In this project, you will learn how to implement Markowitz$$s portfolio optimization with real$-$world datasets.

You should us the R programming language.

$($a$)$ Data Collection

$$ Select $10$ stocks listed on the U$.$S$.$ exchanges. You will construct a portfolio from these $10$ stocks.

Tabulate your selection with the corresponding tickers. You may wish to explain the rationale

behind your selection.

$$ Download the price data of the selected stocks and SPY $($which is the ETF that tracks the S&P

$500$ index$)$ in the past $36$ months. You will compare the performance of your portfolio to the

S&P $500$ index and the $1/$N strategy. $($You can download the price data at Yahoo Finance. You

can also use the $$quantmod$$ package in R$.)$

$$ Split the dataset into a training set and a test set. For example, the training set could include

the price data in the first $24$ months and the test data covers the last $12$ months.

$($b$)$ Estimation

$$ Calculate the daily returns. $($Hint: If you do not use the adjusted price, then you need to manually

adjust the price for corporate actions.$)$

$$ Estimate the expected return $$ and the covariance matrix $\backslash$Sigma of the selected stocks$$ returns from $$

the training set.

$($c$)$ Construction of portfolios

$$ We have proved that the optimal portfolio is of the form wopt $=$

$1$

$\backslash$gamma

$$ wz $+$ wm$,$ where wm is the

min$-$risk portfolio, wz is the zero$-$covariance portfolio, and $\backslash$gamma is the risk aversion. Calculate the

optimal portfolio for different values of $\backslash$gamma $.$ Because we do not know the true values of $$ and $\backslash$Sigma $,$ we

have to plug in our estimates $$ and $\backslash$Sigma $.$ The resulting portfolio is called the $$plug$-$in$$ portfolio. $$

$($d$)$ In$-$sample performance evaluation

You will evaluate the performance of your portfolios on the training set.

$$ Plot the in$-$sample efficient frontier in the $(\backslash$sigma $,)$ plane. Highlight the min$-$risk portfolio in the

graph.

$$ Calculate the mean return and standard deviations of S&P$500$ on the training set. Highlight it

in the $(\backslash$sigma $,)$ plane.

$$ Calculate the mean return and standard deviations of the $1/$N strategy on the training set.

Highlight it in the $(\backslash$sigma $,)$ plane. $($Hint: the $1/$N strategy invests equally in each asset. If you have

selected $10$ stocks, the $1/$N strategy invests $10\%$ of wealth in each stock.$)$

$$ Choose a value of $\backslash$gamma $.$ Calculate the daily returns of the corresponding optimal portfolios, S&P $500,$

and the $1/$N strategy on the training set. Compare the in$-$sample performance of the portfolios.

$($Hint: the comparison can be made in terms of mean return, standard deviation, cumulative

return, Sharpe ratio, Treynor$$s ratio, and Jensen$$s alpha, etc. You may need to find the risk$-$free

rate during the same period. If you need to estimate the beta of your portfolio, you can use S&P

$500$ as the market portfolio.$)$

BR$3214$ Financial and Risk Analytics II Course Project AY$2023-2024$ S$2$

$$ Comment on your findings.

$($e$)$ Out$-$of$-$sample performance evaluation

You will evaluate the performance of your portfolios on the test set.

$$ Use the same value of $\backslash$gamma as in part $($d$).$ Calculate the daily return of the corresponding optimal

portfolios, S&P $500,$ and the $1/$N strategy on the test set. Compare the out$-$of$-$sample performance

of the portfolios. $($Hint: the betas in the test set can be different from your previous calculations.$)$

$$ Comment on your findings.

$($Optional$)$ Consider a portfolio that is equally allocated between the optimal portfolio $(50\%)$ and

the $1/$N strategy $(50\%).$ Comment on its performance. $10$ stocks are "AAPL", $"$MSFT$",$ "AMZN", $"$JPM$","$V$","$PG$",$ "TSLA", $"$JNJ$",$ "XOM", "NVDA", $"$SPY