  Finding an optimal balance between risk and returns in investment portfolios
is a central challenge in quantitative finance, often addressed through
Markowitz portfolio theory (MPT). While traditional portfolio optimization is
carried out in a continuous fashion, as if stocks could be bought in fractional
increments, practical implementations often resort to approximations, as
fractional stocks are typically not tradeable. While these approximations are
effective for large investment budgets, they deteriorate as budgets decrease.
To alleviate this issue, a discrete Markowitz portfolio theory (DMPT) with
finite budgets and integer stock weights can be formulated, but results in a
non-polynomial (NP)-hard problem. Recent progress in quantum processing units
(QPUs), including quantum annealers, makes solving DMPT problems feasible. Our
study explores portfolio optimization on quantum annealers, establishing a
mapping between continuous and discrete Markowitz portfolio theories. We find
that correctly normalized discrete portfolios converge to continuous solutions
as budgets increase. Our DMPT implementation provides efficient frontier
solutions, outperforming traditional rounding methods, even for moderate
budgets. Responding to the demand for environmentally and socially responsible
investments, we enhance our discrete portfolio optimization with ESG
(environmental, social, governance) ratings for EURO STOXX 50 index stocks. We
introduce a utility function incorporating ESG ratings to balance risk, return,
and ESG-friendliness, and discuss implications for ESG-aware investors.
