PORTFOLIO SELECTION WITH LASSO ALGORITHM

1 Introduction The basis of the modern portfolio theory was developed by Harry Markowitz and published under the title "Portfolio Selection" in 1952 by Journal of Finance. Starting from Markovitz a vast amount of literature about mean-variance optimization of the excess return of a portfolio has been published (see, e.g. Elton et al. (2007), Brandt (2007)). The investors objective function is often defined as a trade-off between the expected portfolio return E(Xp)= and the risk of the portfolio, usually characterized by the portfolio variance V(Xp)= . This approach leads to the Global Minimum Variance Portfolio (GMVP): the portfolio with the smallest variance over all portfolios. It is known that the GMVP vector ω is given by It is evident that the inverse covariance matrix of asset returns (also called precision matrix) Θ = Σ-1, plays a fundamental role in definition of optimal portfolio weights. The problem is that the estimation of the covariance matrix is generally difficult because the number of unknown elements in the covariance matrix grows quickly with the size of the matrix and because of the positive-definiteness constraint. Depending on the applications, the sparsity of the covariance matrix or precision matrix is frequently imposed to strike a balance between biases and variances. Many authors have studied this problem and have proposed different methods to deal directly with the individual elements of the covariance matrix (among others Wong et al. (2003)). In 1996 Tibshirani has proposed the use of the Least Absolute Shrinkage and Selection Operator (Lasso) that allows both variable selection and shrinkage. Afterwards, Meinshausen and Buhlmann (2006) have proposed different Lasso algorithms to select zero elements in the precision matrix. In particular, they have proposed an algorithmic approach to find zeros in the matrix Θ, through a Lasso regression of each variable on all the other variables. They have also shown that the resulting estimator is asymptotically consistent in estimating the set of nonzero elements of Θ. Other authors have proposed algorithms for the exact maximization of the L1 penalized log-likelihood. A very fast algorithm to solve a Lasso problem is the Graphical Lasso algorithm (Glasso) proposed by Friedman et al. (2008). The R package Glasso (Friedman et al., 2008) allows to efficiently build a path of models for different values of the tuning parameter. Among other applications of Lasso in finance we remember Huang and Shi (2011) and Goto and Xu (2013). 2 The role of Lasso algorithm in portfolio optimization Suppose that an investor invests in q risky assets. A portfolio P is a linear combination of these q assets. Let ω ∊ ℝ be the vector of portfolio's weights, and let r = (r1,…, rq) be the q-dimensional random variable of asset returns. We assume that r follows a multivariate normal distribution with mean µ ∊ ℝk and positive definite covariance matrix Σ ∊ ℝkxk. The assumption of normality is very common in the literature. Consequently, the excess return (rp) of a portfolio of assets is a weighted average of the return on the individual asset. In our analysis we will exclude short sales, therefore ωi ≥ 0 for all i. The precision matrix Θ, that plays a fundamental role in definition of optimal portfolio weights, has a specific mathematical interpretation. While zeros in a covariance matrix Σ correspond to marginal independencies between variables, if the ij-th element of Θ, θij, is zero this implies that the corresponding variables Yi and Yj are conditionally independent, given the other variables (Mazumder and Hastie (2012), Banerjee et al. (2008)). Since the conditional independence plays an important role in many applications, some authors focus on the sparsity in Θ, rather than in Σ. In fact, sometimes happens that any of the assets that compose the portfolio are highly correlated, but the partial correlations rev