Robust Portfolio Selection Problems: a Comprehensive Review
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Robust Portfolio Selection Problems: A Comprehensive Review Alireza Ghahtarani∗ 1, Ahmed Saif1, and Alireza Ghasemi1 1Department of Industrial Engineering, Dalhousie University Abstract 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 perspec- tives. 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 ap- proaches, and mathematical formulations. Several open questions and potential future research directions are identified. Key Words: Robust Optimization, Portfolio Selection Problem 1 Introduction The portfolio selection problem (PSP) is a fundamental problem in finance that aims at optimally allocating funds among financial assets. Different versions of the PSP appeared in the literature under different names such as mean-variance, mean absolute deviation, factor models, log-return, mean-Value at Risk, mean-Conditional Value at Risk, multi-period PSP, index tracking, and etc. Decision-makers are interested in PSPs formulation according to their risk attitude. In general, investment strategies are divided in two parts: passive and active. In the passive investment strategy, a decision-maker has more risk averse attitude. However, in the active investment strategy, a decision-maker has more risk tolerance. Besides investment strategies, an important part of PSPs is risk measures that quantify the risk of investment. PSPs are classified according to risk measures and ratios defined by risk measures, investment strategies, and methods used to calculate the asset returns. The first classification is according to the risk measure used to capture the volatility of as- set returns. While variance has been the most widely-used risk measure in both theory and practice since the seminal work of Markowitz (1952), it has its shortcomings. First, it equally considers both positive and negative deviations around the expected return as undesirable risk, despite the desirability of the positive deviations for investors. Alternatively, downside risk mea- arXiv:2103.13806v1 [q-fin.PM] 23 Mar 2021 sures that consider only the negative deviations of asset returns, like the lower partial moment (LPM), can be used. Furthermore, given the nonlinearity of variance, it leads to formulations with higher computational complexity compared to linear risk measures like the mean absolute deviation (MAD), proposed by Konno & Yamazaki (1991). Based on volatility risk measures, Sharpe (1966) defined Sharpe ratio (ratio of return to volatility) and Bernardo & Ledoit (2000) proposed omega ratio (risk-return ratio) to evaluate the performance of portfolios based on risk and return simultaneously. Another important class of risk measure used in PSPs is quintile- based risk measures such as Value-at-Risk (VaR), and Conditional-Value-at-Risk (CVaR). The former calculates the maximum loss at a specific confidence level, while the latter represents the expected value of losses greater than VaR at a confidence level. For more detail about quintile-based risk measures, interested readers are referred to Rockafellar et al. (2000). Besides risk measures, PSPs can be classified based on investment strategies. Index tracking was introduced by Dembo & King (1992) as a passive investment strategy that tries to follow a ∗Dalhousie University, Halifax, Nova Scotia, B3J 1B6, Canada ([email protected]). 1 market index. On the other hand, active investment strategies that involve ongoing buying and selling of assets are optimized by solving multi-period PSPs, proposed by Dantzig & Infanger (1993). The asset liability management (ALM) problem is a practical version of multi-period PSP to manage assets such that liabilities in financial institutions are covered. Additionally, PSPs are also classified according to return calculation methods. Goldfarb & Iyengar (2003) incorporated factors (macroeconomic, fundamental, and statistical) to determine market equi- librium and calculate the required rate of return. While Hull (2003) defined the Log-return as the equivalent, continuously-compounded rate of return of asset returns over a period of time. Finally, hedging gives rise to a popular PSP in which an investment position is intended to off- set potential losses or gains that may be incurred by a companion investment. For more detail about hedging, interested readers are referred to (Lutgens et al. (2006)). Despite being a well-studied problem, a common feature of most PSP formulations in the literature is that the problem parameters are assumed to be known with certainty. Ignoring the inherent uncertainty in parameters and instead use their point estimates often leads to subop- timal solutions that have poor out-of-sample performances. To overcome this issue, different methods have been proposed in the literature, with stochastic programming (SP) and robust optimization (RO) being two widely used frameworks for dealing with uncertainty. SP is an approach that focuses on the long-run performance of the portfolio by finding a solution that optimizes the expected value of the loss function. Despite its intuitive appeal and favorable con- vergence properties, SP requires the distribution function of the random variables to be known. Moreover, its risk-neutral nature does not provide protection from unfavorable scenarios, ren- dering it unsuitable for typically risk-averse investors. On the other hand, RO is a risk-averse approach that minimizes the loss function under a worst-case scenario (within an uncertainty set) and does not require the probability distribution of the uncertain parameter to be known, making it an attractive alternative. Nevertheless, given the challenging nature of robust PSPs, both theoretically and computationally, significant contributions in this area have started to be made only in the last decade. Among the earliest reviews of robust PSP is that of Fabozzi et al. (2010), which concen- trates mostly on the application of robust optimization in basic mean-variance, mean-CVaR, and mean-VaR problems, but does not cover more recent variants of the problem like robust index tracking, robust LPM, robust MAD, robust OMEGA ratio, and robust multi-objective PSPs. Scutell`a& Recchia (2010) and Scutell`a& Recchia (2013) also reviewed robust mean- variance, robust CVaR, and robust VaR problems, but similarly, did not survey other robust PSPs. Likewise, Kim et al. (2014a) concentrate on worst-case formulations, while ignoring other important classes, including relative robust models, robust regularization, net-zero alpha ad- justment and asymmetric uncertainty sets. Another review by Kim et al. (2018a) focused on worst-case frameworks in bond portfolio construction, currency hedging, and option pricing, while covering a small number of references on robust asset-liability management problems, log-robust models, and robust multi-period problems. Finally, Xidonas et al. (2020) provides a categorized bibliographic review which has a broad scope, yet is limited in technical details. Unlike previous reviews that were limited in scope or depth, we present a comprehensive review of the application of RO methods in PSPs. Our review covers a wide range of applications and methods and classifies them from both financial and operational research perspectives, highlighting research gaps and proposing future directions in this area. The remainder of this review paper is organized as follows. The next section provides a brief introduction to RO for non-specialists. Section 3 surveys robust PSP formulations based on volatility measures. Section 4 reviews the log-robust portfolio selection, which is based on a robust log-return PSPs. Section 5 surveys the application of RO in index-tracking. Section 6 reviews quantile-based PSPs, which include Value at Risk (VaR), Conditional Value at Risk (CVaR), and their extensions with worst- case RO methods, relative RO and distributionally robust optimization (DRO). Furthermore, the relationship between uncertainty sets and risk measures, application of soft robust formulation with risk measures, worst-case CVaR and its relationship with uniform investment strategy, and robust arbitrage pricing theory with worst-case CVaR are also discussed in Section 6. Section 7 provides a review of RO in multi-period PSPs and asset-liability management (ALM) 2 problems. Besides these two main problems, robust control formulations of investment problems are reviewed in this section. Section 8 reviews other financial problems that are not covered in the above-mentioned categories like hedging problem, risk-adjusted Sharpe ratio, robust scenario- based formulation, and robust data envelopment analysis. The last section provides conclusions and open issues in this context. 2 Robust Optimization This section is intended to provide a brief introduction to RO, a framework for dealing with the uncertainty of parameters in optimization problems. RO assumes that the parameters belong to an uncertainty set and optimizes over the worst realization in this set. The first RO formulation was developed by Soyster (1973) and used a box (hypercubic) uncertainty set that specifies an interval for each individual uncertain parameter. Even though this approach usually leads to tractable formulations, it is too conservative since it is based on the assumption that all parameters will take their worst possible values simultaneously, which rarely happens