  The mean-variance portfolio that considers the trade-off between expected
return and risk has been widely used in the problem of asset allocation for
multi-asset portfolios. However, since it is difficult to estimate the expected
return and the out-of-sample performance of the mean-variance portfolio is
poor, risk-based portfolio construction methods focusing only on risk have been
proposed, and are attracting attention mainly in practice. In terms of risk,
asset fluctuations that make up the portfolio are thought to have common
factors behind them, and principal component analysis, which is a dimension
reduction method, is applied to extract the factors. In this study, we propose
the Schr\"{o}dinger risk diversification portfolio as a factor risk
diversifying portfolio using Schr\"{o}dinger principal component analysis that
applies the Schr\"{o}dinger equation in quantum mechanics. The Schr\"{o}dinger
principal component analysis can accurately estimate the factors even if the
sample points are unequally spaced or in a small number, thus we can make
efficient risk diversification. The proposed method was verified to outperform
the conventional risk parity and other risk diversification portfolio
constructions.
