Monthly excess return data are presented below for each of three stocks and the S&P index $($corrected for dividends$)$ for a $12-$month period.

$\backslash$table$[[,$Security$],[$Month$,$A$,$B$,$C$,$S&P$],[1,12\mathrm{.}05,25\mathrm{.}2,31\mathrm{.}67,12\mathrm{.}28],[2,15\mathrm{.}27,2\mathrm{.}86,15\mathrm{.}82,5\mathrm{.}99],[3,-4\mathrm{.}12,5\mathrm{.}45,10\mathrm{.}58,2\mathrm{.}41],[4,1\mathrm{.}57,4\mathrm{.}56,-14\mathrm{.}43,4\mathrm{.}48],[5,3\mathrm{.}16,3\mathrm{.}72,31\mathrm{.}98,4\mathrm{.}41],[6,-2\mathrm{.}79,10\mathrm{.}79,-0\mathrm{.}72,4\mathrm{.}43],[7,-8\mathrm{.}97,5\mathrm{.}38,-19\mathrm{.}64,-6\mathrm{.}77],[8,-1\mathrm{.}18,-2\mathrm{.}97,-10,-2\mathrm{.}11],[9,1\mathrm{.}07,1\mathrm{.}52,-11\mathrm{.}51,3\mathrm{.}46],[10,12\mathrm{.}75,10\mathrm{.}75,5\mathrm{.}63,6\mathrm{.}16],[11,7\mathrm{.}48,3\mathrm{.}79,-4\mathrm{.}67,2\mathrm{.}47],[12,-0\mathrm{.}94,1\mathrm{.}32,7\mathrm{.}94,-1\mathrm{.}15]]$

Calculate the following quantities:

a$.$ Alpha for each stock

b$.$ Beta for each stock

c$.$ The standard deviation of the residuals from each regression

d$.$ The correlation coefficient between each security and the market

e$.$ The average return on the market

f$.$ The variance of the market

$4.$ Using the same data in Q$3,$

a$.$ Compute the mean return and variance of return for each stock using

$($i$)$ The single$-$index model

$($ii$)$ The historical data

b$.$ Compute the covariance between each possible pair of stocks using

$($i$)$ The single$-$index model

$($ii$)$ The historical data

c$.$ Compute the return and standard deviation of a portfolio constructed by placing one$-$third of your funds in each stock, using

$($i$)$ The single$-$index model

$($ii$)$ The historical data

d$.$ Explain why the answers to parts a$.($i$)$ and a$.(2)$ are the same, while the answers to parts b$.($i$),$ b$.($ii$)$ and c$.($i$)$ and c$.($ii$)$ are different.