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3. (Principal components 0 Suppose there are n random variables x1,x2,..., Xn and let V be the corresponding covariance matrix. An eigenvector of V is a vector v= (V1, V2, ... , Un) such that Vv = iv for some 2 (called an eigenvalue of V). The random variable v1x1+x2x2+...+Unxn is a principal component. The first principal component is the one corresponding to the largest eigenvalue of V, the second to the second largest, and so forth. A good candidate for the factor in a one-factor model of n asset returns is the first principal component extracted from the n returns themselves; that is, by using the principal eigenvector of the covariance matrix of the returns. Find the first principal component for EXERCISES 233 the data of Example 8.2. Does this factor (when normalized) resemble the return on the market portfolio? [Note: For this part, you need an eigenvector calculator as available in most matrix operations packages.] TABLE 8.2 FACTOR MODEL WITH MARKET Year Stock 1 Stock 2 Stock 3 Stock 4 Market Riskless 1 2 3 4 5 6 7 8 9 10 11.91 18.37 3.64 24.37 30.42 -1.45 20.11 9.28 17.63 15.71 29.59 15.25 3.53 17.67 12.74 -2.56 25.46 6.92 9.73 25.09 23.27 19.47 -6.58 15.08 16.24 -15.05 17.80 18.82 3.05 16.94 27.24 17.05 10.20 20.26 19.84 1.51 12.24 16.12 22.93 3.49 23.00 17.54 2.70 19.34 19.81 -4.39 18.90 12.78 13.34 15.31 6.20 6.70 6.40 5.70 5.90 5.20 4.90 5.50 6.10 5.80 aver 5.84 var COV 15.00 90.28 65.08 .90 1.95 31.54 14.34 107.24 73.62 1.02 .34 32.09 10.90 162.19 100.78 1.40 -6.11 21.37 15.09 68.27 48.99 .68 3.82 34.99 13.83 72.12 72.12 1.00 0.00 B e-var 3. (Principal components 0 Suppose there are n random variables x1,x2,..., Xn and let V be the corresponding covariance matrix. An eigenvector of V is a vector v= (V1, V2, ... , Un) such that Vv = iv for some 2 (called an eigenvalue of V). The random variable v1x1+x2x2+...+Unxn is a principal component. The first principal component is the one corresponding to the largest eigenvalue of V, the second to the second largest, and so forth. A good candidate for the factor in a one-factor model of n asset returns is the first principal component extracted from the n returns themselves; that is, by using the principal eigenvector of the covariance matrix of the returns. Find the first principal component for EXERCISES 233 the data of Example 8.2. Does this factor (when normalized) resemble the return on the market portfolio? [Note: For this part, you need an eigenvector calculator as available in most matrix operations packages.] TABLE 8.2 FACTOR MODEL WITH MARKET Year Stock 1 Stock 2 Stock 3 Stock 4 Market Riskless 1 2 3 4 5 6 7 8 9 10 11.91 18.37 3.64 24.37 30.42 -1.45 20.11 9.28 17.63 15.71 29.59 15.25 3.53 17.67 12.74 -2.56 25.46 6.92 9.73 25.09 23.27 19.47 -6.58 15.08 16.24 -15.05 17.80 18.82 3.05 16.94 27.24 17.05 10.20 20.26 19.84 1.51 12.24 16.12 22.93 3.49 23.00 17.54 2.70 19.34 19.81 -4.39 18.90 12.78 13.34 15.31 6.20 6.70 6.40 5.70 5.90 5.20 4.90 5.50 6.10 5.80 aver 5.84 var COV 15.00 90.28 65.08 .90 1.95 31.54 14.34 107.24 73.62 1.02 .34 32.09 10.90 162.19 100.78 1.40 -6.11 21.37 15.09 68.27 48.99 .68 3.82 34.99 13.83 72.12 72.12 1.00 0.00 B e-var