Sparse precision matrices for minimum variance portfolios

Abstract

Financial crises are typically characterized by highly positively correlated asset returns due to the simultaneous distress on almost all securities, high volatilities and the presence of extreme returns. In the aftermath of the 2008 crisis, investors were prompted even further to look for portfolios that minimize risk and can better deal with estimation error in the inputs of the asset allocation models. The minimum variance portfolio à la Markowitz is considered the reference model for risk minimization in equity markets, due to its simplicity in the optimization as well as its need for just one input estimate: the inverse of the covariance estimate, or the so-called precision matrix. In this paper, we propose a data-driven portfolio framework based on two regularization methods, glasso and tlasso, that provide sparse estimates of the precision matrix by penalizing its $L_1$-norm. Glasso and tlasso rely on asset returns Gaussianity or t-Student assumptions, respectively. Simulation and real-world data results support the proposed methods compared to state-of-art approaches, such as random matrix and Ledoit–Wolf shrinkage.

Appendices

A The glasso algorithm

Here we briefly describe the algorithm proposed by Friedman et al. (2008) to solve (6), the glasso model. For convenience, we define $X_i$ as the ith element of X, and and $X_\backslash i$ as the vector of all the elements of X except the ith. We also define the matrices $\mathbf{G }$ to be the estimate of $\varvec{\varSigma }$, and $\mathbf{S }$ the sample covariance matrix. Furthermore, we identify the following partitions:¹⁰

$$\begin{aligned} \mathbf{G }= \begin{pmatrix} \mathbf{G }_{\backslash i,\backslash i} &{} \mathbf{g }_{\backslash i,i}\\ \mathbf{g }_{\backslash i,i}' &{} g_{i,i} \end{pmatrix}, \qquad \mathbf{S }= \begin{pmatrix} \mathbf{S }_{\backslash i, \backslash i} &{} \mathbf{s }_{\backslash i,i}\\ \mathbf{s }_{\backslash i ,i}' &{} s_{i,i} \end{pmatrix}. \end{aligned}$$

(20)

Banerjee et al. (2008) show that the solution for $w_{\backslash i,i}$ can be computed by solving the following box-constrained quadratic program:

$$\begin{aligned} g_{\backslash i,i} = \arg \min _y \left\{ y'\mathbf{G }_{\backslash i,\backslash i}^{-1}y:||y-\mathbf{s }_{\backslash i,i}||_{\infty } \le \rho \right\} , \end{aligned}$$

(21)

or in an equivalent way, by solving the dual problem

$$\begin{aligned} \min _{\beta ^{(i)}}\left\{ \frac{1}{2}||\mathbf{G }_{\backslash i,\backslash i}^{1/2}\beta ^{(i)}-c||^2+\rho ||\beta ^{(i)}||_1\right\} , \end{aligned}$$

(22)

where $c = \mathbf{G }_{\backslash i,\backslash i}^{-1/2}\mathbf{s }_{\backslash i,i}$ and ${\hat{\beta }}^{(i)} = \mathbf{G }_{\backslash i,\backslash i}^{-1}\mathbf{g }_{\backslash i, i}$. As noted by Friedman et al. (2008), (22) resembles a lasso least square problem (see Tibshirani 1996). The algorithm estimates then the ith variable on the others using as input $\mathbf{G }_{\backslash i,\backslash i}$, where $\mathbf{G }_{\backslash i,\backslash i}$ is the current estimate of the upper left block. The algorithm then updates the corresponding row and column of $\mathbf{G }$ using $\mathbf{g }_{\backslash i, i} = \mathbf{G }_{\backslash i,\backslash i}{\hat{\beta }}^{(i)}$ and cycles across the variables until convergence.

Glasso algorithm

• 1. Start with $\mathbf{G }= \mathbf{S }+ \rho \mathbf{I }$ . The diagonal of $\mathbf{G }$ is unchanged in the next steps.

• 2. For each $i = 1,2,\ldots ,n,1,2,\ldots , n,\ldots$, solve the lasso problem (22), which takes as input $\mathbf{G }_{\backslash i,\backslash i}$ and $\mathbf{s }_{\backslash i,i}$. This gives a $n - 1$ vector solution ${\hat{\beta }}$. Fill in the corresponding row and column of $\mathbf{G }$ using $\mathbf{g }_{\backslash i,i} = \mathbf{G }_{\backslash i,\backslash i}{\hat{\beta }}$.

• 3. Repeat until a convergence criterion is satisfied.

The algorithm has a computational complexity of $O(n^3)$ for dense problems, and considerably less than that for sparse problems (Friedman et al. 2008).

B Alternative covariance estimation methods

Here, we briefly describe the benchmark covariance estimators we use in the comparative analysis. Differently from glasso and tlasso, these approaches provide an estimate for the covariance matrix and not for the precision matrix. Hence, we compute the precision matrix for such methods to be plug-in into the minimum variance portfolio by inverting the covariance.

In particular, we consider the sample covariance and the equally weighted methods (that are commonly regarded as naive approaches) and two state-of-art estimators: random matrix theory and Ledoit Wolf Shrinkage.

The equally weighted (EW) portfolio, a tough benchmark to beat (DeMiguel et al. 2009b), can be interpreted as an extreme shrinkage estimator of the global minimum variance portfolio, obtained using the identity matrix as the estimate of the covariance matrix. Indeed, using (3), we obtain $\hat{\mathbf{w }}_{EW} = \dfrac{\mathbf{I }\mathbf{1 }}{\mathbf{1 }' \mathbf{I }\mathbf{1 }} = \frac{1}{n}\mathbf{1 }$. By assuming zero correlations and equal variances, such approach is very conservative in terms of estimation error and it suitable in case of severe unpredictability of the parameters.

The second naive approach is the sample covariance estimator, defined as:

$$\begin{aligned} \mathbf{S }= \frac{1}{t-1} \sum _{\tau =1}^t (X_{\tau }-{\bar{X}}) (X_{\tau }-{\bar{X}})', \end{aligned}$$

(23)

where t is the length of the estimation period, $X_i$ is the multivariate variate vector of assets’ returns at time $\tau$ and ${\bar{X}}$ is the vector of the average return for the n assets. Such estimator, when computed on datasets with a number of asset close to the length of the window size, is typically characterized by a larger eigenvalue dispersion compared to true covariance matrix, causing the matrix to be ill-conditioned (Meucci 2009). Therefore, when computing the precision matrix by inverting the covariance matrix, estimates are typically not reliable and unstable on different samples as its ill-conditioning nature amplifies the effects of the estimation error in the covariance matrix.

The shrinkage methodology of Ledoit-Wolf (LW) is well-known to better control for the presence of estimation errors, especially for datasets with a large ratio of n / t, where n is the number of assets and t the length of the estimation window. The Ledoit-Wolf shrinkage estimator is defined to be a convex combination of the sample covariance matrix $\mathbf{S }$ and $\widehat{\varvec{\varSigma }}_{T}$, a highly structured target estimator, such that $\widehat{\varvec{\varSigma }}_{LW} = a \mathbf{S }+ (1-a) \widehat{\varvec{\varSigma }}_{T}$ with $a \in [0,1]$. Following Ledoit and Wolf (2004a), we consider as structured estimator $\widehat{\varvec{\varSigma }}_{T}$ the constant correlation matrix, such that all the pairwise correlations are identical and equal to the average of all the sample pairwise correlations. As the target estimator is characterized by good conditioning, the resulting shrinkage estimator $\widehat{\varvec{\varSigma }}_{LW}$ has a smaller eigenvalues dispersion than the sample covariance matrix. In fact, the sample covariance matrix is shrunk towards the structured estimator, with intensity depending on the value of the shrinkage constanta. Ledoit–Wolf estimation of a is based on the minimization of the expected distance between $\widehat{\varvec{\varSigma }}_{LW}$ and $\varvec{\varSigma }$. For further details, the reader is referred to Ledoit and Wolf (2004a).¹¹

The last approach we focus on is the so called random matrix theory (RMT) estimator $\widehat{\varvec{\varSigma }}_{RMT}$, introduced by Laloux et al. (1999). The approach is based on the fact that, in the case of financial time series, the smallest eigenvalues of the correlation matrices are often dominated by noise. From the known distribution of the eigenvalues of a random matrix, it is possible then to filter out the part of spectrum that is likely associated with estimation error and maintain only the eigenvalues that carry useful information (Laloux et al. 1999). In particular, when assuming i.i.d. returns, the eigenvalues of the sample correlation matrix are then distributed according to a Marcenko–Pastur (MP) distribution as a consequence of the estimation error. Therefore, we can compute the eigenvalues that correspond to noise based on the minimum and maximum eigenvalues of the theoretical distribution, such that:

$$\begin{aligned} \lambda _{\min \max } = \sigma ^2 \big (1\pm \sqrt{n/t}\big )^2, \end{aligned}$$

(24)

where $\lambda _{\min }$ and $\lambda _{\max }$ are the theoretical smallest and largest eigenvalues in a $n\times n$ random covariance matrix estimated by a sample of t observations and $\sigma ^2$ is the variance of the i.i.d. asset returns. Only the eigenvalues outside the interval [$\lambda _{\min }$, $\lambda _{\max }$] are then assumed to bring useful information, while the others correspond to noise. Here, we estimate the covariance matrix then by eigenvalue clip**, a technique that consists in substituting the eigenvalues smaller than $\lambda _{\max }$ with their average:

$$\begin{aligned} \widehat{\varvec{\varSigma }}_{RMT} = \mathbf{V }\varvec{\varLambda }_{RMT}\mathbf{V }', \end{aligned}$$

(25)

where $\mathbf{V }$ represents the eigenvectors of the sample covariance matrix and $\varvec{\varLambda }_{RMT}$ is the diagonal matrix with the ordered eigenvalues, where the eigenvalues $\lambda \le \lambda _{\max }$ are substituted by their average (Bouchaud and Potters 2009). The RMT filtering has then the effect of averaging the lowest eigenvalues, improving the conditioning of the matrix and therefore reducing the sensitivity of the precision matrix to estimation errors.

For further details the reader is refereed to Laloux et al. (1999), Bouchaud and Potters (2009) and Bruder et al. (2013).