Portfolio Optimization by Mean-Variance-Kullback-Leibler Divergence measure using the AR-GJR-GARCH Filtration: Optimization by Mean-Variance-Kullback-Leibler Divergence measure

Abstract

The classical Markowitz Mean-Variance optimization framework remains foundational but is often criticized for its sensitivity to estimation error and assumption of normality, leading to poorly diversified and non-robust portfolios. We propose a novel hybrid framework that integrates the Kullback-Leibler divergence measure into the mean-variance objective, regularizing the solution towards an investor-defined target distribution. This paper presents an analytical approach to portfolio optimization by integrating the classical mean-variance framework with the Kullback-Leibler (KL) divergence measure. While the mean-variance method, pioneered by Markowitz, seeks to balance expected return and risk (as measured by variance), it often assumes perfect knowledge of asset return distributions. We utilize AR-GJR-GARCH models to estimate and forecast volatility on all asset returns. To address model uncertainty and distributional robustness, we investigate the KL divergence as a regularization term, penalizing deviations from a reference distribution. This fusion results in a robust optimization framework that accounts for uncertainty in the estimated parameters. We derive closed-form solutions under certain assumptions and explore the impact of the divergence parameter on the efficient frontier. The proposed method enhances stability and reliability in portfolio allocation, particularly in data-scarce or high-volatility environments. Empirical results across global markets show that our Mean-KL (M-KL) model achieves superior diversification and higher absolute returns, though with higher volatility, demonstrating a compelling trade-off for target-oriented investors.