solution

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

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

The variance of the portfolio return is

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

where 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, .
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 and restrict and to be
nonnegative. Then measures the positive deviation from
the mean return in
scenario (i.e., when ).

In the case where the scenario return is below the average
return, , we have . 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.

Solve the Markowitz portfolio optimization model that can be
prepared for above case to use semivariance in the objective
function. Solve the model using either Excel Solver or LINGO. Round
your answers to one decimal place. If your answer is zero, enter
“0” and for subtractive or negative numbers use a minus sign even
if there is a + sign before the blank.