Regularised Estimation of the Precision Matrix for Financial Data- Regularisation Through Portfolio Optimisation

Abstract

A fundamental aspect of quantitative finance is portfolio optimisation, a field of mathematics that is very much governed by the Modern portfolio theory introduced by Harry Markowitz in 1952. The goal is to maximise the expected return for a given pre-determined level of risk. An optimal portfolio solving the problem is directly proportional to the inverse covariance matrix – the precision matrix – of the returns. Since the covariance structure in between markets is unknown, so is the precision matrix. Therefore, it must be estimated out of historical data, something that is not easily done due to the bad conditioning of the problem. There is thus a need for regularisation. This Master’s thesis proposes and derives a new estimator of the precision matrix, intending to minimise the expected distance between a pre-determined target level of risk and the actual risk of a Markowitz optimal portfolio. Since the proposed estimator belongs to the class of rotation-invariant estimators, minimisation is carried out by direct manipulation of its eigenvalues. Optimal parameters of a spectral mapping are found based on historical data. The mapping, defined by the computed optimal parameters, is then used for regularising the sample precision matrix of future data. The performance of the new estimator is compared with a simple l2-penalised sample estimator and with two l1- and l2-penalised maximum likelihood estimators. An estimator is considered to perform well if the risk of its corresponding Markowitz portfolio is close to the target risk, given that the estimator doesn’t underestimate the covariance out-of-sample. The results reveal that the choice of spectral mapping is of great importance for the strategy to be successful. For one of the investigated mappings the risk of the corresponding portfolio is indeed close to the target risk, even though the estimator seems to perform less well out-of-sample than some of the reference estimators. Further investigations of other mappings should be carried out.