Regularised Estimation of the Precision Matrix for Financial Data- Regularisation Through Portfolio Optimisation
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Typ
Examensarbete för masterexamen
Program
Publicerad
2019
Författare
Engström, Linn
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
High-dimensional statistics, rotation-invariant estimators, Modern portfolio theory.