Robust Portfolio Selection Problem Research Articles

Weight bound constraints in mean-variance models: a robust control theory foundation via machine learning

Using an innovative representation of the weight bound constrained Markowitz's (Portfolio selection. J. Finance, 1952, 7, 77–91) mean-variance model, developed using the support vector data description, a machine learning algorithm introduced by Tax and Duin (Support vector data description. Mach. Learn., 2004, 54, 45–66), we provide an innovative interpretation of the robustness of these bound constraints in terms of robust control theory in the sense of Hansen and Sargent (Robust control and model uncertainty. Am. Econ. Rev., 2001, 91, 60–66). Building on these insights, firstly, we detail the method for quantifying the degree of misspecification in Markowitz's (1952) mean-variance model using its counterpart with weight upper bounds. Additionally, we show that this degree of misspecification is a decreasing piecewise linear function of the bound. Secondly, we empirically investigate two simulation-based methods, inspired by Michaud's (The Markowitz optimization enigma: Is ‘optimized’ optimal? Financ. Anal. J., 1989, 45, 31–42) resampling technique, for choosing the bound. Thirdly, we compare the robustness of the weight upper bound constrained mean-variance model with that of Goldfarb and Iyengar's (Robust portfolio selection problems. Math. Oper. Res., 2003, 28, 1–38) robust maximum return model.

Robust portfolio choice with limited attention

<abstract><p>This paper investigates a robust portfolio selection problem with the agent's limited attention. The agent has access to a risk-free asset and a stock in a financial market. But she does not observe perfectly the expected return rate of the stock so she has to estimate this key parameter before making decisions. Besides the general observable financial information, the agent can also acquire a news signal process whose accuracy depends on the agent's attention. We assume that the agent pays limited attention on the signal and she does not trust her estimation model. So it is necessary to consider model ambiguity in this paper as well. The agent maximizes the expected utility of her terminal wealth under the worst-case scenario. Under this setting, we derive the robust optimal strategy explicitly. In the presence of the attention and ambiguity aversion, the myopic term of the strategy, the hedging term of the strategy and the worst-case scenario are all changed. We find that more attention makes the variance of the estimated return smaller. The numerical examples also show that a more attentive agent has a better estimation of the unobservable parameter and is more confident on her estimation. Consequently, the worst-case scenario deviates less from the reference model, which implies a higher expected return rate under the worst-case scenario, thus invests more in the stock.</p></abstract>

Tail Value-at-Risk-Based Expectiles for Extreme Risks and Their Application in Distributionally Robust Portfolio Selections

Empirical evidence suggests that financial risk has a heavy-tailed profile. Motivated by recent advances in the generalized quantile risk measure, we propose the tail value-at-risk (TVaR)-based expectile, which can capture the tail risk compared with the classic expectile. In addition to showing that the risk measure is well-defined, the properties of TVaR-based expectiles as risk measures were also studied. In particular, we give the equivalent characterization of the coherency. For extreme risks, usually modeled by a regularly varying survival function, the asymptotic expansion of a TVaR-based expectile (with respect to quantiles) was studied. In addition, motivated by recent advances in distributionally robust optimization in portfolio selections, we give the closed-form of the worst-case TVaR-based expectile based on moment information. Based on this closed form of the worst-case TVaR-based expectile, the distributionally robust portfolio selection problem is reduced to a convex quadratic program. Numerical results are also presented to illustrate the performance of the new risk measure compared with classic risk measures, such as tail value-at-risk-based expectiles.

A scenario-based mathematical approach to a robust project portfolio selection problem under fuzzy uncertainty

As a major concern of chief managers in each organization, project portfolio selection has a special place in their responsibilities. To assist managers in making decisions, applicable optimization models play an essential role in such processes. In this regard, this paper provides a stochastic optimization model for a project portfolio selection problem under different scenarios. Providing the novelty in the model along with making it closer to reality, the interdependency between revenue and cost of projects is considered. Due to the inherent uncertainty of parameters, the revenue and cost of each project, as well as contributed capital, follow triangular fuzzy parameters. Contrary to the previous model, the appreciation of assets is considered in the proposed model as the other novelty of the proposed model. To tackle the uncertainty of parameters, a robust possibilistic approach is used, which has been first-ever devised in such problems. Being both optimistic and pessimistic approaches available for decision-makers, a new measure is introduced to make the model inclusive. Moreover, by considering the confidence level as both parameter and decision variables, the robust possibilistic programming approach is adopted to solve the proposed model. Using the new proposed measure, the optimal average value of robust model are obtained under different confidence level. Finally, solving the optimization model, the results are provided by implementing the realization for uncertain parameters, and regarding the obtained results, discussions are made to provide some insights to the managers.

Robust portfolio selection problems: a comprehensive review

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.

An Extended McKean — Vlasov Dynamic Programming Approach to Robust Equilibrium Controls under Ambiguous Covariance Matrix

This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time-inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman's principle of optimality. Moreover, we model the state by a McKean — Vlasov dynamics under a set of non-dominated probability measures induced by the ambiguous covariance matrix of the noises. We apply a game-theoretic concept of subgame perfect Nash equilibrium to develop a robust equilibrium control approach, which can yield robust time-consistent decisions. We characterize the robust equilibrium control and equilibrium value function by an extended optimality principle and then we further deduce a system of Bellman — Isaacs equations to determine the equilibrium solution on the Wasserstein space of probability measures. The proposed analytical framework is illustrated with its applications to robust continuous-time mean-variance portfolio selection problems with risk aversion coefficient being constant or state-dependent, under the ambiguity stemming from ambiguous volatilities of multiple assets or ambiguous correlation between two risky assets. The explicit equilibrium portfolio solutions are represented in terms of the probability law.

An application of sparse-group lasso regularization to equity portfolio optimization and sector selection

In this paper, we propose a modified mean-variance portfolio selection model that incorporates the sparse-group lasso (abbreviated as SGLasso) regularization in machine learning. This new model essentially has three merits: first, it allows investors to incorporate their preference over equity sectors when constructing portfolios; second, it helps investors select sectors based on assets’ past performances as it encourages sparsity among sectors; third, it has stabilizing and sparsifying effect on the entire portfolio. We connect our model to a robust portfolio selection problem, and investigate effects of the SGLasso regularization on the optimal strategy both theoretically and empirically. We develop an efficient algorithm to find the optimal portfolio and prove its global convergence. We demonstrate the efficiency of the algorithm through simulated experiments under large datasets and evaluate the out-of-sample performance of our model via empirical tests across different datasets.

Time Consistent Multi-period Worst-Case Risk Measure in Robust Portfolio Selection

In this paper, we first construct a time consistent multi-period worst-case risk measure, which measures the dynamic investment risk period-wise from a distributionally robust perspective. Under the usually adopted uncertainty set, we derive the explicit optimal investment strategy for the multi-period robust portfolio selection problem under the multi-period worst-case risk measure. Empirical results demonstrate that the portfolio selection model under the proposed risk measure is a good complement to existing multi-period robust portfolio selection models using the adjustable robust approach.

Time consistent multi-period robust risk measures and portfolio selection models with regime-switching

To better describe the time-varying property of the dynamic investment risk and the ambiguity of the random return process, we propose two multi-period robust risk measures under the regime switching framework. Using regime-dependent dynamic uncertainty sets, we show that the multi-period robust portfolio selection problems under the two multi-period robust risk measures with regime switching can be transformed into second order cone programs, which can thus be efficiently solved in polynomial time. To show the generality of the dynamic uncertainty sets under the regime switching framework, we further consider multi-period robust risk measures under time-varying uncertainty sets with moments uncertainty and discuss the tractability of the corresponding multi-period robust portfolio selection problems. A series of empirical results demonstrate that the robust portfolio selection models with regime switching can flexibly help the investor make superior and robust investment strategies according to the switching of the market environment.

Closed-Form Optimal Portfolios of Distributionally Robust Mean-CVaR Problems with Unknown Mean and Variance

In this paper, we consider both one-period and multi-period distributionally robust mean-CVaR portfolio selection problems. We adopt an uncertainty set which considers the uncertainties in terms of both the distribution and the first two order moments. We use the parametric method and the dynamic programming technique to come up with the closed-form optimal solutions for both the one-period and the multi-period robust portfolio selection problems. Finally, we show that our approaches are efficient when compared with both normal based portfolio selection models, and robust approaches based on known moments.

Robust Multi-Period Portfolio Selection with Asymmetrically Distributed Uncertainty Set and Return Predictability

This paper considers a robust multi-period portfolio selection problem with asymmetrically distributed uncertainty set. We introduce the mean-LPM (lower partial moment) risk measure and establish a multi-period portfolio selection model, in which the mean-LPM is used to limit the loss of portfolio. To add practicality, we allow the uncertainty set to be asymmetric and include transaction cost in the model. A computationally tractable approximation approach based on second order cone optimization is used for solving the proposed model. Comprehensive numerical comparisons with simulated data and real market data are reported. The numerical results indicate that the proposed model can obtain better expected returns and Sharpe ratios, while reduce the standard deviation and turnover ratios, compared with some well-known models in the literature.

Robust portfolio selection under norm uncertainty

In this paper, we consider the robust portfolio selection problem which has a data uncertainty described by the $(p,w)$ -norm in the objective function. We show that the robust formulation of this problem is equivalent to a linear optimization problem. Moreover, we present some numerical results concerning our robust portfolio selection problem.

Robust portfolio selection problem under temperature uncertainty

In this paper, we consider a portfolio selection problem under temperature uncertainty. Weather derivatives based on different temperature indices are used to protect against undesirable temperature events. We introduce stochastic and robust portfolio optimization models using weather derivatives. The investors’ different risk preferences are incorporated into the portfolio allocation problem. The robust investment decisions are derived in view of discrete and continuous sets that the underlying uncertain data in temperature model belong. We illustrate main features of the robust approach and performance of the portfolio optimization models using real market data. In particular, we analyze impact of various model parameters on different robust investment decisions.

A robust set-valued scenario approach for handling modeling risk in portfolio optimization

For portfolio optimization under downside risk measures, such as conditional value-at-risk or lower partial moments, we often invoke a scenario approach to approximate the high-dimensional integral involved when calculating risk. Consequently, two types of modeling risk may arise from this procedure: uncertainty in determining the distribution of asset returns and the error caused by approximating a given distribution with scenarios. To handle these two types of modeling risk within a unified framework, we propose a mathematically tractable set-valued scenario approach. More specifically, when short selling is not permitted, the robust portfolio selection problems modeled within a minimum-maximum decision framework using several types of set-valued scenarios can be translated into linear programs or second-order cone programs. These can be efficiently solved by the interior point method. The proposed set-valued scenario approach can be used not only as a methodology to alleviate the modeling risk but also as a useful tool for evaluating the impact of modeling risk. Our simulation analysis and empirical study show that robustness does not necessarily imply conservativeness, portfolio performance is affected by the investment style characterized by the return-risk tradeoff to a large degree and modeling risk only becomes significant when an aggressive strategy is adopted.

Portfolio management with robustness in both prediction and decision: A mixture model based learning approach

We develop in this paper a novel portfolio selection framework with a feature of double robustness in both return distribution modeling and portfolio optimization. While predicting the future return distributions always represents the most compelling challenge in investment, any underlying distribution can be always well approximated by utilizing a mixture distribution, if we are able to ensure that the component list of a mixture distribution includes all possible distributions corresponding to the scenario analysis of potential market modes. Adopting a mixture distribution enables us to (1) reduce the problem of distribution prediction to a parameter estimation problem in which the mixture weights of a mixture distribution are estimated under a Bayesian learning scheme and the corresponding credible regions of the mixture weights are obtained as well and (2) harmonize information from different channels, such as historical data, market implied information and investors׳ subjective views. We further formulate a robust mean-CVaR portfolio selection problem to deal with the inherent uncertainty in predicting the future return distributions. By employing the duality theory, we show that the robust portfolio selection problem via learning with a mixture model can be reformulated as a linear program or a second-order cone program, which can be effectively solved in polynomial time. We present the results of simulation analyses and primary empirical tests to illustrate a significance of the proposed approach and demonstrate its pros and cons.

A Closed-Form Solution for Robust Portfolio Selection with Worst-Case CVaR Risk Measure

With the uncertainty probability distribution, we establish the worst-case CVaR (WCCVaR) risk measure and discuss a robust portfolio selection problem with WCCVaR constraint. The explicit solution, instead of numerical solution, is found and two-fund separation is proved. The comparison of efficient frontier with mean-variance model is discussed and finally we give numerical comparison with VaR model and equally weighted strategy. The numerical findings indicate that the proposed WCCVaR model has relatively smaller risk and greater return and relatively higher accumulative wealth than VaR model and equally weighted strategy.

Portfolio Management with Dual Robustness in Prediction and Optimization: A Mixture Model Based Learning Approach

We develop in this paper a novel portfolio selection framework with a feature of dual robustness in both return distribution modeling and portfolio optimization. While predicting the return distributions of the future market always represents the most compelling challenge in investment, any underlying distribution can be always well approximated by using a mixture distribution, if we are able to ensure that the component list of a mixture distribution includes all distributions corresponding to scenario analysis of the potential market modes. Adopting a mixture distribution enables us not only to reduce the prediction problem for distributions to a parameter estimation problem of specifying the mixture weights of the component distributions using a Bayesian learning scheme and estimating the corresponding credible regions of the estimations, but also to harmonize information from different channels, such as historical data, market implied information and the investor subjective views. We establish further a robust mean-CVaR portfolio selection problem formulation to deal with the inherent probability uncertainty. By using duality theory, we show that the robust portfolio selection problem via learning with a mixture model can be reformulated as linear program or second-order cone program, which can be effectively solved in polynomial time. We present simulation analysis and some primary empirical results to illustrate the significance of the proposed approach and demonstrate the pros and cons of the method.

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

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

Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

In typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called “ marginal+joint” ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under a “marginal + joint” ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of the Chinese stock market and simulated options confirm the property of both the models. Test results show that (1) under the “ marginal+joint” uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar’s model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs.

Robust Portfolio Selection Involving Options Under a 'Marginal Joint' Ellipsoidal Uncertainty Set

In typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called marginal ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under marginal ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of China stock market and simulated options confirm the property of both the models. Test results show that (1) under the marginal uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar's model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs.