771 research outputs found
Sparse and stable Markowitz portfolios
We consider the problem of portfolio selection within the classical Markowitz meanvariance optimizing framework, which has served as the basis for modern portfolio theory for more than 50 years. Efforts to translate this theoretical foundation into a viable portfolio construction algorithm have been plagued by technical difficulties stemming from the instability of the original optimization problem with respect to the available data. Often, instabilities of this type disappear when a regularizing constraint or penalty term is incorporated in the optimization procedure. This approach seems not to have been used in portfolio design until very recently. To provide such a stabilization, we propose to add to the Markowitz objective function a penalty which is proportional to the sum of the absolute values of the portfolio weights. This penalty stabilizes the optimization problem, automatically encourages sparse portfolios, and facilitates an effective treatment of transaction costs. We implement our methodology using as our securities two sets of portfolios constructed by Fama and French: the 48 industry portfolios and 100 portfolios formed on size and book-to-market. 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 naĂŻve portfolio comprising equal investments in each available asset. In addition to their excellent performance, these portfolios have only a small number of active positions, a desirable feature for small investors, for whom the fixed overhead portion of the transaction cost is not negligible. JEL Classification: G11, C00Penalized Regression, Portfolio Choice, Sparse Portfolio
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
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