Homework 12: Due Tuesday August 11, 2020

The economy of a certain country consists of just three sectors. In terms of market capitalization, Sector 1 makes up 20% of the entire economy, (that is, 20% of the market portfolio), and the balance is accounted for by Sectors 2 and 3. The market capitalization of Sector 3 is four times as much as that of Sector 2. The standard deviation of returns on investments is 35% per year for Sector 1, 50% per year for Sector 2, and 25% per year for Sector 3. Returns on Sector 2 are uncorrelated with those on the other two Sectors. The correlation between returns on Sectors 1 and 3 is 0.4325.

1. Work out the variance of the market portfolio.
2. Work out the covariance and the correlation between the returns on investments in Sector 1 and returns on the market portfolio.
3. Work out the covariance and the correlation between the returns on investments in Sector 2 and returns on the market portfolio.
4. Work out the covariance and the correlation between the returns on investments in Sector 3 and returns on the market portfolio.
5. Work out the beta between the returns on investments in Sector 1 and returns on the market portfolio.
6. Work out the beta between the returns on investments in Sector 2 and returns on the market portfolio.
7. Work out the beta between the returns on investments in Sector 3 and returns on the market portfolio.
8. Show that the average beta of the three sectors, that is an average of the beta of the three sectors weighted with the fraction that each sector forms of the market portfolio, is one.