  The investment risk minimization problem with budget and return constraints
has been the subject of research using replica analysis but there are
shortcomings in the extant literature. With respect to Tobin's separation
theorem and the capital asset pricing model, it is necessary to investigate the
implications of a risk-free asset and examine its influence on the optimal
portfolio. Accordingly, in this work, we explore the investment risk
minimization problem in the presence of a risk-free asset with budget and
return constraints. Moreover, we discuss opportunity loss, the Pythagorean
theorem of the Sharpe ratio, and Tobin's separation theorem.
