Markov's Trilemma Working

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W1 W2 W3 Portfolio S Expected Return

-0.4482 0.5702 0.8780 0.314 0.50

-0.3474 0.5194 0.8280 0.285 0.48

-0.2467 0.4687 0.7780 0.259 0.46

-0.1459 0.4179 0.7280 0.236 0.44

-0.0452 0.3672 0.6780 0.219 0.42

0.0556 0.3164 0.6280 0.207 0.400.1563 0.2657 0.5780 0.203 0.38

0.2571 0.2149 0.5280 0.206 0.36

0.3578 0.1641 0.4780 0.216 0.34

0.4586 0.1134 0.4280 0.233 0.32

0.5593 0.0626 0.3780 0.254 0.30

0.6601 0.0119 0.3280 0.280 0.28

0.7608 -0.0389 0.2780 0.308 0.26

0.8616 -0.0896 0.2281 0.339 0.24

0.9623 -0.1404 0.1781 0.371 0.22

0.0631 -0.1911 0.1281 0.405 0.20

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.000 0.050 0.100 0.150


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Microsoft Excel 12.0 Answer Report

Worksheet: [Markov's Trilemma - TN.xlsx]Sheet1

Report Created: 11/27/2008 4:06:48 PM

Target Cell (Max)

Cell Name Original Value Final Value$E$79 AOL Weight 10% 100%

Adjustable Cells

Cell Name Original Value Final Value

$E$78 INTC Weight 10% 0%

$E$79 AOL Weight 10% 100%

$E$80 IBM Weight 10% 0%

$E$81 GM Weight 10% 0%

$E$82 AA Weight 10% 0%

$E$83 GE Weight 10% 0%

$E$84 IP Weight 20% 0%

$E$85 MRK Weight 20% 0%

Constraints

Cell Name Cell Value Formula Status Slack

$E$86 Weight 100% $E$86=1 Not Binding 0

$E$78 INTC Weight 0% $E$78>=0 Binding 0%

$E$79 AOL Weight 100% $E$79>=0 Not Binding 100%

$E$80 IBM Weight 0% $E$80>=0 Binding 0%$E$81 GM Weight 0% $E$81>=0 Binding 0%

$E$82 AA Weight 0% $E$82>=0 Binding 0%

$E$83 GE Weight 0% $E$83>=0 Binding 0%

$E$84 IP Weight 0% $E$84>=0 Binding 0%

$E$85 MRK Weight 0% $E$85>=0 Binding 0%


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Microsoft Excel 12.0 Sensitivity Report

Worksheet: [Markov's Trilemma - TN.xlsx]Sheet1


Adjustable Cells

Final ReducedCell Name Value Gradient

$E$78 INTC Weight 0% -100%

$E$79 AOL Weight 100% 0%

$E$80 IBM Weight 0% -100%

$E$81 GM Weight 0% -100%

$E$82 AA Weight 0% -100%

$E$83 GE Weight 0% -100%

$E$84 IP Weight 0% -100%

$E$85 MRK Weight 0% -100%

Constraints

Final Lagrange

Cell Name Value Multiplier

$E$86 Weight 100% 100%


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Microsoft Excel 12.0 Limits Report

Worksheet: [Markov's Trilemma - TN.xlsx]Limits Report 1


Target

Cell Name Value$E$79 AOL Weight 100%

Adjustable Lower Target Upper Target

Cell Name Value Limit Result Limit Result

$E$78 INTC Weight 0% 0% 100% 0% 100%

$E$79 AOL Weight 100% 100% 100% 100% 100%

$E$80 IBM Weight 0% 0% 100% 0% 100%

$E$81 GM Weight 0% 0% 100% 0% 100%

$E$82 AA Weight 0% 0% 100% 0% 100%

$E$83 GE Weight 0% 0% 100% 0% 100%

$E$84 IP Weight 0% 0% 100% 0% 100%

$E$85 MRK Weight 0% 0% 100% 0% 100%


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INTC AOL IBM GM AA GE IP MRK INTC AOL IBM

Dec-97 35.05 5.66 51.99 47.51 34.15 71.46 41.38 51.40

Jan-98 40.41 5.98 49.07 45.31 37.06 75.47 43.84 56.91 15.29% 0.05654 -0.056

Feb-98 44.75 7.59 51.89 54.34 35.78 75.72 44.97 61.85 10.74% 0.26923 0.0575

Mar-98 38.95 8.54 51.61 53.40 33.56 84.27 45.21 62.38 -12.96% 0.12516 -0.005

Apr-98 40.32 9.99 57.58 53.11 37.79 83.29 50.34 58.63 3.52% 0.16979 0.1157

May-98 35.64 10.41 58.38 57.06 33.94 81.52 44.58 56.93 -11.61% 0.04204 0.0139

Jun-98 36.98 13.14 57.05 53.04 32.26 88.85 41.68 65.34 3.76% 0.26225 -0.023

Jul-98 42.13 14.64 65.84 57.40 33.91 87.73 43.25 60.36 13.93% 0.11416 0.1541

Aug-98 35.52 10.24 55.96 46.47 29.41 78.47 36.07 56.64 -15.69% -0.3005 -0.15Sep-98 42.78 13.95 63.85 43.87 34.87 78.33 45.45 63.57 20.44% 0.3623 0.141

Oct-98 44.50 15.92 73.79 50.51 38.92 86.14 45.27 66.27 4.02% 0.14122 0.1557

Nov-98 53.72 21.89 82.17 56.28 36.58 88.97 42.58 76.12 20.72% 0.375 0.1136

Dec-98 59.18 38.78 91.75 57.64 36.74 100.77 43.93 72.62 10.16% 0.77159 0.1166

5.19% 19.91% 5.28%

RETURNS 68.8% 585.6% 76.5% 21.3% 7.6% 41.0% 6.2% 41.3%Sd 12.65% 25.33% 9.72%

AnnualisedSD 0.438349848 0.87745 0.3368

GeometricMean 0.688445078 5.85159 0.7648

Correlation

INTC AOL IBM GM AA GE IP MRK

SD 0.4383 0.87744539 0.3368 0.3762 0.3487 0.24 0.382750252 0.301336048

INTC 0.43835 1.00 0.59 0.61 0.36 0.60 0.20 0.62 0.52

AOL 0.87745 0.59 1.00 0.62 0.36 0.27 0.64 0.43 0.25

IBM 0.33682 0.61 0.62 1.00 0.65 0.62 0.29 0.58 0.00

GM 0.37623 0.36 0.36 0.65 1.00 0.10 0.26 0.07 0.12

AA 0.34873 0.60 0.27 0.62 0.10 1.00 0.17 0.88 0.13

GE 0.24 0.20 0.64 0.29 0.26 0.17 1.00 0.16 0.30

IP 0.38275 0.62 0.43 0.58 0.07 0.88 0.16 1.00 0.19

MRK 0.30134 0.52 0.25 0.00 0.12 0.13 0.30 0.19 1

Covarinace Matrix INTC AOL IBM GM AA GE IP MRK

INTC 0.1922 0.22596566 0.0899 0.0586 0.0913 0.0207 0.103330696 0.068734565


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AOL 0.226 0.76991041 0.1833 0.1201 0.0836 0.1339 0.145741408 0.06592451

IBM 0.0899 0.18333833 0.1134 0.082 0.0729 0.0233 0.074472563 0.000494603

GM 0.0586 0.12009992 0.082 0.1415 0.0138 0.0233 0.009369196 0.013539101

AA 0.0913 0.08363793 0.0729 0.0138 0.1216 0.0143 0.116927916 0.013416477

GE 0.0207 0.13386423 0.0233 0.0233 0.0143 0.0576 0.015097232 0.021747476

IP 0.1033 0.14574141 0.0745 0.0094 0.1169 0.0151 0.146497755 0.022041715

MRK 0.0687 0.06592451 0.0005 0.0135 0.0134 0.0217 0.022041715 0.090803414

Weight_Covariance ###### 232764147% ###### ###### ###### ###### -368386787% 254890915%INTC AOL IBM GM AA GE IP MRK

########## INTC 1E+12 -1.4633E+12 3E+11 -9E+10 -1E+12 2E+11 1.05906E+12 -4.87435E+11 -4E+11

########## AOL -1E+12 4.1713E+12 -5E+11 2E+11 1E+12 -1E+12 -1.24969E+12 3.91127E+11 1.6E+12

########## IBM 3E+11 -4.552E+11 1E+11 -5E+10 -4E+11 9E+10 2.92637E+11 -1344745858 -2E+11

56833004% GM -9E+10 1.5888E+11 -5E+10 5E+10 4E+10 -5E+10 -19615843028 19613035369 6E+10

########## AA -1E+12 1.082E+12 -4E+11 4E+10 4E+12 -3E+11 -2.39409E+12 1.90069E+11 5.6E+11

########## GE 2E+11 -1.0813E+12 9E+10 -5E+10 -3E+11 7E+11 1.92996E+11 -1.92358E+11 -4E+11

########## IP 1E+12 -1.2497E+12 3E+11 -2E+10 -2E+12 2E+11 1.9881E+12 -2.06968E+11 -3E+11

########## MRK -5E+11 3.9113E+11 -1E+09 2E+10 2E+11 -2E+11 -2.06968E+11 5.89944E+11 3E+11

SUM ###### ########### ###### ###### ###### ###### ############# ############## #######

Variance #######

SD 1052936

INTC AOL IBM GM AA GE IP MRK

15.29% 0.0565 -0.05616465 -0.0463 0.0852 0.0561 0.0594 0.107198444

10.74% 0.2692 0.05746892 0.1993 -0.0345 0.0033 0.0258 0.086803725

-12.96% 0.1252 -0.00539603 -0.0173 -0.062 0.1129 0.0053 0.008569119

3.52% 0.1698 0.11567526 -0.0054 0.126 -0.0116 0.1135 -0.060115422

-11.61% 0.042 0.01389371 0.0744 -0.1019 -0.0213 -0.1144 -0.028995395

3.76% 0.2622 -0.02278177 -0.0705 -0.0495 0.0899 -0.0651 0.147725277

13.93% 0.1142 0.15407537 0.0822 0.0511 -0.0126 0.0377 -0.076216713


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-15.69% -0.3005 -0.15006075 -0.1904 -0.1327 -0.1056 -0.166 -0.061630219

20.44% 0.3623 0.14099357 -0.056 0.1857 -0.0018 0.26 0.122351695

4.02% 0.1412 0.15567737 0.1514 0.1161 0.0997 -0.004 0.042472865

20.72% 0.375 0.11356552 0.1142 -0.0601 0.0329 -0.0594 0.148634375

10.16% 0.7716 0.11658756 0.0242 0.0044 0.1326 0.0317 -0.045980032

Return 68.8% 585.6% 76.5% 21.3% 7.6% 41.0% 6.2% 41.3%

Portfolio Return Sd Weight WxR Weight 12% 15%INTC 68.8% 2.3833 ########### -2E+06 Implied Return 0.10976 0.2319

AOL 585.6% 20.286 232764147% 1E+07 0.19215 0.226

IBM 76.5% 2.65 ########### -816000 0.22597 0.7699

GM 21.3% 0.7379 56833004% 121054 0.08993 0.1833

AA 7.6% 0.2633 555798702% 422407 0.05858 0.1201

GE 41.0% 1.4203 ########### -1E+06 0.09128 0.0836

IP 6.2% 0.2148 ########### -228400 0.02075 0.1339

MRK 41.3% 1.4307 254890915% 1E+06 0.10333 0.1457

1 0.06873 0.0659

Portfolio Return 1E+07

PortfolioSD 1E+06

Return/Risk 10.3


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GM AA GE IP MRK INTC AOL IBM

0.000% 100.000% 0.000%

-0.04631 0.0852 0.0561 0.0594 0.1072 5.7%

0.199294 -0.035 0.0033 0.0258 0.0868 26.9%

-0.0173 -0.062 0.1129 0.0053 0.0086 12.5%

-0.00543 0.126 -0.012 0.1135 -0.06 17.0%

0.074374 -0.102 -0.021 -0.114 -0.029 4.2%

-0.07045 -0.049 0.0899 -0.065 0.1477 26.2%

0.082202 0.0511 -0.013 0.0377 -0.076 11.4%

-0.19042 -0.133 -0.106 -0.166 -0.062 -30.1%-0.05595 0.1857 -0.002 0.26 0.1224 36.2%

0.151356 0.1161 0.0997 -0.004 0.0425 14.1%

0.114235 -0.06 0.0329 -0.059 0.1486 37.5%

0.024165 0.0044 0.1326 0.0317 -0.046 77.2%

2.16% 1.06% 3.12% 1.04% 3.26% 19.91%

10.86% 10.07% 6.93% 11.05% 8.70% 25.33%

0.376227 0.3487 0.24 0.3828 0.3013 0.877445

0.213218 0.0758 0.4102 0.0616 0.4128 585.159%

6.589116757

0.016013 0.0188 0.0075 0.0049 0.0076 0.001729156 0.008610891 0.00572788


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0.01883 0.0642 0.0153 0.01 0.007 0.011155353 0.012145117 0.005493709

0.007494 0.0153 0.0095 0.0068 0.0061 0.001938491 0.006206047 4.1217E-05

0.004882 0.01 0.0068 0.0118 0.0011 0.001941645 0.000780766 0.001128258

0.007606 0.007 0.0061 0.0011 0.0101 0.00119427 0.009743993 0.00111804

0.001729 0.0112 0.0019 0.0019 0.0012 0.004800047 0.001258103 0.00181229

0.008611 0.0121 0.0062 0.0008 0.0097 0.001258103 0.012208146 0.00183681

0.005728 0.0055 4E-05 0.0011 0.0011 0.00181229 0.00183681 0.007566951


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12% 12% 12% 12% 12% 12%0.082934 0.0596 0.0665 0.0415 0.0811 0.037911641 1

0.089932 0.0586 0.0913 0.0207 0.1033 0.068734565

0.183338 0.1201 0.0836 0.1339 0.1457 0.06592451

0.113448 0.082 0.0729 0.0233 0.0745 0.000494603

0.081979 0.1415 0.0138 0.0233 0.0094 0.013539101

0.07292 0.0138 0.1216 0.0143 0.1169 0.013416477

0.023262 0.0233 0.0143 0.0576 0.0151 0.021747476

0.074473 0.0094 0.1169 0.0151 0.1465 0.022041715

0.000495 0.0135 0.0134 0.0217 0.022 0.090803414


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Demonstration of the Portfolio Standard Deviation Calculation RiskFreeRa 0.07

equal risk return - changing correlation

Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.688Esd3 0.438

W1 -0.05704 W2 0.431444 W3 0.625601 1.00000

Expected Return 0.596449

Correlation12 0.12

Correlation23 0.52

Correlation13 0.35

Covar_12_23_13 0.0136 0.0686 0.0576

Portfolio SD 0.353

Sharpe Ratio 1.491


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Demonstration of the Portfolio Standard Deviation C RiskFreeR 0.07


Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.3

W1 0.023592 W2 0.492837 W3 0.48357 1.00000

Expected 0.406831

Correlatio 0.12

Correlatio 0.3

Correlatio 0.26

Covar_12_ 0.0136 0.0271 0.0293

Portfolio S 0.239

Sharpe Rat 1.410


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.3Esd3 0.24

W1 0.052848 W2 0.493583 W3 0.453568 1.00000

Expected 0.351177

Correlatio 0.12

Correlatio 0.3

Correlatio 0.26

Covar_12_ 0.0136 0.0217 0.0235

Portfolio S 0.214

Sharpe Rat 1.313

0.141376


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0.003085 W2 0.345454 W3 0.65146 1.00000

Expected 0.410428

Correlatio 0.12

Correlatio 0.3

Correlatio 0.26

Covar_12_ 0.0136 0.0217 0.0235

Portfolio S 0.212

Sharpe Rat 1.696


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Demonstration of the Portfolio Standard Deviation Calcul RiskFreeR 0.07


Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0.359557 W2 0 W3 0.640442 1.00000

Expected Retur 0.339167

Correlation12 0.12

Correlation23 0.30

Correlation13 -0.8

Covar_12_23_1 0.0136 0.0217 -0.0722

Portfolio SD 0.093

Sharpe Ratio 2.893


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0 W2 0.346437 W3 0.653562 1.00000

Expected 0.411039

Correlatio 0.12

Correlatio 0.30

Correlatio 0.8

Covar_12_ 0.0136 0.0217 0.0722

Portfolio S 0.213

Sharpe Rat 1.602


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0.052357 W2 0.447643 W3 0.5 1.00000

Expected 0.401029

Correlatio 0.12

Correlatio 0.3

Correlatio 0.26

Covar_12_ 0.0136 0.0217 0.0235

Portfolio S 0.211

Sharpe Rat 1.569


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0 W2 0.346437 W3 0.653562 1.00000

Expected 0.411039

Correlatio 0.12

Correlatio 0.3

Correlatio 0.26

Covar_12_ 0.0136 0.0217 0.0235

Portfolio S 0.213

Sharpe Rat 1.602


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0.408901789 W2 -0.13116 W3 0.722254 1.00000


Correlation12 0.12

Correlation23 0.3

Correlation13 -0.8

Covar_12_23_1 0.0136 0.0217 -0.0722

Portfolio SD 0.084

Sharpe Ratio 3.087


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Demonstration of the Portfolio Standard Deviation Calcula RiskFreeR 0.07


Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0.541535 W2 0 W3 0.458464 1.00000


Correlation12 0.12

Correlation23 0.30

Correlation13 -0.8

Covar_12_23_13 0.0136 0.0217 -0.0722

Portfolio SD 0.000

Sharpe Ratio 1240.111


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Demonstration of the Portfolio Standard Deviation Calculation RiskFreeR 0.07


Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 -0.61359992 W2 0.213351 W3 1.400249 1.00000


Correlation12 0.12

Correlation23 0.3

Correlation13 0.8

Covar_12_23_13 0.0136 0.0217 0.0722

Portfolio SD 0.236

Sharpe Ratio 1.957


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Er1 0.213

Esd1 0.376

Er2 0.413

Esd2 0.301

Er3 0.41Esd3 0.24

W1 0 W2 1 W3 0 1.00000


Correlation12 0.12

Correlation23 0.3

Correlation13 0.8

Covar_12_23_13 0.0136 0.0217 0.0722

Portfolio SD 0.232

Sharpe Ratio 1.479

When shorting is not allowed


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3 Asset Allocation Model

Expecte

d Standard Sharpe

Return Deviation Ratios GM MRK GE

GM 33.0% 21.3% 37.6% 0.38 1.00 0.12 0.26

MRK 34.0% 41.3% 30.1% 1.14 0.12 1.00 0.30GE 33.0% 41.0% 24.0% 1.42 0.26 0.30 1.00

100.0%

7% Risk-free rate

34.6% Expected Return

21.3% Expected standard deviation

1.30 Sharpe Ratio

Asset

Weighting

Correlation Coefficient with:


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Demonstration of the Portfolio Standard Deviation Calculation RiskFreeR 0.07


GM 0.213

Esd1 0.376 0.566489

MRK 0.413

Esd2 0.301 1.372093

GE 0.41Esd3 0.24 1.708333

GM 0.33 MRK 0.33 GE 0.34 1.00000


Correlation12 0.12

Correlation23 0.30

Correlation13 0.26

Covar_12_23_13 0.0136 0.0217 0.0235

Portfolio SD 0.21215

Sharpe Ratio 1.301

Maximising Sharpe RatioDemonstration of the Portfolio Standard Deviation Calculation RiskFreeR 0.07 10

equal risk return - changing correlation 20

Er1 0.213 10

Esd1 0.376 0.566489 20

Er2 0.413 10

Esd2 0.301 1.372093 20

Er3 0.41 10

Esd3 0.24 1.708333 -1

W1 0.04 W2 0.348443 W3 0.659561 1.04800

Expected Return 0.422847088 -10

Correlation12 0.12 10Correlation23 0.30 -10

Correlation13 0.26 10

Covar_12_23_13 0.0136 0.0217 0.0235 -10

Portfolio SD 0.219 10

Sharpe Ratio 1.613 -10

10

-10

-1


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Markov's Trilemma Working

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Transcript of Markov's Trilemma Working