Theory and Applications of Robust Optimization

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 page

Calibration of Distributionally Robust Empirical Optimization Models

We study the out-of-sample properties of robust empirical optimization problems with smooth $\phi$-divergence penalties and smooth concave objective functions, and develop a theory for data-driven calibration of the non-negative "robustness parameter" $\delta$ that controls the size of the deviations from the nominal model. Building on the intuition that robust optimization reduces the sensitivity of the expected reward to errors in the model by controlling the spread of the reward distribution, we show that the first-order benefit of ``little bit of robustness" (i.e., $\delta$ small, positive) is a significant reduction in the variance of the out-of-sample reward while the corresponding impact on the mean is almost an order of magnitude smaller. One implication is that substantial variance (sensitivity) reduction is possible at little cost if the robustness parameter is properly calibrated. To this end, we introduce the notion of a robust mean-variance frontier to select the robustness parameter and show that it can be approximated using resampling methods like the bootstrap. Our examples show that robust solutions resulting from "open loop" calibration methods (e.g., selecting a $90\%$ confidence level regardless of the data and objective function) can be very conservative out-of-sample, while those corresponding to the robustness parameter that optimizes an estimate of the out-of-sample expected reward (e.g., via the bootstrap) with no regard for the variance are often insufficiently robust.Comment: 51 page

Portfolio Decisions with Higher Order Moments

In this paper, we address the global optimization of two interesting nonconvex problems in finance. We relax the normality assumption underlying the classical Markowitz mean-variance portfolio optimization model and consider the incorporation of skewness (third moment) and kurtosis (fourth moment). The investor seeks to maximize the expected return and the skewness of the portfolio and minimize its variance and kurtosis, subject to budget and no short selling constraints. In the first model, it is assumed that asset statistics are exact. The second model allows for uncertainty in asset statistics. We consider rival discrete estimates for the mean, variance, skewness and kurtosis of asset returns. A robust optimization framework is adopted to compute the best investment portfolio maximizing return, skewness and minimizing variance, kurtosis, in view of the worst-case asset statistics. In both models, the resulting optimization problems are nonconvex. We introduce a computational procedure for their global optimization.Mean-variance portfolio selection, Robust portfolio selection, Skewness, Kurtosis, Decomposition methods, Polynomial optimization problems

Ambiguity in asset pricing and portfolio choice: a review of the literature

A growing body of empirical evidence suggests that investors’ behavior is not well described by the traditional paradigm of (subjective) expected utility maximization under rational expectations. A literature has arisen that models agents whose choices are consistent with models that are less restrictive than the standard subjective expected utility framework. In this paper we conduct a survey of the existing literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choice and equilibrium asset prices. We conclude that the ambiguity literature has led to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions, such as the empirical failure of the two-fund separation theorem in portfolio decisions, the modest exposure to risky securities observed for a majority of investors, the home equity preference in international portfolio diversification, the excess volatility of asset returns, the equity premium and the risk-free rate puzzles, and the occurrence of trading break-downs.Capital assets pricing model ; Investments

Robust Portfolio Optimization with Derivative Insurance Guarantees

Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust portfolio optimization model that provides additional strong performance guarantees for all possible realizations of the asset returns. This insurance is provided via optimally chosen derivatives on the assets in the portfolio. The resulting model constitutes a convex second- order cone program, which is amenable to efficient numerical solution. We evaluate the model using simulated and empirical backtests and conclude that it can out- perform standard robust portfolio optimization as well as classical mean-variance optimization.robust optimization, portfolio optimization, portfolio insurance, second order cone programming

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