$1.$ Comparing US and International Stock ETFs Please download daily data $(2019-2023)$ from Yahoo Finance using the below codes and calculate the daily simple returns. We use VTI for the US bond and VXUS for the Ex$-$US international stock ETF portfolios. You need to practice your R codes for answering the below questions.

library$($quantmod$)$ # install.packages$("$quantmod$")$ if needed

getSymbols$($VTI$,$ from $="2018-12-31",$ to$="2023-12-31")$

getSymbols$($VXUS$,$ from $="2018-12-31",$ to$="2023-12-31")$

# calculate the simple returns

VTI $=$ VTI$VTI.Adjusted; VXUS $=$ VXUS$VXUS.Adjusted

L $=$ length$($VTI$)$

rtn$.$VTI $=$ as$.$numeric$($VTI$[-1])/$as$.$numeric$($VTI$[-$L$])-1$

rtn$.$VXUS $=$ as$.$numeric$($VXUS$[-1])/$as$.$numeric$($VXUS$[-$L$])-1$

$($a$)$ What are the maximum and minimum returns for VTI and VXUS, respectively?

$($b$)$ Could you plot the histograms for VTI and VXUS, respectively? Do they look like normal distributions?$[$HINT: use hist$()$ and set breaks$=100]$

$($c$)$ What are the percentages of observations within two standard deviations from the average for VTI and VXUS, respectively? Do we have a $95\%$ coverage for the regression fit? $[$HINT: we use $2$ to approximate $1\mathrm{.}96$ for the $95\%$ confidence interval.$]$

$($d$)$ Could you plot the cumulative returns for these two ETFs in the same figure with different colors.$[$HINT: check codes on the notes$]$

$($e$)$ Please run a simple regression for VXUS on VTI, report the coefficients, plot the scatter plot with the fitted regression line. Please provide a short summary for explaining the regression results. $[$HINT: check codes on the notes$]$

$($f$)$ Finally, please run a simple regression for VTI on VXUS, and report the coefficients. What is the difference between this model and the previous one?