We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e., portfolios with only few active positions), and allows accounting for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naïve evenly weighted portfolio.penalized regression | portfolio choice | sparsity I n 1951, Harry Markowitz ushered in the modern era of portfolio theory by applying simple mathematical ideas to the problem of formulating optimal investment portfolios (1). He argued that single-minded pursuit of high returns constitutes a poor strategy, and suggested that rational investors must, instead, balance their desires for high returns and for low risk, as measured by variability of returns.It is not trivial, however, to translate Markowitz's conceptual framework into a portfolio selection algorithm in a real-world context. The recent survey (2) examined several portfolio construction algorithms inspired by the Markowitz framework. Given a reasonable amount of training data, the authors found none of the surveyed algorithms able to significantly or consistently outperform the naïve strategy where each available asset is given an equal weight in the portfolio. This disappointing performance is partly due to the structure of Markowitz's optimization framework. Specifically, the optimization at the core of the Markowitz scheme is empirically unstable: small changes in assumed asset returns, volatilities, or correlations can have large effects on the output of the optimization procedure. In this sense, the classic Markowitz portfolio optimization is an ill-posed (or ill-conditioned) inverse problem. Such problems are frequently encountered in other fields; a variety of regularization procedures have been proposed to tame the troublesome instabilities (3).In this article, we discuss a regularization of Markowitz's portfolio construction. We will restrict ourselves to the traditional Markowitz mean-variance approach. (Similar ideas could also be applied to different portfolio construction frameworks considered in the literature.) Moreover, we focus on one particular regularization method, and highlight some very special properties of the regularized portfolios obtained through its use.Our proposal consists of augmenting the original Markowitz objective function by adding a penalty term proportional to the sum of the absolute values of the portfolio weigh...

Regularization of ill-posed linear inverse problems via 1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an 1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implementation to 1 -constraints, using a gradient method, with projection on 1 -balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.

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