基于随机参照点的均值-风险分析

本文依据参照依赖偏好模型提出了基于随机参照点的风险度量方法,进而构建了均值-风险模型,并讨论了该决策方法与随机占优之间的一致性。研究发现,该决策方法不仅与一级随机占优是一致的而且与二级随机占优也是一致的。由于二级随机占优与期望效用理论的一致性,因而所构建的均值-风险模型与期望效用理论也是一致的。     

Multiple criteria linear programming model for portfolio selection

The portfolio selection problem is usually considered as a bicriteria optimization problem where a reasonable trade-off between expected rate of return and risk is sought. In the classical Markowitz model the risk is measured with variance, thus generating a quadratic programming model. The Markowitz model is frequently criticized as not consistent with axiomatic models of preferences for choice under risk. Models consistent with the preference axioms are based on the relation of stochastic dominance or on expected utility theory. The former is quite easy to implement for pairwise comparisons of given portfolios whereas it does not offer any computational tool to analyze the portfolio selection problem. The latter, when used for the portfolio selection problem, is restrictive in modeling preferences of investors. In this paper, a multiple criteria linear programming model of the portfolio selection problem is developed. The model is based on the preference axioms for choice under risk. Nevertheless, it allows one to employ the standard multiple criteria procedures to analyze the portfolio selection problem. It is shown that the classical mean-risk approaches resulting in linear programming models correspond to specific solution techniques applied to our multiple criteria model. This revised version was published online in June 2006 with corrections to the Cover Date.     

On consistency of stochastic dominance and mean–semideviation models

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically. Received: September 15, 1998 / Accepted: October 1, 2000?Published online December 15, 2000     

On Extending the LP Computable Risk Measures to Account Downside Risk

A mathematical model of portfolio optimization is usually quantified with mean-risk models offering a lucid form of two criteria with possible trade-off analysis. In the classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. Following Sharpe’s work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving linear programming (LP) problems. Typical LP computable risk measures, like the mean absolute deviation (MAD) or the Gini’s mean absolute difference (GMD) are symmetric with respect to the below-mean and over-mean performances. The paper shows how the measures can be further combined to extend their modeling capabilities with respect to enhancement of the below-mean downside risk aversion. The relations of the below-mean downside stochastic dominance are formally introduced and the corresponding techniques to enhance risk measures are derived.The resulting mean-risk models generate efficient solutions with respect to second degree stochastic dominance, while at the same time preserving simplicity and LP computability of the original models. The models are tested on real-life historical data.The research was supported by the grant PBZ-KBN-016/P03/99 from The State Committee for Scientific Research.     

Reverse logistics network design and planning utilizing conditional value at risk

Nowadays, due to some social, legal, and economical reasons, dealing with reverse supply chain is an unavoidable issue in many industries. Besides, regarding real-world volatile parameters, lead us to use stochastic optimization techniques. In location–allocation type of problems (such as the presented design and planning one), two-stage stochastic optimization techniques are the most appropriate and popular approaches. Nevertheless, traditional two-stage stochastic programming is risk neutral, which considers the expectation of random variables in its objective function. In this paper, a risk-averse two-stage stochastic programming approach is considered in order to design and planning a reverse supply chain network. We specify the conditional value at risk (CVaR) as a risk evaluator, which is a linear, convex, and mathematically well-behaved type of risk measure. We first consider return amounts and prices of second products as two stochastic parameters. Then, the optimum point is achieved in a two-stage stochastic structure regarding a mean-risk (mean-CVaR) objective function. Appropriate numerical examples are designed, and solved in order to compare the classical versus the proposed approach. We comprehensively discuss about the effectiveness of incorporating a risk measure in a two-stage stochastic model. The results prove the capabilities and acceptability of the developed risk-averse approach and the affects of risk parameters in the model behavior.     

Convexity and decomposition of mean-risk stochastic programs

Traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion. A common approach to addressing risk in decision making problems is to consider a weighted mean-risk objective, where some dispersion statistic is used as a measure of risk. We investigate the computational suitability of various mean-risk objective functions in addressing risk in stochastic programming models. We prove that the classical mean-variance criterion leads to computational intractability even in the simplest stochastic programs. On the other hand, a number of alternative mean-risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms. We propose decomposition-based parametric cutting plane algorithms to generate mean-risk efficient frontiers for two particular classes of mean-risk objectives.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Gains from diversification on convex combinations: A majorization and stochastic dominance approach

By incorporating both majorization theory and stochastic dominance theory, this paper presents a general theory and a unifying framework for determining the diversification preferences of risk-averse investors and conditions under which they would unanimously judge a particular asset to be superior. In particular, we develop a theory for comparing the preferences of different convex combinations of assets that characterize a portfolio to give higher expected utility by second-order stochastic dominance. Our findings also provide an additional methodology for determining the second-order stochastic dominance efficient set.     

A multistage linear stochastic programming model for optimal corporate debt management

Large corporations fund their capital and operational expenses by issuing bonds with a variety of indexations, denominations, maturities and amortization schedules. We propose a multistage linear stochastic programming model that optimizes bond issuance by minimizing the mean funding cost while keeping leverage under control and insolvency risk at an acceptable level. The funding requirements are determined by a fixed investment schedule with uncertain cash flows. Candidate bonds are described in a detailed and realistic manner. A specific scenario tree structure guarantees computational tractability even for long horizon problems. Based on a simplified example, we present a sensitivity analysis of the first stage solution and the stochastic efficient frontier of the mean-risk trade-off. A realistic exercise stresses the importance of controlling leverage. Based on the proposed model, a financial planning tool has been implemented and deployed for Brazilian oil company Petrobras.     

TSD-consistent performance assessment of mutual funds

This paper presents three new data envelopment analysis-based approaches to assess the relative efficiency of mutual funds (MFs). Each model considers an appropriate risk measure as input and an appropriate return measure as output. The risk and return measures have been chosen so that the proposed models are consistent with third-order stochastic dominance (TSD) rules. This means that the MFs found efficient by the proposed models are also, in a necessity condition sense, TSD efficient and therefore of highest consideration for all non-satiated, risk averse investors that also have decreasing absolute risk aversion. The proposed approach is illustrated with real data on a set of Spanish MFs and compared with existing approaches from the literature based on Mean–Variance and Mean–Variance–Skewness models.     

Mean-risk analysis with enhanced behavioral content

We study a mean-risk model derived from a behavioral theory of Disappointment with multiple reference points. One distinguishing feature of the risk measure is that it is based on mutual deviations of outcomes, not deviations from a specific target. We prove necessary and sufficient conditions for strict first and second order stochastic dominance, and show that the model is, in addition, a Convex Risk Measure. The model allows for richer, and behaviorally more plausible, risk preference patterns than competing models with equal degrees of freedom, including Expected Utility (EU), Mean–Variance (M-V), Mean-Gini (M-G), and models based on non-additive probability weighting, such as Dual Theory (DT). In asset allocation, the model allows a decision-maker to abstain from diversifying in a positive expected value risky asset if its performance does not meet a certain threshold, and gradually invest beyond this threshold, which appears more acceptable than the extreme solutions provided by either EU and M-V (always diversify) or DT and M-G (always plunge). In asset trading, the model provides no-trade intervals, like DT and M-G, in some, but not all, situations. An illustrative application to portfolio selection is presented. The model can provide an improved criterion for mean-risk analysis by injecting a new level of behavioral realism and flexibility, while maintaining key normative properties.     

Data envelopment analysis of mutual funds based on second-order stochastic dominance

Although data envelopment analysis (DEA) has been extensively used to assess the performance of mutual funds (MF), most of the approaches overestimate the risk associated to the endogenous benchmark portfolio. This is because in the conventional DEA technology the risk of the target portfolio is computed as a linear combination of the risk of the assessed MF. This neglects the important effects of portfolio diversification. Other approaches based on mean–variance or mean–variance–skewness are non-linear. We propose to combine DEA with stochastic dominance criteria. Thus, in this paper, six distinct DEA-like linear programming (LP) models are proposed for computing relative efficiency scores consistent (in the sense of necessity) with second-order stochastic dominance (SSD). The aim is that, being SSD efficient, the obtained target portfolio should be an optimal benchmark for any rational risk-averse investor. The proposed models are compared with several related approaches from the literature.     

Time consistent behavioral portfolio policy for dynamic mean–variance formulation

When one considers an optimal portfolio policy under a mean-risk formulation, it is essential to correctly model investors’ risk aversion which may be time variant or even state dependent. In this paper, we propose a behavioral risk aversion model, in which risk aversion is a piecewise linear function of the current excess wealth level with a reference point at the discounted investment target (either surplus or shortage), to reflect a behavioral pattern with both house money and break-even effects. Due to the time inconsistency of the resulting multi-period mean–variance model with adaptive risk aversion, we investigate the time consistent behavioral portfolio policy by solving a nested mean–variance game formulation. We derive a semi-analytical time consistent behavioral portfolio policy which takes a piecewise linear feedback form of the current excess wealth level with respect to the discounted investment target. Finally, we extend the above results to time consistent behavioral portfolio selection for dynamic mean–variance formulation with a cone constraint.     

A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

Orderings and Probability Functionals Consistent with Preferences

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First, we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally, we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.     

Sufficient conditions under which SSD- and MR-efficient sets are identical

Three approaches are commonly used for analyzing decisions under uncertainty: expected utility (EU), second-degree stochastic dominance (SSD), and mean-risk (MR) models, with the mean–standard deviation (MS) being the best-known MR model. Because MR models generally lead to different efficient sets and thus are a continuing source of controversy, the specific concern of this article is not to suggest another MR model. Instead, we show that the SSD- and MR-efficient sets are identical, as long as (a) the risk measure satisfies both positive homogeneity and consistency with respect to the Rothschild and Stiglitz (1970) definition(s) of increasing risk and (b) the choice set includes the riskless asset and satisfies a generalized location and scale property, which can be interpreted as a market model. Under these conditions, there is no controversy among MR models and they all have a decision-theoretic foundation. They also offer a convenient way to compare the estimation error related to the empirical implementation of different MR models.     

A risk-averse newsvendor with law invariant coherent measures of risk

For general law invariant coherent measures of risk, we derive an equivalent representation of a risk-averse newsvendor problem as a mean-risk model. We prove that the higher the weight of the risk functional, the smaller the order quantity. Our theoretical results are confirmed by sample-based optimization.     

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.     

Mean-risk model for uncertain portfolio selection

This paper discusses the uncertain portfolio selection problem when security returns cannot be well reflected by historical data. It is proposed that uncertain variable should be used to reflect the experts’ subjective estimation of security returns. Regarding the security returns as uncertain variables, the paper introduces a risk curve and develops a mean-risk model. In addition, the crisp form of the model is provided. The presented numerical examples illustrate the application of the mean-risk model and show the disaster result of mistreating uncertain returns as random returns.     

Stochastic goal programming: A mean–variance approach

We propose a stochastic goal programming (GP) model leading to a structure of mean–variance minimisation. The solution to the stochastic problem is obtained from a linkage between the standard expected utility theory and a strictly linear, weighted GP model under uncertainty. The approach essentially consists in specifying the expected utility equation corresponding to every goal. Arrow's absolute risk aversion coefficients play their role in the calculation process. Once the model is defined and justified, an illustrative example is developed.