  In portfolio optimization problems, the minimum expected investment risk is
not always smaller than the expected minimal investment risk. That is, using a
well-known approach from operations research, it is possible to derive a
strategy that minimizes the expected investment risk, but this strategy does
not always result in the best rate of return on assets. Prior to making
investment decisions, it is important to an investor to know the potential
minimal investment risk (or the expected minimal investment risk) and to
determine the strategy that will maximize the return on assets. We use the
self-averaging property to analyze the potential minimal investment risk and
the concentrated investment level for the strategy that gives the best rate of
return. We compare the results from our method with the results obtained by the
operations research approach and with those obtained by a numerical simulation
using the optimal portfolio. The results of our method and the numerical
simulation are in agreement, but they differ from that of the operations
research approach.
