Investment Portfolio Optimization Based on Modern Portfolio Theory and Deep Learning Models

Financial time series forecasting and portfolio optimization have long relied on static models like sample covariance and GARCH. Recent advances in deep learning, especially LSTM and transformers, have shown promise in univariate price prediction. However, extending these models to multivariate risk structures remains challenging due to high dimensionality and the need for positive definite covariance matrices. Prior works have explored deep models for stock prices or volatility, but few have systematically integrated probabilistic deep models with matrix factorization techniques for dynamic covariance forecasting. This gap limits the ability to adapt risk estimates in real-time, especially in volatile markets. The current study builds on these developments, proposing a comprehensive framework that combines deep recurrent models with Cholesky decomposition to produce stable, accurate covariance forecasts for portfolio optimization.

The core challenge is to accurately and dynamically estimate the covariance matrix of multiple assets, which is essential for risk-aware portfolio optimization. Traditional estimators often produce singular or unstable matrices in high dimensions, impairing optimization. Existing deep learning approaches lack systematic integration of matrix positivity constraints, risking invalid covariance matrices. Moreover, capturing long-term dependencies and nonlinear relationships among assets over varying market conditions remains difficult. Addressing these issues requires models that can learn complex temporal patterns, ensure mathematical validity, and adapt to market changes, thereby enabling robust risk management and improved investment performance.

The main innovations include: 1) employing LSTM, DeepVAR, and GPVAR models for multivariate covariance forecasting, capturing complex temporal and cross-asset dependencies; 2) integrating Cholesky decomposition to guarantee positive definiteness of forecasted matrices; 3) systematically analyzing the effects of observation window length and rebalancing frequency on model performance; 4) demonstrating superior empirical results over classical estimators across diverse asset classes and market conditions. These innovations enable more accurate, stable, and real-time risk estimates, advancing the state-of-the-art in dynamic portfolio risk management.

• �� Collect daily asset prices and compute historical covariance matrices over windows of 30, 60, 90, 120 days. • Train deep models (LSTM, DeepVAR, GPVAR) on these historical data, optimizing parameters via maximum likelihood. • Forecast one-day-ahead covariance matrices by predicting the Cholesky factors of the matrices, ensuring positive definiteness. • Use the predicted Cholesky factors to reconstruct covariance matrices for each future date. • Apply the matrices in a Markowitz minimum variance portfolio optimization, incorporating transaction costs. • Rebalance portfolios at specified intervals, adjusting weights based on forecasted risks. • Evaluate performance using metrics like information ratio, annualized return, maximum drawdown, and recovery duration.

The dataset includes daily prices of stocks from the S&P 500 and cryptocurrencies like Bitcoin and Ethereum. The models are trained on historical windows of 30, 60, 90, and 120 days, with forecasts made for the next day. The performance is compared against classical estimators like sample covariance, EWMA, and Ledoit-Wolf shrinkage. Portfolio optimization uses a minimum variance criterion, with transaction costs modeled at 50 basis points per trade. The evaluation spans multiple rebalancing periods (30, 60, 90, 120 days), assessing metrics such as information ratio, annualized return, maximum drawdown, and recovery time. Cross-validation ensures robustness, and hyperparameters are tuned via grid search.

Deep models, especially LSTM, outperform classical methods, with IR improvements of about 15% and annualized returns reaching 12%. Longer observation windows (e.g., 90 days) significantly improve forecast accuracy and portfolio stability. Less frequent rebalancing (every 90 days) yields better risk-adjusted outcomes, balancing transaction costs and market fluctuations. The probabilistic models (DeepVAR, GPVAR) provide richer distributional forecasts, enabling better risk quantification. Results are consistent across stock and crypto assets, demonstrating model robustness. These findings confirm the advantage of deep learning in dynamic covariance estimation for portfolio management.

This framework can be adopted by asset managers, hedge funds, and robo-advisors to enhance risk-aware portfolio construction. It requires only historical price data and computational resources for model training. The approach is suitable for multi-asset portfolios across equities, cryptocurrencies, and other asset classes, especially in volatile markets. It supports real-time risk monitoring and adaptive rebalancing, improving decision-making and performance. Future integration with automated trading systems could further optimize asset allocation in response to market dynamics.

Deep models demand high computational power and large datasets, which may limit real-time deployment. The assumption of market stability may not hold during crises, reducing robustness. The models do not explicitly incorporate transaction costs or liquidity constraints, potentially overestimating strategy profitability. Further research is needed to address extreme market conditions, reduce computational complexity, and incorporate practical trading considerations for broader applicability.