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 Microsoft.
Tasas de retorno
Microsoft Corp., tasas mensuales de retorno
| Microsoft Corp. (MSFT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioMSFT,t1 | DividendoMSFT,t1 | RMSFT,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 jul 2018 | $106.08 | 2,816.29 | ||||
| 1. | 31 ago 2018 | $112.33 | $0.42 | 6.29% | 2,901.52 | 3.03% |
| 2. | 30 sept 2018 | $114.37 | 1.82% | 2,913.98 | 0.43% | |
| 3. | 31 oct 2018 | $106.81 | -6.61% | 2,711.74 | -6.94% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 may 2024 | $415.13 | $0.75 | 6.82% | 5,277.51 | 4.80% |
| 71. | 30 jun 2024 | $446.95 | 7.67% | 5,460.48 | 3.47% | |
| Promedio (R): | 2.33% | 1.07% | ||||
| Desviación estándar: | 6.27% | 5.24% | ||||
| Microsoft Corp. (MSFT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioMSFT,t1 | DividendoMSFT,t1 | RMSFT,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 jul 2018 | $106.08 | 2,816.29 | ||||
| 1. | 31 ago 2018 | $112.33 | $0.42 | 6.29% | 2,901.52 | 3.03% |
| 2. | 30 sept 2018 | $114.37 | 1.82% | 2,913.98 | 0.43% | |
| 3. | 31 oct 2018 | $106.81 | -6.61% | 2,711.74 | -6.94% | |
| 4. | 30 nov 2018 | $110.89 | $0.46 | 4.25% | 2,760.17 | 1.79% |
| 5. | 31 dic 2018 | $101.57 | -8.40% | 2,506.85 | -9.18% | |
| 6. | 31 ene 2019 | $104.43 | 2.82% | 2,704.10 | 7.87% | |
| 7. | 28 feb 2019 | $112.03 | $0.46 | 7.72% | 2,784.49 | 2.97% |
| 8. | 31 mar 2019 | $117.94 | 5.28% | 2,834.40 | 1.79% | |
| 9. | 30 abr 2019 | $130.60 | 10.73% | 2,945.83 | 3.93% | |
| 10. | 31 may 2019 | $123.68 | $0.46 | -4.95% | 2,752.06 | -6.58% |
| 11. | 30 jun 2019 | $133.96 | 8.31% | 2,941.76 | 6.89% | |
| 12. | 31 jul 2019 | $136.27 | 1.72% | 2,980.38 | 1.31% | |
| 13. | 31 ago 2019 | $137.86 | $0.46 | 1.50% | 2,926.46 | -1.81% |
| 14. | 30 sept 2019 | $139.03 | 0.85% | 2,976.74 | 1.72% | |
| 15. | 31 oct 2019 | $143.37 | 3.12% | 3,037.56 | 2.04% | |
| 16. | 30 nov 2019 | $151.38 | $0.51 | 5.94% | 3,140.98 | 3.40% |
| 17. | 31 dic 2019 | $157.70 | 4.17% | 3,230.78 | 2.86% | |
| 18. | 31 ene 2020 | $170.23 | 7.95% | 3,225.52 | -0.16% | |
| 19. | 29 feb 2020 | $162.01 | $0.51 | -4.53% | 2,954.22 | -8.41% |
| 20. | 31 mar 2020 | $157.71 | -2.65% | 2,584.59 | -12.51% | |
| 21. | 30 abr 2020 | $179.21 | 13.63% | 2,912.43 | 12.68% | |
| 22. | 31 may 2020 | $183.25 | $0.51 | 2.54% | 3,044.31 | 4.53% |
| 23. | 30 jun 2020 | $203.51 | 11.06% | 3,100.29 | 1.84% | |
| 24. | 31 jul 2020 | $205.01 | 0.74% | 3,271.12 | 5.51% | |
| 25. | 31 ago 2020 | $225.53 | $0.51 | 10.26% | 3,500.31 | 7.01% |
| 26. | 30 sept 2020 | $210.33 | -6.74% | 3,363.00 | -3.92% | |
| 27. | 31 oct 2020 | $202.47 | -3.74% | 3,269.96 | -2.77% | |
| 28. | 30 nov 2020 | $214.07 | $0.56 | 6.01% | 3,621.63 | 10.75% |
| 29. | 31 dic 2020 | $222.42 | 3.90% | 3,756.07 | 3.71% | |
| 30. | 31 ene 2021 | $231.96 | 4.29% | 3,714.24 | -1.11% | |
| 31. | 28 feb 2021 | $232.38 | $0.56 | 0.42% | 3,811.15 | 2.61% |
| 32. | 31 mar 2021 | $235.77 | 1.46% | 3,972.89 | 4.24% | |
| 33. | 30 abr 2021 | $252.18 | 6.96% | 4,181.17 | 5.24% | |
| 34. | 31 may 2021 | $249.68 | $0.56 | -0.77% | 4,204.11 | 0.55% |
| 35. | 30 jun 2021 | $270.90 | 8.50% | 4,297.50 | 2.22% | |
| 36. | 31 jul 2021 | $284.91 | 5.17% | 4,395.26 | 2.27% | |
| 37. | 31 ago 2021 | $301.88 | $0.56 | 6.15% | 4,522.68 | 2.90% |
| 38. | 30 sept 2021 | $281.92 | -6.61% | 4,307.54 | -4.76% | |
| 39. | 31 oct 2021 | $331.62 | 17.63% | 4,605.38 | 6.91% | |
| 40. | 30 nov 2021 | $330.59 | $0.62 | -0.12% | 4,567.00 | -0.83% |
| 41. | 31 dic 2021 | $336.32 | 1.73% | 4,766.18 | 4.36% | |
| 42. | 31 ene 2022 | $310.98 | -7.53% | 4,515.55 | -5.26% | |
| 43. | 28 feb 2022 | $298.79 | $0.62 | -3.72% | 4,373.94 | -3.14% |
| 44. | 31 mar 2022 | $308.31 | 3.19% | 4,530.41 | 3.58% | |
| 45. | 30 abr 2022 | $277.52 | -9.99% | 4,131.93 | -8.80% | |
| 46. | 31 may 2022 | $271.87 | $0.62 | -1.81% | 4,132.15 | 0.01% |
| 47. | 30 jun 2022 | $256.83 | -5.53% | 3,785.38 | -8.39% | |
| 48. | 31 jul 2022 | $280.74 | 9.31% | 4,130.29 | 9.11% | |
| 49. | 31 ago 2022 | $261.47 | $0.62 | -6.64% | 3,955.00 | -4.24% |
| 50. | 30 sept 2022 | $232.90 | -10.93% | 3,585.62 | -9.34% | |
| 51. | 31 oct 2022 | $232.13 | -0.33% | 3,871.98 | 7.99% | |
| 52. | 30 nov 2022 | $255.14 | $0.68 | 10.21% | 4,080.11 | 5.38% |
| 53. | 31 dic 2022 | $239.82 | -6.00% | 3,839.50 | -5.90% | |
| 54. | 31 ene 2023 | $247.81 | 3.33% | 4,076.60 | 6.18% | |
| 55. | 28 feb 2023 | $249.42 | $0.68 | 0.92% | 3,970.15 | -2.61% |
| 56. | 31 mar 2023 | $288.30 | 15.59% | 4,109.31 | 3.51% | |
| 57. | 30 abr 2023 | $307.26 | 6.58% | 4,169.48 | 1.46% | |
| 58. | 31 may 2023 | $328.39 | $0.68 | 7.10% | 4,179.83 | 0.25% |
| 59. | 30 jun 2023 | $340.54 | 3.70% | 4,376.86 | 4.71% | |
| 60. | 31 jul 2023 | $335.92 | -1.36% | 4,588.96 | 4.85% | |
| 61. | 31 ago 2023 | $327.76 | $0.68 | -2.23% | 4,507.66 | -1.77% |
| 62. | 30 sept 2023 | $315.75 | -3.66% | 4,288.05 | -4.87% | |
| 63. | 31 oct 2023 | $338.11 | 7.08% | 4,193.80 | -2.20% | |
| 64. | 30 nov 2023 | $378.91 | $0.75 | 12.29% | 4,567.80 | 8.92% |
| 65. | 31 dic 2023 | $376.04 | -0.76% | 4,769.83 | 4.42% | |
| 66. | 31 ene 2024 | $397.58 | 5.73% | 4,845.65 | 1.59% | |
| 67. | 29 feb 2024 | $413.64 | $0.75 | 4.23% | 5,096.27 | 5.17% |
| 68. | 31 mar 2024 | $420.72 | 1.71% | 5,254.35 | 3.10% | |
| 69. | 30 abr 2024 | $389.33 | -7.46% | 5,035.69 | -4.16% | |
| 70. | 31 may 2024 | $415.13 | $0.75 | 6.82% | 5,277.51 | 4.80% |
| 71. | 30 jun 2024 | $446.95 | 7.67% | 5,460.48 | 3.47% | |
| Promedio (R): | 2.33% | 1.07% | ||||
| Desviación estándar: | 6.27% | 5.24% | ||||
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 MSFT 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
Microsoft Corp., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RMSFT,t | RS&P 500,t | (RMSFT,t–RMSFT)2 | (RS&P 500,t–RS&P 500)2 | (RMSFT,t–RMSFT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 31 ago 2018 | 6.29% | 3.03% | 15.68 | 3.81 | 7.73 |
| 2. | 30 sept 2018 | 1.82% | 0.43% | 0.26 | 0.41 | 0.33 |
| 3. | 31 oct 2018 | -6.61% | -6.94% | 79.89 | 64.22 | 71.63 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | 31 may 2024 | 6.82% | 4.80% | 20.17 | 13.90 | 16.75 |
| 71. | 30 jun 2024 | 7.67% | 3.47% | 28.49 | 5.73 | 12.78 |
| Total (Σ): | 2,752.47 | 1,922.79 | 1,747.58 | |||
| t | Fecha | RMSFT,t | RS&P 500,t | (RMSFT,t–RMSFT)2 | (RS&P 500,t–RS&P 500)2 | (RMSFT,t–RMSFT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 31 ago 2018 | 6.29% | 3.03% | 15.68 | 3.81 | 7.73 |
| 2. | 30 sept 2018 | 1.82% | 0.43% | 0.26 | 0.41 | 0.33 |
| 3. | 31 oct 2018 | -6.61% | -6.94% | 79.89 | 64.22 | 71.63 |
| 4. | 30 nov 2018 | 4.25% | 1.79% | 3.70 | 0.51 | 1.37 |
| 5. | 31 dic 2018 | -8.40% | -9.18% | 115.19 | 105.08 | 110.02 |
| 6. | 31 ene 2019 | 2.82% | 7.87% | 0.24 | 46.17 | 3.32 |
| 7. | 28 feb 2019 | 7.72% | 2.97% | 29.05 | 3.61 | 10.24 |
| 8. | 31 mar 2019 | 5.28% | 1.79% | 8.69 | 0.52 | 2.12 |
| 9. | 30 abr 2019 | 10.73% | 3.93% | 70.67 | 8.17 | 24.03 |
| 10. | 31 may 2019 | -4.95% | -6.58% | 52.91 | 58.54 | 55.66 |
| 11. | 30 jun 2019 | 8.31% | 6.89% | 35.81 | 33.87 | 34.82 |
| 12. | 31 jul 2019 | 1.72% | 1.31% | 0.36 | 0.06 | -0.14 |
| 13. | 31 ago 2019 | 1.50% | -1.81% | 0.68 | 8.31 | 2.37 |
| 14. | 30 sept 2019 | 0.85% | 1.72% | 2.19 | 0.42 | -0.95 |
| 15. | 31 oct 2019 | 3.12% | 2.04% | 0.63 | 0.94 | 0.77 |
| 16. | 30 nov 2019 | 5.94% | 3.40% | 13.07 | 5.44 | 8.43 |
| 17. | 31 dic 2019 | 4.17% | 2.86% | 3.41 | 3.19 | 3.30 |
| 18. | 31 ene 2020 | 7.95% | -0.16% | 31.56 | 1.53 | -6.94 |
| 19. | 29 feb 2020 | -4.53% | -8.41% | 47.02 | 89.95 | 65.03 |
| 20. | 31 mar 2020 | -2.65% | -12.51% | 24.82 | 184.56 | 67.68 |
| 21. | 30 abr 2020 | 13.63% | 12.68% | 127.80 | 134.82 | 131.26 |
| 22. | 31 may 2020 | 2.54% | 4.53% | 0.04 | 11.94 | 0.73 |
| 23. | 30 jun 2020 | 11.06% | 1.84% | 76.18 | 0.59 | 6.68 |
| 24. | 31 jul 2020 | 0.74% | 5.51% | 2.53 | 19.68 | -7.06 |
| 25. | 31 ago 2020 | 10.26% | 7.01% | 62.89 | 35.20 | 47.05 |
| 26. | 30 sept 2020 | -6.74% | -3.92% | 82.22 | 24.96 | 45.30 |
| 27. | 31 oct 2020 | -3.74% | -2.77% | 36.78 | 14.75 | 23.29 |
| 28. | 30 nov 2020 | 6.01% | 10.75% | 13.53 | 93.73 | 35.61 |
| 29. | 31 dic 2020 | 3.90% | 3.71% | 2.47 | 6.96 | 4.15 |
| 30. | 31 ene 2021 | 4.29% | -1.11% | 3.85 | 4.78 | -4.29 |
| 31. | 28 feb 2021 | 0.42% | 2.61% | 3.63 | 2.36 | -2.93 |
| 32. | 31 mar 2021 | 1.46% | 4.24% | 0.76 | 10.05 | -2.76 |
| 33. | 30 abr 2021 | 6.96% | 5.24% | 21.46 | 17.38 | 19.31 |
| 34. | 31 may 2021 | -0.77% | 0.55% | 9.59 | 0.28 | 1.63 |
| 35. | 30 jun 2021 | 8.50% | 2.22% | 38.08 | 1.32 | 7.08 |
| 36. | 31 jul 2021 | 5.17% | 2.27% | 8.09 | 1.44 | 3.42 |
| 37. | 31 ago 2021 | 6.15% | 2.90% | 14.63 | 3.33 | 6.98 |
| 38. | 30 sept 2021 | -6.61% | -4.76% | 79.92 | 33.99 | 52.12 |
| 39. | 31 oct 2021 | 17.63% | 6.91% | 234.13 | 34.12 | 89.38 |
| 40. | 30 nov 2021 | -0.12% | -0.83% | 6.01 | 3.64 | 4.67 |
| 41. | 31 dic 2021 | 1.73% | 4.36% | 0.35 | 10.81 | -1.95 |
| 42. | 31 ene 2022 | -7.53% | -5.26% | 97.27 | 40.09 | 62.45 |
| 43. | 28 feb 2022 | -3.72% | -3.14% | 36.58 | 17.72 | 25.46 |
| 44. | 31 mar 2022 | 3.19% | 3.58% | 0.74 | 6.27 | 2.15 |
| 45. | 30 abr 2022 | -9.99% | -8.80% | 151.65 | 97.40 | 121.53 |
| 46. | 31 may 2022 | -1.81% | 0.01% | 17.14 | 1.14 | 4.42 |
| 47. | 30 jun 2022 | -5.53% | -8.39% | 61.78 | 89.59 | 74.40 |
| 48. | 31 jul 2022 | 9.31% | 9.11% | 48.75 | 64.61 | 56.12 |
| 49. | 31 ago 2022 | -6.64% | -4.24% | 80.48 | 28.27 | 47.70 |
| 50. | 30 sept 2022 | -10.93% | -9.34% | 175.68 | 108.43 | 138.02 |
| 51. | 31 oct 2022 | -0.33% | 7.99% | 7.07 | 47.79 | -18.38 |
| 52. | 30 nov 2022 | 10.21% | 5.38% | 62.06 | 18.51 | 33.89 |
| 53. | 31 dic 2022 | -6.00% | -5.90% | 69.43 | 48.59 | 58.08 |
| 54. | 31 ene 2023 | 3.33% | 6.18% | 1.01 | 26.03 | 5.12 |
| 55. | 28 feb 2023 | 0.92% | -2.61% | 1.97 | 13.58 | 5.17 |
| 56. | 31 mar 2023 | 15.59% | 3.51% | 175.84 | 5.91 | 32.25 |
| 57. | 30 abr 2023 | 6.58% | 1.46% | 18.05 | 0.15 | 1.66 |
| 58. | 31 may 2023 | 7.10% | 0.25% | 22.76 | 0.68 | -3.94 |
| 59. | 30 jun 2023 | 3.70% | 4.71% | 1.88 | 13.25 | 4.99 |
| 60. | 31 jul 2023 | -1.36% | 4.85% | 13.58 | 14.23 | -13.90 |
| 61. | 31 ago 2023 | -2.23% | -1.77% | 20.74 | 8.09 | 12.96 |
| 62. | 30 sept 2023 | -3.66% | -4.87% | 35.91 | 35.35 | 35.62 |
| 63. | 31 oct 2023 | 7.08% | -2.20% | 22.60 | 10.70 | -15.55 |
| 64. | 30 nov 2023 | 12.29% | 8.92% | 99.22 | 61.54 | 78.14 |
| 65. | 31 dic 2023 | -0.76% | 4.42% | 9.52 | 11.22 | -10.33 |
| 66. | 31 ene 2024 | 5.73% | 1.59% | 11.56 | 0.27 | 1.76 |
| 67. | 29 feb 2024 | 4.23% | 5.17% | 3.61 | 16.80 | 7.79 |
| 68. | 31 mar 2024 | 1.71% | 3.10% | 0.38 | 4.11 | -1.25 |
| 69. | 30 abr 2024 | -7.46% | -4.16% | 95.82 | 27.40 | 51.24 |
| 70. | 31 may 2024 | 6.82% | 4.80% | 20.17 | 13.90 | 16.75 |
| 71. | 30 jun 2024 | 7.67% | 3.47% | 28.49 | 5.73 | 12.78 |
| Total (Σ): | 2,752.47 | 1,922.79 | 1,747.58 | |||
Mostrar todo
VarianzaMSFT = Σ(RMSFT,t–RMSFT)2 ÷ (71 – 1)
= 2,752.47 ÷ (71 – 1)
= 39.32
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,922.79 ÷ (71 – 1)
= 27.47
CovarianzaMSFT, S&P 500 = Σ(RMSFT,t–RMSFT)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 1,747.58 ÷ (71 – 1)
= 24.97
Estimación sistemática del riesgo (β)
| VarianzaMSFT | 39.32 |
| VarianzaS&P 500 | 27.47 |
| CovarianzaMSFT, S&P 500 | 24.97 |
| Coeficiente de correlaciónMSFT, S&P 5001 | 0.76 |
| βMSFT2 | 0.91 |
| αMSFT3 | 1.35% |
Cálculos
1 Coeficiente de correlaciónMSFT, S&P 500
= CovarianzaMSFT, S&P 500 ÷ (Desviación estándarMSFT × Desviación estándarS&P 500)
= 24.97 ÷ (6.27% × 5.24%)
= 0.76
2 βMSFT
= CovarianzaMSFT, S&P 500 ÷ VarianzaS&P 500
= 24.97 ÷ 27.47
= 0.91
3 αMSFT
= PromedioMSFT – βMSFT × PromedioS&P 500
= 2.33% – 0.91 × 1.07%
= 1.35%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.91% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 14.72% |
| Riesgo sistemático de Microsoft acciones ordinarias | βMSFT | 0.91 |
| Tasa de rendimiento esperada de las acciones ordinarias de Microsoft3 | E(RMSFT) | 13.82% |
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(RMSFT) = RF + βMSFT [E(RM) – RF]
= 4.91% + 0.91 [14.72% – 4.91%]
= 13.82%