Regularized covariance estimation with applications to portfolio optimization

Abstract

• Department of Economics and Statistics, University of Salerno, Italy [pcoretto@unisa.it]

• Department of Economics and Statistics, University of Salerno, Italy [storti@unisa.it]

Keywords: Covariance matrix estimation – Robustness – Regularization

1 Introduction

The Markowitz portfolio optimization framework and its modern evolutions are central to understanding financial markets. Portfolio optimization techniques rely heavily on covariance matrix estimation. Suppose we have $N$ tradable financial assets and let ${𝐫}_t\in {ℝ}^N$ denote the random returns of the assets at time $t$. When returns are observed over short periods, based on the efficient-market hypothesis of Fama [1970], it is common to assume the so-called iid model. The returns are simply modeled as ${𝐫}_t=𝝁+{\mathit{ϵ}}_t$, where $𝝁\in {ℝ}^N$ denotes the expected return and ${\mathit{ϵ}}_t\in {ℝ}^N$ a zero-mean random fluctuation with covariance matrix $𝚺\in {ℝ}^{N\times N}$. Let ${𝒑}_t\in {ℝ}^N$ denote the vector of assets prices at time $t$, the previous model corresponds to assume the classical random walk model for the log-prices, that is ${𝐫}_t=\log {𝐩}_t-\log {𝐩}_{t-1}$. ($𝝁$, $𝚺$) They are at the heart of most portfolio optimization techniques and have to be estimated using historical data $\{{𝐫}_1,\mathrm{\dots },{𝐫}_T\}$. The traditional estimator is the sample mean and covariance matrix pair; however, they are extremely noisy and inefficient for three main reasons: (i) lack of stationarity in large samples; (ii) large aspect ratio $\gamma =N/T$; (iii) returns often exhibit large abnormal variations. Financial data is not stationary over long periods of time. Therefore, the sampling window length $T$ cannot be arbitrarily large. Therefore, with large modern portfolios, covariance estimation is often performed in the context of a large aspect ratio, leading to large estimation errors and often close to numerical non-invertibility. The invertibility of the estimated covariance matrix is crucial for portfolio optimization. Moreover, abnormal variations in a sampling window are common due to the intrinsic heavy-tailed nature of the asset returns distribution and the presence of outlying returns due to sharp changes caused by strong market shocks. The major approach to solve the excessive noise due to the large $\gamma$ in the financial literature has been the shrinkage regularization proposed by Ledoit and Wolf [2004] and the large body of literature following their paper [see Ledoit and Wolf, 2020, for a recent survey of the literature]. MLE under heavy-tailed distributions and classical robust estimators have been considered to overcome the problem of the presence of large variations in the log-returns data. Most resistant scatter estimators are not designed to cope with situations where the aspect ratio $\gamma \ge 1$. Recently, there has been an increasing interest in regularized robust covariance estimators. However, they look for sparse estimates, which may not be appropriate in the financial context. We refer to the reading of Tyler et al. [2023] for recent references.

2 Contribution

We propose an estimator that can deal with heavy tails, outliers, and an aspect ratio $\gamma >1$. Our proposal is an MLE for elliptical-symmetric distributions, including heavy-tailed models such as the Student-t distribution, with a constraint that bounds the condition number under the spectral norm of the covariance matrix. The condition number is the ratio between the covariance matrix’s maximum and minimum eigenvalue. The centrality and scatter parameter are estimated jointly; the resulting covariance matrix estimate is positive definite even if $\gamma >1$ and it does not pursue sparsity. The proposed estimator is a generalization of the MLE under the Gaussian model with condition number regularization proposed by Won et al. [2012]. We develop a feasible EM algorithm that almost only relies on closed-form calculations in each step. We provide sufficient conditions for the convergence of the algorithm. The condition number is treated as a hyper-parameter. We explore the sensitivity of the estimator to the hyper-parameter setting via numerical experiments, and we explore data-driven strategies for its optimal tuning. Finally, we compare the performance of our proposal with state-of-the-art alternatives in terms of optimization of high-dimensional portfolios.