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Chapter 12

Chapter 12 - A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18...

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Unformatted text preview: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Mean Sigma Initial investment Cutoff B C D E F G H I 20% 30% 100 80 9.12% <-- =NORMDIST(B6,(1+B3)*B5,B5*B4,TRUE) Data table header: =NORMDIST(B6,(1+B3)*B5,B5*B4,FALSE) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 0.005467 4.46E-006 1.60E-005 5.14E-005 0.0001477 0.0003799 0.0008741 0.0017997 0.0033159 0.005467 0.0080657 0.0106483 0.0125794 0.0132981 0.0125794 0.0106483 0.0080657 0.005467 0.0033159 0.0017997 0.0008741 0.0003799 0.0001477 5.14E-005 1.60E-005 4.46E-006 1.11E-006 Normal Densit y Function 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 30 60 90 120 150 180 210 24 0 270 J A B C D E F G H I CALCULATING THE QUANTILES Mean Sigma Initial investment Cutoff 20% 30% 100 50.209564 <-- =NORMINV(0.01,(1+B3)*B5,B5*B4) 1.00% <-- =NORMDIST(B6,(1+B3)*B5,B5*B4,TRUE) VaR at 1.00% level 49.790436 <-- =B5-B6 Contains string formula which changes with contents of cell B6: ="VaR at "&TEXT(B7,"0.00%")&" level" Data table header: =B7 Cumulat ive Normal Dist ribut ion (only part--so we can see t he 1% quint ile) 0.012 0.01 Pro ba bilit y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 0.008 1% quantile point 0.006 0.004 0.002 0 -30 -20 -10 0 10 20 30 4 0 50 Port folio value (million \$) 60 70 1.00% -20 1.53E-006 -10 7.34E-006 0 3.17E-005 10 0.0001229 15 0.0002326 20 0.0004291 25 0.000771 30 0.0013499 35 0.0023033 40 0.0038304 45 0.0062097 46 0.0068189 47 0.0074803 48 0.0081975 49 0.0089745 50 0.0098153 50.20974 0.0100002 52 0.0117053 53 0.0127635 54 0.0139034 55 0.0151301 56 0.0164487 57 0.0178644 58 0.0193828 59 0.0210094 60 0.0227501 61 0.0246108 62 0.0265976 J A 1 2 3 4 5 6 7 8 9 10 11 12 13 Initial value, V0 Mean, µ Sigma, σ Time period, T B C D E 100 10% 30% 1 <-- in years Parameters of normal distribution of ln(V T) Mean 4.6602 <-- =LN(B3)+(B4-B5^2/2)*B6 Sigma 0.3000 <-- =B5*B6 Cutoff VaR at 1% level 52.57632 <-- =LOGINV(0.01,B9,B10) 47.42368 <-- =B3-B12 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 B C D E VaR FOR 3 ASSET PROBLEM Mean returns 10% 12% 13% Asset 1 Asset 2 Asset 3 Initial investment Mean return Portfolio sigma Variance-covariance matrix 0.1 0.04 0.04 0.2 0.03 -0.04 F 0.03 -0.04 0.6 G H Portfolio proportions 0.30 0.25 0.45 100 0.1185 <-- {=MMULT(TRANSPOSE(B5:B7),H5:H7)} 0.3848 <-- {=SQRT(MMULT(MMULT(TRANSPOSE(H5:H7),D5:F7),H5:H7))} Mean investment value Sigma of investment value 111.8500 38.4838 Cutoff Cumulative PDF VaR at 1.00% level 22.32338 <-- =NORMINV(0.01,(1+B10)*B9,B11*B9) 0.01 <-- =NORMDIST(B16,B13,B14,TRUE) 77.67662 <-- =B9-B16 Note that the functions in cells B10 and B11 are array functions: You must press [Ctrl]+[Shift]+[Enter] when after you write the function in the cell. The curly brackets {} are not written--they appear automatically. A 1 2 3 Units of Index held 4 Bond maturity 5 6 7 8 9 10 11 12 13 14 15 B C D E F G H I J 2 May 8, 2000 Index Value Date 2/10/1997 Ruble Ruble interest exchange rate rate 293 5.30% 3.40 Total Ruble Dollar Portfolio index bond bond value value value value 586.00 -84.2166 -286.3365 299.66 =B3*C9 =H9*E9 =-100*EXP(-(B4-B9)/365*D9) =G9+I9 K A B C D 1 2 2 3 Units of Index held May 8, 2000 4 Bond maturity 5 Foreign Exchange 6 Index interest rate rate 7 Day 1/2/1997 462.71 5.28% 3.50 8 1/3/1997 514.71 5.26% 3.47 9 1/4/1997 456.5 5.23% 3.46 10 1/5/1997 487.39 5.24% 3.45 11 1/6/1997 470.42 5.25% 3.45 12 2/8/1997 467.14 5.31% 3.44 45 2/9/1997 562.06 5.32% 3.41 46 2/10/1997 481.61 5.30% 3.40 47 E F G H I J K L M N O Return data (to be bootstrapped) Daily rates of return over 39 (40?) days Portfolio value 632.13 738.41 622.49 685.17 651.28 644.75 837.17 676.88 Day Day Q R S Data with random series Index P Interest Exchange 1 2 0.1065031 -0.003795 -0.008608 3 -0.120015 -0.00572 -0.002886 4 0.0654759 0.0019102 -0.002894 5 -0.035439 0.0019066 0 38 0.0292149 0.001885 0.0058309 39 0.1849796 0.0018815 -0.008759 40 -0.154474 -0.003766 -0.002937 Index Random T U V W X Y Z AA Data with random series, sorted Interest Random Exchange Random 1 2 0.0035398 0.6691484 -0.003795 0.1374893 -0.008608 0.3609355 3 0.0070423 0.6747233 -0.00572 0.3833547 -0.002886 0.7953518 4 -0.0177 0.7329664 0.0019102 0.5645221 -0.002894 0.5979619 5 0.0071175 0.8677855 0.0019066 0.3780015 0 0.5653586 38 0 0.9667468 0.001885 0.9765719 0.0058309 0.0482257 39 -0.003419 0.4986005 0.0018815 0.9585343 -0.008759 0.810797 40 0.0034188 0.4933599 -0.003766 0.8702513 -0.002937 0.9176926 Day Index Random Interest AB AC AD AE AF AG AH Bootstrapped data Random Exchange Random 1 2 -0.003466 0.035423 -0.024553 0.0980677 -0.002928 0.0092541 3 0.0173917 0.0491389 0.0018921 0.1055221 0.0028777 0.024295 4 -0.006897 0.101026 -0.001871 0.1077552 0.011628 0.0292456 5 -0.006944 0.1287819 0.0037807 0.1243543 0.0058309 0.0482257 38 0.0068493 0.9241847 0.0018815 0.9585343 0.0059172 0.8940201 39 0.0034783 0.9282329 0 0.9631887 -0.002937 0.9176926 40 0 0.9667468 0.001885 0.9765719 -0.008734 0.9280298 Day Index Interest Exchange 2/10/1997 481.61 5.30% 3.4 2/11/1997 479.94353 5.17% 3.3900585 2/12/1997 488.36359 5.18% 3.3998281 2/13/1997 485.00714 5.17% 3.4395922 2/14/1997 481.65069 5.19% 3.4597067 3/19/1997 498.63508 5.31% 3.3416291 3/20/1997 500.37249 5.31% 3.3318296 3/21/1997 500.37249 5.32% 3.3028571 Portfolio Value 676.88 673.16 689.22 679.01 670.74 714.42 718.69 721.19 B C D E F G 1,000 Sta rt time -0.13 Ela psed H 18:26 :4 0 0:00:14 I J Ma x return Min return K L M N 1.15121 <-- =MAX(J:J) -0.539 73 <-- =MIN(J:J) O P Q R S T U V =K4 =H 48/H9 -1 Exchange rate Exchange rand 3.4 1 0.0029 3.4 2 0.0238 3.4 6 0.0319 3.4 5 0.0339 3.37 0.04 4 7 3.4 0 0.04 9 2 3.4 6 0.086 6 3.52 0.09 04 3.4 5 0.09 81 3.4 4 0.1002 3.4 2 0.109 0 3.4 9 0.134 7 3.4 8 0.16 17 3.4 1 0.1703 3.4 3 0.3106 3.4 8 0.36 81 3.4 2 0.3725 3.4 6 0.3854 3.4 4 0.4 372 3.4 9 0.4 4 9 2 3.4 1 0.4 539 3.4 0 0.4 6 00 3.4 1 0.4 86 9 3.39 0.4 889 3.4 2 0.4 9 59 3.4 5 0.5233 3.4 2 0.6 19 4 3.6 8 0.6 29 0 3.4 2 0.6 6 21 3.4 2 0.6 74 5 3.50 0.6 9 56 3.4 4 0.7230 3.4 4 0.8086 3.4 1 0.824 5 3.4 3 0.8855 3.4 6 0.89 6 7 3.4 1 0.9 008 3.4 7 0.9 06 3 3.4 2 0.9 103 3.4 7 0.9 14 8 Return data -0.4 5188 0.4 6 84 9 -0.39 532 -0.2509 1 0.6 054 6 -0.056 72 0.13372 -0.519 9 1 -0.14 737 -0.4 8379 -0.4 729 3 -0.17124 -0.24 021 -0.204 15 0.6 9 6 72 -0.04 06 3 0.02104 -0.3804 3 -0.016 86 -0.0236 4 -0.009 9 0 -0.36 4 50 0.6 9 811 0.6 724 5 -0.0184 7 -0.36 570 -0.029 6 3 -0.39 59 5 0.216 21 -0.4 5322 0.2229 1 -0.539 73 -0.14 24 1 0.11820 0.89 4 4 0 -0.34 84 4 -0.25058 -0.284 59 0.26 253 0.07127 -0.29 9 9 7 0.2776 1 0.014 85 -0.39 736 0.6 79 71 0.379 9 6 -0.19 9 72 0.1589 3 -0.15728 -0.37512 0.07532 1.12718 0.33875 0.02736 0.6 516 2 -0.04 34 3 -0.26 4 33 -0.359 22 0.0879 6 0.534 75 0.02284 0.13802 0.274 59 0.12179 -0.4 9 4 26 -0.4 2524 0.1126 9 -0.4 7815 -0.16 107 0.04 204 -0.22103 -0.09 303 0.01870 0.19 4 56 -0.37522 -0.16 018 -0.4 016 4 0.21221 1.0504 4 0.24 34 0 0.4 8725 0.004 09 0.0279 0 0.53512 -0.07752 -0.4 04 27 -0.10713 -0.17854 -0.0836 2 -0.37718 0.229 6 8 -0.13152 0.4 729 8 -0.12180 -0.06 6 12 0.4 0758 0.36 26 6 0.06 19 2 0.054 86 0.6 0704 -0.34 54 2 0.10319 0.13328 0.30122 0.02387 0.104 57 0.4 8776 1.009 38 0.22816 -0.4 9 079 -0.04 171 -0.316 4 8 -0.4 29 29 0.26 302 -0.4 6 6 72 -0.4 09 09 -0.34 9 80 -0.4 0700 -0.04 717 -0.114 05 0.119 35 0.58128 -0.04 6 26 0.52524 -0.0354 0 0.39 4 6 1 -0.334 83 -0.36 177 -0.5086 9 -0.276 32 0.9 09 30 -0.4 4 86 6 0.05712 0.079 50 0.0384 0 0.08715 -0.3239 0 0.1304 3 -0.17235 -0.35101 0.79 36 8 -0.219 6 8 0.06 4 84 0.0736 1 -0.13876 -0.16 9 72 -0.19 34 9 0.5576 1 0.09 081 -0.376 14 -0.08333 0.10209 0.014 4 5 -0.5104 1 0.706 38 0.219 9 4 -0.38135 0.4 2750 0.6 6 308 0.0124 3 -0.26 9 70 0.30009 0.57130 -0.4 9 9 27 0.06 6 52 -0.05815 -0.22827 0.014 83 0.359 01 -0.304 15 0.4 1825 -0.03505 -0.34 386 0.10552 0.09 217 0.86 6 4 4 -0.0324 0 -0.19 74 3 0.704 9 8 -0.0526 7 0.14 6 9 2 -0.04 4 57 0.02859 -0.37583 -0.22075 -0.4 7529 -0.509 10 0.036 4 1 -0.4 4 4 22 -0.0279 4 0.02358 -0.156 29 0.18305 -0.1039 7 0.026 4 1 -0.0724 6 0.29 6 9 7 -0.1816 5 -0.20781 0.72522 -0.11589 0.184 25 -0.074 9 1 0.529 20 -0.13757 0.71336 -0.14 4 26 0.3229 2 -0.0359 1 0.14 74 5 -0.19 025 -0.35220 0.57319 -0.30732 0.10080 0.01881 -0.18759 -0.32234 -0.28558 -0.4 1501 0.29 554 0.52029 -0.0219 1 0.14 053 0.1356 8 0.039 10 -0.5074 8 -0.22029 -0.174 17 -0.00186 0.03117 -0.05054 -0.336 35 0.28827 0.1774 3 -0.50024 -0.09 338 0.34 6 30 0.184 25 -0.26 86 7 0.09 531 0.04 284 -0.286 4 1 -0.076 33 0.586 77 -0.04 59 2 0.039 6 7 0.6 1173 -0.0529 3 0.14 6 71 -0.19 816 -0.2356 2 0.8109 1 -0.4 84 70 0.284 4 6 0.10108 0.27788 0.04 535 0.12077 -0.12189 -0.12333 -0.4 7719 0.389 11 -0.4 2839 0.09 273 -0.09 557 0.0216 8 0.17709 -0.4 1583 -0.059 6 9 -0.06 9 80 -0.21733 -0.279 27 0.0354 0 -0.08871 0.24 9 04 -0.129 73 0.4 2529 0.4 06 39 -0.08876 -0.1839 2 -0.239 06 -0.06 79 2 -0.019 02 0.6 0113 -0.2784 4 -0.0256 1 0.0319 5 0.6 6 29 8 0.4 8327 -0.536 15 -0.19 6 9 4 -0.07009 0.08855 -0.21711 -0.19 4 75 -0.4 26 9 4 -0.114 16 -0.15054 0.27076 -0.17780 -0.4 9 04 8 -0.054 71 1.014 73 0.36 324 -0.4 1288 1.054 9 4 0.9 3508 0.0034 4 -0.19 724 0.206 9 0 0.039 4 6 0.26 576 -0.09 56 9 0.529 13 0.15131 -0.37378 0.6 7886 0.86 817 0.82015 0.00888 0.6 306 1 -0.236 13 -0.34 506 -0.089 52 0.286 84 -0.34 9 30 0.26 9 6 9 -0.35751 0.27029 -0.4 9 524 -0.31311 0.6 3383 -0.4 339 8 0.809 9 8 0.07325 0.01524 -0.4 9 59 2 -0.03559 -0.319 85 -0.534 82 0.0739 8 -0.39 351 0.9 0272 -0.04 326 0.6 2229 0.539 78 -0.4 109 8 0.39 6 58 0.4 0839 0.73138 -0.00534 -0.32838 0.31314 -0.06 9 70 0.359 54 -0.10779 0.21733 -0.234 6 6 -0.119 52 -0.389 01 0.16 073 -0.08874 -0.03819 -0.35078 -0.14 89 3 -0.25012 -0.176 78 0.12702 -0.04 6 33 -0.4 4 135 -0.33533 0.2276 6 0.30279 -0.4 1000 0.6 54 14 -0.024 81 0.4 6 803 -0.35028 0.21182 0.78200 -0.036 6 3 0.87856 -0.4 3779 0.25285 -0.0754 8 -0.4 36 19 -0.326 6 2 -0.07158 0.6 8300 -0.05520 0.4 8278 0.3589 2 -0.00054 1.06 232 -0.3114 4 0.109 82 -0.04 4 4 1 0.37587 0.36 058 -0.3109 1 -0.51308 -0.21575 -0.4 1287 -0.036 04 -0.04 176 -0.324 57 -0.17818 -0.19 4 14 -0.4 39 6 6 -0.09 4 54 0.06 9 27 0.59 26 1 0.37006 -0.15205 0.29 770 -0.4 9 250 -0.02024 0.58001 0.286 17 0.4 7354 -0.03178 -0.3139 7 -0.4 6 585 0.004 58 -0.4 4 876 0.9 056 5 0.86 9 27 0.4 306 7 -0.314 38 0.14 082 -0.189 73 -0.01250 -0.1256 0 -0.15284 0.204 82 -0.06 76 4 1.0284 2 -0.51277 -0.03075 -0.07223 0.256 28 -0.51005 -0.09 6 87 -0.034 33 -0.16 24 7 -0.25385 0.6 074 4 -0.32073 0.1276 9 -0.384 4 4 0.22156 0.55019 -0.02016 -0.4 4 857 -0.22270 -0.14 6 22 0.9 7372 0.72873 -0.06 878 0.28856 0.37504 0.6 5218 0.04 6 03 0.0379 3 0.1224 3 0.14 170 -0.3384 4 0.30712 -0.00886 -0.379 6 7 -0.18579 -0.37109 1.02250 0.1536 3 0.1284 2 1.09 6 70 -0.284 24 0.0139 3 0.739 4 6 -0.386 05 -0.14 4 14 -0.38257 -0.4 86 10 0.51788 -0.06 278 0.29 726 -0.17733 -0.376 4 0 1.06 527 0.6 29 52 -0.06 6 72 -0.2716 9 -0.19 6 4 5 0.6 39 78 0.4 6 39 2 0.05324 0.106 78 0.56 54 2 -0.359 9 5 -0.05308 0.04 513 -0.23834 -0.39 4 6 3 0.786 17 0.05832 0.04 713 0.05852 0.9 8506 -0.24 736 0.06 704 0.129 56 0.6 8153 -0.36 882 -0.4 6 9 9 7 0.4 384 0 0.05101 0.34 6 87 0.0109 2 0.01835 0.6 886 9 0.03772 -0.3174 7 0.05789 0.4 6 4 13 0.4 5152 0.02582 0.53386 -0.36 4 4 6 0.216 03 0.6 126 7 0.229 21 -0.4 8080 -0.35787 0.13333 0.6 59 37 0.124 9 2 -0.4 0880 0.05500 0.14 256 0.19 072 0.4 0579 0.084 76 -0.4 16 4 3 0.6 7501 0.56 9 05 0.139 75 0.4 0039 -0.07720 0.26 6 00 0.18309 0.029 13 -0.16 9 6 2 1.03585 -0.11233 -0.4 079 9 -0.02312 0.0524 2 0.214 57 0.04 14 7 -0.03731 -0.11803 -0.00529 0.126 56 0.04 4 04 0.17811 0.6 124 4 -0.19 24 5 0.38385 0.11105 0.56 778 0.84 29 7 -0.36 809 0.06 6 6 6 -0.4 06 9 5 -0.076 58 -0.186 6 8 -0.4 3751 -0.16 29 3 -0.2354 9 -0.4 9 84 3 0.5786 7 0.9 29 56 -0.28808 -0.4 4 6 86 0.6 7883 0.536 9 3 0.084 31 -0.0106 4 -0.04 115 0.12808 0.9 6 6 02 0.4 3837 0.11334 -0.06 255 1.05289 -0.37004 -0.054 6 6 0.9 3022 0.0039 5 -0.0034 0 0.6 09 6 8 0.139 19 0.4 839 8 0.27135 -0.08587 -0.09 381 -0.16 583 -0.29 04 9 -0.35003 0.55880 0.586 9 2 -0.5109 1 -0.3576 4 0.0026 1 0.12825 0.1554 8 -0.06 839 0.27136 -0.324 13 0.30017 0.54 586 -0.04 56 6 -0.004 05 0.5116 1 0.00515 -0.18031 0.01211 0.19 09 4 -0.34 530 -0.34 86 9 0.216 78 0.09 26 5 -0.02205 -0.13136 -0.35335 -0.204 26 -0.37307 0.8016 3 -0.39 558 0.076 9 3 0.04 6 39 0.306 4 6 1.11189 -0.00382 0.326 4 7 -0.32277 -0.19 4 25 -0.17518 0.58122 -0.19 285 0.22717 -0.1326 3 -0.2876 9 0.206 77 0.18351 -0.18137 -0.304 01 0.14 720 -0.12510 -0.06 6 73 -0.09 781 -0.35720 0.27278 0.026 4 3 0.07810 0.36 14 7 0.4 6 24 3 0.13730 0.16 39 1 0.6 6 4 13 0.6 26 78 0.34 6 4 0 0.03837 -0.056 81 -0.1216 0 -0.4 1258 0.3739 9 0.6 7285 -0.04 04 8 0.2856 1 -0.39 518 0.4 39 4 2 -0.05009 -0.23132 1.0036 0 -0.304 83 -0.3884 0 0.06 09 1 0.24 070 0.09 9 04 -0.22776 -0.39 9 9 3 0.006 84 0.12522 -0.4 6 6 22 -0.21706 -0.04 9 59 -0.23376 0.056 76 0.75054 -0.1524 5 0.36 19 2 0.20718 0.2736 5 0.19 09 1 -0.1034 0 -0.4 29 37 -0.24 279 -0.35851 0.01000 0.20376 0.2224 1 0.04 536 0.16 115 0.88078 -0.04 4 51 0.836 01 -0.3816 9 1.03231 0.386 16 1.03887 0.016 4 8 0.13535 -0.2576 1 -0.306 4 6 -0.4 04 4 9 -0.33724 -0.3376 4 -0.5129 8 0.09 086 -0.14 9 50 -0.4 9 288 0.03036 -0.0076 7 0.04 89 4 -0.4 89 23 0.71789 -0.2029 3 -0.24 001 -0.4 6 9 03 -0.00219 0.39 6 05 0.9 5532 0.6 039 8 -0.009 9 9 0.4 4 059 -0.10839 1.016 9 2 -0.11771 0.154 81 0.03004 0.174 35 -0.04 79 3 0.14 084 0.20220 0.03130 -0.179 4 2 0.2539 4 0.38155 -0.19 36 6 -0.08211 -0.32311 -0.084 39 0.26 6 39 -0.35351 0.034 81 0.356 6 1 0.31538 -0.119 9 0 -0.02233 0.6 9 574 -0.0014 9 -0.38885 -0.21323 0.024 4 6 -0.04 74 8 -0.28228 -0.326 73 0.50139 -0.09 351 -0.06 055 -0.3526 9 -0.034 32 0.6 1237 0.519 07 -0.30814 -0.4 0588 -0.4 1752 0.21359 -0.0886 8 -0.20507 -0.09 4 25 -0.06 003 0.289 25 -0.07031 -0.05159 -0.3159 0 1.124 54 -0.14 389 -0.34 6 50 -0.14 4 55 0.03159 -0.04 237 -0.51205 -0.386 29 0.59 807 -0.214 07 -0.19 713 -0.19 4 32 -0.3374 4 -0.37758 0.254 6 1 -0.04 288 -0.386 21 -0.27758 -0.05384 0.52726 0.4 6 777 -0.1314 8 -0.086 84 0.109 81 -0.22051 0.4 26 6 8 -0.11152 0.08376 -0.2079 8 -0.104 26 -0.09 172 0.38381 0.26 6 87 0.03050 0.254 08 -0.03730 0.039 34 -0.07075 -0.219 14 0.26 4 26 -0.03336 -0.53124 0.56 750 -0.4 5884 -0.214 6 6 -0.229 16 0.25758 0.059 9 4 -0.26 4 86 -0.236 87 -0.4 9 177 -0.009 33 -0.0214 5 0.00338 0.74 6 52 0.19 6 9 4 -0.29 9 77 0.87500 -0.01318 0.06 6 9 4 -0.4 0171 0.09 9 01 -0.05574 0.9 074 4 -0.009 37 -0.189 52 -0.22872 -0.21036 -0.34 6 05 -0.08327 -0.18726 -0.4 9 103 -0.0526 2 0.01179 1.15121 0.86 9 34 -0.176 83 1.039 26 -0.28804 0.6 0216 0.9 29 6 7 -0.019 6 2 -0.3019 8 0.6 6 507 -0.216 71 0.17139 -0.15718 -0.24 4 79 -0.4 7234 0.879 31 0.279 18 -0.12230 -0.1519 1 0.59 16 8 0.0019 6 0.274 22 -0.4 59 33 0.4 76 58 0.12002 -0.35823 0.036 59 0.01254 -0.34 14 0 -0.04 6 76 -0.4 19 9 6 -0.0314 2 -0.37314 -0.07206 -0.26 306 0.35516 0.4 8251 0.086 85 0.1784 6 -0.359 09 -0.25213 0.0209 6 -0.04 9 03 -0.4 5220 -0.26 022 0.276 39 -0.354 4 9 0.0539 0 0.04 789 -0.14 4 22 -0.13380 0.9 7309 -0.3554 3 0.18120 -0.09 856 0.6 6 9 6 6 0.016 19 -0.21271 0.27827 0.5256 6 -0.24 719 0.14 04 4 0.04 781 -0.30801 -0.36 9 24 -0.20813 -0.16 159 -0.384 6 3 0.6 5327 -0.07200 0.119 9 7 0.1836 5 0.28313 -0.029 83 -0.4 6 188 0.21239 0.01870 -0.12156 -0.0354 6 0.13755 0.5089 6 -0.26 155 1.0224 2 -0.4 354 8 0.59 752 0.01726 0.01221 0.214 6 3 0.89 707 -0.04 89 9 -0.18185 -0.2309 6 0.07803 0.57536 0.6 504 2 -0.0579 7 -0.26 536 0.2179 8 0.04 24 8 -0.03217 1.024 53 -0.25774 0.81372 -0.24 589 0.71551 -0.36 176 -0.4 9 4 73 -0.09 532 -0.01127 0.5339 4 0.13002 0.70836 -0.36 028 0.05272 -0.04 254 0.0576 0 0.29 9 85 -0.18781 -0.159 26 0.54 703 0.04 6 84 -0.28534 -0.28089 0.21138 0.11226 0.086 6 9 0.109 04 -0.13171 Cumula tive percent 0.1% 1.4 % 2.3% 1.8% 2.7% 3.5% 4 .5% 3.1% 2.0% 2.7% 3.2% 4 .2% 2.7% 3.0% 4 .7% 5.9 % 4 .1% 5.1% 4 .7% 3.2% 3.6 % 2.3% 2.0% 2.5% 2.1% 2.6 % 0.7% 1.4 % 1.5% 1.1% 0.9 % 0.8% 1.6 % 1.2% 1.7% 1.0% 1.4 % 1.2% 0.7% 0.2% 0.6 % 0.4 % 0.7% 0.7% 0.4 % 0.4 % 0.4 % 0.9 % 0.5% 0.2% 0.2% 0.0% 0.0% Mu ch more ou tp ut--in this ca se 0.0% d own to cel l J500 9 0.0% W graph title: Err:508 =(\$K\$ 3-\$K\$ 4)/50+L9 Portfolio value 715.4 8 837.80 6 54 .77 6 27.29 6 58.28 1,125.4 2 6 34 .9 4 1,276 .56 1,019 .00 1,29 3.00 9 9 3.76 6 08.51 9 9 8.88 9 85.37 6 86 .39 1,222.37 9 4 4 .4 2 1,050.9 0 1,085.39 79 6 .53 6 05.15 883.04 876 .06 839 .29 6 19 .70 1,04 8.6 6 6 4 6 .52 6 4 0.89 1,015.85 779 .39 9 15.9 8 1,19 2.30 6 73.4 3 6 01.14 74 0.86 6 26 .85 9 9 3.03 6 4 0.4 7 6 36 .16 6 21.24 Bins -0.54 -0.506 -0.4 72 -0.4 38 -0.4 04 -0.371 -0.337 -0.303 -0.26 9 -0.235 -0.202 -0.16 8 -0.134 -0.1 -0.06 6 -0.032 0.0014 0.0352 0.06 9 0.1028 0.136 6 0.1705 0.204 3 0.2381 0.2719 0.3057 0.339 6 0.3734 0.4 072 0.4 4 1 0.4 74 8 0.5087 0.54 25 0.576 3 0.6 101 0.6 4 39 0.6 777 0.7116 0.74 54 0.779 2 0.813 0.84 6 8 0.8807 0.9 14 5 0.9 4 83 0.9 821 1.0159 1.04 9 8 1.0836 1.1174 1.1512 1 14 23 18 27 35 45 31 20 27 32 42 27 30 47 59 41 51 47 32 36 23 20 25 21 26 7 14 15 11 9 8 16 12 17 10 14 12 7 2 6 4 7 7 4 4 4 9 5 2 2 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 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