Improving portfolios global performance using a cleaned and robust covariance matrix estimate

Abstract

This paper presents how the use of a cleaned and robust covariance matrix estimate can improve significantly the overall performance of maximum variety and minimum variance portfolios. We assume that the asset returns are modelled through a multi-factor model where the error term is a multivariate and correlated elliptical symmetric noise extending the classical Gaussian assumptions. The factors are supposed to be unobservable and we focus on a recent method of model order selection, based on the random matrix theory to identify the most informative subspace and then to obtain a cleaned (or de-noised) covariance matrix estimate to be used in the maximum variety and minimum variance portfolio allocation processes. We apply our methodology on real market data and show the improvements it brings if compared with other techniques especially for non-homogeneous asset returns.

Appendix: Brief description of alternative covariance matrix estimators

Here, we briefly introduce some well-known covariance matrix estimators. In the following, $c = m/N$ and $\widehat{\mathbf {E}} = \widetilde{\mathbf {R}}\widetilde{\mathbf {R}}^{\mathrm{T}} / N$ is the standardized SCM where $\widetilde{\mathbf {R}} = (\widetilde{\mathbf {r}}_i)_{i \in [1,m]}$ as defined in Sect. 4.4.

1.1 A.1 Eigenvalue clipping (or RMT-SCM)

Laloux et al. (2000) proposed Eigenvalue clipping in order to separate signal and noise subspaces using Marčenko and Pastur (1967) boundary properties of the eigenvalues. The Eigenvalue clipping estimator of $\widehat{\mathbf {E}}$ is as follows:

$$\begin{aligned} \widehat{\mathbf {E}}_{clip} = \sum _{k=1}^m\lambda _k^{clip}\mathbf {u}_k\mathbf {u}_k^{\mathrm{T}} \end{aligned}$$

with $\mathbf {u}_k$ the eigenvector associated to the eigenvalue $\lambda _k$ of $\widehat{\mathbf {E}}$, and $\lambda _k^{clip}$ defined as follows:

$$\begin{aligned} \lambda _k^{clip} = {\left\{ \begin{array}{ll} \lambda _k, \text { if } \lambda _k \ge (1 + \sqrt{c})^2 \\ {\widetilde{\lambda }}, \text { otherwise} \end{array}\right. } \end{aligned}$$

(9)

where ${\widetilde{\lambda }}$ is chosen such that $Tr(\widehat{\mathbf {E}}_{clip}) = Tr(\widehat{\mathbf {E}})$.

1.2 A.2 Ledoit and Wolf shrinkage (or LW)

Ledoit and Wolf (2003) introduced some shrinkage estimators particularly adapted to financial asset returns and based on the single factor model of Sharpe (1964), where the factor is a market index. LW is a linear combination of the SCM and the covariance matrix containing the market information. This model can be written as follows:

$$\begin{aligned} r_{i,t} = \alpha _i + \beta _i \, F_t + \epsilon _{i,t}, \, \forall i \in [1,m] \text { and } \forall t \in [1,N] \end{aligned}$$

(10)

where $r_{i,t}$ is the return of stock i at time t , $\alpha _i$ is the active return of the asset i, $F_t$ is the market index return at time t, $\beta _i$ is the asset sensitivity to the market index return, and $\epsilon _{i,t}$ is the idiosyncratic return for asset i at t. This latter term is assumed to be uncorrelated to the market index. Then the covariance matrix writes:

$$\begin{aligned} \mathbf {M}_r = \sigma ^2_F \, \varvec{\beta } \, \varvec{\beta }^{\mathrm{T}} + \varvec{\Omega }_\epsilon \end{aligned}$$

with $\varvec{\beta } = [\beta _1, \ldots , \beta _m]^{\mathrm{T}}$, $\sigma ^2_F$ is the variance of the market returns and $\varvec{\Omega }_\epsilon$ the covariance matrix of the idiosyncratic error.

An estimator for $\mathbf {M}_r$ can be determined:

$$\begin{aligned} \widehat{\mathbf {M}}_r = {\hat{\sigma }}^2_F \, \widehat{\varvec{\beta }} \, \widehat{\varvec{\beta }}^{\mathrm{T}} + \widehat{\varvec{\Omega }}_\epsilon \end{aligned}$$

where each $\hat{\beta _i}$ is estimated individually using the OLS estimator based on Eq. (10) and the $\widehat{\varvec{\Omega }}_\epsilon$ is a diagonal matrix composed of the OLS residual variances. Finally, ${\hat{\sigma }}^2_F$ is the sample variance of the market returns.

The shrinkage-to-market estimator from Ledoit and Wolf is therefore equal to:

$$\begin{aligned} {\widehat{\varvec{\Sigma }}}(\gamma ) = \gamma \, \widehat{\mathbf {M}}_r + (1-\gamma ) \, \mathbf {S} \end{aligned}$$

where $\gamma \in [0, 1]$ is the shrinkage parameter estimated as in Ledoit and Wolf (2003), and $\mathbf {S}$ is the SCM of asset returns.

1.3 A.3 Rotational invariant estimator (or RIE)

Bun et al. (2016, 2017) proposed an optimal rotational invariant estimator for general covariance matrices by computing the overlap between the true and sample eigenvectors introduced first by Ledoit and Péché (2011). For large m, the optimal rotational invariant estimator (RIE) of $\widehat{\mathbf {E}}$ is as follows:

$$\begin{aligned} \widehat{\mathbf {E}}_{RIE} = \sum _{k=1}^m\lambda _k^{RIE} \, \mathbf {u}_k \, \mathbf {u}_k^{\mathrm{T}} \end{aligned}$$

with $\mathbf {u}_k$ the eigenvector associated to the eigenvalue $\lambda _k$ of $\widehat{\mathbf {E}}$, and $\lambda _k^{RIE}$ defined as follows:

$$\begin{aligned} \lambda _k^{RIE} = \dfrac{\lambda _k}{|1 - c + c \, z_k \, s(z_k)|^2} \end{aligned}$$

where $z_k = \lambda _k - i \, N^{-1/2}$ is a complex number and s(z) denotes the discrete form of the limiting Stieltjes transform

$$\begin{aligned} s(z) = \dfrac{1}{m}\sum _{j=1}^m\dfrac{1}{z - \lambda _j} \end{aligned}$$

We also ensure that $Tr(\widehat{\mathbf {E}}_{RIE}) = Tr(\widehat{\mathbf {E}})$. For this purpose, we multiply each $\lambda _k$ by $\nu$ with $\nu = \sum \nolimits _{k=1}^m \lambda _k / \sum \nolimits _{k=1}^m \lambda _k^{RIE}$.