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Given that not all points on the efficient portfolio frontier in the mean-variance space have constant returns to scale, and investments are determined according to the investor’s preferences, as investors may have conflicting preferences regarding the return and risk. Thus, risk-taking and expected returns are the criteria that determine the utility amount of the investment from selecting a portfolio of investment in assets. As a result, we may have a significant increase (reduction) in the return (risk) of efficient portfolio obtained with a small increase (reduction) in the amount of risk aversion (expected return). Two models presented in this paper are used to investigate this issue. The models are as follows: after obtaining an efficient portfolio, the first model maximises the ratio of the reduction of the risk rate to the reduction of the return rate. Also the second model maximises the ratio of increase in the return rate to the increase of the risk rate. If the optimal value is greater than one, the return (the risk) can be reduced (increased) to the extent obtained from this model to further reduce the risk (increase the return).

Investigating the ratio of risk-to-return changes in efficient portfolio to achieve a new efficient portfolio