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A practical guide to robust portfolio optimization
Quantitative Finance
Abstract
Robust optimization takes into account the uncertainty in expected returns to address the shortcomings of portfolio mean-variance optimization, namely the sensitivity of the optimal portfolio to inputs. We investigate the mechanisms by which robust optimization achieves its goal and give practical guidance when it comes to the choice of uncertainty in form and level. We explain why the quadratic uncertainty set should be preferred to box uncertainty based on the literature review, we show that a diagonal uncertainty matrix with only variances should be used, and that the level of uncertainty can be chosen as a function of the asset Sharpe ratios. Finally, we use practical examples to show that, with the proposed parametrization, robust optimization does overcome the weaknesses of mean-variance optimization and can be applied in real investment problems such as the management of multi-asset portfolios or in robo-advising.
... In order to take into account the uncertainty in the return estimation, it would be interesting to reformulate the mean-variance optimization problem by taking into consideration the estimation errors which may help to reduce the sensitivity of the portfolios as the estimation of errors increases. This modification has lead to the so-called robust portfolio optimization [3], [8], [9], [10], [11], [12]. More precisely, the mean-variance portfolio (with soft constraint to risk aversion) can be modified as ...
... Equating to zero both partial derivatives of L pµ, Φq relative to µ and Φ and then solving for µ leads to the solution [3], [8] µ "μ´c κ 2 w T Ωw Ωw . ...
... At the end, one obtains the following robust portfolio optimization [3], [8], [9], [10], [11], [12]: ...
In this project, we aim to combine different protfolio allocation techniques with covariance matrix estimators to meet two types of clients' requirements: client A who wants to invest money wisely, not taking too much risk, and not willing to pay too much in rebalancing fees; and client B who wants to make money quickly, benefit from market's short-term volatility, and ready to pay rebalancing fees. Four portfolio techniques are considered (mean-variance, robust portfolio, minimum-variance, and equi-risk budgeting), and four covariance estimators are applied (sample covariance, ordinary least squares (OLS) covariance, cross-validated eigenvalue shrinkage covariance, and eigenvalue clipping). Some comparisons between the covariance estimators in terms of eigenvalue stability and four metrics (i.e. expected risk, gross leverage, Sharpe ratio and effective diversification) exhibit the superiority of the eigenvalue clipping covariance estimator. The experiments on the Russel1000 dataset show that the minimum-variance with eigenvalue clipping is the model suitable for client A, whereas robust portfolio with eigenvalue clipping is the one suitable for client B.
... Assume that asset returns are independent and identically distributed and µ − µ follows a normal distribution with mean value 0 and covariance matrix Ω, where Ω is the covariance matrix of errors in the expected asset return. In Yin et al. [35], they discussed the choice of uncertainty matrix Ω in the quadratic uncertainty set and proposed the selection criteria. In the quadratic uncertainty case, min µ∈U w T µ in problem (5) is equivalent to the following problem: ...
... According to the proposition of Yin et al. [35], we choose Ω ∝ diag(V). By using this uncertainty matrix, it is expected to reduce the sensitivity to the inputs, as well as keep the original volatility unchanged. ...
The robust and sparse portfolio selection problem is one of the most-popular and -frequently studied problems in the optimization and financial literature. By considering the uncertainty of the parameters, the goal is to construct a sparse portfolio with low volatility and decent returns, subject to other investment constraints. In this paper, we propose a new portfolio selection model, which considers the perturbation in the asset return matrix and the parameter uncertainty in the expected asset return. We define three types of stationary points of the penalty problem: the Karush–Kuhn–Tucker point, the strong Karush–Kuhn–Tucker point, and the partial minimizer. We analyze the relationship between these stationary points and the local/global minimizer of the penalty model under mild conditions. We design a penalty alternating-direction method to obtain the solutions. Compared with several existing portfolio models on seven real-world datasets, extensive numerical experiments demonstrate the robustness and effectiveness of our model in generating lower volatility.
... Practitioners often mention that addressing uncertainty in mean returns is sufficient when applying RPO. Yin, Perchet, and Soupé (2021) provide practical guidelines for implementing RPO, also focusing on setting uncertainty sets for mean returns along with related parameters. The importance of RPO in practice is highlighted by Yin, Perchet, and Soupé (2021) and Issaoui et al. (2022) due to increasing demand in automated asset allocation and mass customization of portfolios. ...
... Yin, Perchet, and Soupé (2021) provide practical guidelines for implementing RPO, also focusing on setting uncertainty sets for mean returns along with related parameters. The importance of RPO in practice is highlighted by Yin, Perchet, and Soupé (2021) and Issaoui et al. (2022) due to increasing demand in automated asset allocation and mass customization of portfolios. Pedersen, Babu, and Levine (2021) also share practical guides for performing MVO, and they mention that the major concerns in practice can be addressed using correlation shrinkage and provide insight into its relation to RO and regularization. ...
... Jorion, 1985;Black and Litterman, 1992), and to creating robust optimisation routines (e.g. Brodie et al., 2009;Yin et al., 2021). These themes are ably pursued in Pedersen et al.'s (2021) recent contribution, which advocates shrinking the correlation matrix towards zero. ...
The Gerber Statistic is a recently proposed co-movement measure. The measure is a conditional statistic and due to conditioning bias, will naturally differ from full-sample measures. This contribution makes use of an intuitive two-asset simulation framework to better elucidate the statistical behaviour of the Gerber Statistic, across its three proposed forms. Using graphical
correlation profiles, we explore the measure’s behaviour across return conditioning threshold, sample size and market distribution, detailing its mechanisms of performance but also demonstrating
several caveats around its understanding and use. We conclude that while interesting, the Gerber Statistic is best viewed as an imperfect conditional dependence metric.
... Jorion, 1985;Black and Litterman, 1992), and also to creating more robust optimisation routines (e.g. Brodie et al., 2009;Yin et al., 2021). All of these themes are ably pursued in a recent contribution by Pedersen et al. (2021), which ultimately advocates extreme shrinkage of the correlation matrix towards zero. ...
The Gerber Statistic is a recently proposed co-movement measure. The measure is a conditional statistic and due to conditioning bias, will naturally differ from full-sample measures. This contribution makes use of an intuitive two-asset simulation framework to better elucidate the statistical behaviour of the Gerber Statistic, across its three proposed forms. Using graphical
correlation profiles, we explore the measure’s behaviour across return conditioning threshold, sample size and market distribution, detailing its mechanisms of performance but also demonstrating
several caveats around its understanding and use. We conclude that while interesting, the Gerber Statistic is best viewed as an imperfect conditional dependence metric.
... In addition, the MV model is also said to be sensitive because of the input parameters used. Various kinds of optimal robust portfolio methods have been developed (Supandi, Rosadi, & Abdurakhman, 2017;Yin, Perchet, & Soupé, 2021). However, the MV model is still used as a reference in the framework of development and modification for optimal portfolio modelling. ...
The Capital Asset Pricing Model (CAPM), which has interest rates in its specification, can be deemed non-Shariah compliant. Therefore, the sukuk rate has been proposed to replace these rates in CAPM. This study analyses portfolio modelling by involving two essential elements in Islamic principles, namely zakat and purification. The concept of purification has been applied in the Shariah stock selection process in Indonesia, while at the same time, zakat has been widely socialised in stock investment. The study highlights two models that consider the concept of zakat reduction and the purification factors for portfolios in the Indonesian stock market. According to the robustness tests conducted, the proposed Shariah-compliant asset pricing model developed in this study is valid. Zakat reduction and purification factor integration in mathematical models can be applied in portfolio modelling.
... Unlike mean-variance optimisation, robust optimisation takes into account the uncertainty in the asset return estimates and significantly reduces the sensitivity of the final portfolio to small changes in expected returns. The description of the framework based on robust optimisation can be found in Yin, Perchet, and Soup e (2021). Details about how the allocation of robust portfolios is constructed from the robust optimisation algorithm was investigated by Perchet et al. (2016). ...
We introduce the notion of themes as an additional investment dimension beyond asset classes, regions, sectors and styles, and propose a framework to allocate to thematic investments at a strategic asset allocation level. Allocating to themes requires discipline because thematic investments are not only exposed to the theme but also to the traditional risk factors. Our approach uses a framework based on robust portfolio optimisation, which accounts for the expected excess return from the exposure to the theme and from exposures to traditional risk factors. We provide an example to illustrate how thematic investments fit in traditional multi-asset portfolios.
... Modern Portfolio Theory was introduced by Markowitz (MPT) [5]. The traditional mean-variance strategy, over which Harry Markowitz received the Nobel Prize in Economics in 1990, was the first critical study of a conundrum that every investor faces: the conflicting aims of high return and low risk [6]- [9]. To solve this fundamental difficulty, Markowitz proposed a parametric optimization model that would be both flexible and able for a wide range of real-world circumstances also simple enough in the theoretical modeling and analysis. ...
... The fundamental drawback of Markowitz's model is that the idea assumes the appraisals of each stock returns and the variances are accurate but also tends. This issue is analytically reported and empirically tested by many researchers (see [1], [2], [3]and [4]). While the capital market is affected directly or indirectly by numerous factors, making estimation of assets' value is uncertain. ...
... El conjunto de incertidumbre elipsoidal, también conocido como conjunto de incertidumbre cuadrático, permite incluir información del segundo momento de la distribución de los parámetros inciertos Yin et al., 2021). De esta forma, para los retornos se tiene la siguiente configuración del conjunto de tipo elipsoidal: ...
Los modelos de optimización robusta (OR) han permitido superar las limitaciones del modelo media-varianza (MV), que comprende el enfoque tradicional para la selección de portafolios óptimos de inversión, al incorporar la incertidumbre de los parámetros del modelo (retornos esperados y covarianzas). En este trabajo se presentan los desarrollos de la OR en la teoría de portafolio mediante el enfoque del peor de los casos, a partir del cual se incorporan las formulaciones robustas para el modelo MV, teniendo en cuenta los trabajos de Markowitz y Sharpe. A partir de estas formulaciones, se lleva a cabo una sencilla aplicación en la que se resaltan las ventajas y bondades de las contrapartes robustas frente al modelo MV original. Al final, se presenta una breve discusión de formulaciones adicionales en materia de conjuntos de incertidumbre y otras medidas de desempeño.
The majority of standard approaches to financial portfolio optimization (PO) are based on the mean-variance (MV) framework. Given a risk aversion coefficient, the MV procedure yields a single portfolio that represents the optimal trade-off between risk and return. However, the resulting optimal portfolio is known to be highly sensitive to the input parameters, i.e., the estimates of the return covariance matrix and the mean return vector. It has been shown that a more robust and flexible alternative lies in determining the entire region of near-optimal portfolios. In this paper, we present a novel approach for finding a diverse set of such portfolios based on quality-diversity (QD) optimization. More specifically, we employ the CVT-MAP-Elites algorithm, which is scalable to high-dimensional settings with potentially hundreds of behavioral descriptors and/or assets. The results highlight the promising features of QD as a novel tool in PO.
Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order.
The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes.
This paper reviews recent advances in robust portfolio selection problems and their extensions, from both operational research and financial perspectives. A multi-dimensional classification of the models and methods proposed in the literature is presented, based on the types of financial problems, uncertainty sets, robust optimization approaches, and mathematical formulations. Several open questions and potential future research directions are identified.
In this paper we introduce the notion of themes as an additional investment dimension beyond asset classes, regions, sectors and styles, and we propose a framework to allocate to thematic investments at a strategic asset allocation level. The goal of thematic investments is to provide the means to invest in assets that have their returns significantly impacted by the structural changes underlying the theme. Such changes come about through megatrends that shape societies: Demographic shifts, social or attitudinal changes, environmental impact, resource scarcity, economic imbalances, transfer of power, technological advances and regulatory or political changes. Allocating to themes requires discipline because thematic investments are not only exposed to the theme but also to the traditional risk factors. Our approach to allocating to thematic investments uses a framework based on robust portfolio optimisation, which takes into account the expected excess return derived from the exposure to the theme as well as exposures to traditional risk factors. As an illustration, we provide an example where thematic investments in energy transition, environmental sustainability, healthcare innovation, consumer innovation and disruptive tech are added to a traditional multi-asset portfolio.
Robust portfolio optimization refers to finding an asset allocation strategy whose behavior under the worst possible realizations of the uncertain inputs, e.g., returns and covariances, is optimized. The robust approach is in contrast to the classical approach, where one estimates the inputs to a portfolio allocation problem and then treats them as certain and accurate. In this paper we provide a categorized bibliography on the application of robust mathematical programming to the portfolio selection problem. With no similar surveys available, one of the aims of this review is to provide quick access for those interested, but maybe not yet in the area, so they know what the area is about, what has been accomplished and where everything can be found. Toward this end, a total of 148 references have been compiled and classified in various ways. Additionally, the number of Scopus© citations by contribution and journal is recorded. Finally, a brief discussion of the review’s major findings is provided and some solid leads on future directions are given.
The covariation among financial asset returns is often a key ingredient used in the construction of optimal portfolios. Estimating covariances from data, however, is challenging due to the potential influence of estimation error, specially in high-dimensional problems, which can impact negatively the performance of the resulting portfolios. We address this question by putting forward a simple approach to disentangle the role of variance and covariance information in the case of mean-variance efficient portfolios. Specifically, mean-variance portfolios can be represented as a two-fund rule: one fund is a fully invested portfolio that depends on diagonal covariance elements, whereas the other is a long-short, self financed portfolio associated with the presence of non-zero off-diagonal covariance elements. We characterize the contribution of each of these two components to the overall performance in terms of out-of-sample returns, risk, risk-adjusted returns and turnover. Finally, we provide an empirical illustration of the proposed portfolio decomposition using both simulated and real market data.
Robust optimization has become a widely implemented approach in investment management for incorporating uncertainty into financial models. The first applications were to asset allocation and equity portfolio construction. Significant advancements in robust portfolio optimization took place since it gained popularity almost two decades ago for improving classical models on portfolio optimization. Recently, studies applying the worst-case framework to bond portfolio construction, currency hedging, and option pricing have appeared in the practitioner-oriented literature. Our focus in this paper is on recent advancements to categorize robust optimization models into asset allocation at the asset class level and portfolio selection at the individual asset level, and we further separate robust portfolio selection approaches specific to each asset class. This organization provides a clear overview on how robust optimization is extensively implemented in investment management.
The mean-variance optimization (MVO) theory of Markowitz (1952) for portfolio selection is one of the most important methods used in quantitative finance. This portfolio allocation needs two input parameters, the vector of expected returns and the covariance matrix of asset returns. This process leads to estimation errors, which may
have a large impact on portfolio weights. In this paper we review different methods which aim to stabilize the mean-variance allocation. In particular, we consider recent results from machine learning theory to obtain more robust allocation.
Although portfolio management didn’t change much during the 40 years after the seminal works of Markowitz and Sharpe, the development of risk budgeting techniques marked an important milestone in the deepening of the relationship between risk and asset management. Risk parity then became a popular financial model of investment after the global financial crisis in 2008. Today, pension funds and institutional investors are using this approach in the development of smart indexing and the redefinition of long-term investment policies. Introduction to Risk Parity and Budgeting provides an up-to-date treatment of this alternative method to Markowitz optimization. It builds financial exposure to equities and commodities, considers credit risk in the management of bond portfolios, and designs long-term investment policy. This book contains the solutions of tutorial exercices which are included in Introduction to Risk Parity and Budgeting.
The indifference of many investment practitioners to mean-variance optimization technology, despite its theoretical appeal, is understandable in many cases. The major problem with MV optimization is its tendency to maximize the effects of errors in the input assumptions. Unconstrained MV optimization can yield results that are inferior to those of simple equal-weighting schemes.
Nevertheless, MV optimization is superior to many ad hoc techniques in terms of integration of portfolio objectives with client constraints and efficient use of information. Its practical value may be enhanced by the sophisticated adjustment of inputs and the imposition of constraints based on fundamental investment considerations and the importance of priors. The operating principle should be that, to the extent that reliable information is available, it should be included as part of the definition of the optimization procedure.
We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficient algorithms such as polynomial time interior point methods.
The authors explore the negative effect that estimation error has on mean-variance optimal portfolios. It is shown that asset weights in mean-variance optimal portfolios are very sensitive to slight changes in input parameters. This instability is magnified by the presence of constrains that asset managers typically impose on their portfolios. The authors propose to use robust mean variance, a new technique which is based on robust optimisation, a deterministic framework designed to explicitly consider parameter uncertainty in optimisation problems. Alternative uncertainty regions that create a less conservative robust problem are introduced. In fact, the authors' proposed approach does not assume that all estimation errors will negatively affect the portfolios, as is the case in traditional robust optimisation, but rather that there are as many errors with negative consequences as there are errors with positive consequences. The authors demonstrate through extensive computational experiments that portfolios generated with their proposed robust mean variance methodolgy typically outperform traditional mean variance portfolios in a variety of investment scenarios. Additionally, robust mean variance portfolios are usually less sensitive to input parameters.
In portfolio analysis, uncertainty about parameter values leads to suboptimal portfolio choices. The resulting loss in the investor's utility is a function of the particular estimator chosen for expected returns. So, this is a problem of simultaneous estimation of normal means under a well-specified loss function. In this situation, as Stein has shown, the classical sample mean is inadmissible. This paper presents a simple empirical Bayes estimator that should outperform the sample mean in the context of a portfolio. Simulation analysis shows that these Bayes-Stein estimators provide significant gains in portfolio selection problems.
Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed.
We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization
methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are
in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in
optimality.
In this paper we provide a survey of recent contributions to robust portfolio strategies from operations research and finance
to the theory of portfolio selection. Our survey covers results derived not only in terms of the standard mean-variance objective,
but also in terms of two of the most popular risk measures, mean-VaR and mean-CVaR developed recently. In addition, we review
optimal estimation methods and Bayesian robust approaches.
In this paper we show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce "uncertainty structures" for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. Moreover, we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters. Finally, we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence.
A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library. Subject classifications: Programming, stochastic: robust approach for solving LP/MIP with data uncertainties. Area of review: Financial Services. History: Received September 2001; revision received August 2002; accepted December 2002.
We illustrate the correspondence between uncertainty sets in robust optimization and some pop- ular risk measures in flnance, and show how robust optimization can be used to generalize the concepts of these risk measures. We also show that by using properly deflned uncertainty sets in robust optimization models, one can construct coherent risk measures, and address the issue of the computational tractability of the resulting formulations. Our results have implications for e-cient portfolio optimization under difierent measures of risk.
In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance9 203--228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.
In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering. Comment: 50 pages
Robust optimization has been receiving increased attention in the recent few years due to the possibility of considering the problem of estimation error in the portfolio optimization problem. A question addressed so far by very few works is whether this approach is able to outperform traditional portfolio optimization techniques in terms of out-of-sample performance. Moreover, it is important to know whether this approach is able to deliver stable portfolio compositions over time, thus reducing management costs and facilitating practical implementation. We provide empirical evidence by assessing the out-of-sample performance and the stability of optimal portfolio compositions obtained with robust optimization and with traditional optimization techniques. The results indicated that, for simulated data, robust optimization performed better (both in terms of Sharpe ratios and portfolio turnover) than Markowitz's mean-variance portfolios and similarly to minimum-variance portfolios. The results for real market data indicated that the differences in risk-adjusted performance were not statistically different, but the portfolio compositions associated to robust optimization were more stable over time than traditional portfolio selection techniques.
This paper investigates the sensitivity of mean-variance(MV)-efficient portfolios to changes in the means of individual assets.
When only a budget constraint is imposed on the investment problem, the analytical results indicate that an MV-efficient portfolio’s
weights, mean, and variance can be extremely sensitive to changes in asset means. When nonnegativity constraints are also
imposed on the problem, the computational results confirm that a positively weighted MV-efficient portfolio’s weights are
extremely sensitive to changes in asset means, but the portfolio’s returns are not. A surprisingly small increase in the mean
of just one asset drives half the securities from the portfolio. Yet the portfolio’s expected return and standard deviation
are virtually unchanged.
Robust optimization has been receiving increased attention in the recent few years due to the possibility of considering the problem of estimation error in the portfolio optimization problem. A question addressed so far by very few works is whether this approach is able to outperform traditional portfolio optimization techniques in terms of out-of-sample performance. Moreover, it is important to know whether this approach is able to deliver stable portfolio compositions over time, thus reducing management costs and facilitating practical implementation. We provide empirical evidence by assessing the out-of-sample performance and the stability of optimal portfolio compositions obtained with robust optimization and with traditional optimization techniques. The results indicated that, for simulated data, robust optimization performed better (both in terms of Sharpe ratios and portfolio turnover) than Markowitz's mean-variance portfolios and similarly to minimum-variance portfolios. The results for real market data indicated that the differences in risk-adjusted performance were not statistically different, but the portfolio compositions associated to robust optimization were more stable over time than traditional portfolio selection techniques.
Cambridge Core - Mathematical Finance - Optimization Methods in Finance - by Gérard Cornuéjols
The indifference of many investment practitioners to mean-variance optimization technology, despite its theoretical appeal, is understandable in many cases. The major problem with MV optimization is its tendency to maximize the effects of errors in the input assumptions. Unconstrained MV optimization can yield results that are inferior to those of simple equal-weighting schemes. Nevertheless, MV optimization is superior to many ad hoc techniques in terms of integration of portfolio objectives with client constraints and efficient use of information. Its practical value may be enhanced by the sophisticated adjustment of inputs and the imposition of constraints based on fundamental investment considerations and the importance of priors. The operating principle should be that, to the extent that reliable information is available, it should be included as part of the definition of the optimization procedure.
This note corrects an error in the proof of Proposition 2 of “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraint Helps” that appeared in the Journal of Finance, August 2003.
This paper presents theoretical and empirical results concerned with the question of mis-specification in normally distributed portfolio selection problems. These mis-specifications can occur in three areas: the investor’s utility function, and the mean vector and covariance matrix of the return distribution. Our results suggest that only the second type of error will create significant problems in applications.
This comprehensive reference delivers a toolkit for harvesting market rewards from a wide range of investments. Written by a world-renowned industry expert, the reference discusses how to forecast returns under different parameters. Expected returns of major asset classes, investment strategies, and the effects of underlying risk factors such as growth, inflation, liquidity, and different risk perspectives, are also explained. Judging expected returns requires balancing historical returns with both theoretical considerations and current market conditions. Expected Returns provides extensive empirical evidence, surveys of risk-based and behavioral theories, and practical insights.
In asset allocation problem, the distribution of the assets is usually assumed to be known in order to identify the optimal portfolio. In practice, we need to estimate their distribution. The estimations are not necessarily accurate and it is known as the uncertainty problem. Many researches show that most people are uncertainty aversion and this affects their investment strategy. In this article, we consider risk and information uncertainty under a common asset allocation framework. The effects of risk premium and covariance uncertainty are demonstrated by the worst scenario in a set of measures generated by a relative entropy constraint. The nature of the uncertainty and its impacts on the asset allocation are discussed. © 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS).
As quantitative techniques have become commonplace in the investment industry, the mitigation of estimation and model risk in portfolio management has grown in importance. Robust optimization, which incorporates estimation error directly into the portfolio optimization process, is typically used with conventional robust statistical estimation methods. This perspective on the robust optimization approach reviews useful practical extensions and discusses potential applications for robust portfolio optimization.
This paper provides an overview of developments in robust optimization since 2007. It seeks to give a representative picture of the research topics most explored in recent years, highlight common themes in the investigations of independent research teams and highlight the contributions of rising as well as established researchers both to the theory of robust optimization and its practice. With respect to the theory of robust optimization, this paper reviews recent results on the cases without and with recourse, i.e., the static and dynamic settings, as well as the connection with stochastic optimization and risk theory, the concept of distributionally robust optimization, and findings in robust nonlinear optimization. With respect to the practice of robust optimization, we consider a broad spectrum of applications, in particular inventory and logistics, finance, revenue management, but also queueing networks, machine learning, energy systems and the public good. Key developments in the period from 2007 to present include: (i) an extensive body of work on robust decision-making under uncertainty with uncertain distributions, i.e., “robustifying” stochastic optimization, (ii) a greater connection with decision sciences by linking uncertainty sets to risk theory, (iii) further results on nonlinear optimization and sequential decision-making and (iv) besides more work on established families of examples such as robust inventory and revenue management, the addition to the robust optimization literature of new application areas, especially energy systems and the public good.
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.
Estimation error has always been acknowledged as a substantial problem in portfolio construction. Various approaches exist that range from Bayesian methods with a very strong rooting in decision theory to practitioner-based heuristics with no rooting in decision theory at all as portfolio resampling. Robust optimisation is the latest attempt to address estimation error directly in the portfolio construction process. It will be shown that robust optimisation is equivalent to Bayesian shrinkage estimators and offer no marginal value relative to the former. The implied shrinkage that comes with robust optimisation is difficult to control. Consistent with the ad hoc treatment of uncertainty aversion in robust optimisation, it can be seen that out of sample performance largely depends on the appropriate choice of uncertainty aversion, with no guideline on how to calibrate this parameter or how to make it consistent with the more well-known risk aversion.Journal of Asset Management (2007) 7, 374–387. doi:10.1057/palgrave.jam.2250049
This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$ \end{document} where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x. When the res...
In this article we demonstrate that the optimal portfolios generated by the Black-Litterman asset allocation model have a very simple, intuitive property. The unconstrained optimal portfolio in the Black-Litterman model is the scaled market equilibrium portfolio (reflecting the uncertainty in the equilibrium expected returns) plus a weighted sum of portfolios representing the investor's views. The weight on a portfolio representing a view is positive when the view is more bullish than the one implied by the equilibrium and the other views. The weight increases as the investor becomes more bullish on the view, and the magnitude of the weight also increases as the investor becomes more confident about the view.
We considered five risk-based strategies: equally-weighted, equal-risk budget, equal-risk contribution, minimum variance and maximum diversification. All five can be well described by exposure to the market-cap index and to four simple factors: low-beta, small-cap, low-residual volatility and value. This is, in our view, a major contribution to the understanding of such strategies and provides a simple framework to compare them. All except equally-weighted are defensive with lower volatility than the market-cap index. Equally-weighted is exposed to small-cap stocks. Equal-risk budget and equal-risk contribution are exposed to small-cap and to low-beta stocks. These three have a high correlation of excess returns and their portfolio largely overlap. They invest in all stocks available and have both a low turnover and low tracking error relative to market-cap index. Minimum variance and maximum diversification are essentially exposed to low-beta stocks. They are the most defensive, invest in much the same stocks and have high tracking error and turnover.
This article addresses the problem of finding an optimal allocation of funds among different asset classes in a robust manner
when the estimates of the structure of returns are unreliable. Instead of point estimates used in classical mean-variance
optimization, moments of returns are described using uncertainty sets that contain all, or most, of their possible realizations. The approach presented here takes a conservative viewpoint and
identifies asset mixes that have the best worst-case behavior. Techniques for generating uncertainty sets from historical
data are discussed and numerical results that illustrate the stability of robust optimal asset mixes are reported.
Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predicted by random matrix theory (RMT) and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom–up collective effect of the underlying time series rather than a top–down impact of factors. Our results, in excellent agreement with previous results obtained on large financial correlation matrices, show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner.
In the proposed research, our objective is to provide a general framework for identifying portfolios that perform well out-of-sample even in the presence of estimation error. This general framework relies on solving the traditional minimum-variance problem (based on the sample covariance matrix) but subject to the additional constraint that the p-norm of the portfolio-weight vector be smaller than a given threshold. In particular, we consider the 1-norm constraint, which is that the sum of the absolute values of the weights be smaller than a given threshold, and the 2-norm constraint that the sum of the squares of the portfolio weights be smaller than a given threshold. Our contribution will be to show that our unifying theoretical framework nests as special cases the shrinkage approaches of Jagannathan and Ma (2003) and Ledoit and Wolf (2004), and the 1/N portfolio studied in DeMiguel, Garlappi, and Uppal (2007). We also use our general framework to propose several new portfolio strategies. For these new portfolios, we provide a moment-shrinkage interpretation and a Bayesian interpretation where the investor has a prior belief on portfolio weights rather than on moments of asset returns. Finally, we compare empirically (in terms of portfolio variance, Sharpe ratio, and turnover), the out-of-sample performance of the new portfolios we propose to nine strategies in the existing literature across five datasets. Our preliminary results indicate that the norm-constrained portfolios we propose have a lower variance and a higher Sharpe ratio than the portfolio strategies in Jagannathan and Ma (2003) and Ledoit and Wolf (2004), the 1/N portfolio, and also other strategies in the literature such as factor portfolios and the parametric portfolios in Brandt, Santa-Clara, and Valkanov (2005).
We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satised, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Coun- terpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explic- itly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.
We show that results from the theory of random matrices are potentially of great interest when trying to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). Finally, we give a specific example to show how this idea can be successfully implemented for improving risk management.
The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization. It contains estimation error of the kind most likely to perturb a mean-variance optimizer. In its place, we suggest using the matrix obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients towards more central values, thereby systematically reducing estimation error where it matters most. Statistically, the challenge is to know the optimal shrinkage intensity, and we give the formula for that. Without changing any other step in the portfolio optimization process, we show on actual stock market data that shrinkage reduces tracking error relative to a benchmark index, and substantially increases the realized information ratio of the active portfolio manager.