Consider an investor who has $$20,000$ for investment, and is investing in stocks X$1,$ X$2,$ X$3,$ X$4.$ The

prices of these stocks during the last $12$ months are provided in the table below. The investor also has an

option of keeping the money and not investing $($Cash$).$ Suppose that the objective is to maximize total

value of assets in the portfolio while also penalizing variability.

Month

$123456789101112$

X$1404244404648504148384040$

X$2120118120116122115123124117119118116$

X$3606158525449553738403945$

X$412131415109121314161314$

Probability $0\mathrm{.}10\mathrm{.}050\mathrm{.}050\mathrm{.}030\mathrm{.}030\mathrm{.}030\mathrm{.}010\mathrm{.}20\mathrm{.}30\mathrm{.}060\mathrm{.}070\mathrm{.}07$

The prices $($these are not net returns$)$ shown are the closing prices on the last business day of each month.

Assume that you are currently in the end of month $12.$ i$.$e$.$ purchasing prices for stocks are those of month

$12.$

Assume also that someone gave you the probability distribution for the $12$ prices observed in the past as a

proxy for the likelihood of their occurrence in the future. This information is provided in the last row of

the table above.

Suppose that for ethical purposes the investor wishes to make sure that his combined allocation of funds

for X$1$ and X$2$ does not exceed combined allocation for X$3$ and X$4.$

For risk aversion coefficient assume the values of $0,0\mathrm{.}0001,0\mathrm{.}0005,0\mathrm{.}001,0\mathrm{.}004,0\mathrm{.}007,0\mathrm{.}01,0\mathrm{.}05,0\mathrm{.}2,$

$0\mathrm{.}3.$

$1.$ Turn in the mathematical formulation of this problem $($NO GAMS$).$ Carefully, cleanly, and

clearly define all your sets and variables.

$2.$ Use expected prices, based on past observations, for forecasting and determine the EV frontier

$($Expected value$-$ Variance graph, otherwise also known as locus of efficient portfolios$).$ Turn in the

graph of the EV frontier.

$3.$ Hand in your GAMS code which should include a report writing parameter. Report writing

parameter should display the table of results in the $.$lst file with risk aversion scenarios organized in

columns and corresponding outcomes in rows as shown below.

R$1$ R$2$ R$3$ R$4$ R$5$ R$6$ R$7$ R$8$ R$9$ R$10$

X$1$

X$2$

X$3$

X$4$

RAP

Cash

Expected Payoffs

Variance

Shadow price of funds