Robust Portfolio Optimization Research Articles

Robust International Portfolio Optimization with Worst-Case Mean-LPM

This paper proposes a robust international portfolio optimization model with the consideration of worst-case lower partial moment (LPM) and worst-case mean return. In our model, we assume that the distributions and the first- and second-order moments of distributions of returns of assets and exchange rates are all ambiguous. The proposed model can be reformulated into an equivalent semidefinite programming (SDP) problem, which is computationally tractable. For investigation of the performance of our model, we also give two benchmark models. The first benchmark model is a scenario-based model which uses historical observations of returns to approximate the future distributions. The second benchmark model only considers the ambiguity of distributions but does not consider the ambiguity of the first- and second-order moments of distributions. We conduct empirical experiments in a rolling forward way to evaluate the out-of-sample performances of our proposed model, the two benchmark models, and an equally weighted model using the return measures and various risk-adjusted return measures. The result shows that our model has the best performance. It verifies that investors can obtain benefits when employing the robust model and considering the ambiguity of the first- and second-order moments of distributions.

Worst-Case Higher Moment Coherent Risk Based on Optimal Transport with Application to Distributionally Robust Portfolio Optimization

The tail risk management is of great significance in the investment process. As an extension of the asymmetric tail risk measure—Conditional Value at Risk (CVaR), higher moment coherent risk (HMCR) is compatible with the higher moment information (skewness and kurtosis) of probability distribution of the asset returns as well as capturing distributional asymmetry. In order to overcome the difficulties arising from the asymmetry and ambiguity of the underlying distribution, we propose the Wasserstein distributionally robust mean-HMCR portfolio optimization model based on the kernel smoothing method and optimal transport, where the ambiguity set is defined as a Wasserstein “ball” around the empirical distribution in the weighted kernel density estimation (KDE) distribution function family. Leveraging Fenchel’s duality theory, we obtain the computationally tractable DCP (difference-of-convex programming) reformulations and show that the ambiguity version preserves the asymmetry of the HMCR measure. Primary empirical test results for portfolio selection demonstrate the efficiency of the proposed model.

Distributionally robust portfolio optimization with linearized STARR performance measure

We study the distributionally robust linearized stable tail adjusted return ratio (DRLSTARR) portfolio optimization problem, in which the objective is to maximize the worst-case linearized stable tail adjusted return ratio (LSTARR) performance measure under data-driven Wasserstein ambiguity. We consider two types of imperfectly known uncertainties, named uncertain probabilities and continuum of realizations, associated with the losses of assets. We account for two typical combinatorial trading constraints, called buy-in threshold and diversification constraints, to reflect stock market restrictions. Leveraging conic duality theory to tackle the distributionally robust worst-case expectation, the proposed problems are reformulated into mixed-integer linear programming problems. We carry out a series of empirical tests to illustrate the scalability and effectiveness of the proposed solution framework, and to evaluate the performance of the DRLSTARR-constructed portfolios. The cross-validation results obtained using a rolling-horizon procedure show the superior out-of-sample performance of the DRLSTARR portfolios under an uncertain continuum of realizations.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

Robust mean-risk portfolio optimization using machine learning-based trade-off parameter

Conservatism is the notorious problem of the worst-case robust portfolio optimization, and this issue has raised broad discussion in academia. To this end, we propose the hybrid robust portfolio models under ellipsoidal uncertainty sets in this paper, where both the best-case and the worst-case counterparts are involved. In the suggested models, we introduce a trade-off parameter to adjust the portfolio optimism level. Machine learning algorithms including Long Short-Term Memory (LSTM) and eXtreme Gradient Boosting (XGBoost) are used to evaluate the potential market movements and provide forecasting information to generate the hyperparameter for modeling. Additionally, we develop a clustering-based algorithm for properly constructing joint ellipsoidal uncertainty sets to reduce conservatism further. In the modeling phase, we design the hybrid portfolios based on variance (HRMV) and value at risk (VaR) and prove the equivalent relationship between the hybrid robust mean-VaR model (HRMVaR) and the hybrid robust mean-CVaR (conditional value at risk) according to the existing research. The US 12 industry portfolio data set retrieved from Kenneth R. French is employed for the in-sample and out-of-sample numerical experiments. The experimental results demonstrate the effectiveness and robustness of the proposed portfolios, where HRMV models have better Sharpe ratios and Calmar ratios than the corresponding nominal mean–variance model, and HRMVaR models outperform the baseline VaR-based portfolios in terms of returns. Sensitivity analysis supports the superiority of the joint ellipsoidal uncertainty set Uδ2, where the proposed portfolios constrained with Uδ2 show stable risk characteristics.

KDE distributionally robust portfolio optimization with higher moment coherent risk

KDE distributionally robust portfolio optimization with higher moment coherent risk

Providing an Optimal Robust Portfolio Model with Mean- CVaR Approach

The portfolio selection problem is one of the main investment management prob-lems. In the portfolio selection problem, robustness is sought against uncertainty or variability in the value of the parameters of the problem. This paper has been conducted for Robust portfolio optimization based on the mean-cvar approach. And introduces the linear mean-cvar model as a criterion for calculating risk and provides an optimal Robust mean-cvar model. Robust approach used in this research is the Bertsimas and Sim. In this approach, Robust counterpart presented for a linear programming model remains linear, maintaining the advantages of the linear programming model in the optimal model. The model developed in this research is randomly selected by real data of 20 stocks of the S&P 500 index for three years, this development help portfolio selection problem to consider uncertainty. Interval optimization is modeling approach to consider parameters uncertainty in this paper. Considering uncertainty make model more realistic. The results of model show that this approach has computational efficiency and on the other hand proposed model produce better solution in risk and portfolio rate of return point of view

Bi-level optimization approach for robust mean-variance problems

Portfolio Optimization is based on the efficient allocation of several assets, which can get heavily affected by the uncertainty in input parameters. So we must look for such solutions which can give us steady results in uncertain conditions too. Recently, the uncertainty based optimization problems are being dealt with robust optimization approach. With this development, the interest of researchers has been shifted toward the robust portfolio optimization. In this paper, we study the robust counterparts of the uncertain mean-variance problems under box and ellipsoidal uncertainties. We convert those uncertain problems into bi-level optimization models and then derive their robust counterparts. We also solve a problem using this methodology and compared the optimal results of box and ellipsoidal uncertainty models with the nominal model.

Distributionally robust end-to-end portfolio construction

We propose an end-to-end distributionally robust system for portfolio construction that integrates the asset return prediction model with a distributionally robust portfolio optimization model. We also show how to learn the risk-tolerance parameter and the degree of robustness directly from data. End-to-end systems have an advantage in that information can be communicated between the prediction and decision layers during training, allowing the parameters to be trained for the final task rather than solely for predictive performance. However, existing end-to-end systems are not able to quantify and correct for the impact of model risk on the decision layer. Our proposed distributionally robust end-to-end portfolio selection system explicitly accounts for the impact of model risk. The decision layer chooses portfolios by solving a minimax problem where the distribution of the asset returns is assumed to belong to an ambiguity set centered around a nominal distribution. Using convex duality, we recast the minimax problem in a form that allows for efficient training of the end-to-end system.

COHERENT RISK MEASURES AND NORMAL MIXTURE DISTRIBUTIONS WITH APPLICATIONS IN PORTFOLIO OPTIMIZATION

This paper investigates the coherent risk measure of a class of normal mixture distributions which are widely-used in finance. The main result shows that the mean-risk portfolio optimization problem with these normal mixture distributions can be reduced to a quadratic programming problem which has closed form of solution by fixing the location parameter and skewness parameter. In addition, we show that the efficient frontier of the portfolio optimization problem can be extended to three dimensions in this case. The worst-case value-at-risk in the robust portfolio optimization can also be calculated directly. Finally, the conditional value-at-risk (CVaR) is considered as an example of coherent risk measure. We obtain the marginal contribution to risk for a portfolio based on the normal mixture model.

Robust conditional expectation reward–risk performance measures

In this paper, we develop robust portfolio optimization models for conditional expectation type reward–risk performance measures that significantly improve upon conventional portfolio selection techniques. In particular, we directly address estimation error in the portfolio optimization process by adopting a robust optimization method that is typically used with conventional robust statistical estimation techniques. Alongside this robust optimization, we propose the use of an early-warning system based on moving averages to predict market crises. Empirical analyses based on the US stock market validate the proposed robust approaches and highlight the implications of financial crises for portfolio selection problems. The results confirm that the proposed robust portfolio optimization models substantially improve upon their conventional counterparts for out-of-sample portfolios, providing valuable managerial insights.

Sparse portfolio selection with uncertain probability distribution

Designed as remedies for uncertain parameters and tiny optimal weights in the portfolio selection problem, we consider a class of distributionally robust portfolio optimization models with cardinality constraints. For considering the statistical significance and tractability, we construct two kinds of ambiguity sets based on L1-norm and moment information, respectively. The nominal distribution, as the core of the first ambiguity set, is determined by non-parametric estimation method. To reduce the disturbing error of the second ambiguity set, we apply a shrinkage estimation method to determine the moment information based on historical data. By introducing a binary variable, the proposed sparse portfolio optimization model can be converted equivalently to a tractable mixed-integer 0-1 programming problem, which can be dealt with efficiently by a modified primal-dual Benders’ decomposition method. Through the actual market data, we test the proposed models and show their validity. Furthermore, performances measured by net portfolio return, Sharpe ratio, and cumulative return are superior to the classical portfolio selection models in the back-testing of out-of-sample data.

A study of robust portfolio optimization with European options using polyhedral uncertainty sets

We consider the problem of maximizing the worst-case return of a portfolio when the manager can invest in stocks as well as European options on those stocks, and the stock returns are modeled using an uncertainty set approach. Specifically, the manager knows a range forecast for each factor driving the returns and a budget of uncertainty limiting the scaled deviations of these factors from their nominal values. Our goal is to understand the impact of options on the optimal portfolio allocation. We present theoretical results regarding the structure of that optimal allocation, in particular with respect to portfolio diversification. Specifically, we show that the presence of options only leads to limited diversification across the financial instruments available. We compare our robust portfolio to several benchmarks in numerical experiments and analyze how the optimal allocation varies with the budget of uncertainty. Our results indicate that our approach performs very well in practice.

A New Data-Driven Distributionally Robust Portfolio Optimization Method Based on Wasserstein Ambiguity Set

Since optimal portfolio strategy depends heavily on the distribution of uncertain returns, this article proposes a new method for the portfolio optimization problem with respect to distribution uncertainty. When the distributional information of the uncertain return rate is only observable through a finite sample dataset, we model the portfolio selection problem with a robust optimization method from the data-driven perspective. We first develop an ambiguous mean-CVaR portfolio optimization model, where the ambiguous distribution set employed in the distributionally robust model is a Wasserstein ball centered within the empirical distribution. In addition, the computationally tractable equivalent model of the worst-case expectation under the uncertainty set of a cone is derived, and some theoretical conclusions of the box, budget and ellipsoid uncertainty set are obtained. Finally, to demonstrate the effectiveness of our mean-CVaR portfolio optimization method, two practical examples concerning the Chinese stock market and United States stock market are considered. Furthermore, some numerical experiments are carried out under different uncertainty sets. The proposed data-driven distributionally robust portfolio optimization method offers some advantages over the ambiguity-free stochastic optimization method. The numerical experiments illustrate that the new method is effective.

Tailor-Made Asset Allocation: A Robust Framework to Implement Active Views

Asset managers publish tactical asset allocation views regularly. The implementation of such (usually qualitative) views, in portfolios is often over-simplistic. We propose a robust framework to industrialize the construction of tailored portfolios consistent with the views. First, an unconstrained unique tactical portfolio is created by relating the conviction in each view to the allocation of risk budget to the assets underlying the view. Second, the tailored portfolios with investor-specific constraints and targets are constructed using robust portfolio optimization based on implied active returns derived from the unique unconstrained tactical portfolio. The implied returns are derived from reverse optimization using the same robust approach. Robust optimization is the core engine for the industrialization process. It produces portfolios consistent with the views while complying with constraints without requiring human intervention. Finally, a factor-based risk model endows the framework with transparency, by allowing for comparison of risk-factor exposures in portfolios with those in the original views’ exposures.

A novel two-phase robust portfolio selection and optimization approach under uncertainty: A case study of Tehran stock exchange.

Portfolio construction is one of the most critical problems in financial markets. In this paper, a new two-phase robust portfolio selection and optimization approach is proposed to deal with the uncertainty of the data, increasing the robustness of investment process against uncertainty, decreasing computational complexity, and comprehensive assessments of stocks from different financial aspects and criteria are provided. In the first phase of this approach, all candidate stocks’ efficiency is measured using a robust data envelopment analysis (RDEA) method. Then in the second phase, by applying robust mean-semi variance-liquidity (RMSVL) and robust mean-absolute deviation-liquidity (RMADL) models, the amount of investment in each qualified stock is determined. Finally, the proposed approach is implemented in a real case study of the Tehran stock exchange (TSE). Additionally, a sensitivity analysis of all robust models of this study is examined. Illustrative results show that the proposed approach is effective for portfolio selection and optimization in the presence of uncertain data.

Robust Omega ratio optimization using regular vines

We study the robust portfolio optimization model for the Omega ratio when the joint ambiguity in the returns distributions is modeled utilizing copulas. We propose the copula formulation of the Omega ratio and use it to formulate the worst-case robust optimization model. Furthermore, we propose a Markov chain predicated filtering strategy to filter a set of fewer assets from a large pool of available assets in the market to amend the performance of the conventional Omega ratio model. We propose an R-vine copula Autoregressive Moving Average Generalized Autoregressive Conditional Heteroskedasticity (ARMA-GARCH) model for the joint distribution of assets returns. We obtain the standardized residuals for each return series of the filtered assets using the ARMA-GARCH model and, subsequently, exploit the regular vine copulas to model the joint dependence among the transformed residuals. The tree structure in the regular vines is accomplished using Kendall’s tau. The dependence structure so obtained is used to simulate scenarios for the returns of assets. The simulated data is used to obtain optimal portfolios in the worst-case Copula Omega ratio model. Empirical evidence shows the aptness of the proposed filtering strategy and analyzes the performance of the worst-case copula Omega portfolios on several datasets using a rolling window approach. The optimal portfolios from the worst-case copula Omega model are found to outperform the portfolios from the Gaussian copula Omega ratio model by having a higher information ratio, Value-at-risk ratio, and Rachev ratio.

Worst-Case Expected Shortfall with Univariate and Bivariate Marginals

Computing and minimizing the worst-case bound on the expected shortfall risk of a portfolio given partial information on the distribution of the asset returns is an important problem in risk management. One such bound that been proposed is for the worst-case distribution that is “close” to a reference distribution where closeness in distance among distributions is measured using [Formula: see text]-divergence. In this paper, we advocate the use of such ambiguity sets with a tree structure on the univariate and bivariate marginal distributions. Such an approach has attractive modeling and computational properties. From a modeling perspective, this provides flexibility for risk management applications where there are many more choices for bivariate copulas in comparison with multivariate copulas. Bivariate copulas form the basis of the nested tree structure that is found in vine copulas. Because estimating a vine copula is fairly challenging, our approach provides robust bounds that are valid for the tree structure that is obtained by truncating the vine copula at the top level. The model also provides flexibility in tackling instances when the lower dimensional marginal information is inconsistent that might arise when multiple experts provide information. From a computational perspective, under the assumption of a tree structure on the bivariate marginals, we show that the worst-case expected shortfall is computable in polynomial time in the input size when the distributions are discrete. The corresponding distributionally robust portfolio optimization problem is also solvable in polynomial time. In contrast, under the assumption of independence, the expected shortfall is shown to be #P-hard to compute for discrete distributions. We provide numerical examples with simulated and real data to illustrate the quality of the worst-case bounds in risk management and portfolio optimization and compare it with alternate probabilistic models such as vine copulas and Markov tree distributions.

Robust Portfolio Optimization with Multi-Factor Stochastic Volatility

This paper studies a robust portfolio optimization problem under a multi-factor volatility model. We derive optimal strategies analytically under the worst-case scenario with or without derivative trading in complete and incomplete markets and for assets with jump risk. We extend our study to the case with correlated volatility factors and propose an analytical approximation for the robust optimal strategy. To illustrate the effects of ambiguity, we compare our optimal robust strategy with the strategies that ignore the information of uncertainty, and provide the welfare analysis. We also discuss how derivative trading affects the optimal strategies. Finally, numerical experiments are provided to demonstrate the behavior of the optimal strategy and the utility loss.

Robust portfolio optimization: a categorized bibliographic review

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.