4.The economy of a certain country consists of four sectors. In terms of market capitalization, Sector1 makes up 15% and Sector 2 makes up 5% of the entire economy, (that is, 15% and 5% of themarket portfolio). The market capitalization of Sector4is three times that ofSector3.All securities in Sector 1 are risk-free.The standard deviation of returns on investments is 30% per year for Sector2, and 45% per year for Sector3. Variance of returns on Sector 4 is 0.36 per year. Returns on Sector2 are uncorrelated with those on all the other Sectors. The correlation between returns on Sectors3 and 4 is 0.36.

[1.5 points each for a total of 15 points]

(a):Work out the variance of the market portfolio.

(b):Work out the covariance and the correlation between the returns on investments in Sector 1 and returns on the market portfolio.

(c):Work out the covariance and the correlation between the returns on investments in Sector 2 and returns on the market portfolio.

(d):Work out the covariance and the correlation between the returns on investments in Sector 3 and returns on the market portfolio.

(e):Work out the covariance and the correlation between the returns on investments in Sector 4 and returns on the market portfolio.

(f):Work out the beta between the returns on investments in Sector 1 and returns on the market portfolio.

(g):Work out the beta between the returns on investments in Sector 2 and returns on the market portfolio.

(h):Work out the beta between the returns on investments in Sector 3 and returns on the market portfolio.

(i):Work out the beta between the returns on investments in Sector 4 and returns on the market portfolio.

(j):Show that the average beta of the four sectors, that is an average of the beta of the four sectors weighted with the fraction that each sector forms of the market portfolio, is one.