Problem $1(2$ points$).$ Portfolio Optimization

Suppose you currently have a portfolio of three stocks, A$,$ B$,$ and C$.$ You own $500$ shares of A$,300$ of B$,$ and $200$ of C$.$ The current share prices are $$42\mathrm{.}76,$ $$81\mathrm{.}33,$ and $$58\mathrm{.}22,$ respectively. The expected returns and the covariances are provided in the table below.$$Stock A

Stock B

Stock C

Expected Return

$12\mathrm{.}00\%$

$9\mathrm{.}40\%$

$14\mathrm{.}00\%$

Covariance

Stock A

$0\mathrm{.}09164319$

$-0\mathrm{.}0188809$

$-0\mathrm{.}0022386$

Stock B

$-0\mathrm{.}01888085$

$0\mathrm{.}03885393$

$-0\mathrm{.}0005562$

Stock C

$-0\mathrm{.}00223864$

$-0\mathrm{.}0005562$

$0\mathrm{.}00807181.$ Compute the expected return of your current portfolio.

$2.$ You want to trade stocks so that your new portfolio$$s return can be no less than $13\%$ while minimizing new portfolio$$s risk $($standard deviation$).$ There is a $1\%$ fee on selling or trading each stock. For example, if you sell $10$ shares of Stock A and buy $5$ shares of Stock B$,$ the total dollar amount traded is $10*$$$42\mathrm{.}76+5*$$$81\mathrm{.}33=$$$834\mathrm{.}25$ and you have to pay a transaction fee of $$8\mathrm{.}34.$ These two transactions generate a net amount of $10*$$$42\mathrm{.}76-5*$$$81\mathrm{.}33-$ $$8\mathrm{.}34=$$$12\mathrm{.}61$ in cash. You would like this net amount to be always non$-$negative but no more than $$5.$ Also, you cannot have short position in any stock, i$.$e$.,$ you cannot sell more than what you have.

Formulate a nonlinear optimization model that determines the number of shares of each stock to buy and sell. First solve the model without integer constraints. Then solve it with integer constraints.

$1$

There are two ways to define the decision variables. One has three variables and the other has six variables. With three decision variables, you will have to use either the absolute function or the max function in your formulation. They are acceptable for this assignment.