Sparse and stable Markowitz portfolios

Abstract

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...

Optimal execution with non-linear transient market impact

“…Finally, the effect of a high spread cost is a substantial reduction in negative trading volume similar to what is observed by Brodie et al [16] for Markowitz portfolio weights. In that case a penalization term proportional to the sum of absolute values of weights leads to optimal solutions where the resulting portfolio is sparse with few assets and no-short positions.…”

“…From the point of view of the numerical cost minimization, we impose the additional constraints v i ≥ 0, ∀i (for a buy program). This additional constraint is analogous to the no-short-selling constraint in portfolio optimization [16].…”

“…C S [Π] = δ S X, for strategies where all trades have the same sign. This penalty is a form of L 1 or LASSO regularization widely used in computer science and used by Brodie et al [16] to penalize short positions in Markowitz portfolio optimization. Busseti and Lillo [19] have calibrated optimal execution strategies on real data by regularizing them with a spread cost of this form.…”

Efficient Cardinality/Mean-Variance Portfolios

“…(This time we needed a budget of the order of 10 7 function evaluations, see Appendix A.) We remark that FF48 and FF100 are the data sets also used in [8]. In this paper, as we said before, the authors focus on a modification to the Markowitz classical model by the incorporation of a term involving a multiple of the 1 norm of the vector of portfolio positions.…”

“…DeMiguel et al [13] constrained the Markowitz classical model by imposing a bound on the 1 norm of the vector of portfolio positions, among other possibilities. Brodie et al [8] focus on a modification to the Markowitz mean-variance classical model by the incorporation of a term involving a multiple of the 1 norm of the vector of portfolio positions. Inspired by sparse reconstruction (see, for instance, [7]), they also proposed an heuristic for the solution of the problem.…”