$($a$)$ Find the eigenvalues of $.$ Is this matrix Positive Semi$-$Definite $($PSD$)?$

$($b$)$ Argue whether this problem is convex or not.

$($c$)$ Write a piece of code in Python using CVXPY package ${?}_{-}^1$ to solve the above problem

given the values of $=\{4,5,$dots,$13,14\}{10}^{-1}.$

For each solution $w_{}^{***},$ compute the following quantities

PortfolioExpectedReturn $()=w_{}^{**{*}^{TT}}$

PortfolioVolatility $()=\sqrt[\phantom{2}]{w_{}^{**{*}^{TT}}{}^{***}{w}_{}^{***}}$

$($d$)$ We say the constraint $w^{TT}w$ is active if equality holds. Find the values of $$

for which this constraint is active.

Hint: plot the value of risk versus $.$

$($e$)$ Plot the diagram of PortfolioExpectedReturn versus PortfolioVolatility.

For which value of $,$ is the ratio of

PortfolioExpectedReturn$/$PortfolioVolatility $($known as Sharpe Ratio$)$

maximum?

$($f$)$ Plot a curve including all the elements of $w_{}^{***}$ versus $.$

$($g$)$ Explain why the portfolio has this changing pattern when $$ increases; and what will

happen when $$ is pretty large.