A numerical example is given to illustrate the behavior of the proposed model and algorithm.     

Tracking error: a multistage portfolio model

We study multistage tracking error problems. Different tracking error measures, commonly used in static models, are discussed as well as some problems which arise when we move from static to dynamic models. We are interested in dynamically replicating a benchmark using only a small subset of assets, considering transaction costs due to rebalancing and introducing a liquidity component in the portfolio. We formulate and solve a multistage tracking error model in a stochastic programming framework. We numerically test our model by dynamically replicating the MSCI Euro index. We consider an increasing number of scenarios and assets and show the superior performance of the dynamically optimized tracking portfolio over static strategies.