Markowitz portfolio optimization: Harry Markowitz received the 1990 Nobel Prize for his path-breaking work in portfolio optimization. One version of the Markowitz model is based on minimizing the variance of the portfolio subject to a constraint on return. The below table shows the annual return (%) for five 1-year periods for the six mutual funds with the last row that gives the S&P 500 return for each planning scenario. Scenario 1 represents a year in which the annual returns are good for all the mutual funds. Scenario 2 is also a good year for most of the mutual funds. But scenario 3 is a bad year for the small-cap value fund; scenario 4 is a bad year for the intermediate-term bond fund; and scenario 5 is a bad year for four of the six mutual funds. MUTUAL FUND PERFORMANCE IN FIVE SELECTED YEARLY SCENARIOS Planning Scenarios Mutual Fund Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Foreign Stock Intermediate-Term Bond Large-Cap Growth Large-Cap Value Small-Cap Growth Small-Cap Value S&P 500 Return 10.06 17.64 32.41 32.36 33.44 24.56 13.12 3.25 18.71 20.61 19.40 25.32 13.47 7.51 33.28 12.93 3.85 -6.70 45.42 -1.33 41.46 7.06 58.68 5.43 -21.93 7.36 -23.26 -5.37 -9.02 17.31 25.00 20.00 8.00 30.00 - 10.00 If each of the scenarios is equally likely and occurs with probability 1/5, then the mean return or expected return of the portfolio is R = { R, S-1 The variance of the portfolio return is Vai (R,- R2 Using the scenario return data given in Table above, the Markowitz mean-variance model can be formulated. The objective function is the variance of the portfolio and should be minimized. Assume that the required return on the portfolio is 10%. There is also a unity constraint that all of the money must be invested in mutual funds. 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 given above, the objective function is to minimize variance, which is given by Min (R, - R2 where Rs is the portfolio return under scenario s and R 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, R. However, most investors are happy when Rs > R, but unhappy when Rs R, but unhappy when Rs