Problem $1(50$ points$)$

Consider the log$-$returns of $8$ popular stocks including 'AAPL', 'AMZN', 'GOOGL', 'META',

$\text{'}$MSFT$\text{'},\text{'}$NFLX$\text{'},$ 'NVDA' and TSLA', from $2015-81-01$ to $2023-12-31.$

The price data can be acquired using the following code:

from pandas$\_$datareader inport data as pdr

import yfinance as yf

yf$.$pdr$\_$override$()$

stocks $=[\text{'}$AAPL$\text{'},$ 'AMZN', 'GOOGL', 'META', $\text{'}$MSFT$\text{'},\text{'}$NFLX$\text{'},$ 'NVDA', 'TSLA'$]$

prices $=$ pdr$\_$get$\_$data$\_$yahoo$($stocks$,$ start$="2015-81-01",$ end$="2023-12-31")[[\text{'}$Adj C$]$

$($a$)$ Compute the log$-$returns of the stocks and divide the data into $70\%$ training and $30\%$ test.

Consider the Markowitz's Mean$-$Variance Portfolio $($MVP$)$ formulated as

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

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

where $$ is the expected value $($mean$),$ and $$ is the covariance matrix of the log$-$returns $($on

the training data$).$ Choose $100$ evenly spaced values of $in[0\mathrm{.}1,100]$ in logarithmic scale

$($using np$.$logapace$)$ and plot the efficient frontier, or the curve of expected return $w_{}^{***}{?}^{TT}$

versus volatility $\sqrt[\phantom{2}]{w_{}^{***TT}{w}_{}^{***}}($where $w_{}^{***}$ is the solution to $(1)).$

$($b$)$ Design the IVP, GMVP$,$ MDCP and MSRP$-$ portfolios based on the training $($in$-$sample$)$

data. You may use the cvxpy package for portfolio optimization.

$($c$)$ Compute the $($in$-$sample$)$ expected retum and volatility for each of the designed portfolios

and plot them on the $($return$,$ volatility$)$ trade$-$off curve. Which portfolios are on the efficient

frontier? Which portfolio has the highest $($in$-$sample$)$ Sharpe ratio? Can you choose a proper

value of $$ via the MVP portfolio to achieve the same performance?

$($d$)$ Plot the weight allocation of the assets $($on training data$)$ for each portfolio $($similar to the

figure below$)$ and compare with the EWP $(\frac{1}{N})$ portfolio.

$($e$)$ Compute the out$-$of$-$sample annualized Sharpe ratio, annualized return and annualized

volatility for the designed portfolios and compare with the EWP. Which portfolio scheme

would you choose then? $($You may use the empyrical module.$)$