Consider two uncorrelated assets with volatility a and 02, denote by w the weight in asset 1 and 1-w the weight in asset 2, and show that the weights of the GMV portfolio are inversely proportional to each asset variance, that is: WGMY 1 +02 0 +0 Exercise 2 Consider 3 assets with expected return vector = 12% , volatility vector a = 10% 6% triz C = -2 by == 1 0.6 -0.2 -0.1 1 1. Compute the covariance matriz for these 3 assets. 2. Compute the mean return and volatility for the equally- weighted (EW) portfolio of these 3 assets. 1-GMV = WGMV = 0.6 -0.2 1 -0.1 3. Compute the mean return and volatility for a portfolio given w = your day of birth divided by 31 =your month of birth divided by 12 w=1-W-W Exercise 3 Consider the same 3 assets. 1. Compute the weights of the global minimum variance (GMV) portfolio (respectively denoted by Pew and EW): 20% 16% and correlation ma- 6% -e e'E-le WMSR= 2. Also compute the mean and volatility of that portfolio (re- spectively denoted by HGMV and GMV). Exercise 4 Consider the same 3 assets and further assume that the risk-free rate is r = 1.5%. 1. Compute the weights of the maximum Sharpe ratio (MSR) portfolio: - (-re) e'- (-re) 2. Also compute the mean and volatility of that portfolio (re- spectively denoted by PMSR and MSR). 3. Compare the volatility of the MSR, GMV and EW portfo- lios. What do you conclude? Exercise 5 Consider the same 3 assets and assume again that the risk-free rate is r = 1.5%. 1. Compute the Sharpe ratio of the MSR (denoted by AMSR), GMV (denoted by AMSR) and EW (denoted by AEW) port- folios. 2. Compare the Sharpe ratio of the MSR, GMV and EW port- folios. What do you conclude? 3. Compare the ratio to the ratio us. What do you conclude? AGMY Exercise 6 Consider a general investment universe with n as- sets, and use the standard notation for the expected return and volatility of these assets. 1. Find an explicit expression for the weights of the MSR, GMV and risk parity portfolios in case the n assets are uncorrelated. 2. Find an explicit expression for the weights of the MSR port- folio in case the n assets have the same expected return. 3. Find an explicit expression for the weights of the MSR port- folio in case the n assets are indistinguishable (same volatil- ity, same expected return and same pairwise correlations, not necessarily zero). Exercise 7 Consider again a general investment universe with n assets. Find an explicit expression for the weights of the MSR in case Sharpe's (1964) CAPM is the holds true as the asset pricing model that explains cross-sectional differences in expected returns.