Portfolio optimization based on downside risk: a mean-semivariance efficient frontier from Dow Jones blue chips

To create efficient funds appealing to a sector of bank clients, the objective of minimizing downside risk is relevant to managers of funds offered by the banks. In this paper, a case focusing on this objective is developed. More precisely, the scope and purpose of the paper is to apply the mean-semivariance efficient frontier model, which is a recent approach to portfolio selection of stocks when the investor is especially interested in the constrained minimization of downside risk measured by the portfolio semivariance. Concerning the opportunity set and observation period, the mean-semivariance efficient frontier model is applied to an actual case of portfolio choice from Dow Jones stocks with daily prices observed over the period 2005–2009. From these daily prices, time series of returns (capital gains weekly computed) are obtained as a piece of basic information. Diversification constraints are established so that each portfolio weight cannot exceed 5 per cent. The results show significant differences between the portfolios obtained by mean-semivariance efficient frontier model and those portfolios of equal expected returns obtained by classical Markowitz mean-variance efficient frontier model. Precise comparisons between them are made, leading to the conclusion that the results are consistent with the objective of reflecting downside risk.     

期望和残差收益估计不可靠的鲁棒模型

投资优化问题的最优策略会随着输入参数的扰动而出现敏感的变化,针对投资优化问题中出现的随机变量的参数估计不可靠的情况,本文引入不确定集合描述随机收益的有关矩信息,提出了投资优化问题的一个鲁棒性模型,并采用数学规划的理论和方法,给出了该模型的最优策略和有效前沿的解析表示。本方法能够为采用保守策略的、对不确定性厌恶的投资者提供一种最优的投资策略。     

基于不确定性期望效用模型的投资优化

针对资产的收益的分布不确切知道,并且所获得的矩信息也不是准确值的问题,提出了最大化最坏情形期望效用的鲁棒性方法.引入了凹凸类效用函数来度量模型不确定情形下投资者的效用,用一个不确定性结构来刻画资产收益的所有可能的分布和收益的矩信息,通过把具有不确定性结构的鲁棒性模型转化成参数二次规划问题,得到了最优投资策略、有效前沿和均衡价格的解析表示.方法为采用保守策略并且厌恶不确定性的投资者提供了一种有效的投资决策方案.     

Making inefficient market indices efficient

This paper uses the concept of Marginal Conditional Stochastic Dominance and a generalization of the 50% Portfolio Rule to develop a tractable and parsimonious methodology for constructing a second degree Stochastic Dominance (SSD) efficient portfolio from a given, inefficient index. Because the SSD approach considers the entire probability distributions of asset returns, the resulting portfolios are efficient with respect to all risk-averse, utility-maximizing investors regardless of the form of their utility functions or the distributions of asset returns.     

Optimal portfolios using linear programming models

The classical quadratic programming (QP) formulation of the well-known portfolio selection problem has traditionally been regarded as cumbersome and time consuming. This paper formulates two additional models: (i) maximin, and (ii) minimization of mean absolute deviation. Data from 67 securities over 48 months are used to examine to what extent all three formulations provide similar portfolios. As expected, the maximin formulation yields the highest return and risk, while the QP formulation provides the lowest risk and return, which also creates the efficient frontier. The minimization of mean absolute deviation is close to the QP formulation. When the expected returns are confronted with the true ones at the end of a 6-month period, the maximin portfolios seem to be the most robust of all.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

t-分布与不同风险度量下有效投资组合及其边缘的相互关系研究

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges

In modern portfolio theory, it is common practice to first compute the risk-reward efficient frontier and then to support an individual investor in selecting a portfolio that meets his/her preferences for profitability and risk. Potential flaws include (a) the assumption that past data provide sufficient evidence for predicting the future performances of the securities under consideration and (b) the necessity to mathematically determine or approximate the investor’s utility function. In this paper, we propose a methodology whose initial phase filters portfolios that are inefficient from a historical perspective. While this is consistent with traditional approaches, the second phase differs from the standard approach as it uses a decision table constructed by considering multiple scenarios assuming strict uncertainty. The table cells measure consequences by a multi-criteria linear performance index of simulated future returns, which avoids difficulties with performance ratios. The real world applicability is illustrated through two studies based on data from the stock exchanges in Frankfurt and Vienna.     

Mean–variance–skewness efficient surfaces,Stein’s lemma and the multivariate extended skew-Student distribution

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.     

Quantitative portfolio selection: Using density forecasting to find consistent portfolios

In the knowledge that the ex-post performance of Markowitz efficient portfolios is inferior to that implied ex-ante, we make two contributions to the portfolio selection literature. Firstly, we propose a methodology to identify the region of risk-expected return space where ex-post performance matches ex-ante estimates. Secondly, we extend ex-post efficient set mathematics to overcome the biases in the estimation of the ex-ante efficient frontier. A density forecasting approach is used to measure the accuracy of ex-ante estimates using the Berkowitz statistic, we develop this statistic to increase its sensitivity to changes in the data generating process. The area of risk-expected return space where the density forecasts are accurate, where ex-post performance matches ex-ante estimates, is termed the consistency region. Under the 'laboratory' conditions of a simulated multivariate normal data set, we compute the consistency region and the estimated ex-post frontier. Over different sample sizes used for estimation, the behaviour of the consistency region is shown to be both intuitively reasonable and to enclose the estimated ex-post frontier. Using actual data from the constituents of the US Dow Jones 30 index, we show that the size of the consistency region is time dependent and, in volatile conditions, may disappear. Using our development of the Berkowitz statistic, we demonstrate the superior performance of an investment strategy based on consistent rather than efficient portfolios.     

不存在无风险资产的投资组合灵敏度分析

本文研究了M－V证券投资组合灵敏度分析方法。考虑了不存在无险资产时证券预睡益率和协方差矩阵存在扰动的情形，给出了最优投资组合有效边缘的漂移方程及组合扩展路径。     

Asset pricing and portfolio selection based on the multivariate extended skew-Student-<Emphasis Type="Italic">t</Emphasis> distribution

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar mean-variance frontier, and which may be implemented using quadratic programming.     

基于均值-VaR的投资组合最优化

利用均值-VaR方法,提出了有交易费用存在时的最优投资组合模型。通过求解均值-方差模型来研究均值-VaR模型的有效前沿,并指出在收益率的分布为正态分布的假设下,均值-VaR模型的有效集是均值-方差有效前沿的子集。有关全局最小VaR的存在性的分析显示在选择VaR的置信水平时必须非常小心。最后给出了应用均值-VaR模型的实例分析。     

选择资产组合的EP-MV模型及最优解的解析表示

本文提出了存在无风险资产贷出或借入时的有效投资组合模型(EP-MV模型)，研究了不允许卖空(投资比例非负)约束条件下，EP-MV优化模型的算法，给出了有效投资组合投资比例的解析表示．在资产收益由多因素模型产生的基础上，得到了资产与有效投资组合的期望收益及风险的估计，便于实际应用．     

Statistical foundations for assessing the difference between the classical and weighted-Gini betas

The ‘beta’ is one of the key quantities in the capital asset pricing model (CAPM). In statistical language, the beta can be viewed as the slope of the regression line fitted to financial returns on the market against the returns on the asset under consideration. The insurance counterpart of CAPM, called the weighted insurance pricing model (WIPM), gives rise to the so-called weighted-Gini beta. The aforementioned two betas may or may not coincide, depending on the form of the underlying regression function, and this has profound implications when designing portfolios and allocating risk capital. To facilitate these tasks, in this paper we develop large-sample statistical inference results that, in a straightforward fashion, imply confidence intervals for, and hypothesis tests about, the equality of the two betas.     

Finding DEA-efficient hyperplanes using MOLP efficient faces

This paper suggests a method for finding efficient hyperplanes with variable returns to scale the technology in data envelopment analysis (DEA) by using the multiple objective linear programming (MOLP) structure. By presenting an MOLP problem for finding the gradient of efficient hyperplanes, We characterize the efficient faces. Thus, without finding the extreme efficient points of the MOLP problem and only by identifying the efficient faces of the MOLP problem, we characterize the efficient hyperplanes which make up the DEA efficient frontier. Finally, we provide an algorithm for finding the efficient supporting hyperplanes and efficient defining hyperplanes, which uses only one linear programming problem.     

Geometric representation of the mean–variance–skewness portfolio frontier based upon the shortage function

The literature suggests that investors prefer portfolios based on mean, variance and skewness rather than portfolios based on mean–variance (MV) criteria solely. Furthermore, a small variety of methods have been proposed to determine mean–variance–skewness (MVS) optimal portfolios. Recently, the shortage function has been introduced as a measure of efficiency, allowing to characterize MVS optimal portfolios using non-parametric mathematical programming tools. While tracing the MV portfolio frontier has become trivial, the geometric representation of the MVS frontier is an open challenge. A hitherto unnoticed advantage of the shortage function is that it allows to geometrically represent the MVS portfolio frontier. The purpose of this contribution is to systematically develop geometric representations of the MVS portfolio frontier using the shortage function and related approaches.     

Finding the efficiency score and RTS characteristic of DMUs by means of identifying the efficient frontier in DEA

Data envelopment analysis (DEA) is basically a linear programming based technique used for measuring the relative performance of organizational units, referred to as decision-making units (DMUs), where the presence of multiple inputs and outputs makes comparisons difficult. The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, a method for identifying the efficient frontier is introduced. Then, the efficiency score and returns to scale (RTS) characteristic of DMUs will be produced by means of the equation of efficient frontier.     

The Efficiency Distribution Approach in Data Envelopment Analysis: An Application

A method is developed here for characterizing the empirical distribution of the efficient units in data envelopment analysis. Two empirical applications illustrate the various uses of the distribution approach. One involves the cost frontier which exhibits increasing returns to scale and the other involves a dynamic production frontier, where technological change causes a shift of the production frontier over time.