Problem $1(70$ points$)$

In Portfolio Omtimization, one is interested in optimally allocating a certain amount of

money into a set of $N$ assets $($stocks$),$ where the portfolio usually denoted via

$win{R}^N$ is the normalized money invested in each asset $($such that the sum

${\displaystyle \underset{i=1}{\overset{N}{}}}{w}_i$ equals $1).$ In Modern Portfolio Theory known after Harry Markowitz, the

portfolio design is based on maximizing the return while minimizing the risk. This

scheme, also known as Mean$-$Variance Portfolio $($MVP$),$ can be formulated via the

following optimization problem:

$mamiz{e}_w,w^{TT}$

subject $tow0,w^{TT}1=1,w^{TT}w$

where $$ is the expected $($mean$)$ returns of the stocks, and $$ is the covariance matrix

of the stock returns. The term $w^{TT}w$ represents the portfolio risk $($variance$),$ the

square root of which is called volatility. Hence, the objective is to maximize the profit

while minimizing the risk, with the parameter $$ controlling the accepted level of risk.

Here, we have

$=\left[\begin{array}{ccccccc}1\mathrm{.}0& 0\mathrm{.}02& -0\mathrm{.}01& 1\mathrm{.}0& -0\mathrm{.}1& -0\mathrm{.}1& 1\mathrm{.}0\end{array}\right]$

$-0\mathrm{.}01$

$-0\mathrm{.}01$

$0\mathrm{.}02$

$=[0\mathrm{.}06,0\mathrm{.}01,0\mathrm{.}03{]}^{TT}$