Mean-tail Gini framework for optimal portfolio selection

2025-09-21

AI Summary

The classical mean–variance (MV) framework fails for heavy-tailed, infinite-variance financial assets—such as cryptocurrencies—due to its reliance on normality assumptions and inability to capture downside risk via variance. Method: We propose a novel mean–tail Gini (MTG) portfolio optimization paradigm, using tail Gini as a coherent downside risk measure. We derive the MTG efficient frontier and, under left-tail exchangeability, obtain closed-form optimal portfolio weights—thereby avoiding the distortion amplification inherent in L₂-based risk measures. Tail indices are estimated via the generalized Pareto distribution. Results: Empirical analysis confirms infinite-variance characteristics in cryptocurrency markets. Relative to mean–tail variance (MTV), the MTG framework significantly reduces extreme losses and portfolio volatility, mitigates over-concentration in aggressive assets, and enhances risk-adjusted returns.

Technology Category

• Game Theory and Economic Paradigms: Behavioral Game Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies
• Security and Privacy: Cryptocurrency and smart contracts
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The limitations of the traditional mean-variance (MV) efficient frontier, as introduced by Markowitz (1952), have been extensively documented in the literature. Specifically, the assumptions of normally distributed returns or quadratic investor preferences are often unrealistic in practice. Moreover, variance is not always an appropriate risk measure, particularly for heavy-tailed and highly volatile distributions, such as those observed in insurance claims and cryptocurrency markets, which may exhibit infinite variance. To address these issues, Shalit and Yitzhaki (2005) proposed a mean-Gini (MG) framework for portfolio selection, which requires only finite first moments and accommodates non-normal return distributions. However, downside risk measures - such as tail variance - are generally considered more appropriate for capturing risk managers' risk preference than symmetric measures like variance or Gini. In response, we introduce a novel portfolio optimization framework based on a downside risk metric: the tail Gini. In the first part of the paper, we develop the mean-tail Gini (MTG) efficient frontier. Under the assumption of left-tail exchangeability, we derive closed-form solutions for the optimal portfolio weights corresponding to given expected returns. In the second part, we conduct an empirical study of the mean-tail variance (MTV) and MTG frontiers using data from equity and cryptocurrency markets. By fitting the empirical data to a generalized Pareto distribution, the estimated tail indices provide evidence of infinite-variance distributions in the cryptocurrency market. Additionally, the MTG approach demonstrates superior performance over MTV strategy by mitigating the amplification distortions induced by $mathrm{L}^2$-norm risk measures. The MTG framework helps avoid overly aggressive investment strategies, thereby reducing exposure to unforeseen losses.

Problem

• Addresses limitations of mean-variance framework for non-normal distributions
• Proposes tail Gini metric to better capture downside risk preferences
• Develops mean-tail Gini framework for optimal portfolio selection

Innovation

• Mean-tail Gini framework for downside risk optimization
• Closed-form portfolio weights under tail exchangeability
• Superior performance over mean-tail variance by mitigating distortions