Robust Portfolio Selection Model Research Articles

Dynamic robust portfolio selection under market distress

This article proposes a dynamic robust portfolio selection model that is based on minimizing portfolio’s worst case scenarios using the Conditional Value at Risk as relevant risk measure. Our proposed empirical model for the dynamics of portfolio constituents has three main features: i) accommodates tail dependence between assets by means of a mixture of copula functions; ii) conditional heteroscedasticity and leverage effects are considered through the implementation of a GJR-GARCH model; and iii) extreme events are taken into account by considering parametric and semiparametric hybrid models for the marginal distribution of asset returns. We illustrate the performance of this portfolio before and during the COVID-19 pandemic using statistical measures such as the Sharpe ratio, cumulative returns, and volatility. The results show the outperformance of our WCVaR portfolio during the turmoil period against benchmark portfolios commonly used by practitioners. The method also exhibits good performance during calm periods.

Sparse and robust portfolio selection via semi-definite relaxation

In investment management, especially for automated investment services, it is critical for portfolios to have a manageable number of assets and robust performance. First, portfolios should not contain too many assets in order to reduce the management fees, transaction costs, and taxes. Second, portfolios should be robust as investment environments change rapidly. In this study, therefore, we propose two convex portfolio selection models that provide portfolios that are sparse and robust. We first perform semi-definite relaxation to develop a sparse mean-variance portfolio selection model, and further extend the model by using -norm regularization and worst-case optimization to formulate two sparse and robust portfolio selection models. Empirical analyses with historical stock returns demonstrate the effectiveness of the proposed models in forming sparse and robust portfolios.

Robust Portfolio Selection Based on Copula Change Analysis

ABSTRACT In this article, we construct a robust portfolio selection model based on dynamic copulas. We first use a type of dynamic copula, which contains copulas with time-varying parameters or sequence of copulas, to characterize the dynamic dependence between financial assets. Then, we use it for portfolio selection based on worst-case Conditional Value-at-Risk (WCVaR). In the empirical part we choose four representative assets from Chinese market to construct a macro asset allocation of portfolio and make the performance analysis. Results show that our method performs the best in out-of-sample tests when considering the dynamic dependence between assets and the uncertainty in the estimated model.

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.

Multi-period Robust Portfolio Selection Model with Bat Algorithm

Multi-period Robust Portfolio Selection Model with Bat Algorithm

Robust mean variance portfolio selection model in the jump-diffusion financial market with an intractable claim

In this paper, a continuous-time robust mean variance model in the jump-diffusion financial market with an intractable claim is considered, in which the price processes of the assets not only are driven by the Brownian motion, but also have the Poisson jumps. By combining the martingale representation theorem and the quantile formulation method, an explicit closed-form solution of the robust mean-variance portfolio selection model is given under some suitable assumptions.

Investor-friendly and robust portfolio selection model integrating forecasts for financial tendency and risk-averse

This paper proposes an investor-friendly random fuzzy portfolio selection model considering both robustness and adjustment of future asset returns derived from investor’s forecasts for financial tendency using a fuzzy inference method. It is important to predict the price or the return of each asset appropriately considering current market trends in portfolio optimization. In this paper, a standard multi-factor model, namely Arbitrage Pricing Theory (APT), is introduced as an asset pricing model. In addition, in order to extend standard APT by integrating important rules of current markets trends derived from technical analysis and fundamental analysis, each factor of APT is assumed to be a random fuzzy variable whose mean is adjusted by the fuzzy reasoning method, particularly product–sum-gravity method. Furthermore, it is also important for the investor to reduce the worst case of the total loss in terms of risk-averse. Therefore, worst-case conditional Value-at-Risk, which is a robust programming approach without assuming some specific random distribution, is considered. Since the proposed model is formulated as a biobjective programming problem both minimizing the value of worst-case conditional Value-at-Risk and maximizing the total expected return, it is equivalently transformed into the deterministic nonlinear programming problem using the satisficing trade-off method, and the efficient algorithm to obtain the optimal portfolio is developed. By solving our proposed model, the investor can obtain the risk-averse optimal portfolio with the large total return complying with current market trends.

Regime-dependent robust portfolio selection model

A multi-period mean-variance model is developed where the model parameters change according to a market with Markov random regime switching. Robust optimization is used to solve the problem which, in turn, gives the explicit expression of the efficient frontier of the mean-variance formulation. Furthermore, a numerical example based on real data from the Chinese stock market is provided to demonstrate the application of this paper.

Robust Portfolio Optimization with Transaction Cost for Risky and Risk-Free Assets Coexisting

A robust mean-variance portfolio selection model with transaction cost is presented for the case that both risky and risk-free assets exist in the market and expected returns of assets are uncertain and belong to a convex polyhedron. The model helps investors to identify such portfolios that expectations of investors are ensured even if the worst case in the expected returns of assets occurs. Analytical expression of the optimal portfolio determined by the proposed model is derived based on the Lagrange method for constrained optimization. Empirical analysis with three real stocks is performed to give the efficient frontier of portfolios.

A Numerical Study for Robust Active Portfolio Management with Worst-Case Downside Risk Measure

Recently, active portfolio management problems are paid close attention by many researchers due to the explosion of fund industries. We consider a numerical study of a robust active portfolio selection model with downside risk and multiple weights constraints in this paper. We compare the numerical performance of solutions with the classical mean-variance tracking error model and the naive1/Nportfolio strategy by real market data from China market and other markets. We find from the numerical results that the tested active models are more attractive and robust than the compared models.

Regime-dependent Robust Risk Measures with Application in Portfolio Selection

Current robust risk measures or portfolio selection models are usually derived under the worst-case analysis, which makes the investment decision too conservative and could not reflect the change of uncertainty sets with respect to different market environ- ments. We use the regime switching technique to describe the time-varying uncertainty set of the first and second order moments, and propose two kinds of robust risk measures: worst regime risk measure and mixed worst-case risk measure. These new risk measures have good properties and the robust portfolio selection models derived from them can be efficiently solved in polynomial time. Empirical results show the reasonability and efficiency of our new models.

Optimization of Tracking Error for Robust Portfolio of Risk Assets with Transaction Cost

Based on the optimization of robust portfolio with tracking error, a robust mean-variance portfolio selection model of tracking error with transaction cost is presented for the case that only risky assets exist and expected returns of assets are uncertain and belong to a convex polyhedron. This paper aims to solve the problem of the portfolio with the selection of the ratio on the condition of maximumimum fluctuation of the tracking error, making the expectation of the return to be the maximumimum. It also makes the portfolio’s practical choice by the function of the linear transaction cost as the same time of construction and application of the model. Empirical analysis with five real stocks is performed by the method of LMI (Linear Matrix Inequality) to show the efficiency of the model.

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.

Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection

In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed, the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio that maximizes (respectively, minimizes) the expected utility (respectively, disutility) values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations. First, we show that the single-stage portfolio selection problem within this framework, whenever the disutility function is in the form of lower partial moments (LPM), or conditional value-at-risk (CVaR), or value-at-risk (VaR), can be solved analytically. The results lead to the solutions for single-stage robust portfolio selection models. Furthermore, the results also lead to a multistage adjustable robust optimization (ARO) solution when the disutility function is the second-order LPM. Exploring beyond the confines of convex optimization, we also consider the so-called S-shaped value function, which plays a key role in the prospect theory of Kahneman and Tversky. The nonrobust version of the problem is shown to be NP-hard in general. However, we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem. In this particular case, the consideration of the robustness actually helps to reduce the computational complexity. Finally, we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold. That information helps to further reduce the ambiguity in the robust model. We show that the robust optimization problem in that case can be solved by means of semidefinite programming (SDP), if no more than two additional chance inequalities are to be incorporated.

Robust portfolio selection based on a joint ellipsoidal uncertainty set

‘Separable’ uncertainty sets have been widely used in robust portfolio selection models (e.g. see [E. Erdoğan, D. Goldfarb, and G. Iyengar, Robust portfolio management, manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004; D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38; R.H. Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]). For these uncertainty sets, each type of uncertain parameter (e.g. mean and covariance) has its own uncertainty set. As addressed in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-z], these ‘separable’ uncertainty sets typically share two common properties: (1) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set, is unknown, and it can be much higher than the desired one; and (2) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified, as observed in the computational experiments conducted in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z; R.H.Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]. To combat these drawbacks, we consider a factor model for random asset returns. For this model, we introduce a ‘joint’ ellipsoidal uncertainty set for the model parameters and show that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. We further show that the robust maximum risk-adjusted return (RMRAR) problem with this uncertainty set can be reformulated and solved as a cone programming problem. The computational results reported in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z] demonstrate that the robust portfolio determined by the RMRAR model with our ‘joint’ uncertainty set outperforms that with Goldfarb and Iyengar’s ‘separable’ uncertainty set proposed in the seminal paper [D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38] in terms of wealth growth rate and transaction cost; moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Robust portfolio selection for index tracking

We develop a robust portfolio selection model for tracking a market index using a subset of its assets. The model is a 0–1 integer program that seeks to maximize similarity between selected assets and the assets of the target index. We allow uncertainty in the objective function by using a computationally tractable robust framework that can control the conservativeness of the solution. This protects against worst-case realizations of potential estimation errors and other deviations. Out-of-sample experiments using the S&P 100 demonstrate the advantages of the robust model. Compared to portfolios constructed with the nominal model, moderately conservative robust portfolios are shown to have lower tracking error and risk profiles that are more similar to the target index.

A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set

The “separable” uncertainty sets have been widely used in robust portfolio selection models [e.g., see Erdogan et al. (Robust portfolio management. manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004), Goldfarb and Iyengar (Math Oper Res 28:1–38, 2003), Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004)]. For these uncertainty sets, each type of uncertain parameters (e.g., mean and covariance) has its own uncertainty set. As addressed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008), these “separable” uncertainty sets typically share two common properties: (i) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set is unknown, and it can be much higher than the desired one; and (ii) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified as observed in the computational experiments conducted in this paper and Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004). To combat these drawbacks, the author of this paper introduced a “joint” ellipsoidal uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and showed that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. For this uncertainty set, we showed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) that the corresponding robust maximum risk-adjusted return (RMRAR) model can be reformulated and solved as a cone programming problem. In this paper, we conduct computational experiments to compare the performance of the robust portfolios determined by the RMRAR models with our “joint” uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and Goldfarb and Iyengar’s “separable” uncertainty set proposed in the seminal paper (Goldfarb and Iyengar in Math Oper Res 28:1–38, 2003). Our computational results demonstrate that our robust portfolio outperforms Goldfarb and Iyengar’s in terms of wealth growth rate and transaction cost, and moreover, ours is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Robust Portfolio Selection with and without Relative Entropy

We analyze two robust portfolio selection models, where a mean-variance investor considers possible deviations from a reference distribution of asset returns, adopting a maxmin criterion. The two models differ in the metric used to measure the distance between the reference distribution of asset returns and the alternative probability distributions. In the first model, where relative entropy is used as a measure of distance between distributions, an observational equivalence result obtains, whereby introducing robustness is equivalent to increasing risk aversion and, therefore, the percentage composition of the optimal portfolio of risky assets is equal to that of the optimal portfolio held by an investor without concerns for robustness. In the second model, introducing an alternative measure of distance between distributions, we show that observational equivalence ceases to hold and the proportions between risky assets are altered. We exploit the natural game-theoretic interpretation of the maxmin setting to illustrate the differences between the two models.