Noisy Covariance Matrices and Portfolio Optimization II

TL;DR

This paper investigates how noise in empirical covariance matrices affects portfolio optimization, revealing that the impact varies with the ratio of portfolio size to data length and the application context, thus resolving apparent contradictions in previous findings.

Contribution

It demonstrates that the influence of noise depends on the ratio of portfolio size to data length and the specific financial application, providing a nuanced understanding of covariance matrix reliability.

Findings

Noise impact increases with higher ratio r (e.g., 0.6)

For smaller r (around 0.2), noise effects are acceptable

Using covariance matrices for risk measurement reduces noise impact

Abstract

Recent studies inspired by results from random matrix theory [1,2,3] found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random. This seems, however, to be in contradiction with the fundamental role played by covariance matrices in finance, which constitute the pillars of modern investment theory and have also gained industry-wide applications in risk management. Our paper is an attempt to resolve this embarrassing paradox. The key observation is that the effect of noise strongly depends on the ratio r = n/T, where n is the size of the portfolio and T the length of the available time series. On the basis of numerical experiments and analytic results for some toy portfolio models we show that for relatively large values of r (e.g. 0.6) noise does, indeed, have the pronounced effect suggested by [1,2,3] and illustrated later by [4,5] in a portfolio optimization context, while for smaller r (around 0.2 or below), the error due to noise drops to acceptable levels. Since the length of available time series is for obvious reasons limited in any practical application, any bound imposed on the noise-induced error translates into a bound on the size of the portfolio. In a related set of experiments we find that the effect of noise depends also on whether the problem arises in asset allocation or in a risk measurement context: if covariance matrices are used simply for measuring the risk of portfolios with a fixed composition rather than as inputs to optimization, the effect of noise on the measured risk may become very small.