Portfolio Optimization Under Parameter Uncertainty Using the Risk Aversion Formula (original) (raw)

Robust optimization approaches for portfolio selection: a comparative analysis

Annals of Operations Research, 2021

Robust optimization (RO) models have attracted a lot of interest in the area of portfolio selection. RO extends the framework of traditional portfolio optimization models, incorporating uncertainty through a formal and analytical approach into the modeling process. Although several RO models have been proposed in the literature, comprehensive empirical assessments of their performance are rather lacking. The objective of this study is to fill in this gap in the literature. To this end, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005–2020. For the analysis, two different robust versions of the mean–variance model are considered, together with robust models for conditional value-at-risk and the Omega ratio. The robust versions are compared against the nominal ones through various portfolio performance metrics, focusing on out-of-sample results.

Robust Optimization Approaches for Portfolio Selection: A Computational and Comparative Analysis

arXiv, 2020

The field of portfolio selection is an active research topic, which combines elements and methodologies from various fields, such as optimization, decision analysis, risk management, data science, forecasting, etc. The modeling and treatment of deep uncertainties for the future asset returns is a major issue for the success of analytical portfolio selection models. Recently, robust optimization (RO) models have attracted a lot of interest in this area. RO provides a computationally tractable framework for portfolio optimization based on relatively general assumptions on the probability distributions of the uncertain risk parameters. Thus, RO extends the framework of traditional linear and non-linear models (e.g., the well-known mean-variance model), incorporating uncertainty through a formal and analytical approach into the modeling process. Robust counterparts of existing models can be considered as worst-case re-formulations as far as deviations of the uncertain parameters from their nominal values are concerned. Although several RO models have been proposed in the literature focusing on various risk measures and different types of uncertainty sets about asset returns, analytical empirical assessments of their performance have not been performed in a comprehensive manner. The objective of this study is to fill in this gap in the literature. More specifically, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005-2016. For the analysis, three different robust versions of the mean-variance model are considered, together with two other robust models for conditional value-at-risk and the omega ratio. The robust versions are compared against standard (non-robust) models through various portfolio performance metrics, focusing on out-of-sample results. The analysis is based on a rolling window approach.

Stability advances in robust portfolio optimization under parallelepiped uncertainty

Central European Journal of Operations Research, 2017

In financial markets with high uncertainties, the trade-off between maximizing expected return and minimizing the risk is one of the main challenges in modeling and decision making. Since investors mostly shape their invested amounts towards certain assets and their risk aversion level according to their returns, scientists and practitioners have done studies on that subject since the beginning of the stock markets' establishment. In this study, we model a Robust Optimization problem based on data. We found a robust optimal solution to our portfolio optimization problem. This approach includes the use of Robust Conditional Value-at-Risk under Parallelepiped Uncertainty, an evaluation and a numerical finding of the robust optimal portfolio allocation. Then, we trace back our robust linear programming model to the Standard Form of a Linear Programming model; consequently, we solve it by a well-chosen algorithm and software package. Uncertainty in parameters, based on uncertainty in the prices, and a risk-return analysis are crucial parts of this study. A numerical experiment and a comparison (back testing) application are presented, containing real-world data from stock markets as well as a simulation study. Our approach increases the stability of portfolio allocation and reduces the portfolio risk.

Robust portfolio optimization with a hybrid heuristic algorithm

Computational Management Science, 2012

Estimation errors in both the expected returns and the covariance matrix hamper the constructing of reliable portfolios within the Markowitz framework. Robust techniques that incorporate the uncertainty about the unknown parameters are suggested in the literature. We propose a modification as well as an extension of such a technique and compare both with another robust approach. In order to eliminate oversimplifications of Markowitz' portfolio theory, we generalize the optimization framework to better emulate a more realistic investment environment. Because the adjusted optimization problem is no longer solvable with standard algorithms, we employ a hybrid heuristic to tackle this problem. Our empirical analysis is conducted with a moving time window for returns of the German stock index DAX100. The results of all three robust approaches yield more stable portfolio compositions than those of the original Markowitz framework. Moreover, the out-of-sample risk of the robust approaches is lower and less volatile while their returns are not necessarily smaller.

Recent developments in robust portfolios with a worst-case approach

Robust models have a major role in portfolio optimization for resolving the sensitivity issue of the classical mean-variance model. In this paper, we survey developments of worst-case optimization while focusing on approaches for constructing robust portfolios. In addition to the robust formulations for the Markowitz model, we review work on deriving robust counterparts for valueat-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.

Robust portfolio selection problems: a comprehensive review

Operational Research

In this paper, we provide a comprehensive review of 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.

Robust Portfolio Selection Problems

Mathematics of Operations Research, 2003

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 secondorder 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.

Robust Mean-Conditional Value at Risk Portfolio Optimization

In the portfolio optimization, the goal is to distribute the fixed capital on a set of investment opportunities to maximize return while managing risk. Risk and return are quantities that are used as input parameters for the optimal allocation of the capital in the suggested models. But these quantities are not known at the time of the formulation and solving the problem. Thus they should be estimated to solve the problem which might lead to large error. One of the widely used approaches to deal with such a situation, is robust optimization. In this paper we study the robust models of the Mean-Conditional Value at Risk (Mean-CVaR) portfolio selection problem under the estimation risk in mean return for both interval and ellipsoidal uncertainty sets. The corresponding robust models are a linear programming problem and a second order conic programming problem (SOCP) respectively. At the end an example is given to demonstrate the impact of uncertainty.

1 Robust Modeling of Multistage Portfolio Problems

2000

In the paper, we develop, discuss and illustrate by simulated nu- merical results a new model of multi-stage asset allocation problem. The model is given by a new methodology for optimization under uncertainty { the Robust Counterpart approach.