3. Assume an investor's universe consists of three stocks, Stock 1, 2 and 3. The return of each stock is denoted as r where i E 1,2,3. The weight of each stock in the market portfolio is denoted as Wi. The standard deviation of each stock is o and lastly, the covariance between two stocks is given by Oi,j. Let w be a 3 x 1 matrix of weights and be a 3 x 3 variance-covariance matrix. a) Show that the variance of the market portfolio o = w'Ew is given by the expression below. o = wo? + wo + wzo3 + 2(W,W201,2 + w W301,3 + W2W302,3) = b) Confirm that rm = w'r = Ewiri = wr + w2r2 + Wzrz. Also note that the covariance between the return of asset i and the market (which consists of these three assets) is given by Cov(ri, rm) = 0i,M = Cov(ri,wir1 + w2r2 + wzr3) Using the above show that the market variance ox = Ewidi,M IM c) What is the relationship between Oi,m and o? Can we think of the ratio as the contribution or of a stock to the risk of the market portfolio? 3. Assume an investor's universe consists of three stocks, Stock 1, 2 and 3. The return of each stock is denoted as r where i E 1,2,3. The weight of each stock in the market portfolio is denoted as Wi. The standard deviation of each stock is o and lastly, the covariance between two stocks is given by Oi,j. Let w be a 3 x 1 matrix of weights and be a 3 x 3 variance-covariance matrix. a) Show that the variance of the market portfolio o = w'Ew is given by the expression below. o = wo? + wo + wzo3 + 2(W,W201,2 + w W301,3 + W2W302,3) = b) Confirm that rm = w'r = Ewiri = wr + w2r2 + Wzrz. Also note that the covariance between the return of asset i and the market (which consists of these three assets) is given by Cov(ri, rm) = 0i,M = Cov(ri,wir1 + w2r2 + wzr3) Using the above show that the market variance ox = Ewidi,M IM c) What is the relationship between Oi,m and o? Can we think of the ratio as the contribution or of a stock to the risk of the market portfolio