Sampling distributions of optimal portfolio weights and characteristics in low and large dimensions
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Date
2019
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Abstract
Optimal portfolio selection problems are determined by the (unknown) parameters of
the data generating process. If an investor want to realise the position suggested by the
optimal portfolios he/she needs to estimate the unknown parameters and to account the
parameter uncertainty into the decision process. Most often, the parameters of interest
are the population mean vector and the population covariance matrix of the asset re
turn distribution. In this paper we characterise the exact sampling distribution of the
estimated optimal portfolio weights and their characteristics by deriving their sampling
distribution which is present in terms of a stochastic representation. This approach pos
sesses several advantages, like (i) it determines the sampling distribution of the estimated
optimal portfolio weights by expressions which could be used to draw samples from this
distribution efficiently; (ii) the application of the derived stochastic representation pro
vides an easy way to obtain the asymptotic approximation of the sampling distribution.
The later property is used to show that the high-dimensional asymptotic distribution
of optimal portfolio weights is a multivariate normal and to determine its parameters.
Moreover, a consistent estimator of optimal portfolio weights and their characteristics
is derived under the high-dimensional settings. Via an extensive simulation study, we
investigate the finite-sample performance of the derived asymptotic approximation and
study its robustness to the violation of the model assumptions used in the derivation of
the theoretical results.
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Keywords
sampling distribution, high-dimensional asymptotics, stochastic rep resentation, parameter uncertainty, optimal portfolio