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 ITW.
Tasas de retorno
Illinois Tool Works Inc., tasas mensuales de retorno
| Illinois Tool Works Inc. (ITW) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioITW,t1 | DividendoITW,t1 | RITW,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2017 | $127.20 | 2,278.87 | ||||
| 1. | 28 feb. 2017 | $132.01 | 3.78% | 2,363.64 | 3.72% | |
| 2. | 31 mar. 2017 | $132.47 | $0.65 | 0.84% | 2,362.72 | -0.04% |
| 3. | 30 abr. 2017 | $138.09 | 4.24% | 2,384.20 | 0.91% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2021 | $232.15 | 1.88% | 4,567.00 | -0.83% | |
| 59. | 31 dic. 2021 | $246.80 | $1.22 | 6.84% | 4,766.18 | 4.36% |
| Promedio (R): | 1.51% | 1.36% | ||||
| Desviación estándar: | 5.99% | 4.48% | ||||
| Illinois Tool Works Inc. (ITW) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioITW,t1 | DividendoITW,t1 | RITW,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2017 | $127.20 | 2,278.87 | ||||
| 1. | 28 feb. 2017 | $132.01 | 3.78% | 2,363.64 | 3.72% | |
| 2. | 31 mar. 2017 | $132.47 | $0.65 | 0.84% | 2,362.72 | -0.04% |
| 3. | 30 abr. 2017 | $138.09 | 4.24% | 2,384.20 | 0.91% | |
| 4. | 31 may. 2017 | $141.22 | 2.27% | 2,411.80 | 1.16% | |
| 5. | 30 jun. 2017 | $143.25 | $0.65 | 1.90% | 2,423.41 | 0.48% |
| 6. | 31 jul. 2017 | $140.71 | -1.77% | 2,470.30 | 1.93% | |
| 7. | 31 ago. 2017 | $137.51 | -2.27% | 2,471.65 | 0.05% | |
| 8. | 30 sept. 2017 | $147.96 | $0.78 | 8.17% | 2,519.36 | 1.93% |
| 9. | 31 oct. 2017 | $156.52 | 5.79% | 2,575.26 | 2.22% | |
| 10. | 30 nov. 2017 | $169.25 | 8.13% | 2,647.58 | 2.81% | |
| 11. | 31 dic. 2017 | $166.85 | $0.78 | -0.96% | 2,673.61 | 0.98% |
| 12. | 31 ene. 2018 | $173.67 | 4.09% | 2,823.81 | 5.62% | |
| 13. | 28 feb. 2018 | $161.44 | -7.04% | 2,713.83 | -3.89% | |
| 14. | 31 mar. 2018 | $156.66 | $0.78 | -2.48% | 2,640.87 | -2.69% |
| 15. | 30 abr. 2018 | $142.02 | -9.35% | 2,648.05 | 0.27% | |
| 16. | 31 may. 2018 | $143.70 | 1.18% | 2,705.27 | 2.16% | |
| 17. | 30 jun. 2018 | $138.54 | $0.78 | -3.05% | 2,718.37 | 0.48% |
| 18. | 31 jul. 2018 | $143.33 | 3.46% | 2,816.29 | 3.60% | |
| 19. | 31 ago. 2018 | $138.88 | -3.10% | 2,901.52 | 3.03% | |
| 20. | 30 sept. 2018 | $141.12 | $1.00 | 2.33% | 2,913.98 | 0.43% |
| 21. | 31 oct. 2018 | $127.57 | -9.60% | 2,711.74 | -6.94% | |
| 22. | 30 nov. 2018 | $139.05 | 9.00% | 2,760.17 | 1.79% | |
| 23. | 31 dic. 2018 | $126.69 | $1.00 | -8.17% | 2,506.85 | -9.18% |
| 24. | 31 ene. 2019 | $137.31 | 8.38% | 2,704.10 | 7.87% | |
| 25. | 28 feb. 2019 | $144.08 | 4.93% | 2,784.49 | 2.97% | |
| 26. | 31 mar. 2019 | $143.53 | $1.00 | 0.31% | 2,834.40 | 1.79% |
| 27. | 30 abr. 2019 | $155.63 | 8.43% | 2,945.83 | 3.93% | |
| 28. | 31 may. 2019 | $139.64 | -10.27% | 2,752.06 | -6.58% | |
| 29. | 30 jun. 2019 | $150.81 | $1.00 | 8.72% | 2,941.76 | 6.89% |
| 30. | 31 jul. 2019 | $154.23 | 2.27% | 2,980.38 | 1.31% | |
| 31. | 31 ago. 2019 | $149.86 | -2.83% | 2,926.46 | -1.81% | |
| 32. | 30 sept. 2019 | $156.49 | $1.07 | 5.14% | 2,976.74 | 1.72% |
| 33. | 31 oct. 2019 | $168.58 | 7.73% | 3,037.56 | 2.04% | |
| 34. | 30 nov. 2019 | $174.33 | 3.41% | 3,140.98 | 3.40% | |
| 35. | 31 dic. 2019 | $179.63 | $1.07 | 3.65% | 3,230.78 | 2.86% |
| 36. | 31 ene. 2020 | $174.98 | -2.59% | 3,225.52 | -0.16% | |
| 37. | 29 feb. 2020 | $167.78 | -4.11% | 2,954.22 | -8.41% | |
| 38. | 31 mar. 2020 | $142.12 | $1.07 | -14.66% | 2,584.59 | -12.51% |
| 39. | 30 abr. 2020 | $162.50 | 14.34% | 2,912.43 | 12.68% | |
| 40. | 31 may. 2020 | $172.46 | 6.13% | 3,044.31 | 4.53% | |
| 41. | 30 jun. 2020 | $174.85 | $1.07 | 2.01% | 3,100.29 | 1.84% |
| 42. | 31 jul. 2020 | $184.99 | 5.80% | 3,271.12 | 5.51% | |
| 43. | 31 ago. 2020 | $197.55 | 6.79% | 3,500.31 | 7.01% | |
| 44. | 30 sept. 2020 | $193.21 | $1.14 | -1.62% | 3,363.00 | -3.92% |
| 45. | 31 oct. 2020 | $195.88 | 1.38% | 3,269.96 | -2.77% | |
| 46. | 30 nov. 2020 | $211.09 | 7.76% | 3,621.63 | 10.75% | |
| 47. | 31 dic. 2020 | $203.88 | $1.14 | -2.88% | 3,756.07 | 3.71% |
| 48. | 31 ene. 2021 | $194.21 | -4.74% | 3,714.24 | -1.11% | |
| 49. | 28 feb. 2021 | $202.18 | 4.10% | 3,811.15 | 2.61% | |
| 50. | 31 mar. 2021 | $221.52 | $1.14 | 10.13% | 3,972.89 | 4.24% |
| 51. | 30 abr. 2021 | $230.46 | 4.04% | 4,181.17 | 5.24% | |
| 52. | 31 may. 2021 | $231.76 | 0.56% | 4,204.11 | 0.55% | |
| 53. | 30 jun. 2021 | $223.56 | $1.14 | -3.05% | 4,297.50 | 2.22% |
| 54. | 31 jul. 2021 | $226.67 | 1.39% | 4,395.26 | 2.27% | |
| 55. | 31 ago. 2021 | $232.86 | 2.73% | 4,522.68 | 2.90% | |
| 56. | 30 sept. 2021 | $206.63 | $1.22 | -10.74% | 4,307.54 | -4.76% |
| 57. | 31 oct. 2021 | $227.87 | 10.28% | 4,605.38 | 6.91% | |
| 58. | 30 nov. 2021 | $232.15 | 1.88% | 4,567.00 | -0.83% | |
| 59. | 31 dic. 2021 | $246.80 | $1.22 | 6.84% | 4,766.18 | 4.36% |
| Promedio (R): | 1.51% | 1.36% | ||||
| Desviación estándar: | 5.99% | 4.48% | ||||
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 ITW 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
Illinois Tool Works Inc., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RITW,t | RS&P 500,t | (RITW,t–RITW)2 | (RS&P 500,t–RS&P 500)2 | (RITW,t–RITW)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2017 | 3.78% | 3.72% | 5.17 | 5.58 | 5.37 |
| 2. | 31 mar. 2017 | 0.84% | -0.04% | 0.45 | 1.95 | 0.93 |
| 3. | 30 abr. 2017 | 4.24% | 0.91% | 7.47 | 0.20 | -1.23 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2021 | 1.88% | -0.83% | 0.14 | 4.80 | -0.81 |
| 59. | 31 dic. 2021 | 6.84% | 4.36% | 28.38 | 9.02 | 16.00 |
| Total (Σ): | 2,078.90 | 1,164.17 | 1,283.40 | |||
| t | Fecha | RITW,t | RS&P 500,t | (RITW,t–RITW)2 | (RS&P 500,t–RS&P 500)2 | (RITW,t–RITW)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2017 | 3.78% | 3.72% | 5.17 | 5.58 | 5.37 |
| 2. | 31 mar. 2017 | 0.84% | -0.04% | 0.45 | 1.95 | 0.93 |
| 3. | 30 abr. 2017 | 4.24% | 0.91% | 7.47 | 0.20 | -1.23 |
| 4. | 31 may. 2017 | 2.27% | 1.16% | 0.57 | 0.04 | -0.15 |
| 5. | 30 jun. 2017 | 1.90% | 0.48% | 0.15 | 0.77 | -0.34 |
| 6. | 31 jul. 2017 | -1.77% | 1.93% | 10.77 | 0.33 | -1.89 |
| 7. | 31 ago. 2017 | -2.27% | 0.05% | 14.31 | 1.70 | 4.93 |
| 8. | 30 sept. 2017 | 8.17% | 1.93% | 44.33 | 0.33 | 3.81 |
| 9. | 31 oct. 2017 | 5.79% | 2.22% | 18.29 | 0.74 | 3.68 |
| 10. | 30 nov. 2017 | 8.13% | 2.81% | 43.88 | 2.10 | 9.61 |
| 11. | 31 dic. 2017 | -0.96% | 0.98% | 6.08 | 0.14 | 0.92 |
| 12. | 31 ene. 2018 | 4.09% | 5.62% | 6.65 | 18.15 | 10.99 |
| 13. | 28 feb. 2018 | -7.04% | -3.89% | 73.12 | 27.59 | 44.91 |
| 14. | 31 mar. 2018 | -2.48% | -2.69% | 15.89 | 16.37 | 16.13 |
| 15. | 30 abr. 2018 | -9.35% | 0.27% | 117.81 | 1.18 | 11.79 |
| 16. | 31 may. 2018 | 1.18% | 2.16% | 0.11 | 0.64 | -0.26 |
| 17. | 30 jun. 2018 | -3.05% | 0.48% | 20.76 | 0.76 | 3.98 |
| 18. | 31 jul. 2018 | 3.46% | 3.60% | 3.80 | 5.04 | 4.37 |
| 19. | 31 ago. 2018 | -3.10% | 3.03% | 21.28 | 2.78 | -7.70 |
| 20. | 30 sept. 2018 | 2.33% | 0.43% | 0.68 | 0.86 | -0.77 |
| 21. | 31 oct. 2018 | -9.60% | -6.94% | 123.44 | 68.86 | 92.20 |
| 22. | 30 nov. 2018 | 9.00% | 1.79% | 56.10 | 0.18 | 3.21 |
| 23. | 31 dic. 2018 | -8.17% | -9.18% | 93.67 | 111.00 | 101.97 |
| 24. | 31 ene. 2019 | 8.38% | 7.87% | 47.25 | 42.39 | 44.75 |
| 25. | 28 feb. 2019 | 4.93% | 2.97% | 11.71 | 2.61 | 5.53 |
| 26. | 31 mar. 2019 | 0.31% | 1.79% | 1.43 | 0.19 | -0.52 |
| 27. | 30 abr. 2019 | 8.43% | 3.93% | 47.91 | 6.62 | 17.81 |
| 28. | 31 may. 2019 | -10.27% | -6.58% | 138.84 | 62.97 | 93.51 |
| 29. | 30 jun. 2019 | 8.72% | 6.89% | 51.93 | 30.64 | 39.89 |
| 30. | 31 jul. 2019 | 2.27% | 1.31% | 0.58 | 0.00 | -0.03 |
| 31. | 31 ago. 2019 | -2.83% | -1.81% | 18.85 | 10.03 | 13.75 |
| 32. | 30 sept. 2019 | 5.14% | 1.72% | 13.17 | 0.13 | 1.31 |
| 33. | 31 oct. 2019 | 7.73% | 2.04% | 38.65 | 0.47 | 4.26 |
| 34. | 30 nov. 2019 | 3.41% | 3.40% | 3.62 | 4.19 | 3.89 |
| 35. | 31 dic. 2019 | 3.65% | 2.86% | 4.60 | 2.25 | 3.22 |
| 36. | 31 ene. 2020 | -2.59% | -0.16% | 16.79 | 2.31 | 6.23 |
| 37. | 29 feb. 2020 | -4.11% | -8.41% | 31.62 | 95.43 | 54.94 |
| 38. | 31 mar. 2020 | -14.66% | -12.51% | 261.30 | 192.37 | 224.20 |
| 39. | 30 abr. 2020 | 14.34% | 12.68% | 164.64 | 128.29 | 145.33 |
| 40. | 31 may. 2020 | 6.13% | 4.53% | 21.35 | 10.05 | 14.65 |
| 41. | 30 jun. 2020 | 2.01% | 1.84% | 0.25 | 0.23 | 0.24 |
| 42. | 31 jul. 2020 | 5.80% | 5.51% | 18.41 | 17.24 | 17.82 |
| 43. | 31 ago. 2020 | 6.79% | 7.01% | 27.89 | 31.91 | 29.83 |
| 44. | 30 sept. 2020 | -1.62% | -3.92% | 9.79 | 27.89 | 16.52 |
| 45. | 31 oct. 2020 | 1.38% | -2.77% | 0.02 | 17.01 | 0.52 |
| 46. | 30 nov. 2020 | 7.76% | 10.75% | 39.14 | 88.30 | 58.79 |
| 47. | 31 dic. 2020 | -2.88% | 3.71% | 19.22 | 5.54 | -10.32 |
| 48. | 31 ene. 2021 | -4.74% | -1.11% | 39.08 | 6.11 | 15.45 |
| 49. | 28 feb. 2021 | 4.10% | 2.61% | 6.73 | 1.57 | 3.25 |
| 50. | 31 mar. 2021 | 10.13% | 4.24% | 74.32 | 8.33 | 24.88 |
| 51. | 30 abr. 2021 | 4.04% | 5.24% | 6.39 | 15.09 | 9.82 |
| 52. | 31 may. 2021 | 0.56% | 0.55% | 0.89 | 0.65 | 0.76 |
| 53. | 30 jun. 2021 | -3.05% | 2.22% | 20.75 | 0.75 | -3.93 |
| 54. | 31 jul. 2021 | 1.39% | 2.27% | 0.01 | 0.84 | -0.11 |
| 55. | 31 ago. 2021 | 2.73% | 2.90% | 1.49 | 2.38 | 1.88 |
| 56. | 30 sept. 2021 | -10.74% | -4.76% | 150.04 | 37.39 | 74.90 |
| 57. | 31 oct. 2021 | 10.28% | 6.91% | 76.92 | 30.87 | 48.73 |
| 58. | 30 nov. 2021 | 1.88% | -0.83% | 0.14 | 4.80 | -0.81 |
| 59. | 31 dic. 2021 | 6.84% | 4.36% | 28.38 | 9.02 | 16.00 |
| Total (Σ): | 2,078.90 | 1,164.17 | 1,283.40 | |||
Mostrar todo
VarianzaITW = Σ(RITW,t–RITW)2 ÷ (59 – 1)
= 2,078.90 ÷ (59 – 1)
= 35.84
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,164.17 ÷ (59 – 1)
= 20.07
CovarianzaITW, S&P 500 = Σ(RITW,t–RITW)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,283.40 ÷ (59 – 1)
= 22.13
Estimación sistemática del riesgo (β)
| VarianzaITW | 35.84 |
| VarianzaS&P 500 | 20.07 |
| CovarianzaITW, S&P 500 | 22.13 |
| Coeficiente de correlaciónITW, S&P 5001 | 0.82 |
| βITW2 | 1.10 |
| αITW3 | 0.01% |
Cálculos
1 Coeficiente de correlaciónITW, S&P 500
= CovarianzaITW, S&P 500 ÷ (Desviación estándarITW × Desviación estándarS&P 500)
= 22.13 ÷ (5.99% × 4.48%)
= 0.82
2 βITW
= CovarianzaITW, S&P 500 ÷ VarianzaS&P 500
= 22.13 ÷ 20.07
= 1.10
3 αITW
= PromedioITW – βITW × PromedioS&P 500
= 1.51% – 1.10 × 1.36%
= 0.01%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.80% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.50% |
| Riesgo sistemático de ITW acciones ordinarias | βITW | 1.10 |
| Tasa de rendimiento esperada de las acciones ordinarias de ITW3 | E(RITW) | 14.39% |
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(RITW) = RF + βITW [E(RM) – RF]
= 4.80% + 1.10 [13.50% – 4.80%]
= 14.39%