Critical Finance Review, 2015, 4: 117–138

（不）可能边界：评论

作者：Moshe Levy (The Hebrew University), Richard Roll (UCLA)

摘要：完全正的均值方差有效总体市场组合，即资产的权重均不为负的组合，是资本资产定价模型（CAPM）的重要均衡特征。Brennan和Lo（2010）指出，当每个有效组合都至少有一个负权重的资产时，有效边界是“不可能的”。他们证明了，对于随机抽取的协方差矩阵，随着资产数量的增长，不可能边界出现的概率接近于1。不管取样方法如何，实证样本参数也总是得到不可能边界。这些结果看似是对CAPM的致命打击。然而，我们在本文中指出，在估计误差范围内，样本参数的微小变化可以形成具有一部分正的投资组合的边界。产生可能边界的参数有点类似于实数轴上的有理数：它们在参数空间不占据位置，但其附近总会存在一组参数。因此，从不可能边界出发，随着资产价格趋于经济均衡，资产价格的轻微变化提供了与CAPM相符的可能边界。

关键词：均值方差分析，资本资产定价模型，投资组合最优化，卖空，反向优化

(Im)Possible Frontiers: A Comment

Moshe Levy (The Hebrew University), Richard Roll (UCLA)

ABSTRACT

A totally positive mean-variance efficient aggregate market portfolio—one with no negative weights—is the key equilibrium feature of the Capital Asset Pricing Model (CAPM). Brennan and Lo (2010) designate an efficient frontier as “impossible” when every efficient portfolio has at least one negative weight. For randomly drawn covariance matrices, they prove that the probability of an impossible frontier approaches 1 as the number of assets grows. Impossible frontiers are also invariably found with empirical sample parameters, regardless of the sampling method. These results might seem like a deadly blow to the CAPM. However, we show here that slight variations in sample parameters, well within estimation error bounds, can lead to frontiers with positive portfolio segments. Parameters producing possible frontiers are somewhat like rational numbers on the real line: they occupy a zero-measure of parameter space, but there is always one close by. Thus, starting from an impossible frontier, slight changes in asset prices, as they converge to an economic equilibrium, deliver a possible frontier, consistent with the CAPM.

Keywords: Mean-variance analysis, CAPM, portfolio optimization, short selling, reverse optimization

原文链接：http://cfr.ivo-welch.info/readers/pub/cfr-015.pdf

翻译：任兆月