Modelo de fijación de precios de activos de capital (CAPM)
El modelo de fijación de precios de activos de capital (CAPM) indica cuál debería ser la tasa de rendimiento esperada o requerida de los activos de riesgo, como las acciones ordinarias de Apple.
Tasas de retorno
Apple Inc., tasas de rendimiento mensuales
| Apple Inc. (AAPL) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioAAPL,t1 | DividendoAAPL,t1 | RAAPL,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 oct. 2015 | $29.88 | 2,079.36 | ||||
| 1. | 30 nov. 2015 | $29.58 | $0.13 | -0.57% | 2,080.41 | 0.05% |
| 2. | 31 dic. 2015 | $26.32 | -11.02% | 2,043.94 | -1.75% | |
| 3. | 31 ene. 2016 | $24.34 | -7.52% | 1,940.24 | -5.07% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 ago. 2021 | $151.83 | $0.22 | 4.24% | 4,522.68 | 2.90% |
| 71. | 30 sept. 2021 | $141.50 | -6.80% | 4,307.54 | -4.76% | |
| Promedio (R): | 2.69% | 1.12% | ||||
| Desviación estándar: | 8.49% | 4.19% | ||||
| Apple Inc. (AAPL) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioAAPL,t1 | DividendoAAPL,t1 | RAAPL,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 oct. 2015 | $29.88 | 2,079.36 | ||||
| 1. | 30 nov. 2015 | $29.58 | $0.13 | -0.57% | 2,080.41 | 0.05% |
| 2. | 31 dic. 2015 | $26.32 | -11.02% | 2,043.94 | -1.75% | |
| 3. | 31 ene. 2016 | $24.34 | -7.52% | 1,940.24 | -5.07% | |
| 4. | 29 feb. 2016 | $24.17 | $0.13 | -0.16% | 1,932.23 | -0.41% |
| 5. | 31 mar. 2016 | $27.25 | 12.74% | 2,059.74 | 6.60% | |
| 6. | 30 abr. 2016 | $23.44 | -13.98% | 2,065.30 | 0.27% | |
| 7. | 31 may. 2016 | $24.97 | $0.1425 | 7.14% | 2,096.95 | 1.53% |
| 8. | 30 jun. 2016 | $23.90 | -4.29% | 2,098.86 | 0.09% | |
| 9. | 31 jul. 2016 | $26.05 | 9.00% | 2,173.60 | 3.56% | |
| 10. | 31 ago. 2016 | $26.53 | $0.1425 | 2.39% | 2,170.95 | -0.12% |
| 11. | 30 sept. 2016 | $28.26 | 6.52% | 2,168.27 | -0.12% | |
| 12. | 31 oct. 2016 | $28.39 | 0.46% | 2,126.15 | -1.94% | |
| 13. | 30 nov. 2016 | $27.63 | $0.1425 | -2.18% | 2,198.81 | 3.42% |
| 14. | 31 dic. 2016 | $28.96 | 4.81% | 2,238.83 | 1.82% | |
| 15. | 31 ene. 2017 | $30.34 | 4.77% | 2,278.87 | 1.79% | |
| 16. | 28 feb. 2017 | $34.25 | $0.1425 | 13.36% | 2,363.64 | 3.72% |
| 17. | 31 mar. 2017 | $35.92 | 4.88% | 2,362.72 | -0.04% | |
| 18. | 30 abr. 2017 | $35.91 | -0.03% | 2,384.20 | 0.91% | |
| 19. | 31 may. 2017 | $38.19 | $0.1575 | 6.79% | 2,411.80 | 1.16% |
| 20. | 30 jun. 2017 | $36.01 | -5.71% | 2,423.41 | 0.48% | |
| 21. | 31 jul. 2017 | $37.18 | 3.25% | 2,470.30 | 1.93% | |
| 22. | 31 ago. 2017 | $41.00 | $0.1575 | 10.70% | 2,471.65 | 0.05% |
| 23. | 30 sept. 2017 | $38.53 | -6.02% | 2,519.36 | 1.93% | |
| 24. | 31 oct. 2017 | $42.26 | 9.68% | 2,575.26 | 2.22% | |
| 25. | 30 nov. 2017 | $42.96 | $0.1575 | 2.03% | 2,647.58 | 2.81% |
| 26. | 31 dic. 2017 | $42.31 | -1.51% | 2,673.61 | 0.98% | |
| 27. | 31 ene. 2018 | $41.86 | -1.06% | 2,823.81 | 5.62% | |
| 28. | 28 feb. 2018 | $44.53 | $0.1575 | 6.75% | 2,713.83 | -3.89% |
| 29. | 31 mar. 2018 | $41.95 | -5.79% | 2,640.87 | -2.69% | |
| 30. | 30 abr. 2018 | $41.32 | -1.50% | 2,648.05 | 0.27% | |
| 31. | 31 may. 2018 | $46.72 | $0.1825 | 13.51% | 2,705.27 | 2.16% |
| 32. | 30 jun. 2018 | $46.28 | -0.94% | 2,718.37 | 0.48% | |
| 33. | 31 jul. 2018 | $47.57 | 2.79% | 2,816.29 | 3.60% | |
| 34. | 31 ago. 2018 | $56.91 | $0.1825 | 20.02% | 2,901.52 | 3.03% |
| 35. | 30 sept. 2018 | $56.44 | -0.83% | 2,913.98 | 0.43% | |
| 36. | 31 oct. 2018 | $54.72 | -3.05% | 2,711.74 | -6.94% | |
| 37. | 30 nov. 2018 | $44.65 | $0.1825 | -18.07% | 2,760.17 | 1.79% |
| 38. | 31 dic. 2018 | $39.44 | -11.67% | 2,506.85 | -9.18% | |
| 39. | 31 ene. 2019 | $41.61 | 5.50% | 2,704.10 | 7.87% | |
| 40. | 28 feb. 2019 | $43.29 | $0.1825 | 4.48% | 2,784.49 | 2.97% |
| 41. | 31 mar. 2019 | $47.49 | 9.70% | 2,834.40 | 1.79% | |
| 42. | 30 abr. 2019 | $50.17 | 5.64% | 2,945.83 | 3.93% | |
| 43. | 31 may. 2019 | $43.77 | $0.1925 | -12.37% | 2,752.06 | -6.58% |
| 44. | 30 jun. 2019 | $49.48 | 13.05% | 2,941.76 | 6.89% | |
| 45. | 31 jul. 2019 | $53.26 | 7.64% | 2,980.38 | 1.31% | |
| 46. | 31 ago. 2019 | $52.19 | $0.1925 | -1.65% | 2,926.46 | -1.81% |
| 47. | 30 sept. 2019 | $55.99 | 7.28% | 2,976.74 | 1.72% | |
| 48. | 31 oct. 2019 | $62.19 | 11.07% | 3,037.56 | 2.04% | |
| 49. | 30 nov. 2019 | $66.81 | $0.1925 | 7.74% | 3,140.98 | 3.40% |
| 50. | 31 dic. 2019 | $73.41 | 9.88% | 3,230.78 | 2.86% | |
| 51. | 31 ene. 2020 | $77.38 | 5.41% | 3,225.52 | -0.16% | |
| 52. | 29 feb. 2020 | $68.34 | $0.1925 | -11.43% | 2,954.22 | -8.41% |
| 53. | 31 mar. 2020 | $63.57 | -6.98% | 2,584.59 | -12.51% | |
| 54. | 30 abr. 2020 | $73.45 | 15.54% | 2,912.43 | 12.68% | |
| 55. | 31 may. 2020 | $79.49 | $0.205 | 8.50% | 3,044.31 | 4.53% |
| 56. | 30 jun. 2020 | $91.20 | 14.73% | 3,100.29 | 1.84% | |
| 57. | 31 jul. 2020 | $106.26 | 16.51% | 3,271.12 | 5.51% | |
| 58. | 31 ago. 2020 | $129.04 | $0.205 | 21.63% | 3,500.31 | 7.01% |
| 59. | 30 sept. 2020 | $115.81 | -10.25% | 3,363.00 | -3.92% | |
| 60. | 31 oct. 2020 | $108.86 | -6.00% | 3,269.96 | -2.77% | |
| 61. | 30 nov. 2020 | $119.05 | $0.205 | 9.55% | 3,621.63 | 10.75% |
| 62. | 31 dic. 2020 | $132.69 | 11.46% | 3,756.07 | 3.71% | |
| 63. | 31 ene. 2021 | $131.96 | -0.55% | 3,714.24 | -1.11% | |
| 64. | 28 feb. 2021 | $121.26 | $0.205 | -7.95% | 3,811.15 | 2.61% |
| 65. | 31 mar. 2021 | $122.15 | 0.73% | 3,972.89 | 4.24% | |
| 66. | 30 abr. 2021 | $131.46 | 7.62% | 4,181.17 | 5.24% | |
| 67. | 31 may. 2021 | $124.61 | $0.22 | -5.04% | 4,204.11 | 0.55% |
| 68. | 30 jun. 2021 | $136.96 | 9.91% | 4,297.50 | 2.22% | |
| 69. | 31 jul. 2021 | $145.86 | 6.50% | 4,395.26 | 2.27% | |
| 70. | 31 ago. 2021 | $151.83 | $0.22 | 4.24% | 4,522.68 | 2.90% |
| 71. | 30 sept. 2021 | $141.50 | -6.80% | 4,307.54 | -4.76% | |
| Promedio (R): | 2.69% | 1.12% | ||||
| Desviación estándar: | 8.49% | 4.19% | ||||
Mostrar todo
1 Datos en US$ por acción de acciones ordinarias, ajustados por escisiones y dividendos de acciones.
2 Tasa de rendimiento de las acciones ordinarias de AAPL 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
Apple Inc., cálculo de varianza y covarianza de rendimientos
| t | Fecha | RAAPL,t | RS&P 500,t | (RAAPL,t–RAAPL)2 | (RS&P 500,t–RS&P 500)2 | (RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 30 nov. 2015 | -0.57% | 0.05% | 10.62 | 1.14 | 3.48 |
| 2. | 31 dic. 2015 | -11.02% | -1.75% | 187.98 | 8.24 | 39.36 |
| 3. | 31 ene. 2016 | -7.52% | -5.07% | 104.29 | 38.33 | 63.23 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 ago. 2021 | 4.24% | 2.90% | 2.42 | 3.17 | 2.77 |
| 71. | 30 sept. 2021 | -6.80% | -4.76% | 90.12 | 34.51 | 55.77 |
| Total (Σ): | 5,051.17 | 1,226.33 | 1,589.97 | |||
| t | Fecha | RAAPL,t | RS&P 500,t | (RAAPL,t–RAAPL)2 | (RS&P 500,t–RS&P 500)2 | (RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 30 nov. 2015 | -0.57% | 0.05% | 10.62 | 1.14 | 3.48 |
| 2. | 31 dic. 2015 | -11.02% | -1.75% | 187.98 | 8.24 | 39.36 |
| 3. | 31 ene. 2016 | -7.52% | -5.07% | 104.29 | 38.33 | 63.23 |
| 4. | 29 feb. 2016 | -0.16% | -0.41% | 8.14 | 2.34 | 4.37 |
| 5. | 31 mar. 2016 | 12.74% | 6.60% | 101.08 | 30.04 | 55.11 |
| 6. | 30 abr. 2016 | -13.98% | 0.27% | 277.93 | 0.72 | 14.14 |
| 7. | 31 may. 2016 | 7.14% | 1.53% | 19.77 | 0.17 | 1.84 |
| 8. | 30 jun. 2016 | -4.29% | 0.09% | 48.64 | 1.05 | 7.16 |
| 9. | 31 jul. 2016 | 9.00% | 3.56% | 39.77 | 5.97 | 15.41 |
| 10. | 31 ago. 2016 | 2.39% | -0.12% | 0.09 | 1.54 | 0.37 |
| 11. | 30 sept. 2016 | 6.52% | -0.12% | 14.68 | 1.54 | -4.76 |
| 12. | 31 oct. 2016 | 0.46% | -1.94% | 4.97 | 9.37 | 6.82 |
| 13. | 30 nov. 2016 | -2.18% | 3.42% | 23.66 | 5.29 | -11.19 |
| 14. | 31 dic. 2016 | 4.81% | 1.82% | 4.51 | 0.49 | 1.49 |
| 15. | 31 ene. 2017 | 4.77% | 1.79% | 4.31 | 0.45 | 1.39 |
| 16. | 28 feb. 2017 | 13.36% | 3.72% | 113.80 | 6.77 | 27.76 |
| 17. | 31 mar. 2017 | 4.88% | -0.04% | 4.78 | 1.34 | -2.53 |
| 18. | 30 abr. 2017 | -0.03% | 0.91% | 7.38 | 0.04 | 0.57 |
| 19. | 31 may. 2017 | 6.79% | 1.16% | 16.80 | 0.00 | 0.16 |
| 20. | 30 jun. 2017 | -5.71% | 0.48% | 70.52 | 0.41 | 5.35 |
| 21. | 31 jul. 2017 | 3.25% | 1.93% | 0.31 | 0.67 | 0.46 |
| 22. | 31 ago. 2017 | 10.70% | 0.05% | 64.14 | 1.13 | -8.52 |
| 23. | 30 sept. 2017 | -6.02% | 1.93% | 75.93 | 0.66 | -7.08 |
| 24. | 31 oct. 2017 | 9.68% | 2.22% | 48.88 | 1.21 | 7.70 |
| 25. | 30 nov. 2017 | 2.03% | 2.81% | 0.44 | 2.86 | -1.12 |
| 26. | 31 dic. 2017 | -1.51% | 0.98% | 17.66 | 0.02 | 0.57 |
| 27. | 31 ene. 2018 | -1.06% | 5.62% | 14.09 | 20.25 | -16.89 |
| 28. | 28 feb. 2018 | 6.75% | -3.89% | 16.53 | 25.13 | -20.38 |
| 29. | 31 mar. 2018 | -5.79% | -2.69% | 71.97 | 14.49 | 32.29 |
| 30. | 30 abr. 2018 | -1.50% | 0.27% | 17.57 | 0.72 | 3.55 |
| 31. | 31 may. 2018 | 13.51% | 2.16% | 117.09 | 1.09 | 11.28 |
| 32. | 30 jun. 2018 | -0.94% | 0.48% | 13.19 | 0.40 | 2.30 |
| 33. | 31 jul. 2018 | 2.79% | 3.60% | 0.01 | 6.17 | 0.24 |
| 34. | 31 ago. 2018 | 20.02% | 3.03% | 300.27 | 3.64 | 33.07 |
| 35. | 30 sept. 2018 | -0.83% | 0.43% | 12.36 | 0.47 | 2.42 |
| 36. | 31 oct. 2018 | -3.05% | -6.94% | 32.91 | 64.94 | 46.23 |
| 37. | 30 nov. 2018 | -18.07% | 1.79% | 430.92 | 0.45 | -13.87 |
| 38. | 31 dic. 2018 | -11.67% | -9.18% | 206.15 | 106.00 | 147.83 |
| 39. | 31 ene. 2019 | 5.50% | 7.87% | 7.91 | 45.57 | 18.99 |
| 40. | 28 feb. 2019 | 4.48% | 2.97% | 3.19 | 3.44 | 3.31 |
| 41. | 31 mar. 2019 | 9.70% | 1.79% | 49.18 | 0.45 | 4.73 |
| 42. | 30 abr. 2019 | 5.64% | 3.93% | 8.73 | 7.92 | 8.31 |
| 43. | 31 may. 2019 | -12.37% | -6.58% | 226.88 | 59.22 | 115.92 |
| 44. | 30 jun. 2019 | 13.05% | 6.89% | 107.25 | 33.35 | 59.81 |
| 45. | 31 jul. 2019 | 7.64% | 1.31% | 24.50 | 0.04 | 0.96 |
| 46. | 31 ago. 2019 | -1.65% | -1.81% | 18.81 | 8.57 | 12.70 |
| 47. | 30 sept. 2019 | 7.28% | 1.72% | 21.08 | 0.36 | 2.76 |
| 48. | 31 oct. 2019 | 11.07% | 2.04% | 70.29 | 0.86 | 7.76 |
| 49. | 30 nov. 2019 | 7.74% | 3.40% | 25.49 | 5.23 | 11.55 |
| 50. | 31 dic. 2019 | 9.88% | 2.86% | 51.69 | 3.03 | 12.52 |
| 51. | 31 ene. 2020 | 5.41% | -0.16% | 7.39 | 1.64 | -3.48 |
| 52. | 29 feb. 2020 | -11.43% | -8.41% | 199.47 | 90.80 | 134.58 |
| 53. | 31 mar. 2020 | -6.98% | -12.51% | 93.49 | 185.77 | 131.79 |
| 54. | 30 abr. 2020 | 15.54% | 12.68% | 165.19 | 133.78 | 148.66 |
| 55. | 31 may. 2020 | 8.50% | 4.53% | 33.79 | 11.63 | 19.82 |
| 56. | 30 jun. 2020 | 14.73% | 1.84% | 145.01 | 0.52 | 8.68 |
| 57. | 31 jul. 2020 | 16.51% | 5.51% | 191.09 | 19.29 | 60.72 |
| 58. | 31 ago. 2020 | 21.63% | 7.01% | 358.78 | 34.67 | 111.54 |
| 59. | 30 sept. 2020 | -10.25% | -3.92% | 167.50 | 25.41 | 65.24 |
| 60. | 31 oct. 2020 | -6.00% | -2.77% | 75.53 | 15.09 | 33.76 |
| 61. | 30 nov. 2020 | 9.55% | 10.75% | 47.05 | 92.86 | 66.10 |
| 62. | 31 dic. 2020 | 11.46% | 3.71% | 76.88 | 6.73 | 22.75 |
| 63. | 31 ene. 2021 | -0.55% | -1.11% | 10.50 | 4.98 | 7.23 |
| 64. | 28 feb. 2021 | -7.95% | 2.61% | 113.27 | 2.22 | -15.87 |
| 65. | 31 mar. 2021 | 0.73% | 4.24% | 3.82 | 9.77 | -6.11 |
| 66. | 30 abr. 2021 | 7.62% | 5.24% | 24.33 | 17.01 | 20.34 |
| 67. | 31 may. 2021 | -5.04% | 0.55% | 59.80 | 0.32 | 4.40 |
| 68. | 30 jun. 2021 | 9.91% | 2.22% | 52.15 | 1.22 | 7.97 |
| 69. | 31 jul. 2021 | 6.50% | 2.27% | 14.51 | 1.34 | 4.41 |
| 70. | 31 ago. 2021 | 4.24% | 2.90% | 2.42 | 3.17 | 2.77 |
| 71. | 30 sept. 2021 | -6.80% | -4.76% | 90.12 | 34.51 | 55.77 |
| Total (Σ): | 5,051.17 | 1,226.33 | 1,589.97 | |||
Mostrar todo
VarianzaAAPL = Σ(RAAPL,t–RAAPL)2 ÷ (71 – 1)
= 5,051.17 ÷ (71 – 1)
= 72.16
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,226.33 ÷ (71 – 1)
= 17.52
CovarianzaAAPL, S&P 500 = Σ(RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 1,589.97 ÷ (71 – 1)
= 22.71
Estimación sistemática del riesgo (β)
| VarianzaAAPL | 72.16 |
| VarianzaS&P 500 | 17.52 |
| CovarianzaAAPL, S&P 500 | 22.71 |
| Coeficiente de correlaciónAAPL, S&P 5001 | 0.64 |
| βAAPL2 | 1.30 |
| αAAPL3 | 1.24% |
Cálculos
1 Coeficiente de correlaciónAAPL, S&P 500
= CovarianzaAAPL, S&P 500 ÷ (Desviación estándarAAPL × Desviación estándarS&P 500)
= 22.71 ÷ (8.49% × 4.19%)
= 0.64
2 βAAPL
= CovarianzaAAPL, S&P 500 ÷ VarianzaS&P 500
= 22.71 ÷ 17.52
= 1.30
3 αAAPL
= PromedioAAPL – βAAPL × PromedioS&P 500
= 2.69% – 1.30 × 1.12%
= 1.24%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento de LT Treasury Composite1 | RF | 3.26% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.05% |
| Riesgo sistemático de acciones ordinarias Apple | βAAPL | 1.30 |
| Tasa de rendimiento esperada de las acciones ordinarias de Apple3 | E(RAAPL) | 15.96% |
1 Promedio no ponderado de los rendimientos de las ofertas en todos los bonos del Tesoro de los Estados Unidos con cupón fijo pendientes, ni vencidos ni exigibles en menos de 10 años (proxy de tasa de rendimiento libre de riesgo).
3 E(RAAPL) = RF + βAAPL [E(RM) – RF]
= 3.26% + 1.30 [13.05% – 3.26%]
= 15.96%