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 Apple.
Tasas de retorno
Apple Inc., tasas mensuales de retorno
| 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. 2017 | $42.26 | 2,575.26 | ||||
| 1. | 30 nov. 2017 | $42.96 | $0.1575 | 2.03% | 2,647.58 | 2.81% |
| 2. | 31 dic. 2017 | $42.31 | -1.51% | 2,673.61 | 0.98% | |
| 3. | 31 ene. 2018 | $41.86 | -1.06% | 2,823.81 | 5.62% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 ago. 2023 | $187.87 | $0.24 | -4.25% | 4,507.66 | -1.77% |
| 71. | 30 sept. 2023 | $171.21 | -8.87% | 4,288.05 | -4.87% | |
| Promedio (R): | 2.46% | 0.85% | ||||
| Desviación estándar: | 8.90% | 5.13% | ||||
| 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. 2017 | $42.26 | 2,575.26 | ||||
| 1. | 30 nov. 2017 | $42.96 | $0.1575 | 2.03% | 2,647.58 | 2.81% |
| 2. | 31 dic. 2017 | $42.31 | -1.51% | 2,673.61 | 0.98% | |
| 3. | 31 ene. 2018 | $41.86 | -1.06% | 2,823.81 | 5.62% | |
| 4. | 28 feb. 2018 | $44.53 | $0.1575 | 6.75% | 2,713.83 | -3.89% |
| 5. | 31 mar. 2018 | $41.95 | -5.79% | 2,640.87 | -2.69% | |
| 6. | 30 abr. 2018 | $41.32 | -1.50% | 2,648.05 | 0.27% | |
| 7. | 31 may. 2018 | $46.72 | $0.1825 | 13.51% | 2,705.27 | 2.16% |
| 8. | 30 jun. 2018 | $46.28 | -0.94% | 2,718.37 | 0.48% | |
| 9. | 31 jul. 2018 | $47.57 | 2.79% | 2,816.29 | 3.60% | |
| 10. | 31 ago. 2018 | $56.91 | $0.1825 | 20.02% | 2,901.52 | 3.03% |
| 11. | 30 sept. 2018 | $56.44 | -0.83% | 2,913.98 | 0.43% | |
| 12. | 31 oct. 2018 | $54.72 | -3.05% | 2,711.74 | -6.94% | |
| 13. | 30 nov. 2018 | $44.65 | $0.1825 | -18.07% | 2,760.17 | 1.79% |
| 14. | 31 dic. 2018 | $39.44 | -11.67% | 2,506.85 | -9.18% | |
| 15. | 31 ene. 2019 | $41.61 | 5.50% | 2,704.10 | 7.87% | |
| 16. | 28 feb. 2019 | $43.29 | $0.1825 | 4.48% | 2,784.49 | 2.97% |
| 17. | 31 mar. 2019 | $47.49 | 9.70% | 2,834.40 | 1.79% | |
| 18. | 30 abr. 2019 | $50.17 | 5.64% | 2,945.83 | 3.93% | |
| 19. | 31 may. 2019 | $43.77 | $0.1925 | -12.37% | 2,752.06 | -6.58% |
| 20. | 30 jun. 2019 | $49.48 | 13.05% | 2,941.76 | 6.89% | |
| 21. | 31 jul. 2019 | $53.26 | 7.64% | 2,980.38 | 1.31% | |
| 22. | 31 ago. 2019 | $52.19 | $0.1925 | -1.65% | 2,926.46 | -1.81% |
| 23. | 30 sept. 2019 | $55.99 | 7.28% | 2,976.74 | 1.72% | |
| 24. | 31 oct. 2019 | $62.19 | 11.07% | 3,037.56 | 2.04% | |
| 25. | 30 nov. 2019 | $66.81 | $0.1925 | 7.74% | 3,140.98 | 3.40% |
| 26. | 31 dic. 2019 | $73.41 | 9.88% | 3,230.78 | 2.86% | |
| 27. | 31 ene. 2020 | $77.38 | 5.41% | 3,225.52 | -0.16% | |
| 28. | 29 feb. 2020 | $68.34 | $0.1925 | -11.43% | 2,954.22 | -8.41% |
| 29. | 31 mar. 2020 | $63.57 | -6.98% | 2,584.59 | -12.51% | |
| 30. | 30 abr. 2020 | $73.45 | 15.54% | 2,912.43 | 12.68% | |
| 31. | 31 may. 2020 | $79.49 | $0.205 | 8.50% | 3,044.31 | 4.53% |
| 32. | 30 jun. 2020 | $91.20 | 14.73% | 3,100.29 | 1.84% | |
| 33. | 31 jul. 2020 | $106.26 | 16.51% | 3,271.12 | 5.51% | |
| 34. | 31 ago. 2020 | $129.04 | $0.205 | 21.63% | 3,500.31 | 7.01% |
| 35. | 30 sept. 2020 | $115.81 | -10.25% | 3,363.00 | -3.92% | |
| 36. | 31 oct. 2020 | $108.86 | -6.00% | 3,269.96 | -2.77% | |
| 37. | 30 nov. 2020 | $119.05 | $0.205 | 9.55% | 3,621.63 | 10.75% |
| 38. | 31 dic. 2020 | $132.69 | 11.46% | 3,756.07 | 3.71% | |
| 39. | 31 ene. 2021 | $131.96 | -0.55% | 3,714.24 | -1.11% | |
| 40. | 28 feb. 2021 | $121.26 | $0.205 | -7.95% | 3,811.15 | 2.61% |
| 41. | 31 mar. 2021 | $122.15 | 0.73% | 3,972.89 | 4.24% | |
| 42. | 30 abr. 2021 | $131.46 | 7.62% | 4,181.17 | 5.24% | |
| 43. | 31 may. 2021 | $124.61 | $0.22 | -5.04% | 4,204.11 | 0.55% |
| 44. | 30 jun. 2021 | $136.96 | 9.91% | 4,297.50 | 2.22% | |
| 45. | 31 jul. 2021 | $145.86 | 6.50% | 4,395.26 | 2.27% | |
| 46. | 31 ago. 2021 | $151.83 | $0.22 | 4.24% | 4,522.68 | 2.90% |
| 47. | 30 sept. 2021 | $141.50 | -6.80% | 4,307.54 | -4.76% | |
| 48. | 31 oct. 2021 | $149.80 | 5.87% | 4,605.38 | 6.91% | |
| 49. | 30 nov. 2021 | $165.30 | $0.22 | 10.49% | 4,567.00 | -0.83% |
| 50. | 31 dic. 2021 | $177.57 | 7.42% | 4,766.18 | 4.36% | |
| 51. | 31 ene. 2022 | $174.78 | -1.57% | 4,515.55 | -5.26% | |
| 52. | 28 feb. 2022 | $165.12 | $0.22 | -5.40% | 4,373.94 | -3.14% |
| 53. | 31 mar. 2022 | $174.61 | 5.75% | 4,530.41 | 3.58% | |
| 54. | 30 abr. 2022 | $157.65 | -9.71% | 4,131.93 | -8.80% | |
| 55. | 31 may. 2022 | $148.84 | $0.23 | -5.44% | 4,132.15 | 0.01% |
| 56. | 30 jun. 2022 | $136.72 | -8.14% | 3,785.38 | -8.39% | |
| 57. | 31 jul. 2022 | $162.51 | 18.86% | 4,130.29 | 9.11% | |
| 58. | 31 ago. 2022 | $157.22 | $0.23 | -3.11% | 3,955.00 | -4.24% |
| 59. | 30 sept. 2022 | $138.20 | -12.10% | 3,585.62 | -9.34% | |
| 60. | 31 oct. 2022 | $153.34 | 10.96% | 3,871.98 | 7.99% | |
| 61. | 30 nov. 2022 | $148.03 | $0.23 | -3.31% | 4,080.11 | 5.38% |
| 62. | 31 dic. 2022 | $129.93 | -12.23% | 3,839.50 | -5.90% | |
| 63. | 31 ene. 2023 | $144.29 | 11.05% | 4,076.60 | 6.18% | |
| 64. | 28 feb. 2023 | $147.41 | $0.23 | 2.32% | 3,970.15 | -2.61% |
| 65. | 31 mar. 2023 | $164.90 | 11.86% | 4,109.31 | 3.51% | |
| 66. | 30 abr. 2023 | $169.68 | 2.90% | 4,169.48 | 1.46% | |
| 67. | 31 may. 2023 | $177.25 | $0.24 | 4.60% | 4,179.83 | 0.25% |
| 68. | 30 jun. 2023 | $193.97 | 9.43% | 4,376.86 | 4.71% | |
| 69. | 31 jul. 2023 | $196.45 | 1.28% | 4,588.96 | 4.85% | |
| 70. | 31 ago. 2023 | $187.87 | $0.24 | -4.25% | 4,507.66 | -1.77% |
| 71. | 30 sept. 2023 | $171.21 | -8.87% | 4,288.05 | -4.87% | |
| Promedio (R): | 2.46% | 0.85% | ||||
| Desviación estándar: | 8.90% | 5.13% | ||||
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 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 la varianza y covarianza de los 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. 2017 | 2.03% | 2.81% | 0.19 | 3.83 | -0.84 |
| 2. | 31 dic. 2017 | -1.51% | 0.98% | 15.78 | 0.02 | -0.52 |
| 3. | 31 ene. 2018 | -1.06% | 5.62% | 12.41 | 22.72 | -16.79 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 ago. 2023 | -4.25% | -1.77% | 44.95 | 6.88 | 17.59 |
| 71. | 30 sept. 2023 | -8.87% | -4.87% | 128.31 | 32.75 | 64.83 |
| Total (Σ): | 5,545.99 | 1,838.69 | 2,309.32 | |||
| 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. 2017 | 2.03% | 2.81% | 0.19 | 3.83 | -0.84 |
| 2. | 31 dic. 2017 | -1.51% | 0.98% | 15.78 | 0.02 | -0.52 |
| 3. | 31 ene. 2018 | -1.06% | 5.62% | 12.41 | 22.72 | -16.79 |
| 4. | 28 feb. 2018 | 6.75% | -3.89% | 18.45 | 22.52 | -20.38 |
| 5. | 31 mar. 2018 | -5.79% | -2.69% | 68.12 | 12.53 | 29.21 |
| 6. | 30 abr. 2018 | -1.50% | 0.27% | 15.69 | 0.34 | 2.30 |
| 7. | 31 may. 2018 | 13.51% | 2.16% | 122.12 | 1.71 | 14.47 |
| 8. | 30 jun. 2018 | -0.94% | 0.48% | 11.57 | 0.13 | 1.25 |
| 9. | 31 jul. 2018 | 2.79% | 3.60% | 0.11 | 7.57 | 0.90 |
| 10. | 31 ago. 2018 | 20.02% | 3.03% | 308.30 | 4.73 | 38.19 |
| 11. | 30 sept. 2018 | -0.83% | 0.43% | 10.79 | 0.18 | 1.39 |
| 12. | 31 oct. 2018 | -3.05% | -6.94% | 30.33 | 60.71 | 42.91 |
| 13. | 30 nov. 2018 | -18.07% | 1.79% | 421.43 | 0.87 | -19.19 |
| 14. | 31 dic. 2018 | -11.67% | -9.18% | 199.60 | 100.58 | 141.69 |
| 15. | 31 ene. 2019 | 5.50% | 7.87% | 9.26 | 49.24 | 21.35 |
| 16. | 28 feb. 2019 | 4.48% | 2.97% | 4.07 | 4.50 | 4.28 |
| 17. | 31 mar. 2019 | 9.70% | 1.79% | 52.45 | 0.89 | 6.82 |
| 18. | 30 abr. 2019 | 5.64% | 3.93% | 10.14 | 9.49 | 9.81 |
| 19. | 31 may. 2019 | -12.37% | -6.58% | 220.00 | 55.19 | 110.19 |
| 20. | 30 jun. 2019 | 13.05% | 6.89% | 112.06 | 36.50 | 63.96 |
| 21. | 31 jul. 2019 | 7.64% | 1.31% | 26.83 | 0.21 | 2.39 |
| 22. | 31 ago. 2019 | -1.65% | -1.81% | 16.87 | 7.08 | 10.93 |
| 23. | 30 sept. 2019 | 7.28% | 1.72% | 23.25 | 0.75 | 4.18 |
| 24. | 31 oct. 2019 | 11.07% | 2.04% | 74.20 | 1.42 | 10.27 |
| 25. | 30 nov. 2019 | 7.74% | 3.40% | 27.87 | 6.52 | 13.48 |
| 26. | 31 dic. 2019 | 9.88% | 2.86% | 55.05 | 4.03 | 14.90 |
| 27. | 31 ene. 2020 | 5.41% | -0.16% | 8.69 | 1.03 | -2.99 |
| 28. | 29 feb. 2020 | -11.43% | -8.41% | 193.02 | 85.79 | 128.68 |
| 29. | 31 mar. 2020 | -6.98% | -12.51% | 89.10 | 178.57 | 126.14 |
| 30. | 30 abr. 2020 | 15.54% | 12.68% | 171.15 | 140.02 | 154.81 |
| 31. | 31 may. 2020 | 8.50% | 4.53% | 36.52 | 13.52 | 22.22 |
| 32. | 30 jun. 2020 | 14.73% | 1.84% | 150.60 | 0.98 | 12.12 |
| 33. | 31 jul. 2020 | 16.51% | 5.51% | 197.51 | 21.71 | 65.47 |
| 34. | 31 ago. 2020 | 21.63% | 7.01% | 367.54 | 37.89 | 118.00 |
| 35. | 30 sept. 2020 | -10.25% | -3.92% | 161.60 | 22.79 | 60.69 |
| 36. | 31 oct. 2020 | -6.00% | -2.77% | 71.58 | 13.09 | 30.61 |
| 37. | 30 nov. 2020 | 9.55% | 10.75% | 50.26 | 98.08 | 70.21 |
| 38. | 31 dic. 2020 | 11.46% | 3.71% | 80.96 | 8.18 | 25.74 |
| 39. | 31 ene. 2021 | -0.55% | -1.11% | 9.06 | 3.86 | 5.91 |
| 40. | 28 feb. 2021 | -7.95% | 2.61% | 108.42 | 3.09 | -18.30 |
| 41. | 31 mar. 2021 | 0.73% | 4.24% | 2.98 | 11.51 | -5.85 |
| 42. | 30 abr. 2021 | 7.62% | 5.24% | 26.65 | 19.28 | 22.67 |
| 43. | 31 may. 2021 | -5.04% | 0.55% | 56.29 | 0.09 | 2.27 |
| 44. | 30 jun. 2021 | 9.91% | 2.22% | 55.52 | 1.88 | 10.21 |
| 45. | 31 jul. 2021 | 6.50% | 2.27% | 16.31 | 2.03 | 5.75 |
| 46. | 31 ago. 2021 | 4.24% | 2.90% | 3.18 | 4.19 | 3.65 |
| 47. | 30 sept. 2021 | -6.80% | -4.76% | 85.81 | 31.45 | 51.95 |
| 48. | 31 oct. 2021 | 5.87% | 6.91% | 11.60 | 36.76 | 20.65 |
| 49. | 30 nov. 2021 | 10.49% | -0.83% | 64.55 | 2.84 | -13.54 |
| 50. | 31 dic. 2021 | 7.42% | 4.36% | 24.64 | 12.32 | 17.42 |
| 51. | 31 ene. 2022 | -1.57% | -5.26% | 16.25 | 37.33 | 24.63 |
| 52. | 28 feb. 2022 | -5.40% | -3.14% | 61.79 | 15.90 | 31.34 |
| 53. | 31 mar. 2022 | 5.75% | 3.58% | 10.81 | 7.43 | 8.96 |
| 54. | 30 abr. 2022 | -9.71% | -8.80% | 148.17 | 93.06 | 117.43 |
| 55. | 31 may. 2022 | -5.44% | 0.01% | 62.44 | 0.72 | 6.68 |
| 56. | 30 jun. 2022 | -8.14% | -8.39% | 112.41 | 85.44 | 98.00 |
| 57. | 31 jul. 2022 | 18.86% | 9.11% | 269.09 | 68.23 | 135.50 |
| 58. | 31 ago. 2022 | -3.11% | -4.24% | 31.06 | 25.96 | 28.40 |
| 59. | 30 sept. 2022 | -12.10% | -9.34% | 211.91 | 103.85 | 148.35 |
| 60. | 31 oct. 2022 | 10.96% | 7.99% | 72.18 | 50.91 | 60.62 |
| 61. | 30 nov. 2022 | -3.31% | 5.38% | 33.32 | 20.47 | -26.11 |
| 62. | 31 dic. 2022 | -12.23% | -5.90% | 215.70 | 45.54 | 99.11 |
| 63. | 31 ene. 2023 | 11.05% | 6.18% | 73.83 | 28.35 | 45.75 |
| 64. | 28 feb. 2023 | 2.32% | -2.61% | 0.02 | 11.99 | 0.48 |
| 65. | 31 mar. 2023 | 11.86% | 3.51% | 88.46 | 7.04 | 24.96 |
| 66. | 30 abr. 2023 | 2.90% | 1.46% | 0.19 | 0.38 | 0.27 |
| 67. | 31 may. 2023 | 4.60% | 0.25% | 4.59 | 0.36 | -1.29 |
| 68. | 30 jun. 2023 | 9.43% | 4.71% | 48.63 | 14.92 | 26.94 |
| 69. | 31 jul. 2023 | 1.28% | 4.85% | 1.39 | 15.96 | -4.72 |
| 70. | 31 ago. 2023 | -4.25% | -1.77% | 44.95 | 6.88 | 17.59 |
| 71. | 30 sept. 2023 | -8.87% | -4.87% | 128.31 | 32.75 | 64.83 |
| Total (Σ): | 5,545.99 | 1,838.69 | 2,309.32 | |||
Mostrar todo
VarianzaAAPL = Σ(RAAPL,t–RAAPL)2 ÷ (71 – 1)
= 5,545.99 ÷ (71 – 1)
= 79.23
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,838.69 ÷ (71 – 1)
= 26.27
CovarianzaAAPL, S&P 500 = Σ(RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 2,309.32 ÷ (71 – 1)
= 32.99
Estimación sistemática del riesgo (β)
| VarianzaAAPL | 79.23 |
| VarianzaS&P 500 | 26.27 |
| CovarianzaAAPL, S&P 500 | 32.99 |
| Coeficiente de correlaciónAAPL, S&P 5001 | 0.72 |
| βAAPL2 | 1.26 |
| αAAPL3 | 1.39% |
Cálculos
1 Coeficiente de correlaciónAAPL, S&P 500
= CovarianzaAAPL, S&P 500 ÷ (Desviación estándarAAPL × Desviación estándarS&P 500)
= 32.99 ÷ (8.90% × 5.13%)
= 0.72
2 βAAPL
= CovarianzaAAPL, S&P 500 ÷ VarianzaS&P 500
= 32.99 ÷ 26.27
= 1.26
3 αAAPL
= PromedioAAPL – βAAPL × PromedioS&P 500
= 2.46% – 1.26 × 0.85%
= 1.39%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.01% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.80% |
| Riesgo sistemático de Apple acciones ordinarias | βAAPL | 1.26 |
| Tasa de rendimiento esperada de las acciones ordinarias de Apple3 | E(RAAPL) | 16.31% |
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(RAAPL) = RF + βAAPL [E(RM) – RF]
= 4.01% + 1.26 [13.80% – 4.01%]
= 16.31%