The set of all portfolios with the same utility score plots as a risk-indifference curve. An investor will accept any portfolio with a utility score on her risk-indifference curve as being equally acceptable.
However, there are many possible portfolios on many risk-indifference curves that do not yield the highest return for a given risk. All of these portfolios lie below the efficient frontier. The optimal portfolio is a portfolio on the efficient frontier that would yield the best combination of return and risk for a given investor, which would give that investor the most satisfaction.
These risk-indifference curves, calculated with the utility formula with the risk aversion coefficient = 2, but with higher utility values resulting from setting the risk-free rate to successively higher values. Of course, any investor, regardless of risk aversion, would like to receive a higher return for the same risk. The utility of these risk-indifference curves is that they allow the selection of the optimum portfolio out of all those that are attainable by combining these curves with the efficient frontier. Where 1 of the curves intersects the efficient frontier at a single point is the portfolio that will yield the best risk-return trade-off for the risk that the investor is willing to accept.
In the graph below, risk-indifference curves are plotted along with the investment opportunity set of attainable portfolios. Data points outside of the investment opportunity set designate portfolios that are not attainable, while those portfolios that lie along the northwest boundary of the investment opportunity set is the efficient frontier. All portfolios that lie below the efficient frontier have a risk-return trade-off that is inferior to those that lie on the efficient frontier. If a utility curve intersects the efficient frontier at 2 points, there are a number of portfolios on the same curve that lie below the efficient frontier; hence they are not optimal. Remember that all points on a risk-indifference curve are equally attractive to the investor; therefore, if any points on the indifference curve lie below the efficient frontier, then no point on that curve can be an optimum portfolio for the investor. If a utility curve lies wholly above the efficient frontier, then there is no attainable portfolio on that utility curve.
However, there is a utility curve such that it intersects the efficient frontier at a single point this is the optimum portfolio. The only attainable portfolio is on the efficient frontier, and thus, provides the greatest satisfaction to the investor. The optimum portfolio will yield the highest return for the amount of risk that the investor is willing to take.