The midterm contains two questions. Please solve these prob$-$

lems in R and knit a pdf report. The report includes the question

description, your solution and any comments if necessary.

Question $1$: $(50$ points$)$

$1\mathrm{.}1$ Download daily equity data of JPM and WFC $(2015-01-01$ to $2024-01-$

$01).$

$1\mathrm{.}2$ Calculate both the daily log return and weekly log return for each stock.

$1\mathrm{.}3$ Visualize the distribution of these log returns using hist$()$ function. Use

par$()$ function to put the four histogram together into one single graph,

where each histogram is an individual subplot.

$1\mathrm{.}4$ Calculate the first four moments, i$.$e$.$ mean, variance, skewness and kur$-$

tosis, for each stock. Store the calculate result in a data frame and report

the result in a table.

$1\mathrm{.}5$ Draw a scatter plot of JPM daily return against WFC daily return. $($i$.$e$.$

WFC return on x$-$axis and JPM return on y$-$axis$)$

$1\mathrm{.}6$ Build a simple linear regression model using the WFC daily return as ex$-$

planatory variable and the JPM daily return as response variable. Report

the fitted model using summary$()$ function.

$1\mathrm{.}7$ Draw a regression line on the scatter plot using the fitted model above.

Make sure use a different color to draw the regression line.

Question $2$: $(50$ points$)$

$2\mathrm{.}1$ Without using packages, create a function of $2$ variables $"\backslash ($ x $\backslash )"$ and "adjusted" that calculates the sample skewness of $"$x$"$ using the formulas on Lecture $6$ page $20$ and page $21.$ When "adjusted" $=$ TRUE, it returns the adjusted skewness of $"$x$",$ and FALSE returns the unadjusted one.

$2\mathrm{.}2$ Without using packages, create a function of $2$ variables $"\backslash ($ x $\backslash )"$ and "adjusted" that calculates the sample kurtosis of $"$x$"$ using the formulas on Lecture $6$ page $20$ and page $23.$ When "adjusted" $=$ TRUE, it returns the adjusted kurtosis of $"\backslash ($ x $\backslash )",$ and FALSE returns the unadjusted one.

$2\mathrm{.}3$ Download historical price for ticker $"$SPY$"$ for the whole $2012$ and $2013$ years with quantmod package, use its adjusted close price to calculate daily log return $($Note the adjusted close price is different from the "adjusted" for sample moments$).$

$2\mathrm{.}4$ Calculate the adjusted and unadjusted skewness for the daily log return in $2\mathrm{.}3$ using the function you defined. $($both numbers should be close to $-0\mathrm{.}15)$

$2\mathrm{.}5$ Calculate the adjusted and unadjusted kurtosis for the daily log return in $2\mathrm{.}3$ using the function you defined. $($both numbers should be close to $4\mathrm{.}1)$ The midterm contains two questions. Please solve these problems in $\backslash ($ R $\backslash )$ and knit a pdf report. The report includes the question description, your solution and any comments if necessary.

Question $1$: $(50$ points$)$

$1\mathrm{.}1$ Download daily equity data of JPM and WFC $(2015-01-01$ to $2024-0101).$

$1\mathrm{.}2$ Calculate both the daily $\backslash (\backslash$log $\backslash )$ return and weekly log return for each stock.

$1\mathrm{.}3$ Visualize the distribution of these log returns using hist$()$ function. Use $\backslash (\backslash$operatorname$\{$par$\}()\backslash )$ function to put the four histogram together into one single graph, where each histogram is an individual subplot.

$1\mathrm{.}4$ Calculate the first four moments, i$.$e$.$ mean, variance, skewness and kurtosis, for each stock. Store the calculate result in a data frame and report the result in a table.

$1\mathrm{.}5$ Draw a scatter plot of JPM daily return against WFC daily return. $($i$.$e$.$ WFC return on x$-$axis and JPM return on y$-$axis$)$

$1\mathrm{.}6$ Build a simple linear regression model using the WFC daily return as explanatory variable and the JPM daily return as response variable. Report the fitted model using summary$()$ function.

$1\mathrm{.}7$ Draw a regression line on the scatter plot using the fitted model above. Make sure use a different color to draw the regression line.