Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.     

CVaR鲁棒均值-CVaR投资组合模型与求解

传统的均值-风险(包括方差、VaR、CVaR等)组合选择模型在计算最优投资组合时,常假定均值是已知的常值,但在实际资产配置中,收益的均值估计会有偏差,即存在着估计风险.在利用CVaR测度估计风险的基础上,研究了CVaR鲁棒均值-CVaR投资组合选择模型,给出了另外两种不同的求解方法,即对偶法和光滑优化方法,并探讨了它们的相关性质及特征,数值实验表明在求解大样本或者大规模投资组合选择问题上,对偶法和光滑优化方法在计算上是可行且有效的.     

Conditional value-at-risk in portfolio optimization: Coherent but fragile

We evaluate conditional value-at-risk (CVaR) as a risk measure in data-driven portfolio optimization. We show that portfolios obtained by solving mean-CVaR and global minimum CVaR problems are unreliable due to estimation errors of CVaR and/or the mean, which are magnified by optimization. This problem is exacerbated when the tail of the return distribution is made heavier. We conclude that CVaR, a coherent risk measure, is fragile in portfolio optimization due to estimation errors.     

On the role of norm constraints in portfolio selection

Several optimization approaches for portfolio selection have been proposed in order to alleviate the estimation error in the optimal portfolio. Among them are the norm-constrained variance minimization and the robust portfolio models. In this paper, we examine the role of the norm constraint in portfolio optimization from several directions. First, it is shown that the norm constraint can be regarded as a robust constraint associated with the return vector. Second, the reformulations of the robust counterparts of the value-at-risk (VaR) and conditional value-at-risk (CVaR) minimizations contain norm terms and are shown to be highly related to the ν-support vector machine (ν-SVM), a powerful statistical learning method. For the norm-constrained VaR and CVaR minimizations, a nonparametric theoretical validation is posed on the basis of the generalization error bound for the ν-SVM. Third, the norm-constrained approaches are applied to the tracking portfolio problem. Computational experiments reveal that the norm-constrained minimization with a parameter tuning strategy improves on the traditional norm-unconstrained models in terms of the out-of-sample tracking error.     

Sensitivity of portfolio VaR and CVaR to portfolio return characteristics

Risk management through marginal rebalancing is important for institutional investors due to the size of their portfolios. We consider the problem of improving marginally portfolio VaR and CVaR through a marginal change in the portfolio return characteristics. We study the relative significance of standard deviation, mean, tail thickness, and skewness in a parametric setting assuming a Student’s t or a stable distribution for portfolio returns. We also carry out an empirical study with the constituents of DAX30, CAC40, and SMI. Our analysis leads to practical implications for institutional investors and regulators.     

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.     

Minimizing loss probability bounds for portfolio selection

In this paper, we derive a portfolio optimization model by minimizing upper and lower bounds of loss probability. These bounds are obtained under a nonparametric assumption of underlying return distribution by modifying the so-called generalization error bounds for the support vector machine, which has been developed in the field of statistical learning. Based on the bounds, two fractional programs are derived for constructing portfolios, where the numerator of the ratio in the objective includes the value-at-risk (VaR) or conditional value-at-risk (CVaR) while the denominator is any norm of portfolio vector. Depending on the parameter values in the model, the derived formulations can result in a nonconvex constrained optimization, and an algorithm for dealing with such a case is proposed. Some computational experiments are conducted on real stock market data, demonstrating that the CVaR-based fractional programming model outperforms the empirical probability minimization.     

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.     

VaR测度下的风险对冲策略研究

本文分别在正态分布和任意分布设定下讨论最小在险价值(VaR)的风险对冲问题。在正态分布设定下,本文深入讨论最小方差对冲比率和最小VaR对冲比率的性质,并得出最小VaR对冲策略下组合收益率的均值和方差大于最小方差策略下组合收益率的均值和方差。在任意分布设定下,本文构建一种新的VaR对冲模型,该模型引入非参数核估计方法对VaR进行估计,然后基于VaR核估计量建立风险对冲问题,实现风险估计与风险对冲同步进行。实证结果非常稳健地表明,不做任何分布假设下的核估计法得到的风险对冲效果优于最小方差对冲策略和正态分布设定下的最小VaR对冲策略。     

Robust portfolios that do not tilt factor exposure

Robust portfolios reduce the uncertainty in portfolio performance. In particular, the worst-case optimization approach is based on the Markowitz model and form portfolios that are more robust compared to mean–variance portfolios. However, since the robust formulation finds a different portfolio from the optimal mean–variance portfolio, the two portfolios may have dissimilar levels of factor exposure. In most cases, investors need a portfolio that is not only robust but also has a desired level of dependency on factor movement for managing the total portfolio risk. Therefore, we introduce new robust formulations that allow investors to control the factor exposure of portfolios. Empirical analysis shows that the robust portfolios from the proposed formulations are more robust than the classical mean–variance approach with comparable levels of exposure on fundamental factors.     

Robust asset allocation with conditional value at risk using the forward search

The well‐known Markowitz approach to portfolio allocation, based on expected returns and their covariance, seems to provide questionable results in financial management. One motivation for the pitfall is that financial returns have heavier than Gaussian tails, so the covariance of returns, used in the Markowitz model as a measure of portfolio risk, is likely to provide a loose quantification of the effective risk. Additionally, the Markowitz approach is very sensitive to small changes in either the expected returns or their correlation, often leading to irrelevant portfolio allocations. More recent allocation techniques are based on alternative risk measures, such as value at risk (VaR) and conditional VaR (CVaR), which are believed to be more accurate measures of risk for fat‐tailed distributions. Nevertheless, both VaR and CVaR estimates can be influenced by the presence of extreme returns. In this paper, we discuss sensitivity to the presence of extreme returns and outliers when optimizing the allocation, under the constraint of keeping CVaR to a minimum. A robust and efficient approach, based on the forward search, is suggested. A Monte Carlo simulation study shows the advantages of the proposed approach, which outperforms both robust and nonrobust alternatives under a variety of specifications. The performance of the method is also thoroughly evaluated with an application to a set of US stocks.     

基于几种风险测度的多阶段组合优化研究

基于均值-方差(MV)、VaR(Value at Risk)、CVaR(Conditional VaR)、HMCR(p=1,2,3)(Higher Moment Coherent Risk)几种风险测度进行多阶段组合优化研究。首先从一致性公理和随机占优一致性角度分析几种风险测度的风险识别能力,认为HMCR(p=2,3)的风险识别能力最高,然后给出静态和动态下的风险规避型的规划函数及多阶段CVaR和HMCR模型,最后依据单阶段和多阶段优化模型,对上证50指数成份股进行实证分析。对比单阶段和多阶段下几种风险测度优化组合的累计收益率及几种风险测度之间的关系,结合上证50指数收益率发现,多阶段优化组合要整体优于单阶段优化组合,且HMCR(p=2,3)要优于指数收益率和其它几种风险测度。从投资者投资决策方面来分析,HMCR(p=2,3)型积极投资策略比较适用于股市平稳期、顶峰期和下降期,被动投资策略比较适用于股市上升期。     

基于推广多因子模型的资产动态配置研究——以上证A股为例

考虑投资者关于市场的观点,并用椭球分布替代正态分布作为资产收益和投资者观点分布导出推广的多因子模型(EMFM).采集2015年1月5日到2018年9月11日上证A股中分列于11个行业共828只股票的日收益率及其相关指标,采用5个系统风险因子,在3种常用风险指标下应用EMFM对该高数据进行动态建模并做资产配置分析.实证结果显示:从回溯期内的收益表现来看,使用自由度为3的多维t分布,一种特殊的椭球分布,即便合并简单的绝对观点的EMFM在SD风险指标下的组合都优于马科维兹组合的情形,同时也优于其使用VaR和CVaR的情形,且所有组合都优于上证指数的情形.     

Recent Developments in Robust Portfolios with a Worst-Case Approach

Robust models have a major role in portfolio optimization for resolving the sensitivity issue of the classical mean–variance model. In this paper, we survey developments of worst-case optimization while focusing on approaches for constructing robust portfolios. In addition to the robust formulations for the Markowitz model, we review work on deriving robust counterparts for value-at-risk and conditional value-at-risk problems as well as methods for combining uncertainty in factor models. Recent findings on properties of robust portfolios are introduced, and we conclude by presenting our thoughts on future research directions.     

求解带有交易费的CVaR投资组合模型的L-S算法

本文假设投资者是风险厌恶型,用CVaR作为测量投资组合风险的方法.在预算约束的条件下,以最小化CVaR为目标函数,建立了带有交易费用的投资组合模型.将模型转化为两阶段补偿随机优化模型,构造了求解模型的随机L-S算法.为了验证算法的有效性,用中国证券市场中的股票进行数值试验,得到了最优投资组合、VaR和CVaR的值.而且对比分析了有交易费和没有交易费的最优投资组合的不同,给出了相应的有效前沿.     

基于CVaR约束的指数组合优化模型及实证分析

随着指数衍生产品日益受到重视,指数化投资组合常被投资者或机构所采用,而用有限的资金按指数构成比例进行投资显然是不现实的,所以指数的最优误差追踪就显得更加重要。本文将追踪误差定义为证券投资组合收益率与所追踪的指数基准收益率之差,并在分析CvaR(ConditionalValue at Risk)的基础上,在无交易费用和有交易费用的情况下,建立了基于CVaR约束的追踪误差最小化的指数组合优化模型,对指数进行复制,并通过实证分析,得出了基于CVaR约束的追踪误差最小时的样本期内及样本期外的最优投资策略,验证了CVaR约束控制风险的有效性。     

不同均值-风险准则下的资产组合有效前沿比较研究

本文根据V aR和CV aR风险度量方法,对马克维茨的均值-方差资产组合选择模型进行拓展,研究在均值-风险准则下更具有一般性的资产组合选择问题.并在正态分布假设条件下,证明当不存在无风险资产时和存在无风险资产时,基于方差、V aR和CV aR风险度量准则的资产组合有有沿之间的关系,指出根据均值-V aR准则和均值-CV aR准则求解有效资产组合时,置信水平必须满足的条件     

The practice of Delta–Gamma VaR: Implementing the quadratic portfolio model

This paper intends to critically evaluate state-of-the-art methodologies for calculating the value-at-risk (VaR) of non-linear portfolios from the point of view of computational accuracy and efficiency. We focus on the quadratic portfolio model, also known as “Delta–Gamma”, and, as a working assumption, we model risk factor returns as multi-normal random variables. We present the main approaches to Delta–Gamma VaR weighing their merits and accuracy from an implementation-oriented standpoint. One of our main conclusions is that the Delta–Gamma-Normal VaR may be less accurate than even Delta VaR. On the other hand, we show that methods that essentially take into account the non-linearity (hence gammas and third or higher moments) of the portfolio values may present significant advantages over full Monte Carlo revaluations. The role of non-diagonal terms in the Gamma matrix as well as the sensitivity to correlation is considered both for accuracy and computational effort. We also qualitatively examine the robustness of Delta–Gamma methodologies by considering a highly non-quadratic portfolio value function.