📝 SummarXiv

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.