Portfolio Optimization

From Factor Models to Deep Learning: Machine Learning in Reshaping Empirical Asset Pricing ArXiv ID: 2403.06779 “View on arXiv” Authors: Unknown Abstract This paper comprehensively reviews the application of machine learning (ML) and AI in finance, specifically in the context of asset pricing. It starts by summarizing the traditional asset pricing models and examining their limitations in capturing the complexities of financial markets. It explores how 1) ML models, including supervised, unsupervised, semi-supervised, and reinforcement learning, provide versatile frameworks to address these complexities, and 2) the incorporation of advanced ML algorithms into traditional financial models enhances return prediction and portfolio optimization. These methods can adapt to changing market dynamics by modeling structural changes and incorporating heterogeneous data sources, such as text and images. In addition, this paper explores challenges in applying ML in asset pricing, addressing the growing demand for explainability in decision-making and mitigating overfitting in complex models. This paper aims to provide insights into novel methodologies showcasing the potential of ML to reshape the future of quantitative finance. ...

Portfolio Analysis in High Dimensions with TE and Weight Constraints ArXiv ID: 2402.17523 “View on arXiv” Authors: Unknown Abstract This paper explores the statistical properties of forming constrained optimal portfolios within a high-dimensional set of assets. We examine portfolios with tracking error constraints, those with simultaneous tracking error and weight restrictions, and portfolios constrained solely by weight. Tracking error measures portfolio performance against a benchmark (typically an index), while weight constraints determine asset allocation based on regulatory requirements or fund prospectuses. Our approach employs a novel statistical learning technique that integrates factor models with nodewise regression, named the Constrained Residual Nodewise Optimal Weight Regression (CROWN) method. We demonstrate its estimation consistency in large dimensions, even when assets outnumber the portfolio’s time span. Convergence rate results for constrained portfolio weights, risk, and Sharpe Ratio are provided, and simulation and empirical evidence highlight the method’s outstanding performance. ...

Finding Near-Optimal Portfolios With Quality-Diversity ArXiv ID: 2402.16118 “View on arXiv” Authors: Unknown Abstract The majority of standard approaches to financial portfolio optimization (PO) are based on the mean-variance (MV) framework. Given a risk aversion coefficient, the MV procedure yields a single portfolio that represents the optimal trade-off between risk and return. However, the resulting optimal portfolio is known to be highly sensitive to the input parameters, i.e., the estimates of the return covariance matrix and the mean return vector. It has been shown that a more robust and flexible alternative lies in determining the entire region of near-optimal portfolios. In this paper, we present a novel approach for finding a diverse set of such portfolios based on quality-diversity (QD) optimization. More specifically, we employ the CVT-MAP-Elites algorithm, which is scalable to high-dimensional settings with potentially hundreds of behavioral descriptors and/or assets. The results highlight the promising features of QD as a novel tool in PO. ...

Combining Transformer based Deep Reinforcement Learning with Black-Litterman Model for Portfolio Optimization ArXiv ID: 2402.16609 “View on arXiv” Authors: Unknown Abstract As a model-free algorithm, deep reinforcement learning (DRL) agent learns and makes decisions by interacting with the environment in an unsupervised way. In recent years, DRL algorithms have been widely applied by scholars for portfolio optimization in consecutive trading periods, since the DRL agent can dynamically adapt to market changes and does not rely on the specification of the joint dynamics across the assets. However, typical DRL agents for portfolio optimization cannot learn a policy that is aware of the dynamic correlation between portfolio asset returns. Since the dynamic correlations among portfolio assets are crucial in optimizing the portfolio, the lack of such knowledge makes it difficult for the DRL agent to maximize the return per unit of risk, especially when the target market permits short selling (i.e., the US stock market). In this research, we propose a hybrid portfolio optimization model combining the DRL agent and the Black-Litterman (BL) model to enable the DRL agent to learn the dynamic correlation between the portfolio asset returns and implement an efficacious long/short strategy based on the correlation. Essentially, the DRL agent is trained to learn the policy to apply the BL model to determine the target portfolio weights. To test our DRL agent, we construct the portfolio based on all the Dow Jones Industrial Average constitute stocks. Empirical results of the experiments conducted on real-world United States stock market data demonstrate that our DRL agent significantly outperforms various comparison portfolio choice strategies and alternative DRL frameworks by at least 42% in terms of accumulated return. In terms of the return per unit of risk, our DRL agent significantly outperforms various comparative portfolio choice strategies and alternative strategies based on other machine learning frameworks. ...

Sizing the bets in a focused portfolio ArXiv ID: 2402.15588 “View on arXiv” Authors: Unknown Abstract The paper provides a mathematical model and a tool for the focused investing strategy as advocated by Buffett, Munger, and others from this investment community. The approach presented here assumes that the investor’s role is to think about probabilities of different outcomes for a set of businesses. Based on these assumptions, the tool calculates the optimal allocation of capital for each of the investment candidates. The model is based on a generalized Kelly Criterion with options to provide constraints that ensure: no shorting, limited use of leverage, providing a maximum limit to the risk of permanent loss of capital, and maximum individual allocation. The software is applied to an example portfolio from which certain observations about excessive diversification are obtained. In addition, the software is made available for public use. ...

A Note on Optimal Liquidation with Linear Price Impact ArXiv ID: 2402.14100 “View on arXiv” Authors: Unknown Abstract In this note we consider the maximization of the expected terminal wealth for the setup of quadratic transaction costs. First, we provide a very simple probabilistic solution to the problem. Although the problem was largely studied, as far as we know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, we apply the general result for the numerical study of the case where the risky asset is given by a fractional Brownian Motion and the information flow of the investor can be diversified. ...

The Boosted Difference of Convex Functions Algorithm for Value-at-Risk Constrained Portfolio Optimization ArXiv ID: 2402.09194 “View on arXiv” Authors: Unknown Abstract A highly relevant problem of modern finance is the design of Value-at-Risk (VaR) optimal portfolios. Due to contemporary financial regulations, banks and other financial institutions are tied to use the risk measure to control their credit, market, and operational risks. Despite its practical relevance, the non-convexity induced by VaR constraints in portfolio optimization problems remains a major challenge. To address this complexity more effectively, this paper proposes the use of the Boosted Difference-of-Convex Functions Algorithm (BDCA) to approximately solve a Markowitz-style portfolio selection problem with a VaR constraint. As one of the key contributions, we derive a novel line search framework that allows the application of the algorithm to Difference-of-Convex functions (DC) programs where both components are non-smooth. Moreover, we prove that the BDCA linearly converges to a Karush-Kuhn-Tucker point for the problem at hand using the Kurdyka-Lojasiewicz property. We also outline that this result can be generalized to a broader class of piecewise-linear DC programs with linear equality and inequality constraints. In the practical part, extensive numerical experiments under consideration of best practices then demonstrate the robustness of the BDCA under challenging constraint settings and adverse initialization. In particular, the algorithm consistently identifies the highest number of feasible solutions even under the most challenging conditions, while other approaches from chance-constrained programming lead to a complete failure in these settings. Due to the open availability of all data sets and code, this paper further provides a practical guide for transparent and easily reproducible comparisons of VaR-constrained portfolio selection problems in Python. ...

Finding Moving-Band Statistical Arbitrages via Convex-Concave Optimization ArXiv ID: 2402.08108 “View on arXiv” Authors: Unknown Abstract We propose a new method for finding statistical arbitrages that can contain more assets than just the traditional pair. We formulate the problem as seeking a portfolio with the highest volatility, subject to its price remaining in a band and a leverage limit. This optimization problem is not convex, but can be approximately solved using the convex-concave procedure, a specific sequential convex programming method. We show how the method generalizes to finding moving-band statistical arbitrages, where the price band midpoint varies over time. ...

Downside Risk Reduction Using Regime-Switching Signals: A Statistical Jump Model Approach ArXiv ID: 2402.05272 “View on arXiv” Authors: Unknown Abstract This article investigates a regime-switching investment strategy aimed at mitigating downside risk by reducing market exposure during anticipated unfavorable market regimes. We highlight the statistical jump model (JM) for market regime identification, a recently developed robust model that distinguishes itself from traditional Markov-switching models by enhancing regime persistence through a jump penalty applied at each state transition. Our JM utilizes a feature set comprising risk and return measures derived solely from the return series, with the optimal jump penalty selected through a time-series cross-validation method that directly optimizes strategy performance. Our empirical analysis evaluates the realistic out-of-sample performance of various strategies on major equity indices from the US, Germany, and Japan from 1990 to 2023, in the presence of transaction costs and trading delays. The results demonstrate the consistent outperformance of the JM-guided strategy in reducing risk metrics such as volatility and maximum drawdown, and enhancing risk-adjusted returns like the Sharpe ratio, when compared to both hidden Markov model-guided strategy and the buy-and-hold strategy. These findings underline the enhanced persistence, practicality, and versatility of strategies utilizing JMs for regime-switching signals. ...

Optimal portfolio under ratio-type periodic evaluation in incomplete markets with stochastic factors ArXiv ID: 2401.14672 “View on arXiv” Authors: Unknown Abstract This paper studies a type of periodic utility maximization for portfolio management in an incomplete market model, where the underlying price diffusion process depends on some external stochastic factors. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon. For both power and logarithmic utilities, we formulate the auxiliary one-period optimization problems with modified utility functions, for which we develop the martingale duality approach to establish the existence of the optimal portfolio processes and the dual minimizers can be identified as the “least favorable” completion of the market. With the help of the duality results in the auxiliary problems and some fixed point arguments, we further derive and verify the optimal portfolio processes in a periodic manner for the original periodic evaluation problems over an infinite horizon. ...