Modelo de fijación de precios de activos de capital (CAPM)
El modelo de fijación de precios de activos de capital (CAPM, por sus siglas en inglés) indica cuál debería ser la tasa de rendimiento esperada o requerida en activos de riesgo como las acciones ordinarias de Apache.
Tasas de retorno
Apache Corp., tasas mensuales de retorno
| Apache Corp. (APA) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioAPA,t1 | DividendoAPA,t1 | RAPA,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2011 | $119.36 | 1,286.12 | ||||
| 1. | 28 feb 2011 | $124.62 | 4.41% | 1,327.22 | 3.20% | |
| 2. | 31 mar 2011 | $130.92 | 5.06% | 1,325.83 | -0.10% | |
| 3. | 30 abr 2011 | $133.37 | $0.15 | 1.99% | 1,363.61 | 2.85% |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2015 | $49.18 | 4.35% | 2,080.41 | 0.05% | |
| 59. | 31 dic 2015 | $44.47 | -9.58% | 2,043.94 | -1.75% | |
| Promedio (R): | -1.14% | 0.84% | ||||
| Desviación estándar: | 9.27% | 3.40% | ||||
| Apache Corp. (APA) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioAPA,t1 | DividendoAPA,t1 | RAPA,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2011 | $119.36 | 1,286.12 | ||||
| 1. | 28 feb 2011 | $124.62 | 4.41% | 1,327.22 | 3.20% | |
| 2. | 31 mar 2011 | $130.92 | 5.06% | 1,325.83 | -0.10% | |
| 3. | 30 abr 2011 | $133.37 | $0.15 | 1.99% | 1,363.61 | 2.85% |
| 4. | 31 may 2011 | $124.60 | -6.58% | 1,345.20 | -1.35% | |
| 5. | 30 jun 2011 | $123.39 | -0.97% | 1,320.64 | -1.83% | |
| 6. | 31 jul 2011 | $123.72 | $0.15 | 0.39% | 1,292.28 | -2.15% |
| 7. | 31 ago 2011 | $103.07 | -16.69% | 1,218.89 | -5.68% | |
| 8. | 30 sept 2011 | $80.24 | -22.15% | 1,131.42 | -7.18% | |
| 9. | 31 oct 2011 | $99.63 | $0.15 | 24.35% | 1,253.30 | 10.77% |
| 10. | 30 nov 2011 | $99.44 | -0.19% | 1,246.96 | -0.51% | |
| 11. | 31 dic 2011 | $90.58 | -8.91% | 1,257.60 | 0.85% | |
| 12. | 31 ene 2012 | $98.88 | $0.15 | 9.33% | 1,312.41 | 4.36% |
| 13. | 29 feb 2012 | $107.93 | 9.15% | 1,365.68 | 4.06% | |
| 14. | 31 mar 2012 | $100.44 | -6.94% | 1,408.47 | 3.13% | |
| 15. | 30 abr 2012 | $95.94 | $0.17 | -4.31% | 1,397.91 | -0.75% |
| 16. | 31 may 2012 | $81.38 | -15.18% | 1,310.33 | -6.27% | |
| 17. | 30 jun 2012 | $87.89 | 8.00% | 1,362.16 | 3.96% | |
| 18. | 31 jul 2012 | $86.12 | $0.17 | -1.82% | 1,379.32 | 1.26% |
| 19. | 31 ago 2012 | $85.75 | -0.43% | 1,406.58 | 1.98% | |
| 20. | 30 sept 2012 | $86.47 | 0.84% | 1,440.67 | 2.42% | |
| 21. | 31 oct 2012 | $82.72 | $0.17 | -4.14% | 1,412.16 | -1.98% |
| 22. | 30 nov 2012 | $77.09 | -6.81% | 1,416.18 | 0.28% | |
| 23. | 31 dic 2012 | $78.50 | 1.83% | 1,426.19 | 0.71% | |
| 24. | 31 ene 2013 | $83.76 | $0.17 | 6.92% | 1,498.11 | 5.04% |
| 25. | 28 feb 2013 | $74.27 | -11.33% | 1,514.68 | 1.11% | |
| 26. | 31 mar 2013 | $77.16 | 3.89% | 1,569.19 | 3.60% | |
| 27. | 30 abr 2013 | $73.88 | $0.20 | -3.99% | 1,597.57 | 1.81% |
| 28. | 31 may 2013 | $82.13 | 11.17% | 1,630.74 | 2.08% | |
| 29. | 30 jun 2013 | $83.83 | 2.07% | 1,606.28 | -1.50% | |
| 30. | 31 jul 2013 | $80.25 | $0.20 | -4.03% | 1,685.73 | 4.95% |
| 31. | 31 ago 2013 | $85.68 | 6.77% | 1,632.97 | -3.13% | |
| 32. | 30 sept 2013 | $85.14 | -0.63% | 1,681.55 | 2.97% | |
| 33. | 31 oct 2013 | $88.80 | $0.20 | 4.53% | 1,756.54 | 4.46% |
| 34. | 30 nov 2013 | $91.49 | 3.03% | 1,805.81 | 2.80% | |
| 35. | 31 dic 2013 | $85.94 | -6.07% | 1,848.36 | 2.36% | |
| 36. | 31 ene 2014 | $80.26 | $0.20 | -6.38% | 1,782.59 | -3.56% |
| 37. | 28 feb 2014 | $79.29 | -1.21% | 1,859.45 | 4.31% | |
| 38. | 31 mar 2014 | $82.95 | 4.62% | 1,872.34 | 0.69% | |
| 39. | 30 abr 2014 | $86.80 | $0.25 | 4.94% | 1,883.95 | 0.62% |
| 40. | 31 may 2014 | $93.22 | 7.40% | 1,923.57 | 2.10% | |
| 41. | 30 jun 2014 | $100.62 | 7.94% | 1,960.23 | 1.91% | |
| 42. | 31 jul 2014 | $102.66 | $0.25 | 2.28% | 1,930.67 | -1.51% |
| 43. | 31 ago 2014 | $101.83 | -0.81% | 2,003.37 | 3.77% | |
| 44. | 30 sept 2014 | $93.87 | -7.82% | 1,972.29 | -1.55% | |
| 45. | 31 oct 2014 | $77.20 | $0.25 | -17.49% | 2,018.05 | 2.32% |
| 46. | 30 nov 2014 | $64.09 | -16.98% | 2,067.56 | 2.45% | |
| 47. | 31 dic 2014 | $62.67 | -2.22% | 2,058.90 | -0.42% | |
| 48. | 31 ene 2015 | $62.57 | $0.25 | 0.24% | 1,994.99 | -3.10% |
| 49. | 28 feb 2015 | $65.84 | 5.23% | 2,104.50 | 5.49% | |
| 50. | 31 mar 2015 | $60.33 | -8.37% | 2,067.89 | -1.74% | |
| 51. | 30 abr 2015 | $68.40 | $0.25 | 13.79% | 2,085.51 | 0.85% |
| 52. | 31 may 2015 | $59.84 | -12.51% | 2,107.39 | 1.05% | |
| 53. | 30 jun 2015 | $57.63 | -3.69% | 2,063.11 | -2.10% | |
| 54. | 31 jul 2015 | $45.86 | $0.25 | -19.99% | 2,103.84 | 1.97% |
| 55. | 31 ago 2015 | $45.24 | -1.35% | 1,972.18 | -6.26% | |
| 56. | 30 sept 2015 | $39.16 | -13.44% | 1,920.03 | -2.64% | |
| 57. | 31 oct 2015 | $47.13 | $0.25 | 20.99% | 2,079.36 | 8.30% |
| 58. | 30 nov 2015 | $49.18 | 4.35% | 2,080.41 | 0.05% | |
| 59. | 31 dic 2015 | $44.47 | -9.58% | 2,043.94 | -1.75% | |
| Promedio (R): | -1.14% | 0.84% | ||||
| Desviación estándar: | 9.27% | 3.40% | ||||
Mostrar todo
1 Datos en dólares estadounidenses por acción ordinaria, ajustados por divisiones y dividendos de acciones.
2 Tasa de rendimiento de las acciones ordinarias de APA durante el período t.
3 Tasa de rendimiento del S&P 500 (el proxy de la cartera de mercado) durante el período t.
Varianza y covarianza
Apache Corp., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RAPA,t | RS&P 500,t | (RAPA,t–RAPA)2 | (RS&P 500,t–RS&P 500)2 | (RAPA,t–RAPA)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb 2011 | 4.41% | 3.20% | 30.82 | 5.53 | 13.05 |
| 2. | 31 mar 2011 | 5.06% | -0.10% | 38.44 | 0.90 | -5.89 |
| 3. | 30 abr 2011 | 1.99% | 2.85% | 9.80 | 4.02 | 6.28 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2015 | 4.35% | 0.05% | 30.18 | 0.63 | -4.36 |
| 59. | 31 dic 2015 | -9.58% | -1.75% | 71.11 | 6.75 | 21.91 |
| Total (Σ): | 4,988.66 | 670.89 | 1,031.10 | |||
| t | Fecha | RAPA,t | RS&P 500,t | (RAPA,t–RAPA)2 | (RS&P 500,t–RS&P 500)2 | (RAPA,t–RAPA)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb 2011 | 4.41% | 3.20% | 30.82 | 5.53 | 13.05 |
| 2. | 31 mar 2011 | 5.06% | -0.10% | 38.44 | 0.90 | -5.89 |
| 3. | 30 abr 2011 | 1.99% | 2.85% | 9.80 | 4.02 | 6.28 |
| 4. | 31 may 2011 | -6.58% | -1.35% | 29.50 | 4.82 | 11.92 |
| 5. | 30 jun 2011 | -0.97% | -1.83% | 0.03 | 7.13 | -0.46 |
| 6. | 31 jul 2011 | 0.39% | -2.15% | 2.35 | 8.95 | -4.59 |
| 7. | 31 ago 2011 | -16.69% | -5.68% | 241.70 | 42.56 | 101.42 |
| 8. | 30 sept 2011 | -22.15% | -7.18% | 441.24 | 64.34 | 168.48 |
| 9. | 31 oct 2011 | 24.35% | 10.77% | 650.06 | 98.56 | 253.12 |
| 10. | 30 nov 2011 | -0.19% | -0.51% | 0.91 | 1.82 | -1.29 |
| 11. | 31 dic 2011 | -8.91% | 0.85% | 60.30 | 0.00 | -0.07 |
| 12. | 31 ene 2012 | 9.33% | 4.36% | 109.69 | 12.35 | 36.80 |
| 13. | 29 feb 2012 | 9.15% | 4.06% | 106.03 | 10.33 | 33.10 |
| 14. | 31 mar 2012 | -6.94% | 3.13% | 33.59 | 5.24 | -13.26 |
| 15. | 30 abr 2012 | -4.31% | -0.75% | 10.03 | 2.54 | 5.05 |
| 16. | 31 may 2012 | -15.18% | -6.27% | 196.89 | 50.55 | 99.76 |
| 17. | 30 jun 2012 | 8.00% | 3.96% | 83.61 | 9.68 | 28.44 |
| 18. | 31 jul 2012 | -1.82% | 1.26% | 0.46 | 0.17 | -0.28 |
| 19. | 31 ago 2012 | -0.43% | 1.98% | 0.51 | 1.28 | 0.81 |
| 20. | 30 sept 2012 | 0.84% | 2.42% | 3.94 | 2.49 | 3.13 |
| 21. | 31 oct 2012 | -4.14% | -1.98% | 8.97 | 7.97 | 8.46 |
| 22. | 30 nov 2012 | -6.81% | 0.28% | 32.06 | 0.31 | 3.17 |
| 23. | 31 dic 2012 | 1.83% | 0.71% | 8.84 | 0.02 | -0.41 |
| 24. | 31 ene 2013 | 6.92% | 5.04% | 64.99 | 17.62 | 33.84 |
| 25. | 28 feb 2013 | -11.33% | 1.11% | 103.75 | 0.07 | -2.66 |
| 26. | 31 mar 2013 | 3.89% | 3.60% | 25.36 | 7.58 | 13.87 |
| 27. | 30 abr 2013 | -3.99% | 1.81% | 8.11 | 0.93 | -2.74 |
| 28. | 31 may 2013 | 11.17% | 2.08% | 151.56 | 1.52 | 15.16 |
| 29. | 30 jun 2013 | 2.07% | -1.50% | 10.33 | 5.50 | -7.54 |
| 30. | 31 jul 2013 | -4.03% | 4.95% | 8.34 | 16.82 | -11.84 |
| 31. | 31 ago 2013 | 6.77% | -3.13% | 62.58 | 15.80 | -31.44 |
| 32. | 30 sept 2013 | -0.63% | 2.97% | 0.26 | 4.54 | 1.10 |
| 33. | 31 oct 2013 | 4.53% | 4.46% | 32.24 | 13.07 | 20.53 |
| 34. | 30 nov 2013 | 3.03% | 2.80% | 17.42 | 3.84 | 8.18 |
| 35. | 31 dic 2013 | -6.07% | 2.36% | 24.22 | 2.28 | -7.44 |
| 36. | 31 ene 2014 | -6.38% | -3.56% | 27.38 | 19.39 | 23.04 |
| 37. | 28 feb 2014 | -1.21% | 4.31% | 0.00 | 12.02 | -0.22 |
| 38. | 31 mar 2014 | 4.62% | 0.69% | 33.18 | 0.02 | -0.87 |
| 39. | 30 abr 2014 | 4.94% | 0.62% | 37.05 | 0.05 | -1.37 |
| 40. | 31 may 2014 | 7.40% | 2.10% | 72.94 | 1.58 | 10.75 |
| 41. | 30 jun 2014 | 7.94% | 1.91% | 82.49 | 1.13 | 9.64 |
| 42. | 31 jul 2014 | 2.28% | -1.51% | 11.70 | 5.54 | -8.05 |
| 43. | 31 ago 2014 | -0.81% | 3.77% | 0.11 | 8.53 | 0.98 |
| 44. | 30 sept 2014 | -7.82% | -1.55% | 44.52 | 5.74 | 15.99 |
| 45. | 31 oct 2014 | -17.49% | 2.32% | 267.25 | 2.18 | -24.12 |
| 46. | 30 nov 2014 | -16.98% | 2.45% | 250.83 | 2.59 | -25.48 |
| 47. | 31 dic 2014 | -2.22% | -0.42% | 1.15 | 1.60 | 1.35 |
| 48. | 31 ene 2015 | 0.24% | -3.10% | 1.91 | 15.59 | -5.46 |
| 49. | 28 feb 2015 | 5.23% | 5.49% | 40.58 | 21.57 | 29.59 |
| 50. | 31 mar 2015 | -8.37% | -1.74% | 52.19 | 6.68 | 18.67 |
| 51. | 30 abr 2015 | 13.79% | 0.85% | 223.06 | 0.00 | 0.11 |
| 52. | 31 may 2015 | -12.51% | 1.05% | 129.28 | 0.04 | -2.32 |
| 53. | 30 jun 2015 | -3.69% | -2.10% | 6.50 | 8.68 | 7.51 |
| 54. | 31 jul 2015 | -19.99% | 1.97% | 355.14 | 1.28 | -21.29 |
| 55. | 31 ago 2015 | -1.35% | -6.26% | 0.04 | 50.45 | 1.47 |
| 56. | 30 sept 2015 | -13.44% | -2.64% | 151.17 | 12.17 | 42.90 |
| 57. | 31 oct 2015 | 20.99% | 8.30% | 489.97 | 55.56 | 164.99 |
| 58. | 30 nov 2015 | 4.35% | 0.05% | 30.18 | 0.63 | -4.36 |
| 59. | 31 dic 2015 | -9.58% | -1.75% | 71.11 | 6.75 | 21.91 |
| Total (Σ): | 4,988.66 | 670.89 | 1,031.10 | |||
Mostrar todo
VarianzaAPA = Σ(RAPA,t–RAPA)2 ÷ (59 – 1)
= 4,988.66 ÷ (59 – 1)
= 86.01
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 670.89 ÷ (59 – 1)
= 11.57
CovarianzaAPA, S&P 500 = Σ(RAPA,t–RAPA)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,031.10 ÷ (59 – 1)
= 17.78
Estimación sistemática del riesgo (β)
| VarianzaAPA | 86.01 |
| VarianzaS&P 500 | 11.57 |
| CovarianzaAPA, S&P 500 | 17.78 |
| Coeficiente de correlaciónAPA, S&P 5001 | 0.56 |
| βAPA2 | 1.54 |
| αAPA3 | -2.44% |
Cálculos
1 Coeficiente de correlaciónAPA, S&P 500
= CovarianzaAPA, S&P 500 ÷ (Desviación estándarAPA × Desviación estándarS&P 500)
= 17.78 ÷ (9.27% × 3.40%)
= 0.56
2 βAPA
= CovarianzaAPA, S&P 500 ÷ VarianzaS&P 500
= 17.78 ÷ 11.57
= 1.54
3 αAPA
= PromedioAPA – βAPA × PromedioS&P 500
= -1.14% – 1.54 × 0.84%
= -2.44%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.60% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 14.89% |
| Riesgo sistemático de Apache acciones ordinarias | βAPA | 1.54 |
| Tasa de rendimiento esperada de las acciones ordinarias de Apache3 | E(RAPA) | 20.41% |
1 Promedio no ponderado de los rendimientos de las ofertas de todos los bonos del Tesoro de EE. UU. con cupón fijo en circulación que no vencen ni son rescatables en menos de 10 años (proxy de tasa de rendimiento libre de riesgo).
3 E(RAPA) = RF + βAPA [E(RM) – RF]
= 4.60% + 1.54 [14.89% – 4.60%]
= 20.41%