$[6$ points total: $1$ point for each part$]$ Consider weekly data on closing prices of the SP$500$ index from $1/1/1960$ to $4/4/2025.$ The data is in the attached Excel file.

a$.$ Generate a series of weekly rates of return. Then Compute the mean and the standard deviation of the weekly rates of return over the entire sample period, as well as the $95\%$ and $99\%$ confidence intervals. How many standard deviations away from the mean is the rate of return of the week ending on $\frac{4}{4}\frac{?}{2025}?$

b$.$ Plot a histogram of the weekly rates of return, with $100$ bins of equal size.

c$.$ Compute the arithmetic annual rate of return of the SP$500$ price. This is the so called annual percentage rate $($APR$)$:

APR$=n^{**}r(weekly)$

where $n$ is the number of weeks in a year $($number of compounding periods in a year$)$ and week $\frac{?}{y}$ is the average weekly rate of return over the sample. ${?}^1$

d$.$ Compute the geometric annual rate of return of the SP$500$ index. This is the so called effective annual rate $($EAR$)$:

I EAR$=[1r(T){]}^{\frac{1}{T}},$ where