Statistical properties of estimators for the log-optimal portfolio
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Statistical properties of estimators for the log-optimal portfolio Gabriel Frahm1 Received: 1 April 2019 / Revised: 7 January 2020 © The Author(s) 2020
Abstract The best constant re-balanced portfolio represents the standard estimator for the logoptimal portfolio. It is shown that a quadratic approximation of log-returns works very well on a daily basis and a mean-variance estimator is proposed as an alternative to the best constant re-balanced portfolio. It can easily be computed and the numerical algorithm is very fast even if the number of dimensions is high. Some small-sample and the basic large-sample properties of the estimators are derived. The asymptotic results can be used for constructing hypothesis tests and for computing confidence regions. For this purpose, one should apply a finite-sample correction, which substantially improves the large-sample approximation. However, it is shown that the impact of estimation errors concerning the expected asset returns is serious. The given results confirm a general rule, which has become folklore during the last decades, namely that portfolio optimization typically fails on estimating expected asset returns. Keywords Best constant re-balanced portfolio · Estimation risk · Growth-optimal portfolio · Log-optimal portfolio · Mean-variance optimization JEL Classification C13 · G11
1 Motivation During the last decades, the log-optimal portfolio (LOP) has become increasingly important in portfolio theory. There is a significant number of publications related to the LOP—or to the growth-optimal portfolio (GOP), which is often treated synonymously. The reader can find a huge number of articles in MacLean et al. (2011). For an overview on the subject matter see Christensen (2005). In spite of a controversial debate (Merton
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Gabriel Frahm [email protected] Department of Mathematics and Statistics, Chair of Applied Stochastics and Risk Management, Helmut Schmidt University, Hamburg, Germany
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and Samuelson 1974), it cannot be denied that the LOP has a number of nice and beautiful properties. For example, it is asymptotically optimal among all portfolios that share the same constraints on the portfolio weights (Cover and Thomas 1991, Chapter 15). Moreover, the LOP can be considered a discrete-time approximation of the GOP, which serves as a numéraire portfolio and thus plays a major role in financial mathematics (Karatzas and Kardaras 2007; Platen and Heath 2006). The GOP provides a link between financial mathematics, neoclassical finance, and financial econometrics (Frahm 2016). Hence, the LOP is of particular interest for a variety of reasons. In this work, the statistical properties of LOP estimators are investigated. To the best of my knowledge, this is not done so far in the literature. We will consider the standard estimator for the LOP, i.e., the best constant re-balanced portfolio (BCRP), and the mean-variance estimator (MVE), which is based on a quadratic approximation of logreturns. The question of whether or not the BCRP or the MVE outperforms an
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