Critical Finance Review, 2015, 4: 117–138

（不）可能邊界：評論

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

摘要：完全正的均值方差有效總體市場組合，即資産的權重均不為負的組合，是資本資産定價模型（CAPM）的重要均衡特征。Brennan和Lo（2010）指出，當每個有效組合都至少有一個負權重的資産時，有效邊界是“不可能的”。他們證明了，對于随機抽取的協方差矩陣，随着資産數量的增長，不可能邊界出現的概率接近于1。不管取樣方法如何，實證樣本參數也總是得到不可能邊界。這些結果看似是對CAPM的緻命打擊。然而，我們在本文中指出，在估計誤差範圍内，樣本參數的微小變化可以形成具有一部分正的投資組合的邊界。産生可能邊界的參數有點類似于實數軸上的有理數：它們在參數空間不占據位置，但其附近總會存在一組參數。因此，從不可能邊界出發，随着資産價格趨于經濟均衡，資産價格的輕微變化提供了與CAPM相符的可能邊界。

關鍵詞：均值方差分析，資本資産定價模型，投資組合最優化，賣空，反向優化

(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

原文鍊接：http://cfr.ivo-welch.info/readers/pub/cfr-015.pdf

翻譯：任兆月