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The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. In this paper, we provide a new perspective. The key insight is to replace the statistical framework in the original approach with ideas from inverse optimization. This insight allows us to significantly expand the scope and applicability of the BL model. We provide a richer formulation that, unlike the original model, is flexible enough to incorporate investor information on volatility and market dynamics. Equally importantly, our approach allows us to move beyond the traditional mean-variance paradigm of the original model and construct \"BL\"-type estimators for more general notions of risk such as coherent risk measures. Computationally, we introduce and study two new \"BL\"-type estimators and their corresponding portfolios: a mean variance inverse optimization (MV-IO) portfolio and a robust mean variance inverse optimization (RMV-IO) portfolio. These two approaches are motivated by ideas from arbitrage pricing theory and volatility uncertainty. Using numerical simulation and historical backtesting, we show that both methods often demonstrate a better risk-reward trade-off than their BL counterparts and are more robust to incorrect investor views.

Computational finance is an emerging application field of metaheuristic algorithms. In particular, these optimisation methods are becoming the solving approach alternative when dealing with realistic versions of several decision-making problems in finance, such as rich portfolio optimisation and risk management. This paper reviews the scientific literature on the use of metaheuristics for solving NP-hard versions of these optimisation problems and illustrates their capacity to provide high-quality solutions under scenarios considering realistic constraints. The paper contributes to the existing literature in three ways. Firstly, it reviews the literature on metaheuristic optimisation applications for portfolio and risk management in a systematic way. Secondly, it identifies the linkages between portfolio optimisation and risk management and presents a unified view and classification of both problems. Finally, it outlines the trends that have gradually become apparent in the literature and will dominate future research in order to further improve the state-of-the-art in this knowledge area.

Financial Portfolio Optimization Problem (FPOP) is a cornerstone in quantitative investing and financial engineering, focusing on optimizing assets allocation to balance risk and expected return, a concept evolving since Harry Markowitz’s 1952 Mean-Variance model. This paper introduces a novel meta-heuristic approach based on the Black Widow Algorithm for Portfolio Optimization (BWAPO) to solve the FPOP. The new method addresses three versions of the portfolio optimization problems: the unconstrained version, the equality cardinality-constrained version, and the inequality cardinality-constrained version. New features are introduced for the BWAPO to adapt better to the problem, including (1) mating attraction and (2) differential evolution mutation strategy. The proposed BWAPO is evaluated against other metaheuristic approaches used in portfolio optimization from literature, and its performance demonstrates its effectiveness through comparative studies on benchmark datasets using multiple performance metrics, particularly in the unconstrained Mean-Variance portfolio optimization version. Additionally, when encountering cardinality constraint, the proposed approach yields competitive results, especially noticeable with smaller datasets. This leads to a focused examination of the outcomes arising from equality versus inequality cardinality constraints, intending to determine which constraint type is more effective in producing portfolios with higher returns. The paper also presents a comprehensive mathematical model that integrates real-world constraints such as transaction costs, transaction lots, and a dollar-denominated budget, in addition to cardinality and bounding constraints. The model assesses both equality/inequality cardinality constraint versions of the problem, revealing that the inequality constraint tends to offer a wider range of feasible solutions with increased return potential.

This paper presents an improved method of applying entropy as a risk in portfolio optimization. A new family of portfolio optimization problems called the return-entropy portfolio optimization (REPO) is introduced that simplifies the computation of portfolio entropy using a combinatorial approach. REPO addresses five main practical concerns with the mean-variance portfolio optimization (MVPO). Pioneered by Harry Markowitz, MVPO revolutionized the financial industry as the first formal mathematical approach to risk-averse investing. REPO uses a mean-entropy objective function instead of the mean-variance objective function used in MVPO. REPO also simplifies the portfolio entropy calculation by utilizing combinatorial generating functions in the optimization objective function. REPO and MVPO were compared by emulating competing portfolios over historical data and REPO significantly outperformed MVPO in a strong majority of cases.

This paper proposes a transformation of the portfolio selection problem into SAT. SAT was the first problem to be shown tobe NP-complete, and has been widely investigated ever since. We derive the SAT instances from the Portfolio Selection onesusing the concept of cover, and reduce their size via established reduction techniques. The resulting instances are based onthe use of variance as the main risk measure, and are solved via both a standard SAT solver and an adaptive genetic algorithm.Results show that adaptive genetic algorithms are effective in solving these variance-based instances. Further work will bedevoted to investigate other SAT formulations based on different risk measures.

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

Theoretical framework--The research is grounded in modern portfolio theory and robust optimization, drawing on the multifactor asset pricing model of Chen, Roll, and Ross (1986). It leverages the Kalman filter to estimate dynamic betas and their uncertainties, and incorporates analysts' forecasts to assess expected returns and their associated uncertainties. Practical & social implications of research--The findings provide insights for investors, fund managers, and practitioners seeking to improve portfolio stability and performance under uncertainty. However, reliance on analysts' estimates should be approached with caution, as deviations from expected values can still occur.

The participation of household prosumers in wholesale electricity markets is very limited, considering the minimum participation limit imposed by most market participation rules. The generation capacity of households has been increasing since the installation of distributed generation from renewable sources in their facilities brings advantages for themselves and the system. Due to the growth of self-consumption, network operators have been putting aside the purchase of electricity from households, and there has been a reduction in the price of these transactions. This paper proposes an innovative model that uses the aggregation of households to reach the minimum limits of electricity volume needed to participate in the wholesale market. In this way, the Aggregator represents the community of households in market sales and purchases. An electricity transactions portfolio optimization model is proposed to enable the Aggregator reaching the decisions on which markets to participate to maximize the market negotiation outcomes, considering the day-ahead market, intra-day market, and retail market. A case study is presented, considering the Iberian wholesale electricity market and the Portuguese retail market. A community of 50 prosumers equipped with photovoltaic generators and individual storage systems is used to carry out the experiments. A cost reduction of 6–11% is achieved when the community of households buys and sells electricity in the wholesale market through the Aggregator.

Portfolio analysis is widely used by financial investors to find portfolios producing efficient results under various economic conditions. Markowitz started the portfolio optimization approach through mean-variance, whose objective is to minimize risk and maximize the return. This study is called Markowitz Mean-Variance Theory (MVP). An optimal portfolio has a good return and low risk, in addition to being well diversified. In this paper, we proposed a methodology for obtaining an optimal portfolio with the highest expected return and the lowest risk. This methodology uses Mixture Design of Experiments (MDE) as a strategy for building non-linear models of risk and return in portfolio optimization; computational replicas in MDE to capture dynamical evolution of series; Shannon entropy index to handle better portfolio diversification; and desirability function to optimize multiple variables, leading to the maximum expected return and lowest risk. To illustrate this proposal, some time series were simulated by ARMA-GARCH models. The result is compared to the efficient frontier generated by the traditional theory of Markowitz Mean-Variance (MVP). The results show that this methodology facilitates decision making, since the portfolio is obtained in the non-dominated region, in a unique combination. The advantage of using the proposed method is that the replicas improve the model precision.

The portfolio optimization model has limited impact in practice because of estimation issues when applied to real data. To address this, we adapt two machine learning methods, regularization and cross-validation, for portfolio optimization. First, we introduce performance-based regularization (PBR), where the idea is to constrain the sample variances of the estimated portfolio risk and return, which steers the solution toward one associated with less estimation error in the performance. We consider PBR for both mean-variance and mean-conditional value-at-risk (CVaR) problems. For the mean-variance problem, PBR introduces a quartic polynomial constraint, for which we make two convex approximations: one based on rank-1 approximation and another based on a convex quadratic approximation. The rank-1 approximation PBR adds a bias to the optimal allocation, and the convex quadratic approximation PBR shrinks the sample covariance matrix. For the mean-CVaR problem, the PBR model is a combinatorial optimization problem, but we prove its convex relaxation, a quadratically constrained quadratic program, is essentially tight. We show that the PBR models can be cast as robust optimization problems with novel uncertainty sets and establish asymptotic optimality of both sample average approximation (SAA) and PBR solutions and the corresponding efficient frontiers. To calibrate the right-hand sides of the PBR constraints, we develop new, performance-based k -fold cross-validation algorithms. Using these algorithms, we carry out an extensive empirical investigation of PBR against SAA, as well as L1 and L2 regularizations and the equally weighted portfolio. We find that PBR dominates all other benchmarks for two out of three Fama–French data sets. This paper was accepted by Yinyu Ye, optimization .