Use the following data to answer problems $2$ through $4$

State of Probability of State Stock A S&P$500$

Economy of Economy Rate of Return Rate of Return

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Recession $0\mathrm{.}25-0\mathrm{.}150\mathrm{.}02$

Normal $0\mathrm{.}550\mathrm{.}180\mathrm{.}09$

Boom $0\mathrm{.}200\mathrm{.}500\mathrm{.}18$

$2.$ Assume that the S&P$500$ is your market index.

a$.$ Calculate the variance for the S&P $500$s$.$

b$.$ Calculate the covariance between Stock A and the S&P $500.$