The economy of a certain country consists of four sectors. In terms of market capitalization, Sector 1 makes up 15% and Sector 2 makes up 5% of the entire economy, (that is, 15% and 5% of the market portfolio). The market capitalization of Sector 4 is three times that of Sector 3. All securities in Sector 1 are risk-free. The standard deviation of returns on investments is 30% per year for Sector 2, and 45% per year for Sector 3. Variance of returns on Sector 4 is 0.36 per year. Returns on Sector 2 are uncorrelated with those on all the other Sectors. The correlation between returns on Sectors 3 and 4 is 0.36.

Work out the variance of the market portfolio.

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

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

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

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

Work out the beta between the returns on investments in Sector 1 and returns on the market portfolio.

Work out the beta between the returns on investments in Sector 2 and returns on the market portfolio.

Work out the beta between the returns on investments in Sector 3 and returns on the market portfolio.

Work out the beta between the returns on investments in Sector 4 and returns on the market portfolio.

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.