  Managing a large-scale portfolio with many assets is one of the most
challenging tasks in the field of finance. It is partly because estimation of
either covariance or precision matrix of asset returns tends to be unstable or
even infeasible when the number of assets $p$ exceeds the number of
observations $n$. For this reason, most of the previous studies on portfolio
management have focused on the case of $p < n$. To deal with the case of $p >
n$, we propose to use a new Bayesian framework based on adaptive graphical
LASSO for estimating the precision matrix of asset returns in a large-scale
portfolio. Unlike the previous studies on graphical LASSO in the literature,
our approach utilizes a Bayesian estimation method for the precision matrix
proposed by Oya and Nakatsuma (2020) so that the positive definiteness of the
precision matrix should be always guaranteed. As an empirical application, we
construct the global minimum variance portfolio of $p=100$ for various values
of $n$ with the proposed approach as well as the non-Bayesian graphical LASSO
approach, and compare their out-of-sample performance with the equal weight
portfolio as the benchmark. In this comparison, the proposed approach produces
more stable results than the non-Bayesian approach in terms of Sharpe ratio,
portfolio composition and turnover. Furthermore, the proposed approach succeeds
in estimating the precision matrix even if $n$ is much smaller than $p$ and the
non-Bayesian approach fails to do so.
