21 research outputs found
On the Exact Solution of the Multi-Period Portfolio Choice Problem for an Exponential Utility under Return Predictability
In this paper we derive the exact solution of the multi-period portfolio
choice problem for an exponential utility function under return predictability.
It is assumed that the asset returns depend on predictable variables and that
the joint random process of the asset returns and the predictable variables
follow a vector autoregressive process. We prove that the optimal portfolio
weights depend on the covariance matrices of the next two periods and the
conditional mean vector of the next period. The case without predictable
variables and the case of independent asset returns are partial cases of our
solution. Furthermore, we provide an empirical study where the cumulative
empirical distribution function of the investor's wealth is calculated using
the exact solution. It is compared with the investment strategy obtained under
the additional assumption that the asset returns are independently distributed.Comment: 16 pages, 2 figure
Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix
In this work we construct an optimal shrinkage estimator for the precision
matrix in high dimensions. We consider the general asymptotics when the number
of variables and the sample size so
that . The precision matrix is estimated
directly, without inverting the corresponding estimator for the covariance
matrix. The recent results from the random matrix theory allow us to find the
asymptotic deterministic equivalents of the optimal shrinkage intensities and
estimate them consistently. The resulting distribution-free estimator has
almost surely the minimum Frobenius loss. Additionally, we prove that the
Frobenius norms of the inverse and of the pseudo-inverse sample covariance
matrices tend almost surely to deterministic quantities and estimate them
consistently. At the end, a simulation is provided where the suggested
estimator is compared with the estimators for the precision matrix proposed in
the literature. The optimal shrinkage estimator shows significant improvement
and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the
generalized inverse of the sample covariance matrix. The abstract was updated
accordingl
Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions
In this paper we consider the asymptotic distributions of functionals of the
sample covariance matrix and the sample mean vector obtained under the
assumption that the matrix of observations has a matrix-variate location
mixture of normal distributions. The central limit theorem is derived for the
product of the sample covariance matrix and the sample mean vector. Moreover,
we consider the product of the inverse sample covariance matrix and the mean
vector for which the central limit theorem is established as well. All results
are obtained under the large-dimensional asymptotic regime where the dimension
and the sample size approach to infinity such that when the sample covariance matrix does not need to be invertible and
otherwise.Comment: 30 pages, 8 figures, 1st revisio
Spectral analysis of large reflexive generalized inverse and Moore-Penrose inverse matrices
A reflexive generalized inverse and the Moore-Penrose inverse are often
confused in statistical literature but in fact they have completely different
behaviour in case the population covariance matrix is not a multiple of
identity. In this paper, we study the spectral properties of a reflexive
generalized inverse and of the Moore-Penrose inverse of the sample covariance
matrix. The obtained results are used to assess the difference in the
asymptotic behaviour of their eigenvalues.Comment: 13 pages, 1 figure, a letter/short articl
Bayesian Inference of the Multi-Period Optimal Portfolio for an Exponential Utility
We consider the estimation of the multi-period optimal portfolio obtained by
maximizing an exponential utility. Employing Jeffreys' non-informative prior
and the conjugate informative prior, we derive stochastic representations for
the optimal portfolio weights at each time point of portfolio reallocation.
This provides a direct access not only to the posterior distribution of the
portfolio weights but also to their point estimates together with uncertainties
and their asymptotic distributions. Furthermore, we present the posterior
predictive distribution for the investor's wealth at each time point of the
investment period in terms of a stochastic representation for the future wealth
realization. This in turn makes it possible to use quantile-based risk measures
or to calculate the probability of default. We apply the suggested Bayesian
approach to assess the uncertainty in the multi-period optimal portfolio by
considering assets from the FTSE 100 in the weeks after the British referendum
to leave the European Union. The behaviour of the novel portfolio estimation
method in a precarious market situation is illustrated by calculating the
predictive wealth, the risk associated with the holding portfolio, and the
default probability in each period.Comment: 38 pages, 5 figure
Central limit theorems for functionals of large dimensional sample covariance matrix and mean vector in matrix-variate skewed model
In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix variate general skew normal distribution. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of an inverse covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large dimensional asymptotic regime where the dimension p and sample size n approach to infinity such that p/n β c β (0, 1)
Statistical inference for the EU portfolio in high dimensions
In this paper, using the shrinkage-based approach for portfolio weights and
modern results from random matrix theory we construct an effective procedure
for testing the efficiency of the expected utility (EU) portfolio and discuss
the asymptotic behavior of the proposed test statistic under the
high-dimensional asymptotic regime, namely when the number of assets
increases at the same rate as the sample size such that their ratio
approaches a positive constant as . We provide an
extensive simulation study where the power function and receiver operating
characteristic curves of the test are analyzed. In the empirical study, the
methodology is applied to the returns of S\&P 500 constituents.Comment: 27 pages, 5 figures, 2 table