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

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

$-0\mathrm{.}01$

$-0\mathrm{.}01$

$0\mathrm{.}02$

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

$($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.