Problem 2: Following is information for the required returns and standard deviat

Question

Problem 2: Following is information for the required returns and standard deviations of returns for A, B, and C. Here are the expected returns and standard deviations for stocks A, B, and C: Stock ri si A 7.0% 33.11% B 10.0% 53.85% C 20.0% 89.44% Here is the correlation matrix: A B C A 1.0000 0.1571 0.1891 B 0.1571 1.0000 0.1661 C 0.1891 0.1661 1.0000 a. Suppose a portfolio has 30 percent invested in A, 50 percent in B, and 20 percent in C. What are the expected return and standard deviation of the portfolio? wA = 30% wB = 50% wC = 20% rp = Portfolio variance = sp = b. The partial model lists 66 different combinations of portfolio weights. For each combination of weights, find the required return and standard deviation. If you would like a return of 10.50 percent, what is the smallest standard deviation that you must accept? Why? Portoflio # wA wB wC Variance sp rp 1 0.0 0.0 1.0 2 0.0 0.1 0.9 3 0.0 0.2 0.8 4 0.0 0.3 0.7 5 0.0 0.4 0.6 6 0.0 0.5 0.5 7 0.0 0.6 0.4 8 0.0 0.7 0.3 9 0.0 0.8 0.2 10 0.0 0.9 0.1 11 0.0 1.0 0.0 12 0.1 0.0 0.9 13 0.1 0.1 0.8 14 0.1 0.2 0.7 15 0.1 0.3 0.6 16 0.1 0.4 0.5 17 0.1 0.5 0.4 18 0.1 0.6 0.3 19 0.1 0.7 0.2 20 0.1 0.8 0.1 21 0.1 0.9 0.0 22 0.2 0.0 0.8 23 0.2 0.1 0.7 24 0.2 0.2 0.6 25 0.2 0.3 0.5 26 0.2 0.4 0.4 27 0.2 0.5 0.3 28 0.2 0.6 0.2 29 0.2 0.7 0.1 30 0.2 0.8 0.0 31 0.3 0.0 0.7 32 0.3 0.1 0.6 33 0.3 0.2 0.5 34 0.3 0.3 0.4 35 0.3 0.4 0.3 36 0.3 0.5 0.2 37 0.3 0.6 0.1 38 0.3 0.7 0.0 39 0.4 0.0 0.6 40 0.4 0.1 0.5 41 0.4 0.2 0.4 42 0.4 0.3 0.3 43 0.4 0.4 0.2 44 0.4 0.5 0.1 45 0.4 0.6 0.0 46 0.5 0.0 0.5 47 0.5 0.1 0.4 48 0.5 0.2 0.3 49 0.5 0.3 0.2 50 0.5 0.4 0.1 51 0.5 0.5 0.0 52 0.6 0.0 0.4 53 0.6 0.1 0.3 54 0.6 0.2 0.2 55 0.6 0.3 0.1 56 0.6 0.4 0.0 57 0.7 0.0 0.3 58 0.7 0.1 0.2 59 0.7 0.2 0.1 60 0.7 0.3 0.0 61 0.8 0.0 0.2 62 0.8 0.1 0.1 63 0.8 0.2 0.0 64 0.9 0.0 0.1 65 0.9 0.1 0.0 66 1.0 0.0 0.0 Problem 2: Following is information for the required returns and standard deviations of returns for A, B, and C. Here are the expected returns and standard deviations for stocks A, B, and C: Stock ri si A 7.0% 33.11% B 10.0% 53.85% C 20.0% 89.44% Here is the correlation matrix: A B C A 1.0000 0.1571 0.1891 B 0.1571 1.0000 0.1661 C 0.1891 0.1661 1.0000 a. Suppose a portfolio has 30 percent invested in A, 50 percent in B, and 20 percent in C. What are the expected return and standard deviation of the portfolio? wA = 30% wB = 50% wC = 20% rp = Portfolio variance = sp = b. The partial model lists 66 different combinations of portfolio weights. For each combination of weights, find the required return and standard deviation. If you would like a return of 10.50 percent, what is the smallest standard deviation that you must accept? Why? Portoflio # wA wB wC Variance sp rp 1 0.0 0.0 1.0 2 0.0 0.1 0.9 3 0.0 0.2 0.8 4 0.0 0.3 0.7 5 0.0 0.4 0.6 6 0.0 0.5 0.5 7 0.0 0.6 0.4 8 0.0 0.7 0.3 9 0.0 0.8 0.2 10 0.0 0.9 0.1 11 0.0 1.0 0.0 12 0.1 0.0 0.9 13 0.1 0.1 0.8 14 0.1 0.2 0.7 15 0.1 0.3 0.6 16 0.1 0.4 0.5 17 0.1 0.5 0.4 18 0.1 0.6 0.3 19 0.1 0.7 0.2 20 0.1 0.8 0.1 21 0.1 0.9 0.0 22 0.2 0.0 0.8 23 0.2 0.1 0.7 24 0.2 0.2 0.6 25 0.2 0.3 0.5 26 0.2 0.4 0.4 27 0.2 0.5 0.3 28 0.2 0.6 0.2 29 0.2 0.7 0.1 30 0.2 0.8 0.0 31 0.3 0.0 0.7 32 0.3 0.1 0.6 33 0.3 0.2 0.5 34 0.3 0.3 0.4 35 0.3 0.4 0.3 36 0.3 0.5 0.2 37 0.3 0.6 0.1 38 0.3 0.7 0.0 39 0.4 0.0 0.6 40 0.4 0.1 0.5 41 0.4 0.2 0.4 42 0.4 0.3 0.3 43 0.4 0.4 0.2 44 0.4 0.5 0.1 45 0.4 0.6 0.0 46 0.5 0.0 0.5 47 0.5 0.1 0.4 48 0.5 0.2 0.3 49 0.5 0.3 0.2 50 0.5 0.4 0.1 51 0.5 0.5 0.0 52 0.6 0.0 0.4 53 0.6 0.1 0.3 54 0.6 0.2 0.2 55 0.6 0.3 0.1 56 0.6 0.4 0.0 57 0.7 0.0 0.3 58 0.7 0.1 0.2 59 0.7 0.2 0.1 60 0.7 0.3 0.0 61 0.8 0.0 0.2 62 0.8 0.1 0.1 63 0.8 0.2 0.0 64 0.9 0.0 0.1 65 0.9 0.1 0.0 66 1.0 0.0 0.0

Explanation / Answer

standard deviation= square root of variance

Formula:

Expected return of portfolio = wa* E(ra) + wb* E(rb) + wc* E(rc )

Variance= w ^2 *a 2 (ra) + w^ 2 *b 2 (rb) + w ^2 * c 2 (rc ) + 2* wa* wb* cov(ra,rb) + 2*wa* wc* cov(ra,rc ) + 2*wb*wc* cov(rb,rc )

for a expected return of 10.5% Standard deviartion in line 49= 43.63%

Portoflio # wA wB wC rp working rp Variance working Variance sp 1 0 0 1 0.07*0+0.1*0+0.2*1 0.2000 0^2 *0.3311^2 + 0^2 *0.5385^2 +1^2 *0.8944^2 + 2*0 *0*0.1571+ 2 * 0*1*0.1661+ 2 *0*1*0.1891 0.799951 0.8944 2 0 0.1 0.9 0.07*0+0.1*0.1+0.2*0.9 0.1900 0^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.9^2 *0.8944^2 + 2*0 *0.1*0.1571+ 2 * 0.1*0.9*0.1661+ 2 *0*0.9*0.1891 0.680758 0.825081 3 0 0.2 0.8 0.07*0+0.1*0.2+0.2*0.8 0.1800 0^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.8^2 *0.8944^2 + 2*0 *0.2*0.1571+ 2 * 0.2*0.8*0.1661+ 2 *0*0.8*0.1891 0.57672 0.759421 4 0 0.3 0.7 0.07*0+0.1*0.3+0.2*0.7 0.1700 0^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.7^2 *0.8944^2 + 2*0 *0.3*0.1571+ 2 * 0.3*0.7*0.1661+ 2 *0*0.7*0.1891 0.487837 0.698453 5 0 0.4 0.6 0.07*0+0.1*0.4+0.2*0.6 0.1600 0^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.6^2 *0.8944^2 + 2*0 *0.4*0.1571+ 2 * 0.4*0.6*0.1661+ 2 *0*0.6*0.1891 0.414108 0.643512 6 0 0.5 0.5 0.07*0+0.1*0.5+0.2*0.5 0.1500 0^2 *0.3311^2 + 0.5^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0 *0.5*0.1571+ 2 * 0.5*0.5*0.1661+ 2 *0*0.5*0.1891 0.355533 0.596266 7 0 0.6 0.4 0.07*0+0.1*0.6+0.2*0.4 0.1400 0^2 *0.3311^2 + 0.6^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0 *0.6*0.1571+ 2 * 0.6*0.4*0.1661+ 2 *0*0.4*0.1891 0.312114 0.558671 8 0 0.7 0.3 0.07*0+0.1*0.7+0.2*0.3 0.1300 0^2 *0.3311^2 + 0.7^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0 *0.7*0.1571+ 2 * 0.7*0.3*0.1661+ 2 *0*0.3*0.1891 0.283849 0.532775 9 0 0.8 0.2 0.07*0+0.1*0.8+0.2*0.2 0.1200 0^2 *0.3311^2 + 0.8^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0 *0.8*0.1571+ 2 * 0.8*0.2*0.1661+ 2 *0*0.2*0.1891 0.270739 0.520326 10 0 0.9 0.1 0.07*0+0.1*0.9+0.2*0.1 0.1100 0^2 *0.3311^2 + 0.9^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0 *0.9*0.1571+ 2 * 0.9*0.1*0.1661+ 2 *0*0.1*0.1891 0.272783 0.522286 11 0 1 0 0.07*0+0.1*1+0.2*0 0.1000 0^2 *0.3311^2 + 1^2 *0.5385^2 +0^2 *0.8944^2 + 2*0 *1*0.1571+ 2 * 1*0*0.1661+ 2 *0*0*0.1891 0.289982 0.5385 12 0.1 0 0.9 0.07*0.1+0.1*0+0.2*0.9 0.1870 0.1^2 *0.3311^2 + 0^2 *0.5385^2 +0.9^2 *0.8944^2 + 2*0.1 *0*0.1571+ 2 * 0*0.9*0.1661+ 2 *0.1*0.9*0.1891 0.683095 0.826496 13 0.1 0.1 0.8 0.07*0.1+0.1*0.1+0.2*0.8 0.1770 0.1^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.8^2 *0.8944^2 + 2*0.1 *0.1*0.1571+ 2 * 0.1*0.8*0.1661+ 2 *0.1*0.8*0.1891 0.575939 0.758906 14 0.1 0.2 0.7 0.07*0.1+0.1*0.2+0.2*0.7 0.1670 0.1^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.7^2 *0.8944^2 + 2*0.1 *0.2*0.1571+ 2 * 0.2*0.7*0.1661+ 2 *0.1*0.7*0.1891 0.483938 0.695656 15 0.1 0.3 0.6 0.07*0.1+0.1*0.3+0.2*0.6 0.1570 0.1^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.6^2 *0.8944^2 + 2*0.1 *0.3*0.1571+ 2 * 0.3*0.6*0.1661+ 2 *0.1*0.6*0.1891 0.407091 0.638037 16 0.1 0.4 0.5 0.07*0.1+0.1*0.4+0.2*0.5 0.1470 0.1^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0.1 *0.4*0.1571+ 2 * 0.4*0.5*0.1661+ 2 *0.1*0.5*0.1891 0.345399 0.587707 17 0.1 0.5 0.4 0.07*0.1+0.1*0.5+0.2*0.4 0.1370 0.1^2 *0.3311^2 + 0.5^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.1 *0.5*0.1571+ 2 * 0.5*0.4*0.1661+ 2 *0.1*0.4*0.1891 0.298862 0.546683 18 0.1 0.6 0.3 0.07*0.1+0.1*0.6+0.2*0.3 0.1270 0.1^2 *0.3311^2 + 0.6^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.1 *0.6*0.1571+ 2 * 0.6*0.3*0.1661+ 2 *0.1*0.3*0.1891 0.26748 0.517184 19 0.1 0.7 0.2 0.07*0.1+0.1*0.7+0.2*0.2 0.1170 0.1^2 *0.3311^2 + 0.7^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.1 *0.7*0.1571+ 2 * 0.7*0.2*0.1661+ 2 *0.1*0.2*0.1891 0.251252 0.50125 20 0.1 0.8 0.1 0.07*0.1+0.1*0.8+0.2*0.1 0.1070 0.1^2 *0.3311^2 + 0.8^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.1 *0.8*0.1571+ 2 * 0.8*0.1*0.1661+ 2 *0.1*0.1*0.1891 0.250178 0.500178 21 0.1 0.9 0 0.07*0.1+0.1*0.9+0.2*0 0.0970 0.1^2 *0.3311^2 + 0.9^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.1 *0.9*0.1571+ 2 * 0.9*0*0.1661+ 2 *0.1*0*0.1891 0.26426 0.514062 22 0.2 0 0.8 0.07*0.2+0.1*0+0.2*0.8 0.1740 0.2^2 *0.3311^2 + 0^2 *0.5385^2 +0.8^2 *0.8944^2 + 2*0.2 *0*0.1571+ 2 * 0*0.8*0.1661+ 2 *0.2*0.8*0.1891 0.576866 0.759517 23 0.2 0.1 0.7 0.07*0.2+0.1*0.1+0.2*0.7 0.1640 0.2^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.7^2 *0.8944^2 + 2*0.2 *0.1*0.1571+ 2 * 0.1*0.7*0.1661+ 2 *0.2*0.7*0.1891 0.481747 0.69408 24 0.2 0.2 0.6 0.07*0.2+0.1*0.2+0.2*0.6 0.1540 0.2^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.6^2 *0.8944^2 + 2*0.2 *0.2*0.1571+ 2 * 0.2*0.6*0.1661+ 2 *0.2*0.6*0.1891 0.401783 0.633863 25 0.2 0.3 0.5 0.07*0.2+0.1*0.3+0.2*0.5 0.1440 0.2^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0.2 *0.3*0.1571+ 2 * 0.3*0.5*0.1661+ 2 *0.2*0.5*0.1891 0.336973 0.580494 26 0.2 0.4 0.4 0.07*0.2+0.1*0.4+0.2*0.4 0.1340 0.2^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.2 *0.4*0.1571+ 2 * 0.4*0.4*0.1661+ 2 *0.2*0.4*0.1891 0.287318 0.536021 27 0.2 0.5 0.3 0.07*0.2+0.1*0.5+0.2*0.3 0.1240 0.2^2 *0.3311^2 + 0.5^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.2 *0.5*0.1571+ 2 * 0.5*0.3*0.1661+ 2 *0.2*0.3*0.1891 0.252818 0.50281 28 0.2 0.6 0.2 0.07*0.2+0.1*0.6+0.2*0.2 0.1140 0.2^2 *0.3311^2 + 0.6^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.2 *0.6*0.1571+ 2 * 0.6*0.2*0.1661+ 2 *0.2*0.2*0.1891 0.233473 0.48319 29 0.2 0.7 0.1 0.07*0.2+0.1*0.7+0.2*0.1 0.1040 0.2^2 *0.3311^2 + 0.7^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.2 *0.7*0.1571+ 2 * 0.7*0.1*0.1661+ 2 *0.2*0.1*0.1891 0.229282 0.478834 30 0.2 0.8 0 0.07*0.2+0.1*0.8+0.2*0 0.0940 0.2^2 *0.3311^2 + 0.8^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.2 *0.8*0.1571+ 2 * 0.8*0*0.1661+ 2 *0.2*0*0.1891 0.240246 0.490149 31 0.3 0 0.7 0.07*0.3+0.1*0+0.2*0.7 0.1610 0.3^2 *0.3311^2 + 0^2 *0.5385^2 +0.7^2 *0.8944^2 + 2*0.3 *0*0.1571+ 2 * 0*0.7*0.1661+ 2 *0.3*0.7*0.1891 0.481265 0.693732 32 0.3 0.1 0.6 0.07*0.3+0.1*0.1+0.2*0.6 0.1510 0.3^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.6^2 *0.8944^2 + 2*0.3 *0.1*0.1571+ 2 * 0.1*0.6*0.1661+ 2 *0.3*0.6*0.1891 0.398183 0.631017 33 0.3 0.2 0.5 0.07*0.3+0.1*0.2+0.2*0.5 0.1410 0.3^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0.3 *0.2*0.1571+ 2 * 0.2*0.5*0.1661+ 2 *0.3*0.5*0.1891 0.330256 0.574679 34 0.3 0.3 0.4 0.07*0.3+0.1*0.3+0.2*0.4 0.1310 0.3^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.3 *0.3*0.1571+ 2 * 0.3*0.4*0.1661+ 2 *0.3*0.4*0.1891 0.277483 0.526767 35 0.3 0.4 0.3 0.07*0.3+0.1*0.4+0.2*0.3 0.1210 0.3^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.3 *0.4*0.1571+ 2 * 0.4*0.3*0.1661+ 2 *0.3*0.3*0.1891 0.239865 0.48976 36 0.3 0.5 0.2 0.07*0.3+0.1*0.5+0.2*0.2 0.1110 0.3^2 *0.3311^2 + 0.5^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.3 *0.5*0.1571+ 2 * 0.5*0.2*0.1661+ 2 *0.3*0.2*0.1891 0.217402 0.466264 37 0.3 0.6 0.1 0.07*0.3+0.1*0.6+0.2*0.1 0.1010 0.3^2 *0.3311^2 + 0.6^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.3 *0.6*0.1571+ 2 * 0.6*0.1*0.1661+ 2 *0.3*0.1*0.1891 0.210094 0.45836 38 0.3 0.7 0 0.07*0.3+0.1*0.7+0.2*0 0.0910 0.3^2 *0.3311^2 + 0.7^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.3 *0.7*0.1571+ 2 * 0.7*0*0.1661+ 2 *0.3*0*0.1891 0.21794 0.46684 39 0.4 0 0.6 0.07*0.4+0.1*0+0.2*0.6 0.1480 0.4^2 *0.3311^2 + 0^2 *0.5385^2 +0.6^2 *0.8944^2 + 2*0.4 *0*0.1571+ 2 * 0*0.6*0.1661+ 2 *0.4*0.6*0.1891 0.396291 0.629516 40 0.4 0.1 0.5 0.07*0.4+0.1*0.1+0.2*0.5 0.1380 0.4^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0.4 *0.1*0.1571+ 2 * 0.1*0.5*0.1661+ 2 *0.4*0.5*0.1891 0.325246 0.570303 41 0.4 0.2 0.4 0.07*0.4+0.1*0.2+0.2*0.4 0.1280 0.4^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.4 *0.2*0.1571+ 2 * 0.2*0.4*0.1661+ 2 *0.4*0.4*0.1891 0.269356 0.518995 42 0.4 0.3 0.3 0.07*0.4+0.1*0.3+0.2*0.3 0.1180 0.4^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.4 *0.3*0.1571+ 2 * 0.3*0.3*0.1661+ 2 *0.4*0.3*0.1891 0.22862 0.478143 43 0.4 0.4 0.2 0.07*0.4+0.1*0.4+0.2*0.2 0.1080 0.4^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.4 *0.4*0.1571+ 2 * 0.4*0.2*0.1661+ 2 *0.4*0.2*0.1891 0.20304 0.450599 44 0.4 0.5 0.1 0.07*0.4+0.1*0.5+0.2*0.1 0.0980 0.4^2 *0.3311^2 + 0.5^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.4 *0.5*0.1571+ 2 * 0.5*0.1*0.1661+ 2 *0.4*0.1*0.1891 0.192613 0.438877 45 0.4 0.6 0 0.07*0.4+0.1*0.6+0.2*0 0.0880 0.4^2 *0.3311^2 + 0.6^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.4 *0.6*0.1571+ 2 * 0.6*0*0.1661+ 2 *0.4*0*0.1891 0.197342 0.444232 46 0.5 0 0.5 0.07*0.5+0.1*0+0.2*0.5 0.1350 0.5^2 *0.3311^2 + 0^2 *0.5385^2 +0.5^2 *0.8944^2 + 2*0.5 *0*0.1571+ 2 * 0*0.5*0.1661+ 2 *0.5*0.5*0.1891 0.321945 0.567402 47 0.5 0.1 0.4 0.07*0.5+0.1*0.1+0.2*0.4 0.1250 0.5^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.5 *0.1*0.1571+ 2 * 0.1*0.4*0.1661+ 2 *0.5*0.4*0.1891 0.262937 0.512774 48 0.5 0.2 0.3 0.07*0.5+0.1*0.2+0.2*0.3 0.1150 0.5^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.5 *0.2*0.1571+ 2 * 0.2*0.3*0.1661+ 2 *0.5*0.3*0.1891 0.219084 0.468064 49 0.5 0.3 0.2 0.07*0.5+0.1*0.3+0.2*0.2 0.1050 0.5^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.5 *0.3*0.1571+ 2 * 0.3*0.2*0.1661+ 2 *0.5*0.2*0.1891 0.190385 0.436332 50 0.5 0.4 0.1 0.07*0.5+0.1*0.4+0.2*0.1 0.0950 0.5^2 *0.3311^2 + 0.4^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.5 *0.4*0.1571+ 2 * 0.4*0.1*0.1661+ 2 *0.5*0.1*0.1891 0.176841 0.420525 51 0.5 0.5 0 0.07*0.5+0.1*0.5+0.2*0 0.0850 0.5^2 *0.3311^2 + 0.5^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.5 *0.5*0.1571+ 2 * 0.5*0*0.1661+ 2 *0.5*0*0.1891 0.178452 0.422436 52 0.6 0 0.4 0.07*0.6+0.1*0+0.2*0.4 0.1220 0.6^2 *0.3311^2 + 0^2 *0.5385^2 +0.4^2 *0.8944^2 + 2*0.6 *0*0.1571+ 2 * 0*0.4*0.1661+ 2 *0.6*0.4*0.1891 0.258226 0.508159 53 0.6 0.1 0.3 0.07*0.6+0.1*0.1+0.2*0.3 0.1120 0.6^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.6 *0.1*0.1571+ 2 * 0.1*0.3*0.1661+ 2 *0.6*0.3*0.1891 0.211255 0.459625 54 0.6 0.2 0.2 0.07*0.6+0.1*0.2+0.2*0.2 0.1020 0.6^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.6 *0.2*0.1571+ 2 * 0.2*0.2*0.1661+ 2 *0.6*0.2*0.1891 0.179439 0.423603 55 0.6 0.3 0.1 0.07*0.6+0.1*0.3+0.2*0.1 0.0920 0.6^2 *0.3311^2 + 0.3^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.6 *0.3*0.1571+ 2 * 0.3*0.1*0.1661+ 2 *0.6*0.1*0.1891 0.162778 0.403457 56 0.6 0.4 0 0.07*0.6+0.1*0.4+0.2*0 0.0820 0.6^2 *0.3311^2 + 0.4^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.6 *0.4*0.1571+ 2 * 0.4*0*0.1661+ 2 *0.6*0*0.1891 0.161271 0.401586 57 0.7 0 0.3 0.07*0.7+0.1*0+0.2*0.3 0.1090 0.7^2 *0.3311^2 + 0^2 *0.5385^2 +0.3^2 *0.8944^2 + 2*0.7 *0*0.1571+ 2 * 0*0.3*0.1661+ 2 *0.7*0.3*0.1891 0.205135 0.452918 58 0.7 0.1 0.2 0.07*0.7+0.1*0.1+0.2*0.2 0.0990 0.7^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.7 *0.1*0.1571+ 2 * 0.1*0.2*0.1661+ 2 *0.7*0.2*0.1891 0.170201 0.412554 59 0.7 0.2 0.1 0.07*0.7+0.1*0.2+0.2*0.1 0.0890 0.7^2 *0.3311^2 + 0.2^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.7 *0.2*0.1571+ 2 * 0.2*0.1*0.1661+ 2 *0.7*0.1*0.1891 0.150422 0.387843 60 0.7 0.3 0 0.07*0.7+0.1*0.3+0.2*0 0.0790 0.7^2 *0.3311^2 + 0.3^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.7 *0.3*0.1571+ 2 * 0.3*0*0.1661+ 2 *0.7*0*0.1891 0.145798 0.381835 61 0.8 0 0.2 0.07*0.8+0.1*0+0.2*0.2 0.0960 0.8^2 *0.3311^2 + 0^2 *0.5385^2 +0.2^2 *0.8944^2 + 2*0.8 *0*0.1571+ 2 * 0*0.2*0.1661+ 2 *0.8*0.2*0.1891 0.162671 0.403326 62 0.8 0.1 0.1 0.07*0.8+0.1*0.1+0.2*0.1 0.0860 0.8^2 *0.3311^2 + 0.1^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.8 *0.1*0.1571+ 2 * 0.1*0.1*0.1661+ 2 *0.8*0.1*0.1891 0.139775 0.373865 63 0.8 0.2 0 0.07*0.8+0.1*0.2+0.2*0 0.0760 0.8^2 *0.3311^2 + 0.2^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.8 *0.2*0.1571+ 2 * 0.2*0*0.1661+ 2 *0.8*0*0.1891 0.132033 0.363363 64 0.9 0 0.1 0.07*0.9+0.1*0+0.2*0.1 0.0830 0.9^2 *0.3311^2 + 0^2 *0.5385^2 +0.1^2 *0.8944^2 + 2*0.9 *0*0.1571+ 2 * 0*0.1*0.1661+ 2 *0.9*0.1*0.1891 0.130836 0.361712 65 0.9 0.1 0 0.07*0.9+0.1*0.1+0.2*0 0.0730 0.9^2 *0.3311^2 + 0.1^2 *0.5385^2 +0^2 *0.8944^2 + 2*0.9 *0.1*0.1571+ 2 * 0.1*0*0.1661+ 2 *0.9*0*0.1891 0.119976 0.346375 66 1 0 0 0.07*1+0.1*0+0.2*0 0.0700 1^2 *0.3311^2 + 0^2 *0.5385^2 +0^2 *0.8944^2 + 2*1 *0*0.1571+ 2 * 0*0*0.1661+ 2 *1*0*0.1891 0.109627 0.3311

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