Exercise 1 Consider two uncorrelated assets with volatility 1 and 2, denote by w the weight in asset 1 and 1w the weight in asset 2, and show that the weights of the GMV portfolio are inversely proportional to each asset variance, that is: wGMV=12+2212;1wGMV=12+2222 Exercise 2 Consider 3 assets with expected return vector = 12%10%6%, volatility vector =20%16%6% and correlation matrix C=10.60.20.610.10.20.11 1. Compute the covariance matrix for these 3 assets. 2. Compute the mean return and volatility for the equallyweighted ( EW ) portfolio of these 3 assets. 3. Compute the mean return and volatility for a portfolio given by =w1=yourdayofbirthdividedby31w2=yourmonthofbirthdividedby12w1=1w1w27junC. Exercise 3 Consider the same 3 assets. 1. Compute the weights of the global minimum variance (GMV) portfolio (respectively denoted by EW and EW ): wGMV=e1e1e 2. Also compute the mean and volatility of that portfolio (respectively denoted by GMV 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: wMSR=e1(re)1(re) 2. Also compute the mean and volatility of that portfolio (respectively denoted by MSR and MSR ). 3. Compare the volatility of the MSR, GMV and EW portfolios. 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 MSR ), GMV (denoted by MSR) and EW (denoted by EW) portfolios. 2. Compare the Sharpe ratio of the MSR, GMV and EW portfolios. What do you conclude? 3. Compare the ratio GMVMSR to the ratio GMVMSR. What do you conclude? Exercise 6 Consider a general investment universe with n assets, 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 portfolio in case the n assets have the same expected return. 3. Find an explicit expression for the weights of the MSR portfolio in case the n assets are indistinguishable (same volatility, 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