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 IBM.
Tasas de retorno
International Business Machines Corp., tasas mensuales de retorno
| International Business Machines Corp. (IBM) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioIBM,t1 | DividendoIBM,t1 | RIBM,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2017 | $174.52 | 2,278.87 | ||||
| 1. | 28 feb. 2017 | $179.82 | $1.40 | 3.84% | 2,363.64 | 3.72% |
| 2. | 31 mar. 2017 | $174.14 | -3.16% | 2,362.72 | -0.04% | |
| 3. | 30 abr. 2017 | $160.29 | -7.95% | 2,384.20 | 0.91% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2021 | $117.10 | $1.64 | -5.08% | 4,567.00 | -0.83% |
| 59. | 31 dic. 2021 | $133.66 | 14.14% | 4,766.18 | 4.36% | |
| Promedio (R): | 0.21% | 1.36% | ||||
| Desviación estándar: | 7.27% | 4.48% | ||||
| International Business Machines Corp. (IBM) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioIBM,t1 | DividendoIBM,t1 | RIBM,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2017 | $174.52 | 2,278.87 | ||||
| 1. | 28 feb. 2017 | $179.82 | $1.40 | 3.84% | 2,363.64 | 3.72% |
| 2. | 31 mar. 2017 | $174.14 | -3.16% | 2,362.72 | -0.04% | |
| 3. | 30 abr. 2017 | $160.29 | -7.95% | 2,384.20 | 0.91% | |
| 4. | 31 may. 2017 | $152.63 | $1.50 | -3.84% | 2,411.80 | 1.16% |
| 5. | 30 jun. 2017 | $153.83 | 0.79% | 2,423.41 | 0.48% | |
| 6. | 31 jul. 2017 | $144.67 | -5.95% | 2,470.30 | 1.93% | |
| 7. | 31 ago. 2017 | $143.03 | $1.50 | -0.10% | 2,471.65 | 0.05% |
| 8. | 30 sept. 2017 | $145.08 | 1.43% | 2,519.36 | 1.93% | |
| 9. | 31 oct. 2017 | $154.06 | 6.19% | 2,575.26 | 2.22% | |
| 10. | 30 nov. 2017 | $153.97 | $1.50 | 0.92% | 2,647.58 | 2.81% |
| 11. | 31 dic. 2017 | $153.42 | -0.36% | 2,673.61 | 0.98% | |
| 12. | 31 ene. 2018 | $163.70 | 6.70% | 2,823.81 | 5.62% | |
| 13. | 28 feb. 2018 | $155.83 | $1.50 | -3.89% | 2,713.83 | -3.89% |
| 14. | 31 mar. 2018 | $153.43 | -1.54% | 2,640.87 | -2.69% | |
| 15. | 30 abr. 2018 | $144.96 | -5.52% | 2,648.05 | 0.27% | |
| 16. | 31 may. 2018 | $141.31 | $1.57 | -1.43% | 2,705.27 | 2.16% |
| 17. | 30 jun. 2018 | $139.70 | -1.14% | 2,718.37 | 0.48% | |
| 18. | 31 jul. 2018 | $144.93 | 3.74% | 2,816.29 | 3.60% | |
| 19. | 31 ago. 2018 | $146.48 | $1.57 | 2.15% | 2,901.52 | 3.03% |
| 20. | 30 sept. 2018 | $151.21 | 3.23% | 2,913.98 | 0.43% | |
| 21. | 31 oct. 2018 | $115.43 | -23.66% | 2,711.74 | -6.94% | |
| 22. | 30 nov. 2018 | $124.27 | $1.57 | 9.02% | 2,760.17 | 1.79% |
| 23. | 31 dic. 2018 | $113.67 | -8.53% | 2,506.85 | -9.18% | |
| 24. | 31 ene. 2019 | $134.42 | 18.25% | 2,704.10 | 7.87% | |
| 25. | 28 feb. 2019 | $138.13 | $1.57 | 3.93% | 2,784.49 | 2.97% |
| 26. | 31 mar. 2019 | $141.10 | 2.15% | 2,834.40 | 1.79% | |
| 27. | 30 abr. 2019 | $140.27 | -0.59% | 2,945.83 | 3.93% | |
| 28. | 31 may. 2019 | $126.99 | $1.62 | -8.31% | 2,752.06 | -6.58% |
| 29. | 30 jun. 2019 | $137.90 | 8.59% | 2,941.76 | 6.89% | |
| 30. | 31 jul. 2019 | $148.24 | 7.50% | 2,980.38 | 1.31% | |
| 31. | 31 ago. 2019 | $135.53 | $1.62 | -7.48% | 2,926.46 | -1.81% |
| 32. | 30 sept. 2019 | $145.42 | 7.30% | 2,976.74 | 1.72% | |
| 33. | 31 oct. 2019 | $133.73 | -8.04% | 3,037.56 | 2.04% | |
| 34. | 30 nov. 2019 | $134.45 | $1.62 | 1.75% | 3,140.98 | 3.40% |
| 35. | 31 dic. 2019 | $134.04 | -0.30% | 3,230.78 | 2.86% | |
| 36. | 31 ene. 2020 | $143.73 | 7.23% | 3,225.52 | -0.16% | |
| 37. | 29 feb. 2020 | $130.15 | $1.62 | -8.32% | 2,954.22 | -8.41% |
| 38. | 31 mar. 2020 | $110.93 | -14.77% | 2,584.59 | -12.51% | |
| 39. | 30 abr. 2020 | $125.56 | 13.19% | 2,912.43 | 12.68% | |
| 40. | 31 may. 2020 | $124.90 | $1.63 | 0.77% | 3,044.31 | 4.53% |
| 41. | 30 jun. 2020 | $120.77 | -3.31% | 3,100.29 | 1.84% | |
| 42. | 31 jul. 2020 | $122.94 | 1.80% | 3,271.12 | 5.51% | |
| 43. | 31 ago. 2020 | $123.31 | $1.63 | 1.63% | 3,500.31 | 7.01% |
| 44. | 30 sept. 2020 | $121.67 | -1.33% | 3,363.00 | -3.92% | |
| 45. | 31 oct. 2020 | $111.66 | -8.23% | 3,269.96 | -2.77% | |
| 46. | 30 nov. 2020 | $123.52 | $1.63 | 12.08% | 3,621.63 | 10.75% |
| 47. | 31 dic. 2020 | $125.88 | 1.91% | 3,756.07 | 3.71% | |
| 48. | 31 ene. 2021 | $119.11 | -5.38% | 3,714.24 | -1.11% | |
| 49. | 28 feb. 2021 | $118.93 | $1.63 | 1.22% | 3,811.15 | 2.61% |
| 50. | 31 mar. 2021 | $133.26 | 12.05% | 3,972.89 | 4.24% | |
| 51. | 30 abr. 2021 | $141.88 | 6.47% | 4,181.17 | 5.24% | |
| 52. | 31 may. 2021 | $143.74 | $1.64 | 2.47% | 4,204.11 | 0.55% |
| 53. | 30 jun. 2021 | $146.59 | 1.98% | 4,297.50 | 2.22% | |
| 54. | 31 jul. 2021 | $140.96 | -3.84% | 4,395.26 | 2.27% | |
| 55. | 31 ago. 2021 | $140.34 | $1.64 | 0.72% | 4,522.68 | 2.90% |
| 56. | 30 sept. 2021 | $138.93 | -1.00% | 4,307.54 | -4.76% | |
| 57. | 31 oct. 2021 | $125.10 | -9.95% | 4,605.38 | 6.91% | |
| 58. | 30 nov. 2021 | $117.10 | $1.64 | -5.08% | 4,567.00 | -0.83% |
| 59. | 31 dic. 2021 | $133.66 | 14.14% | 4,766.18 | 4.36% | |
| Promedio (R): | 0.21% | 1.36% | ||||
| Desviación estándar: | 7.27% | 4.48% | ||||
Mostrar todo
1 Datos en US$ por acción de acciones ordinarias, ajustados por escisiones y dividendos de acciones.
2 Tasa de rendimiento de las acciones ordinarias de IBM 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
International Business Machines Corp., cálculo de varianza y covarianza de rendimientos
| t | Fecha | RIBM,t | RS&P 500,t | (RIBM,t–RIBM)2 | (RS&P 500,t–RS&P 500)2 | (RIBM,t–RIBM)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2017 | 3.84% | 3.72% | 13.20 | 5.58 | 8.58 |
| 2. | 31 mar. 2017 | -3.16% | -0.04% | 11.32 | 1.95 | 4.70 |
| 3. | 30 abr. 2017 | -7.95% | 0.91% | 66.56 | 0.20 | 3.66 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2021 | -5.08% | -0.83% | 27.98 | 4.80 | 11.59 |
| 59. | 31 dic. 2021 | 14.14% | 4.36% | 194.23 | 9.02 | 41.86 |
| Total (Σ): | 3,065.40 | 1,164.17 | 1,320.75 | |||
| t | Fecha | RIBM,t | RS&P 500,t | (RIBM,t–RIBM)2 | (RS&P 500,t–RS&P 500)2 | (RIBM,t–RIBM)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2017 | 3.84% | 3.72% | 13.20 | 5.58 | 8.58 |
| 2. | 31 mar. 2017 | -3.16% | -0.04% | 11.32 | 1.95 | 4.70 |
| 3. | 30 abr. 2017 | -7.95% | 0.91% | 66.56 | 0.20 | 3.66 |
| 4. | 31 may. 2017 | -3.84% | 1.16% | 16.39 | 0.04 | 0.81 |
| 5. | 30 jun. 2017 | 0.79% | 0.48% | 0.34 | 0.77 | -0.51 |
| 6. | 31 jul. 2017 | -5.95% | 1.93% | 37.94 | 0.33 | -3.55 |
| 7. | 31 ago. 2017 | -0.10% | 0.05% | 0.09 | 1.70 | 0.39 |
| 8. | 30 sept. 2017 | 1.43% | 1.93% | 1.51 | 0.33 | 0.70 |
| 9. | 31 oct. 2017 | 6.19% | 2.22% | 35.81 | 0.74 | 5.15 |
| 10. | 30 nov. 2017 | 0.92% | 2.81% | 0.50 | 2.10 | 1.03 |
| 11. | 31 dic. 2017 | -0.36% | 0.98% | 0.32 | 0.14 | 0.21 |
| 12. | 31 ene. 2018 | 6.70% | 5.62% | 42.19 | 18.15 | 27.67 |
| 13. | 28 feb. 2018 | -3.89% | -3.89% | 16.78 | 27.59 | 21.52 |
| 14. | 31 mar. 2018 | -1.54% | -2.69% | 3.05 | 16.37 | 7.06 |
| 15. | 30 abr. 2018 | -5.52% | 0.27% | 32.78 | 1.18 | 6.22 |
| 16. | 31 may. 2018 | -1.43% | 2.16% | 2.69 | 0.64 | -1.32 |
| 17. | 30 jun. 2018 | -1.14% | 0.48% | 1.81 | 0.76 | 1.17 |
| 18. | 31 jul. 2018 | 3.74% | 3.60% | 12.52 | 5.04 | 7.94 |
| 19. | 31 ago. 2018 | 2.15% | 3.03% | 3.79 | 2.78 | 3.25 |
| 20. | 30 sept. 2018 | 3.23% | 0.43% | 9.14 | 0.86 | -2.81 |
| 21. | 31 oct. 2018 | -23.66% | -6.94% | 569.67 | 68.86 | 198.06 |
| 22. | 30 nov. 2018 | 9.02% | 1.79% | 77.67 | 0.18 | 3.77 |
| 23. | 31 dic. 2018 | -8.53% | -9.18% | 76.30 | 111.00 | 92.03 |
| 24. | 31 ene. 2019 | 18.25% | 7.87% | 325.78 | 42.39 | 117.51 |
| 25. | 28 feb. 2019 | 3.93% | 2.97% | 13.86 | 2.61 | 6.01 |
| 26. | 31 mar. 2019 | 2.15% | 1.79% | 3.78 | 0.19 | 0.85 |
| 27. | 30 abr. 2019 | -0.59% | 3.93% | 0.63 | 6.62 | -2.04 |
| 28. | 31 may. 2019 | -8.31% | -6.58% | 72.55 | 62.97 | 67.59 |
| 29. | 30 jun. 2019 | 8.59% | 6.89% | 70.32 | 30.64 | 46.42 |
| 30. | 31 jul. 2019 | 7.50% | 1.31% | 53.19 | 0.00 | -0.33 |
| 31. | 31 ago. 2019 | -7.48% | -1.81% | 59.08 | 10.03 | 24.34 |
| 32. | 30 sept. 2019 | 7.30% | 1.72% | 50.30 | 0.13 | 2.55 |
| 33. | 31 oct. 2019 | -8.04% | 2.04% | 67.96 | 0.47 | -5.65 |
| 34. | 30 nov. 2019 | 1.75% | 3.40% | 2.39 | 4.19 | 3.16 |
| 35. | 31 dic. 2019 | -0.30% | 2.86% | 0.26 | 2.25 | -0.77 |
| 36. | 31 ene. 2020 | 7.23% | -0.16% | 49.34 | 2.31 | -10.68 |
| 37. | 29 feb. 2020 | -8.32% | -8.41% | 72.70 | 95.43 | 83.29 |
| 38. | 31 mar. 2020 | -14.77% | -12.51% | 224.19 | 192.37 | 207.67 |
| 39. | 30 abr. 2020 | 13.19% | 12.68% | 168.56 | 128.29 | 147.05 |
| 40. | 31 may. 2020 | 0.77% | 4.53% | 0.32 | 10.05 | 1.80 |
| 41. | 30 jun. 2020 | -3.31% | 1.84% | 12.33 | 0.23 | -1.69 |
| 42. | 31 jul. 2020 | 1.80% | 5.51% | 2.53 | 17.24 | 6.61 |
| 43. | 31 ago. 2020 | 1.63% | 7.01% | 2.02 | 31.91 | 8.03 |
| 44. | 30 sept. 2020 | -1.33% | -3.92% | 2.36 | 27.89 | 8.11 |
| 45. | 31 oct. 2020 | -8.23% | -2.77% | 71.11 | 17.01 | 34.78 |
| 46. | 30 nov. 2020 | 12.08% | 10.75% | 141.04 | 88.30 | 111.60 |
| 47. | 31 dic. 2020 | 1.91% | 3.71% | 2.91 | 5.54 | 4.01 |
| 48. | 31 ene. 2021 | -5.38% | -1.11% | 31.17 | 6.11 | 13.80 |
| 49. | 28 feb. 2021 | 1.22% | 2.61% | 1.02 | 1.57 | 1.27 |
| 50. | 31 mar. 2021 | 12.05% | 4.24% | 140.28 | 8.33 | 34.18 |
| 51. | 30 abr. 2021 | 6.47% | 5.24% | 39.23 | 15.09 | 24.33 |
| 52. | 31 may. 2021 | 2.47% | 0.55% | 5.11 | 0.65 | -1.83 |
| 53. | 30 jun. 2021 | 1.98% | 2.22% | 3.16 | 0.75 | 1.53 |
| 54. | 31 jul. 2021 | -3.84% | 2.27% | 16.37 | 0.84 | -3.71 |
| 55. | 31 ago. 2021 | 0.72% | 2.90% | 0.27 | 2.38 | 0.80 |
| 56. | 30 sept. 2021 | -1.00% | -4.76% | 1.46 | 37.39 | 7.40 |
| 57. | 31 oct. 2021 | -9.95% | 6.91% | 103.22 | 30.87 | -56.45 |
| 58. | 30 nov. 2021 | -5.08% | -0.83% | 27.98 | 4.80 | 11.59 |
| 59. | 31 dic. 2021 | 14.14% | 4.36% | 194.23 | 9.02 | 41.86 |
| Total (Σ): | 3,065.40 | 1,164.17 | 1,320.75 | |||
Mostrar todo
VarianzaIBM = Σ(RIBM,t–RIBM)2 ÷ (59 – 1)
= 3,065.40 ÷ (59 – 1)
= 52.85
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,164.17 ÷ (59 – 1)
= 20.07
CovarianzaIBM, S&P 500 = Σ(RIBM,t–RIBM)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,320.75 ÷ (59 – 1)
= 22.77
Estimación sistemática del riesgo (β)
| VarianzaIBM | 52.85 |
| VarianzaS&P 500 | 20.07 |
| CovarianzaIBM, S&P 500 | 22.77 |
| Coeficiente de correlaciónIBM, S&P 5001 | 0.70 |
| βIBM2 | 1.13 |
| αIBM3 | -1.34% |
Cálculos
1 Coeficiente de correlaciónIBM, S&P 500
= CovarianzaIBM, S&P 500 ÷ (Desviación estándarIBM × Desviación estándarS&P 500)
= 22.77 ÷ (7.27% × 4.48%)
= 0.70
2 βIBM
= CovarianzaIBM, S&P 500 ÷ VarianzaS&P 500
= 22.77 ÷ 20.07
= 1.13
3 αIBM
= PromedioIBM – βIBM × PromedioS&P 500
= 0.21% – 1.13 × 1.36%
= -1.34%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento de LT Treasury Composite1 | RF | 3.20% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 12.23% |
| Riesgo sistemático de acciones ordinarias IBM | βIBM | 1.13 |
| Tasa de rendimiento esperada de las acciones ordinarias de IBM3 | E(RIBM) | 13.44% |
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(RIBM) = RF + βIBM [E(RM) – RF]
= 3.20% + 1.13 [12.23% – 3.20%]
= 13.44%