Given two random variables $$ and $$:

Probability of State of Nature $20\%$

$20\%$

$20\%$

$20\%20\%$

State of Nature I

II

III

IV V

V ariable $$ V ariable $180$

$5-3$

$1215$

$41261$

a$.$ Calculate the mean and variance of each of these variables, and the covariance between them.

b$.$ Suppose that $$ and $$ represent the returns from two assets. Calculate the mean and variance for the following portfolios

Proportions Portfolio Portfolio Portfolio Portfolio Portfolio Portfolio Portfolio $1234567$

in$1251007550250-25$ in$-250255075100125$

c$.$ Find the portfolio that has the minimum variance $($MVP$).$

d$.$ Calculate the covariance between portfolio $3$ and portfolio $5.$

e$.$ Calculate the covariance between the minimum variance portfolio $($MVP$)$ and portfolio

$3$ and calculate the covariance between the MVP and portfolio $5.$

f$.$ What is the covariance between the MVP and any other portfolio along the efficient

set$/$frontier$?$

g$.$ What is the relationship between the covariance of the MVP with other efficient

portfolios, and the variance of the minimum variance portfolio?