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Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.

The growth-optimal portfolio is designed to have maximum expected log return over the next rebalancing period. Thus, it can be computed with relative ease by solving a static optimization problem. The growth-optimal portfolio has sparked fascination among finance professionals and researchers because it can be shown to outperform any other portfolio with probability 1 in the long run. In the short run, however, it is notoriously volatile. Moreover, its computation requires precise knowledge of the asset return distribution, which is not directly observable but must be inferred from sparse data. By using methods from distributionally robust optimization, we design fixed-mix strategies that offer similar performance guarantees as the growth-optimal portfolio but for a finite investment horizon and for a whole family of distributions that share the same first- and second-order moments. We demonstrate that the resulting robust growth-optimal portfolios can be computed efficiently by solving a tractable conic program whose size is independent of the length of the investment horizon. Simulated and empirical backtests show that the robust growth-optimal portfolios are competitive with the classical growth-optimal portfolio across most realistic investment horizons and for an overwhelming majority of contaminated return distributions. This paper was accepted by Yinyu Ye, optimization .

Value-at-risk is one of the most popular risk management tools in the financial industry. Over the past 20 years, several attempts to include VaR in the portfolio selection process have been proposed. However, using VaR as a risk measure in portfolio optimization models leads to problems that are computationally hard to solve. In view of this, few practical applications of VaR in portfolio selection have appeared in the literature up to now. In this paper, we propose to add the VaR criterion to the classical Mean-Variance approach in order to better address the typical regulatory constraints of the financial industry. We thus obtain a portfolio selection model characterized by three criteria: expected return, variance, and VaR at a specified confidence level. The resulting optimization problem consists in minimizing variance with parametric constraints on the levels of expected return and VaR. This model can be formulated as a mixed-integer quadratic programming (MIQP) problem. An extensive empirical analysis on seven real-world datasets demonstrates the practical applicability of the proposed approach. Furthermore, the out-of-sample performance of the more binding optimal Mean-Variance-VaR portfolios seems to be generally better than that of the Equally Weighted and of the Mean-Variance-CVaR portfolios.

Purpose This paper aims to examine from commodity portfolio managers’ perspective the performance of liquidity adjusted risk modeling in assessing the market risk parameters of a large commodity portfolio and in obtaining efficient and coherent portfolios under different market circumstances. Design/methodology/approach The implemented market risk modeling algorithm and investment portfolio analytics using reinforcement machine learning techniques can simultaneously handle risk-return characteristics of commodity investments under regular and crisis market settings besides considering the particular effects of the time-varying liquidity constraints of the multiple-asset commodity portfolios. Findings In particular, the paper implements a robust machine learning method to commodity optimal portfolio selection and within a liquidity-adjusted value-at-risk (LVaR) framework. In addition, the paper explains how the adapted LVaR modeling algorithms can be used by a commodity trading unit in a dynamic asset allocation framework for estimating risk exposure, assessing risk reduction alternates and creating efficient and coherent market portfolios. Originality/value The optimization parameters subject to meaningful operational and financial constraints, investment portfolio analytics and empirical results can have important practical uses and applications for commodity portfolio managers particularly in the wake of the 2007–2009 global financial crisis. In addition, the recommended reinforcement machine learning optimization algorithms can aid in solving some real-world dilemmas under stressed and adverse market conditions (e.g. illiquidity, switching in correlations factors signs, nonlinear and non-normal distribution of assets’ returns) and can have key applications in machine learning, expert systems, smart financial functions, internet of things (IoT) and financial technology (FinTech) in big data ecosystems.

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large assortment of examples and assessing which may exhibit superior behavior. Connections are shown with coherent risk measures in the sense of Artzner, Delbaen, Eber and Heath, when those are applied to the difference between a random variable and its expectation, instead of to the random variable itself. However, the correspondence is only one-to-one when both classes are restricted by properties called lower range dominance, on the one hand, and strict expectation boundedness on the other. Dual characterizations in terms of sets called risk envelopes are fully provided. [PUBLICATION ABSTRACT]

Purpose This study aims to examine the theoretical foundations for multivariate portfolio optimization algorithms under illiquid market conditions. In this study, special emphasis is devoted to the application of a risk-engine, which is based on the contemporary concept of liquidity-adjusted value-at-risk (LVaR), to multivariate optimization of investment portfolios. Design/methodology/approach This paper examines the modeling parameters of LVaR technique under event market settings and discusses how to integrate asset liquidity risk into LVaR models. Finally, the authors discuss scenario optimization algorithms for the assessment of structured investment portfolios and present a detailed operational methodology for computer programming purposes and prospective research design with the backing of a graphical flowchart. Findings To that end, the portfolio/risk manager can specify different closeout horizons and dependence measures and calculate the necessary LVaR and resulting investable portfolios. In addition, portfolio managers can compare the return/risk ratio and asset allocation of obtained investable portfolios with different liquidation horizons in relation to the conventional Markowitz´s mean-variance approach. Practical implications The examined optimization algorithms and modeling techniques have important practical applications for portfolio management and risk assessment, and can have many uses within machine learning and artificial intelligence, expert systems and smart financial applications, financial technology (FinTech), and within big data environments. In addition, it provide key real-world implications for portfolio/risk managers, treasury directors, risk management executives, policymakers and financial regulators to comply with the requirements of Basel III best practices on liquidly risk. Originality/value The proposed optimization algorithms can aid in advancing portfolios selection and management in financial markets by assessing investable portfolios subject to meaningful operational and financial constraints. Furthermore, the robust risk-algorithms and portfolio optimization techniques can aid in solving some real-world dilemmas under stressed and adverse market conditions, such as the effect of liquidity when it dries up in financial and commodity markets, the impact of correlations factors when there is a switching in their signs and the integration of the influence of the nonlinear and non-normal distribution of assets’ returns in portfolio optimization and management.

We consider a lender (bank) that determines the optimal loan price (interest rate) to offer to prospective borrowers under uncertain borrower response and default risk. A borrower may or may not accept the loan at the price offered, and both the principal loaned and the interest income become uncertain because of the risk of default. We present a risk-based loan pricing optimization framework that explicitly takes into account the marginal risk contribution, the portfolio risk, and a borrower’s acceptance probability. Marginal risk assesses the incremental risk contribution of a prospective loan to the bank’s overall portfolio risk by capturing the dependencies between the prospective loan and the existing portfolio and is evaluated with respect to the value-at-risk and conditional value-at-risk measures. We examine the properties and computational challenges of the formulations. We design a reformulation method based on the concavifiability concept to transform the nonlinear objective functions and to derive equivalent mixed-integer nonlinear reformulations with convex continuous relaxations. We also extend the approach to multiloan pricing problems, which feature explicit loan selection decisions in addition to pricing decisions. We derive formulations with multiple loans that take the form of mixed-integer nonlinear problems with nonconvex continuous relaxations and develop a computationally efficient algorithmic method. We provide numerical evidence demonstrating the value of the proposed framework, test the computational tractability, and discuss managerial implications. This paper was accepted by Chung Piaw Teo, optimization.

Value-at-Risk (VaR) is one of the most widely accepted risk measures in the financial and insurance industries, yet efficient optimization of VaR remains a very difficult problem. We propose a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques. The method results in the optimization of a modified VaR measure, Asymmetry-Robust VaR (ARVaR), that takes into consideration asymmetries in the distributions of returns and is coherent, which makes it desirable from a financial theory perspective. We show that ARVaR approximates the Conditional VaR of the portfolio as well. Numerical experiments with simulated and real market data indicate that the proposed approach results in lower realized portfolio VaR, better efficient frontier, and lower maximum realized portfolio loss than alternative approaches for quantile-based portfolio risk minimization.

This study proposes a hierarchical signal-to-policy learning framework for risk-aware portfolio optimization that integrates model-based return forecasting, explainable machine learning, and deep reinforcement learning (DRL) within a unified architecture. In the first stage, next-period returns are estimated using gradient-boosted tree models, and SHAP-based feature attributions are extracted to provide transparent, factor-level explanations of the predictive signals. In the second stage, a Proximal Policy Optimization (PPO) agent incorporates both predictive forecasts and explanatory signals into its state representation and learns dynamic allocation policies under a mean–CVaR reward function that explicitly penalizes tail risk while controlling trading frictions. By separating signal extraction from policy learning, the proposed architecture allows the use of economically interpretable predictive signals to incorporate into the policy’s state representation while preserving the flexibility and adaptability of reinforcement learning. Empirical evaluations on U.S. sector ETFs and Dow Jones Industrial Average constituents show that the hierarchical framework delivers higher and stable out-of-sample risk-adjusted returns relative to both a single-layer DRL agent trained solely on technical indicators, a mean–CVaR optimized portfolio using the same parameters used in the proposed hierarchical model and standard equal weight as well as index-based benchmarks. These results demonstrate that integrating explainable predictive signals with risk-sensitive reinforcement learning improves the robustness and stability of data-driven portfolio strategies.

Average Value-at-Risk (AVaR) is a potential alternative to Value-at-Risk in the financial regulation of banking and insurance institutions. To understand how AVaR influences a company’s investment behavior, we study portfolio optimization under the AVaR constraint. Our main contribution is to derive analytical solutions for non-concave portfolio optimization problems under the AVaR constraint in a complete financial market by quantile formulation and the decomposition method, where the non-concavity arises from assuming that the company is surplus-driven. Given the AVaR constraint, the company takes three investment strategies depending on its initial budget constraint. Under each investment strategy, we derive the fair return for the company’s debt holders fulfilling the risk-neutral pricing constraint in closed form. Further, we illustrate the above analytical results in a Black–Scholes market. We find that the fair return varies drastically, e.g., from 4.99% to 37.2% in different situations, implying that the company’s strategy intimately determines the default risk faced by its debt holders. Our analysis and numerical experiment show that the AVaR constraint cannot eliminate the company’s default risk but can reduce it compared with the benchmark portfolio. However, the protection for the debt holders is poor if the company has a low initial budget.