Most investors are happy when their returns are above average, but not so happy when they are below average. In the Markowitz portfolio optimization model

Most investors are happy when their returns are “above average,” but not so happy when they are “below average.” In the Markowitz portfolio optimization model in Problem 14, the objective function is to minimize variance which is given by

Where Rs is the portfolio return under scenario s and is the expected or average return of the portfolio.
With this objective function, we are choosing a portfolio that minimizes deviations both above and below the average,. However, most investors are happy when, but unhappy when. With this preference in mind, an alternative to the variance measure in the objective function for the Markowitz model is the semivariance. The semivariance is calculated by only considering deviations below.
Let Dsp - Dsn = Rs - and restrict Dsp and Dsn to be nonnegative. Then Dsp measures the positive deviation from the mean return in scenario s (i.e., Dsp = Rs - when Rs >). In the case where the scenario return is below the average return, Rs , we have -Dsn = Rs - . Using these new variables we can reformulate the Markowitz model to only minimize the square of negative deviations below the average return. By doing so, we will use the semivariance rather than the variance in the objective function.
Reformulate the Markowitz portfolio optimization model given in Problem 14 to use semivariance in the objective function. Solve the model using either Excel Solver orLINGO.

Min (R, -R