John Alejandro Bedoya VillegasEstudiante de Ingeniería AdministrativaUniversidad Nacional de Colombia sede MedellínMarzo de 2011

Fecha COLCAP HelmBank (P) Ln(ri) COLCAP Ln(ri) HelmBank (P)1/4/2010 1,374.28 4441/5/2010 1,380.62 445 0.004603 0.0022501/6/2010 1,381.35 440 0.000529 -0.0113001/7/2010 1,368.91 440 -0.009046 0.0000001/8/2010 1,374.72 440 0.004235 0.000000

1/12/2010 1,369.53 440 -0.003782 0.0000001/13/2010 1,379.91 440 0.007551 0.0000001/14/2010 1,386.91 440 0.005060 0.0000001/15/2010 1,382.66 440 -0.003069 0.0000001/18/2010 1,382.07 440 -0.000427 0.0000001/19/2010 1,386.01 440 0.002847 0.0000001/20/2010 1,381.80 440 -0.003042 0.0000001/21/2010 1,355.28 438 -0.019379 -0.0045561/22/2010 1,364.33 438 0.006655 0.0000001/25/2010 1,345.21 438 -0.014113 0.0000001/26/2010 1,357.80 438 0.009316 0.0000001/27/2010 1,355.80 438 -0.001474 0.0000001/28/2010 1,353.57 439 -0.001646 0.0022811/29/2010 1,363.67 440 0.007434 0.002275

2/1/2010 1,358.49 440 -0.003806 0.0000002/2/2010 1,363.85 440 0.003938 0.0000002/3/2010 1,359.43 440 -0.003246 0.0000002/4/2010 1,334.80 440 -0.018284 0.0000002/5/2010 1,326.08 440 -0.006554 0.0000002/8/2010 1,327.03 439 0.000716 -0.0022752/9/2010 1,345.54 439 0.013852 0.000000

2/10/2010 1,353.70 440 0.006046 0.0022752/11/2010 1,358.21 440 0.003326 0.0000002/12/2010 1,355.84 445 -0.001746 0.0113002/15/2010 1,357.59 442 0.001290 -0.0067642/16/2010 1,364.53 442 0.005099 0.0000002/17/2010 1,366.11 445 0.001157 0.0067642/18/2010 1,359.64 440 -0.004747 -0.0113002/19/2010 1,372.51 440 0.009421 0.0000002/22/2010 1,372.84 440 0.000240 0.0000002/23/2010 1,379.75 440 0.005021 0.0000002/24/2010 1,398.47 441 0.013476 0.0022702/25/2010 1,385.85 441 -0.009065 0.0000002/26/2010 1,391.02 442 0.003724 0.002265

3/1/2010 1,392.91 442 0.001358 0.0000003/2/2010 1,391.36 442 -0.001113 0.000000

3/3/2010 1,401.53 445 0.007283 0.0067643/4/2010 1,398.81 445 -0.001943 0.0000003/5/2010 1,399.63 441 0.000586 -0.0090293/8/2010 1,408.35 445 0.006211 0.0090293/9/2010 1,403.17 444 -0.003685 -0.002250

3/10/2010 1,403.82 441 0.000463 -0.0067803/11/2010 1,405.26 438 0.001025 -0.0068263/12/2010 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 0.0000004/6/2010 1,460.30 435 0.006437 0.0000004/7/2010 1,472.19 435 0.008109 0.0000004/8/2010 1,472.76 435 0.000387 0.0000004/9/2010 1,468.95 439 -0.002590 0.009153

4/12/2010 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 0.0000005/4/2010 1,481.63 440 -0.005741 0.0000005/5/2010 1,446.55 435 -0.023961 -0.0114295/6/2010 1,403.06 435 -0.030526 0.0000005/7/2010 1,412.80 435 0.006918 0.000000

5/10/2010 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 0.023311

5/13/2010 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 0.0000006/2/2010 1,451.91 441 0.003208 0.0000006/3/2010 1,453.36 430 0.000998 -0.0252606/4/2010 1,430.30 430 -0.015994 0.0000006/8/2010 1,433.27 430 0.002074 0.0000006/9/2010 1,445.53 430 0.008517 0.000000

6/10/2010 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 -0.0118347/2/2010 1,455.04 420 -0.002725 0.0000007/6/2010 1,463.82 420 0.006016 0.0000007/7/2010 1,462.31 420 -0.001032 0.0000007/8/2010 1,474.59 415 0.008363 -0.0119767/9/2010 1,478.49 415 0.002641 0.000000

7/12/2010 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 0.000000

7/27/2010 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 -0.0121218/3/2010 1,586.92 414 0.007533 0.0097098/4/2010 1,595.41 412 0.005336 -0.0048438/5/2010 1,601.01 415 0.003504 0.0072558/6/2010 1,580.41 410 -0.012950 -0.0121218/9/2010 1,582.51 443 0.001328 0.077413

8/10/2010 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 0.0000008/26/2010 1,616.93 451 0.008684 -0.0066308/27/2010 1,639.89 456 0.014100 0.0110258/30/2010 1,642.91 459 0.001840 0.0065578/31/2010 1,664.18 466 0.012863 0.015135

9/1/2010 1,713.81 480 0.029386 0.0296009/2/2010 1,681.03 480 -0.019312 0.0000009/3/2010 1,687.24 475 0.003687 -0.0104719/6/2010 1,719.62 486 0.019009 0.0228949/7/2010 1,715.27 487 -0.002533 0.0020559/8/2010 1,735.68 504 0.011829 0.0343129/9/2010 1,730.15 495 -0.003191 -0.018019

9/10/2010 1,699.91 500 -0.017633 0.0100509/13/2010 1,695.47 507 -0.002615 0.0139039/14/2010 1,703.11 502 0.004496 -0.0099119/15/2010 1,693.44 499 -0.005694 -0.0059949/16/2010 1,691.32 490 -0.001253 -0.0182019/17/2010 1,691.24 485 -0.000047 -0.0102579/20/2010 1,709.46 487 0.010716 0.0041159/21/2010 1,716.15 494 0.003906 0.0142719/22/2010 1,707.34 488 -0.005147 -0.0122209/23/2010 1,693.47 480 -0.008157 -0.0165299/24/2010 1,697.26 476 0.002236 -0.0083689/27/2010 1,711.46 481 0.008332 0.0104499/28/2010 1,744.84 477 0.019316 -0.0083519/29/2010 1,766.22 489 0.012179 0.0248469/30/2010 1,769.50 485 0.001855 -0.00821410/1/2010 1,766.79 490 -0.001533 0.010257

10/4/2010 1,766.42 483 -0.000209 -0.01438910/5/2010 1,794.73 495 0.015900 0.02454110/6/2010 1,825.05 500 0.016753 0.01005010/7/2010 1,808.89 499 -0.008894 -0.00200210/8/2010 1,826.38 490 0.009622 -0.018201

10/11/2010 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 -0.00629611/3/2010 1,923.28 484 0.004899 0.01877011/4/2010 1,942.37 488 0.009877 0.00823011/5/2010 1,940.38 480 -0.001025 -0.01652911/8/2010 1,939.96 475 -0.000216 -0.01047111/9/2010 1,938.49 468 -0.000758 -0.014847

11/10/2010 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 0.01801911/16/2010 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 -0.00700112/2/2010 1,825.55 438 0.017684 0.02543512/3/2010 1,830.26 440 0.002577 0.00455612/6/2010 1,841.85 440 0.006312 0.00000012/7/2010 1,837.18 440 -0.002539 0.00000012/9/2010 1,826.70 440 -0.005721 0.000000

12/10/2010 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 0.013668

12/15/2010 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 0.0000001/4/2011 1,810.27 451 0.004024 -0.0154021/5/2011 1,790.57 445 -0.010942 -0.0133931/6/2011 1,766.77 445 -0.013381 0.0000001/7/2011 1,781.71 450 0.008421 0.011173

1/11/2011 1,763.08 450 -0.010511 0.0000001/12/2011 1,784.10 445 0.011852 -0.0111731/13/2011 1,801.86 443 0.009905 -0.0045051/14/2011 1,796.05 450 -0.003230 0.0156781/17/2011 1,801.66 445 0.003119 -0.0111731/18/2011 1,790.41 446 -0.006264 0.0022451/19/2011 1,766.21 444 -0.013609 -0.0044941/20/2011 1,741.61 435 -0.014026 -0.0204791/21/2011 1,756.48 435 0.008502 0.0000001/24/2011 1,759.99 437 0.001996 0.0045871/25/2011 1,754.92 437 -0.002885 0.0000001/26/2011 1,771.55 440 0.009432 0.0068421/27/2011 1,773.04 453 0.000841 0.0291171/28/2011 1,768.13 446 -0.002773 -0.0155731/31/2011 1,762.43 446 -0.003229 0.000000

2/1/2011 1,759.61 446 -0.001601 0.0000002/2/2011 1,745.24 450 -0.008200 0.0089292/3/2011 1,735.96 448 -0.005332 -0.0044542/4/2011 1,709.06 440 -0.015617 -0.0180192/7/2011 1,691.87 440 -0.010109 0.0000002/8/2011 1,710.54 430 0.010975 -0.0229902/9/2011 1,696.85 427 -0.008036 -0.007001

2/10/2011 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 -0.002372

2/23/2011 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

Estudiante de Ingeniería AdministrativaUniversidad Nacional de Colombia sede Medellín

285Indice-Accion Media Varianza Desv. Estandar(1) COLCAP 0.00085 0.000095 0.009737(2) HelmBank (P) -0.00023 0.000149 0.012202

Matriz de CovarianzaLn(ri) COLCAP Ln(ri) HelmBank (P)

(1) Ln(ri) COLCAP 9.4483366E-05 3.124498711746E-05(2) Ln(ri) HelmBank (P) 3.1244987E-05 0.0001483642022969

Matriz de CorrelacionLn(ri) COLCAP Ln(ri) HelmBank (P)

(1) Ln(ri) COLCAP 1 0.263899221521703(2) Ln(ri) HelmBank (P) 0.26389922152 1

Datos tomados de

ResultadosW* Optimo

W* = (W1) (1-W*) = (W2)Ecuacion 10 0.649 0.351Ecuacion 12 0.649 0.351

Varianza y volatilidad optima del portafolioVarianza Volatilidad

Ecuacion 13 - 17 7.3E-05 8.5E-03Ecuacion 13 - 17 7.3E-05 8.5E-03

Rendimiento medio del PortafolioUp

Ecuacion 16 0.000473

Variasiones w no optimos RentabilidadW 1 W 2 esperada

1 0 0.000850.9 0.1 0.000740.8 0.2 0.000640.7 0.3 0.000530.6 0.4 0.000420.5 0.5 0.00031

http://www.grupoaval.com/portal/page?_pageid=33,54994&_dad=portal&_schema=PORTAL

0.4 0.6 0.000200.3 0.7 0.000100.2 0.8 -0.000010.1 0.9 -0.00012

0 1 -0.00023Puntos optimos 0.000473

Combinacion portafoliosPortafolio optimo

reduccion 0.12547608552

0.00700 0.00800 0.00900 0.01000 0.01100 0.01200 0.01300 0.01400

-0.000400

-0.000200

0.000000

0.000200

0.000400

0.000600

0.000800

0.001000

1; 0.000473

1; 0.00085

2; 0.00074

3; 0.00064

4; 0.00053

5; 0.00042

6; 0.00031

7; 0.00020

8; 0.00010

9; -0.00001

10; -0.00012

11; -0.00023

Frontera de portafolios

Combinacion portafolios

Portafolio optimo

Rent

abili

dad

med

ia

Volatilidad

Desviación estandar0.009740.009160.008750.008540.008540.00875

http://www.grupoaval.com/portal/page?_pageid=33,54994&_dad=portal&_schema=PORTAL

0.009150.009730.010440.011280.012200.00852

0.00700 0.00800 0.00900 0.01000 0.01100 0.01200 0.01300 0.01400

-0.000400

-0.000200

0.000000

0.000200

0.000400

0.000600

0.000800

0.001000

1; 0.000473

1; 0.00085

2; 0.00074

3; 0.00064

4; 0.00053

5; 0.00042

6; 0.00031

7; 0.00020

8; 0.00010

9; -0.00001

10; -0.00012

11; -0.00023

Frontera de portafolios

Combinacion portafolios

Portafolio optimo

Volatilidad