Abstract
This paper discusses the effects of introducing nonlinear interactions and noise-filtering to the covariance matrix used in Markowitz’s portfolio allocation model, evaluating the technique’s performances for daily data from seven financial markets between January 2000 and August 2018. We estimated the covariance matrix by applying Kernel functions, and applied filtering following the theoretical distribution of the eigenvalues based on the Random Matrix Theory. The results were compared with the traditional linear Pearson estimator and robust estimation methods for covariance matrices. The results showed that noise-filtering yielded portfolios with significantly larger risk-adjusted profitability than its non-filtered counterpart for almost half of the tested cases. Moreover, we analyzed the improvements and setbacks of the nonlinear approaches over linear ones, discussing in which circumstances the additional complexity of nonlinear features seemed to predominantly add more noise or predictive performance.
Highlights
Finance can be defined as the research field that studies the management of value—for an arbitrary investor that operates inside the financial market, the value of the assets that he/she chose can be measured in terms of how profitable or risky they are
Regarding applications in quantitative finance, Laloux et al [47] compared the empirical eigenvalues density of major stock market data with their theoretical prediction, assuming that the covariance matrix was random following a Wishart distribution (If a vector of random matrix variables follows a multivariate Gaussian distribution, its Sample covariance matrix will follow a Wishart distribution [48]).The results showed that over 94% of the eigenvalues fell within the theoretical bounds, implying that less than 6% of the eigenvalues contain useful information; the largest eigenvalue is significantly higher than the theoretical upper bound, which is evidence that the covariance matrix estimated via Markowitz is composed of few very informative principal components and many low-valued eigenvalues dominated by noise
In view of the importance of controlling the complexity introduced alongside nonlinearities, in this paper we sought to verify whether the stylized behavior of the top eigenvalues persists after introducing nonlinearities into the covariance matrix, as well as the effect of cleaning the matrix’s noises in the portfolio profitability and consistency over time, in order to obtain insights regarding the cost–benefit relationship between using higher degrees of nonlinearity to estimate the covariance between financial assets and the out-of-sample performance of the resulting portfolios
Summary
Finance can be defined as the research field that studies the management of value—for an arbitrary investor that operates inside the financial market, the value of the assets that he/she chose can be measured in terms of how profitable or risky they are. This paper focused on those questions, investigating whether the use of a nonlinear and nonparametric covariance matrix or the application of noise-filtering techniques can help a financial investor to build better portfolios in terms of cumulative return and risk-adjusted measures, namely Sharpe and Sortino ratios. We analyzed various robust methods for estimating the covariance matrix, and whether nonlinearities and noise-filtering managed to bring improvements to the portfolios’ performance, which can be useful to the construction of portfolio-building strategies for financial investors.
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