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The idea behind the optimal ESG portfolio (OESGP) is to expand the mean–variance theory by adding the portfolio ESG value (PESGV) multiplied by the ESG strength parameter γ (which is the investor’s choice) to the minimizing objective function [26,27]. PESGV is assumed to be the sum of portfolio constituents’ weighted ESG ratings that are offered by several providers. In this work, we analyzed the sensitivity of the OESGP based on the constituents of the Dow Jones Index to the ESG ratings provided by MSCI, S&P Global, and Sustainalytics. We describe discrepancies among various ESG ratings for the same securities and their effects on the OESGP performance. We found that with growing γ, the OESGP diversity and Sharpe ratio may monotonically decrease. However, the ESG-tilted Sharpe ratio has one or two maximums. The 1st maximum exists at moderate values of γ and yields a moderately diversified OESGP, which can serve as a criterion for optimal ESG portfolios. The 2nd maximum at large γ corresponds to highly concentrated OESGPs. It appears as if the portfolio has one or two securities with a lucky combination of high returns and high ESG ratings.