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Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
Ejercicio markowitz dos activos
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Ejercicio markowitz dos activos

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  • 1. John Alejandro Bedoya Villegas Estudiante de Ingeniería Administrativa Universidad Nacional de Colombia sede Medellín Marzo de 2011Fecha COLCAP HelmBank (P) Ln(ri) COLCAP Ln(ri) HelmBank (P) 1/4/2010 1,374.28 444 1/5/2010 1,380.62 445 0.004603 0.002250 1/6/2010 1,381.35 440 0.000529 -0.011300 1/7/2010 1,368.91 440 -0.009046 0.000000 1/8/2010 1,374.72 440 0.004235 0.000000 1/12/2010 1,369.53 440 -0.003782 0.000000 1/13/2010 1,379.91 440 0.007551 0.000000 1/14/2010 1,386.91 440 0.005060 0.000000 1/15/2010 1,382.66 440 -0.003069 0.000000 1/18/2010 1,382.07 440 -0.000427 0.000000 1/19/2010 1,386.01 440 0.002847 0.000000 1/20/2010 1,381.80 440 -0.003042 0.000000 1/21/2010 1,355.28 438 -0.019379 -0.004556 1/22/2010 1,364.33 438 0.006655 0.000000 1/25/2010 1,345.21 438 -0.014113 0.000000 1/26/2010 1,357.80 438 0.009316 0.000000 1/27/2010 1,355.80 438 -0.001474 0.000000 1/28/2010 1,353.57 439 -0.001646 0.002281 1/29/2010 1,363.67 440 0.007434 0.002275 2/1/2010 1,358.49 440 -0.003806 0.000000 2/2/2010 1,363.85 440 0.003938 0.000000 2/3/2010 1,359.43 440 -0.003246 0.000000 2/4/2010 1,334.80 440 -0.018284 0.000000 2/5/2010 1,326.08 440 -0.006554 0.000000 2/8/2010 1,327.03 439 0.000716 -0.002275 2/9/2010 1,345.54 439 0.013852 0.000000 2/10/2010 1,353.70 440 0.006046 0.002275 2/11/2010 1,358.21 440 0.003326 0.000000 2/12/2010 1,355.84 445 -0.001746 0.011300 2/15/2010 1,357.59 442 0.001290 -0.006764 2/16/2010 1,364.53 442 0.005099 0.000000 2/17/2010 1,366.11 445 0.001157 0.006764 2/18/2010 1,359.64 440 -0.004747 -0.011300 2/19/2010 1,372.51 440 0.009421 0.000000 2/22/2010 1,372.84 440 0.000240 0.000000 2/23/2010 1,379.75 440 0.005021 0.000000 2/24/2010 1,398.47 441 0.013476 0.002270 2/25/2010 1,385.85 441 -0.009065 0.000000 2/26/2010 1,391.02 442 0.003724 0.002265 3/1/2010 1,392.91 442 0.001358 0.000000 3/2/2010 1,391.36 442 -0.001113 0.000000 3/3/2010 1,401.53 445 0.007283 0.006764 3/4/2010 1,398.81 445 -0.001943 0.000000 3/5/2010 1,399.63 441 0.000586 -0.009029 3/8/2010 1,408.35 445 0.006211 0.009029 3/9/2010 1,403.17 444 -0.003685 -0.002250 3/10/2010 1,403.82 441 0.000463 -0.006780 3/11/2010 1,405.26 438 0.001025 -0.006826
  • 2. 3/12/2010 1,406.49 435 0.000875 -0.0068733/15/2010 1,404.42 435 -0.001473 0.0000003/16/2010 1,411.71 435 0.005177 0.0000003/17/2010 1,429.40 436 0.012453 0.0022963/18/2010 1,432.30 436 0.002027 0.0000003/19/2010 1,414.10 438 -0.012788 0.0045773/23/2010 1,426.74 438 0.008899 0.0000003/24/2010 1,422.69 438 -0.002843 0.0000003/25/2010 1,427.24 440 0.003193 0.0045563/26/2010 1,423.02 435 -0.002961 -0.0114293/29/2010 1,429.29 435 0.004396 0.0000003/30/2010 1,430.71 435 0.000993 0.0000003/31/2010 1,440.58 435 0.006875 0.000000 4/5/2010 1,450.93 435 0.007159 0.000000 4/6/2010 1,460.30 435 0.006437 0.000000 4/7/2010 1,472.19 435 0.008109 0.000000 4/8/2010 1,472.76 435 0.000387 0.000000 4/9/2010 1,468.95 439 -0.002590 0.0091534/12/2010 1,470.70 438 0.001191 -0.0022814/13/2010 1,470.91 438 0.000143 0.0000004/14/2010 1,478.40 435 0.005079 -0.0068734/15/2010 1,481.14 440 0.001852 0.0114294/16/2010 1,463.27 440 -0.012138 0.0000004/19/2010 1,453.14 442 -0.006947 0.0045354/20/2010 1,467.76 442 0.010011 0.0000004/21/2010 1,475.06 442 0.004961 0.0000004/22/2010 1,478.51 435 0.002336 -0.0159644/23/2010 1,488.05 436 0.006432 0.0022964/26/2010 1,484.44 440 -0.002429 0.0091324/27/2010 1,472.11 435 -0.008341 -0.0114294/28/2010 1,457.28 442 -0.010125 0.0159644/29/2010 1,469.22 440 0.008160 -0.0045354/30/2010 1,467.26 440 -0.001335 0.000000 5/3/2010 1,490.16 440 0.015487 0.000000 5/4/2010 1,481.63 440 -0.005741 0.000000 5/5/2010 1,446.55 435 -0.023961 -0.011429 5/6/2010 1,403.06 435 -0.030526 0.000000 5/7/2010 1,412.80 435 0.006918 0.0000005/10/2010 1,438.18 424 0.017805 -0.0256135/11/2010 1,442.63 424 0.003089 0.0000005/12/2010 1,451.06 434 0.005826 0.0233115/13/2010 1,454.93 435 0.002663 0.0023015/14/2010 1,444.27 434 -0.007354 -0.0023015/18/2010 1,429.49 434 -0.010286 0.0000005/19/2010 1,406.37 434 -0.016306 0.0000005/20/2010 1,395.99 434 -0.007408 0.0000005/21/2010 1,397.39 434 0.001002 0.0000005/24/2010 1,410.93 438 0.009643 0.0091745/25/2010 1,399.25 438 -0.008313 0.0000005/26/2010 1,432.82 438 0.023708 0.0000005/27/2010 1,428.47 437 -0.003041 -0.0022865/28/2010 1,446.48 437 0.012529 0.0000005/31/2010 1,444.54 441 -0.001342 0.009112 6/1/2010 1,447.26 441 0.001881 0.000000 6/2/2010 1,451.91 441 0.003208 0.000000
  • 3. 6/3/2010 1,453.36 430 0.000998 -0.025260 6/4/2010 1,430.30 430 -0.015994 0.000000 6/8/2010 1,433.27 430 0.002074 0.000000 6/9/2010 1,445.53 430 0.008517 0.0000006/10/2010 1,446.34 435 0.000560 0.0115616/11/2010 1,448.08 434 0.001202 -0.0023016/15/2010 1,450.71 430 0.001815 -0.0092596/16/2010 1,461.39 429 0.007335 -0.0023286/17/2010 1,460.56 429 -0.000568 0.0000006/18/2010 1,472.41 420 0.008081 -0.0212026/21/2010 1,466.78 425 -0.003831 0.0118346/22/2010 1,467.67 420 0.000607 -0.0118346/23/2010 1,471.95 420 0.002912 0.0000006/24/2010 1,468.73 415 -0.002190 -0.0119766/25/2010 1,475.04 420 0.004287 0.0119766/28/2010 1,472.02 420 -0.002050 0.0000006/29/2010 1,453.60 415 -0.012592 -0.0119766/30/2010 1,466.78 425 0.009026 0.023811 7/1/2010 1,459.01 420 -0.005311 -0.011834 7/2/2010 1,455.04 420 -0.002725 0.000000 7/6/2010 1,463.82 420 0.006016 0.000000 7/7/2010 1,462.31 420 -0.001032 0.000000 7/8/2010 1,474.59 415 0.008363 -0.011976 7/9/2010 1,478.49 415 0.002641 0.0000007/12/2010 1,496.79 415 0.012302 0.0000007/13/2010 1,507.16 413 0.006904 -0.0048317/14/2010 1,521.17 411 0.009253 -0.0048547/15/2010 1,532.45 411 0.007388 0.0000007/16/2010 1,526.67 411 -0.003779 0.0000007/19/2010 1,528.75 408 0.001362 -0.0073267/21/2010 1,565.50 413 0.023755 0.0121807/22/2010 1,566.33 413 0.000530 0.0000007/23/2010 1,569.86 410 0.002251 -0.0072907/26/2010 1,596.49 410 0.016821 0.0000007/27/2010 1,602.61 408 0.003826 -0.0048907/28/2010 1,591.33 410 -0.007063 0.0048907/29/2010 1,562.05 410 -0.018571 0.0000007/30/2010 1,571.62 415 0.006108 0.012121 8/2/2010 1,575.01 410 0.002155 -0.012121 8/3/2010 1,586.92 414 0.007533 0.009709 8/4/2010 1,595.41 412 0.005336 -0.004843 8/5/2010 1,601.01 415 0.003504 0.007255 8/6/2010 1,580.41 410 -0.012950 -0.012121 8/9/2010 1,582.51 443 0.001328 0.0774138/10/2010 1,577.34 464 -0.003272 0.0463158/11/2010 1,558.12 456 -0.012260 -0.0173928/12/2010 1,557.42 441 -0.000449 -0.0334488/13/2010 1,567.78 447 0.006630 0.0135148/17/2010 1,575.02 439 0.004607 -0.0180598/18/2010 1,581.48 445 0.004093 0.0135758/19/2010 1,586.73 442 0.003314 -0.0067648/20/2010 1,585.60 448 -0.000712 0.0134838/23/2010 1,606.82 441 0.013294 -0.0157488/24/2010 1,605.58 454 -0.000772 0.0290528/25/2010 1,602.95 454 -0.001639 0.000000
  • 4. 8/26/2010 1,616.93 451 0.008684 -0.006630 8/27/2010 1,639.89 456 0.014100 0.011025 8/30/2010 1,642.91 459 0.001840 0.006557 8/31/2010 1,664.18 466 0.012863 0.015135 9/1/2010 1,713.81 480 0.029386 0.029600 9/2/2010 1,681.03 480 -0.019312 0.000000 9/3/2010 1,687.24 475 0.003687 -0.010471 9/6/2010 1,719.62 486 0.019009 0.022894 9/7/2010 1,715.27 487 -0.002533 0.002055 9/8/2010 1,735.68 504 0.011829 0.034312 9/9/2010 1,730.15 495 -0.003191 -0.018019 9/10/2010 1,699.91 500 -0.017633 0.010050 9/13/2010 1,695.47 507 -0.002615 0.013903 9/14/2010 1,703.11 502 0.004496 -0.009911 9/15/2010 1,693.44 499 -0.005694 -0.005994 9/16/2010 1,691.32 490 -0.001253 -0.018201 9/17/2010 1,691.24 485 -0.000047 -0.010257 9/20/2010 1,709.46 487 0.010716 0.004115 9/21/2010 1,716.15 494 0.003906 0.014271 9/22/2010 1,707.34 488 -0.005147 -0.012220 9/23/2010 1,693.47 480 -0.008157 -0.016529 9/24/2010 1,697.26 476 0.002236 -0.008368 9/27/2010 1,711.46 481 0.008332 0.010449 9/28/2010 1,744.84 477 0.019316 -0.008351 9/29/2010 1,766.22 489 0.012179 0.024846 9/30/2010 1,769.50 485 0.001855 -0.008214 10/1/2010 1,766.79 490 -0.001533 0.010257 10/4/2010 1,766.42 483 -0.000209 -0.014389 10/5/2010 1,794.73 495 0.015900 0.024541 10/6/2010 1,825.05 500 0.016753 0.010050 10/7/2010 1,808.89 499 -0.008894 -0.002002 10/8/2010 1,826.38 490 0.009622 -0.01820110/11/2010 1,845.91 494 0.010637 0.00813010/12/2010 1,842.54 495 -0.001827 0.00202210/13/2010 1,873.50 500 0.016663 0.01005010/14/2010 1,855.43 499 -0.009692 -0.00200210/15/2010 1,833.59 494 -0.011841 -0.01007110/19/2010 1,857.75 496 0.013090 0.00404010/20/2010 1,860.52 498 0.001490 0.00402410/21/2010 1,877.98 485 0.009341 -0.02645110/22/2010 1,906.66 490 0.015156 0.01025710/25/2010 1,900.10 489 -0.003447 -0.00204310/26/2010 1,870.75 480 -0.015567 -0.01857610/27/2010 1,862.84 475 -0.004237 -0.01047110/28/2010 1,871.21 470 0.004483 -0.01058210/29/2010 1,901.83 478 0.016231 0.016878 11/2/2010 1,913.88 475 0.006316 -0.006296 11/3/2010 1,923.28 484 0.004899 0.018770 11/4/2010 1,942.37 488 0.009877 0.008230 11/5/2010 1,940.38 480 -0.001025 -0.016529 11/8/2010 1,939.96 475 -0.000216 -0.010471 11/9/2010 1,938.49 468 -0.000758 -0.01484711/10/2010 1,889.62 449 -0.025534 -0.04144511/11/2010 1,841.69 440 -0.025692 -0.02024811/12/2010 1,795.09 448 -0.025628 0.018019
  • 5. 11/16/2010 1,729.16 435 -0.037419 -0.02944711/17/2010 1,786.89 440 0.032841 0.01142911/18/2010 1,814.13 440 0.015129 0.00000011/19/2010 1,778.83 436 -0.019650 -0.00913211/22/2010 1,775.45 428 -0.001902 -0.01851911/23/2010 1,750.36 428 -0.014232 0.00000011/24/2010 1,790.15 435 0.022478 0.01622311/25/2010 1,784.80 435 -0.002993 0.00000011/26/2010 1,770.59 432 -0.007994 -0.00692011/29/2010 1,762.11 427 -0.004801 -0.01164211/30/2010 1,779.94 430 0.010068 0.007001 12/1/2010 1,793.55 427 0.007617 -0.007001 12/2/2010 1,825.55 438 0.017684 0.025435 12/3/2010 1,830.26 440 0.002577 0.004556 12/6/2010 1,841.85 440 0.006312 0.000000 12/7/2010 1,837.18 440 -0.002539 0.000000 12/9/2010 1,826.70 440 -0.005721 0.00000012/10/2010 1,830.86 440 0.002275 0.00000012/13/2010 1,841.24 436 0.005653 -0.00913212/14/2010 1,845.71 442 0.002425 0.01366812/15/2010 1,823.61 447 -0.012046 0.01124912/16/2010 1,826.74 455 0.001715 0.01773912/17/2010 1,823.10 461 -0.001995 0.01310112/20/2010 1,815.55 458 -0.004150 -0.00652912/21/2010 1,824.82 455 0.005093 -0.00657212/22/2010 1,839.59 465 0.008061 0.02174012/23/2010 1,846.50 470 0.003749 0.01069512/24/2010 1,853.24 465 0.003644 -0.01069512/27/2010 1,866.12 464 0.006926 -0.00215312/28/2010 1,849.51 458 -0.008941 -0.01301512/29/2010 1,839.72 457 -0.005307 -0.00218612/30/2010 1,823.70 458 -0.008746 0.002186 1/3/2011 1,803.00 458 -0.011415 0.000000 1/4/2011 1,810.27 451 0.004024 -0.015402 1/5/2011 1,790.57 445 -0.010942 -0.013393 1/6/2011 1,766.77 445 -0.013381 0.000000 1/7/2011 1,781.71 450 0.008421 0.011173 1/11/2011 1,763.08 450 -0.010511 0.000000 1/12/2011 1,784.10 445 0.011852 -0.011173 1/13/2011 1,801.86 443 0.009905 -0.004505 1/14/2011 1,796.05 450 -0.003230 0.015678 1/17/2011 1,801.66 445 0.003119 -0.011173 1/18/2011 1,790.41 446 -0.006264 0.002245 1/19/2011 1,766.21 444 -0.013609 -0.004494 1/20/2011 1,741.61 435 -0.014026 -0.020479 1/21/2011 1,756.48 435 0.008502 0.000000 1/24/2011 1,759.99 437 0.001996 0.004587 1/25/2011 1,754.92 437 -0.002885 0.000000 1/26/2011 1,771.55 440 0.009432 0.006842 1/27/2011 1,773.04 453 0.000841 0.029117 1/28/2011 1,768.13 446 -0.002773 -0.015573 1/31/2011 1,762.43 446 -0.003229 0.000000 2/1/2011 1,759.61 446 -0.001601 0.000000 2/2/2011 1,745.24 450 -0.008200 0.008929 2/3/2011 1,735.96 448 -0.005332 -0.004454
  • 6. 2/4/2011 1,709.06 440 -0.015617 -0.018019 2/7/2011 1,691.87 440 -0.010109 0.000000 2/8/2011 1,710.54 430 0.010975 -0.022990 2/9/2011 1,696.85 427 -0.008036 -0.0070012/10/2011 1,687.92 425 -0.005277 -0.0046952/11/2011 1,694.51 417 0.003897 -0.0190032/14/2011 1,701.65 415 0.004205 -0.0048082/15/2011 1,692.43 415 -0.005433 0.0000002/16/2011 1,689.95 411 -0.001466 -0.0096852/17/2011 1,666.93 408 -0.013715 -0.0073262/18/2011 1,644.66 406 -0.013450 -0.0049142/21/2011 1,658.33 422 0.008277 0.0386522/22/2011 1,661.43 421 0.001868 -0.0023722/23/2011 1,698.63 418 0.022143 -0.0071512/24/2011 1,699.17 420 0.000318 0.0047732/25/2011 1,713.40 420 0.008340 0.0000002/28/2011 1,750.97 416 0.021690 -0.009569
  • 7. dministrativaolombia sede Medellín 285 Indice-Accion Media Varianza Desv. Estandar (1) COLCAP 0.00085 0.000095 0.009737 (2) HelmBank (P) -0.00023 0.000149 0.012202 Matriz de Covarianza Ln(ri) COLCAP Ln(ri) HelmBank (P) (1) Ln(ri) COLCAP 0 0 (2) Ln(ri) HelmBank (P) 0 0 Matriz de Correlacion Ln(ri) COLCAP Ln(ri) HelmBank (P) (1) Ln(ri) COLCAP 1 0.26 (2) Ln(ri) HelmBank (P) 0.26 1 Datos tomados de http://www.grupoaval.com/portal/page?_pageid=33,54994&_dad=portal&_schema=PORTAL Resultados W* Optimo W* = (W1) (1-W*) = (W2) Ecuacion 10 0.649 0.351 Ecuacion 12 0.649 0.351 Varianza y volatilidad optima del portafolio Varianza Volatilidad Ecuacion 13 - 17 7.3E-05 8.5E-03 Ecuacion 13 - 17 7.3E-05 8.5E-03 Rendimiento medio del Portafolio Up Ecuacion 16 0.000473 Variasiones w no optimos Rentabilidad W1 W2 esperada 1 0 0.00085 0.9 0.1 0.00074 0.8 0.2 0.00064 0.7 0.3 0.00053 0.6 0.4 0.00042 0.5 0.5 0.00031 0.4 0.6 0.00020 0.3 0.7 0.00010 0.2 0.8 -0.00001 0.1 0.9 -0.00012 0 1 -0.00023 Puntos optimos 0.000473
  • 8. Combinacion portafoliosPortafolio optimo Frontera de portafolios 0.001000 0.00085 0.000800 0.00074 0.00064 Combinacion porta 0.000600 0.00053 0.000473 0.000400 0.00042 0.00031 Rentabilidad 0.000200 0.00020 media 0.00010 0.000000 -0.00001 -0.0 -0.000200 -0.000400reduccion0.00700 0.00800 0.13 0.00900 Volatilidad 0.01000 0.01100
  • 9. &_schema=PORTAL Desviación estandar 0.00974 0.00916 0.00875 0.00854 0.00854 0.00875 0.00915 0.00973 0.01044 0.01128 0.01220 0.00852
  • 10. de portafolios Combinacion portafolios Portafolio optimo -0.00001 -0.00012 -0.000230 0.01100 0.01200 0.01300 0.01400

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