Alex Shkolnik (UC Santa Barbara): On the Markowitz Enigma for Minimum Variance
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Alex Shkolnik: On the Markowitz Enigma for Minimum Variance
Abstract: The Markowitz enigma entails the observation (by R. Michaud) that risk optimizers are, fundamentally, “estimation-error maximizers”. No exception to this principle, is principal component analysis (PCA), which is often used to construct equity risk models. We show that a PCA constructed minimum variance portfolio displays highly counterintuitive properties as more securities are added. For example, the ratio of the actual to the predicted portfolio risk grows without bound. The cause is the systematic (factor) risk that persists even as the number of securities tends to infinity. We derive a correction formula that adjusts the PCA model in such a way that, as the number of securities grows, this systematic risk vanishes. The resulting minimum variance portfolio achieves zero variance asymptotically. Aside from theorems we explore the results numerically by simulating the security returns from a multi-factor model that incorporates market risk as well as style and industry risk factors. Joint work with H. Gurdogan (UCLA).