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 Linde.
Tasas de retorno
Linde plc, tasas mensuales de retorno
| Linde plc (LIN) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioLIN,t1 | DividendoLIN,t1 | RLIN,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2020 | $203.13 | 3,225.52 | ||||
| 1. | 29 feb 2020 | $191.01 | -5.97% | 2,954.22 | -8.41% | |
| 2. | 31 mar 2020 | $173.00 | $0.963 | -8.92% | 2,584.59 | -12.51% |
| 3. | 30 abr 2020 | $183.99 | 6.35% | 2,912.43 | 12.68% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2024 | $460.99 | 1.06% | 6,032.38 | 5.73% | |
| 59. | 31 dic 2024 | $418.67 | $1.39 | -8.88% | 5,881.63 | -2.50% |
| Promedio (R): | 1.57% | 1.16% | ||||
| Desviación estándar: | 6.65% | 5.28% | ||||
| Linde plc (LIN) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioLIN,t1 | DividendoLIN,t1 | RLIN,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2020 | $203.13 | 3,225.52 | ||||
| 1. | 29 feb 2020 | $191.01 | -5.97% | 2,954.22 | -8.41% | |
| 2. | 31 mar 2020 | $173.00 | $0.963 | -8.92% | 2,584.59 | -12.51% |
| 3. | 30 abr 2020 | $183.99 | 6.35% | 2,912.43 | 12.68% | |
| 4. | 31 may 2020 | $202.34 | 9.97% | 3,044.31 | 4.53% | |
| 5. | 30 jun 2020 | $212.11 | $0.963 | 5.30% | 3,100.29 | 1.84% |
| 6. | 31 jul 2020 | $245.11 | 15.56% | 3,271.12 | 5.51% | |
| 7. | 31 ago 2020 | $249.74 | 1.89% | 3,500.31 | 7.01% | |
| 8. | 30 sept 2020 | $238.13 | $0.963 | -4.26% | 3,363.00 | -3.92% |
| 9. | 31 oct 2020 | $220.34 | -7.47% | 3,269.96 | -2.77% | |
| 10. | 30 nov 2020 | $256.42 | 16.37% | 3,621.63 | 10.75% | |
| 11. | 31 dic 2020 | $263.51 | $0.963 | 3.14% | 3,756.07 | 3.71% |
| 12. | 31 ene 2021 | $245.40 | -6.87% | 3,714.24 | -1.11% | |
| 13. | 28 feb 2021 | $244.27 | -0.46% | 3,811.15 | 2.61% | |
| 14. | 31 mar 2021 | $280.14 | $1.06 | 15.12% | 3,972.89 | 4.24% |
| 15. | 30 abr 2021 | $285.84 | 2.03% | 4,181.17 | 5.24% | |
| 16. | 31 may 2021 | $300.60 | 5.16% | 4,204.11 | 0.55% | |
| 17. | 30 jun 2021 | $289.10 | $1.06 | -3.47% | 4,297.50 | 2.22% |
| 18. | 31 jul 2021 | $307.39 | 6.33% | 4,395.26 | 2.27% | |
| 19. | 31 ago 2021 | $314.59 | 2.34% | 4,522.68 | 2.90% | |
| 20. | 30 sept 2021 | $293.38 | $1.06 | -6.41% | 4,307.54 | -4.76% |
| 21. | 31 oct 2021 | $319.20 | 8.80% | 4,605.38 | 6.91% | |
| 22. | 30 nov 2021 | $318.14 | -0.33% | 4,567.00 | -0.83% | |
| 23. | 31 dic 2021 | $346.43 | $1.06 | 9.23% | 4,766.18 | 4.36% |
| 24. | 31 ene 2022 | $318.68 | -8.01% | 4,515.55 | -5.26% | |
| 25. | 28 feb 2022 | $293.24 | -7.98% | 4,373.94 | -3.14% | |
| 26. | 31 mar 2022 | $319.43 | $1.17 | 9.33% | 4,530.41 | 3.58% |
| 27. | 30 abr 2022 | $311.96 | -2.34% | 4,131.93 | -8.80% | |
| 28. | 31 may 2022 | $324.68 | 4.08% | 4,132.15 | 0.01% | |
| 29. | 30 jun 2022 | $287.53 | $1.17 | -11.08% | 3,785.38 | -8.39% |
| 30. | 31 jul 2022 | $302.00 | 5.03% | 4,130.29 | 9.11% | |
| 31. | 31 ago 2022 | $282.86 | -6.34% | 3,955.00 | -4.24% | |
| 32. | 30 sept 2022 | $269.59 | $1.17 | -4.28% | 3,585.62 | -9.34% |
| 33. | 31 oct 2022 | $297.35 | 10.30% | 3,871.98 | 7.99% | |
| 34. | 30 nov 2022 | $336.48 | 13.16% | 4,080.11 | 5.38% | |
| 35. | 31 dic 2022 | $326.18 | $1.17 | -2.71% | 3,839.50 | -5.90% |
| 36. | 31 ene 2023 | $330.94 | 1.46% | 4,076.60 | 6.18% | |
| 37. | 28 feb 2023 | $348.37 | 5.27% | 3,970.15 | -2.61% | |
| 38. | 31 mar 2023 | $355.44 | $1.275 | 2.40% | 4,109.31 | 3.51% |
| 39. | 30 abr 2023 | $369.45 | 3.94% | 4,169.48 | 1.46% | |
| 40. | 31 may 2023 | $353.66 | -4.27% | 4,179.83 | 0.25% | |
| 41. | 30 jun 2023 | $381.08 | $1.275 | 8.11% | 4,376.86 | 4.71% |
| 42. | 31 jul 2023 | $390.67 | 2.52% | 4,588.96 | 4.85% | |
| 43. | 31 ago 2023 | $387.04 | -0.93% | 4,507.66 | -1.77% | |
| 44. | 30 sept 2023 | $372.35 | $1.275 | -3.47% | 4,288.05 | -4.87% |
| 45. | 31 oct 2023 | $382.16 | 2.63% | 4,193.80 | -2.20% | |
| 46. | 30 nov 2023 | $413.77 | 8.27% | 4,567.80 | 8.92% | |
| 47. | 31 dic 2023 | $410.71 | $1.275 | -0.43% | 4,769.83 | 4.42% |
| 48. | 31 ene 2024 | $404.83 | -1.43% | 4,845.65 | 1.59% | |
| 49. | 29 feb 2024 | $448.82 | 10.87% | 5,096.27 | 5.17% | |
| 50. | 31 mar 2024 | $464.32 | $1.39 | 3.76% | 5,254.35 | 3.10% |
| 51. | 30 abr 2024 | $440.96 | -5.03% | 5,035.69 | -4.16% | |
| 52. | 31 may 2024 | $435.52 | -1.23% | 5,277.51 | 4.80% | |
| 53. | 30 jun 2024 | $438.81 | $1.39 | 1.07% | 5,460.48 | 3.47% |
| 54. | 31 jul 2024 | $453.50 | 3.35% | 5,522.30 | 1.13% | |
| 55. | 31 ago 2024 | $478.25 | 5.46% | 5,648.40 | 2.28% | |
| 56. | 30 sept 2024 | $476.86 | $1.39 | 0.00% | 5,762.48 | 2.02% |
| 57. | 31 oct 2024 | $456.15 | -4.34% | 5,705.45 | -0.99% | |
| 58. | 30 nov 2024 | $460.99 | 1.06% | 6,032.38 | 5.73% | |
| 59. | 31 dic 2024 | $418.67 | $1.39 | -8.88% | 5,881.63 | -2.50% |
| Promedio (R): | 1.57% | 1.16% | ||||
| Desviación estándar: | 6.65% | 5.28% | ||||
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 LIN 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
Linde plc, cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RLIN,t | RS&P 500,t | (RLIN,t–RLIN)2 | (RS&P 500,t–RS&P 500)2 | (RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 29 feb 2020 | -5.97% | -8.41% | 56.83 | 91.63 | 72.16 |
| 2. | 31 mar 2020 | -8.92% | -12.51% | 110.18 | 186.96 | 143.52 |
| 3. | 30 abr 2020 | 6.35% | 12.68% | 22.86 | 132.78 | 55.09 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2024 | 1.06% | 5.73% | 0.26 | 20.87 | -2.33 |
| 59. | 31 dic 2024 | -8.88% | -2.50% | 109.22 | 13.40 | 38.25 |
| Total (Σ): | 2,565.41 | 1,618.62 | 1,515.53 | |||
| t | Fecha | RLIN,t | RS&P 500,t | (RLIN,t–RLIN)2 | (RS&P 500,t–RS&P 500)2 | (RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 29 feb 2020 | -5.97% | -8.41% | 56.83 | 91.63 | 72.16 |
| 2. | 31 mar 2020 | -8.92% | -12.51% | 110.18 | 186.96 | 143.52 |
| 3. | 30 abr 2020 | 6.35% | 12.68% | 22.86 | 132.78 | 55.09 |
| 4. | 31 may 2020 | 9.97% | 4.53% | 70.58 | 11.34 | 28.29 |
| 5. | 30 jun 2020 | 5.30% | 1.84% | 13.93 | 0.46 | 2.53 |
| 6. | 31 jul 2020 | 15.56% | 5.51% | 195.61 | 18.91 | 60.82 |
| 7. | 31 ago 2020 | 1.89% | 7.01% | 0.10 | 34.17 | 1.85 |
| 8. | 30 sept 2020 | -4.26% | -3.92% | 34.05 | 25.85 | 29.67 |
| 9. | 31 oct 2020 | -7.47% | -2.77% | 81.77 | 15.43 | 35.52 |
| 10. | 30 nov 2020 | 16.37% | 10.75% | 219.12 | 92.03 | 142.01 |
| 11. | 31 dic 2020 | 3.14% | 3.71% | 2.46 | 6.51 | 4.00 |
| 12. | 31 ene 2021 | -6.87% | -1.11% | 71.31 | 5.18 | 19.21 |
| 13. | 28 feb 2021 | -0.46% | 2.61% | 4.13 | 2.10 | -2.94 |
| 14. | 31 mar 2021 | 15.12% | 4.24% | 183.51 | 9.50 | 41.76 |
| 15. | 30 abr 2021 | 2.03% | 5.24% | 0.21 | 16.66 | 1.89 |
| 16. | 31 may 2021 | 5.16% | 0.55% | 12.90 | 0.38 | -2.20 |
| 17. | 30 jun 2021 | -3.47% | 2.22% | 25.45 | 1.12 | -5.35 |
| 18. | 31 jul 2021 | 6.33% | 2.27% | 22.61 | 1.24 | 5.29 |
| 19. | 31 ago 2021 | 2.34% | 2.90% | 0.59 | 3.02 | 1.34 |
| 20. | 30 sept 2021 | -6.41% | -4.76% | 63.63 | 35.02 | 47.21 |
| 21. | 31 oct 2021 | 8.80% | 6.91% | 52.26 | 33.10 | 41.59 |
| 22. | 30 nov 2021 | -0.33% | -0.83% | 3.63 | 3.98 | 3.80 |
| 23. | 31 dic 2021 | 9.23% | 4.36% | 58.58 | 10.24 | 24.49 |
| 24. | 31 ene 2022 | -8.01% | -5.26% | 91.82 | 41.21 | 61.52 |
| 25. | 28 feb 2022 | -7.98% | -3.14% | 91.29 | 18.47 | 41.06 |
| 26. | 31 mar 2022 | 9.33% | 3.58% | 60.19 | 5.84 | 18.74 |
| 27. | 30 abr 2022 | -2.34% | -8.80% | 15.29 | 99.14 | 38.94 |
| 28. | 31 may 2022 | 4.08% | 0.01% | 6.28 | 1.34 | -2.90 |
| 29. | 30 jun 2022 | -11.08% | -8.39% | 160.11 | 91.26 | 120.88 |
| 30. | 31 jul 2022 | 5.03% | 9.11% | 11.98 | 63.21 | 27.51 |
| 31. | 31 ago 2022 | -6.34% | -4.24% | 62.56 | 29.22 | 42.75 |
| 32. | 30 sept 2022 | -4.28% | -9.34% | 34.22 | 110.27 | 61.43 |
| 33. | 31 oct 2022 | 10.30% | 7.99% | 76.13 | 46.58 | 59.55 |
| 34. | 30 nov 2022 | 13.16% | 5.38% | 134.27 | 17.76 | 48.83 |
| 35. | 31 dic 2022 | -2.71% | -5.90% | 18.36 | 49.82 | 30.25 |
| 36. | 31 ene 2023 | 1.46% | 6.18% | 0.01 | 25.14 | -0.56 |
| 37. | 28 feb 2023 | 5.27% | -2.61% | 13.65 | 14.23 | -13.94 |
| 38. | 31 mar 2023 | 2.40% | 3.51% | 0.68 | 5.49 | 1.93 |
| 39. | 30 abr 2023 | 3.94% | 1.46% | 5.62 | 0.09 | 0.72 |
| 40. | 31 may 2023 | -4.27% | 0.25% | 34.17 | 0.83 | 5.34 |
| 41. | 30 jun 2023 | 8.11% | 4.71% | 42.80 | 12.62 | 23.24 |
| 42. | 31 jul 2023 | 2.52% | 4.85% | 0.89 | 13.58 | 3.48 |
| 43. | 31 ago 2023 | -0.93% | -1.77% | 6.26 | 8.60 | 7.34 |
| 44. | 30 sept 2023 | -3.47% | -4.87% | 25.38 | 36.40 | 30.39 |
| 45. | 31 oct 2023 | 2.63% | -2.20% | 1.13 | 11.28 | -3.57 |
| 46. | 30 nov 2023 | 8.27% | 8.92% | 44.88 | 60.17 | 51.97 |
| 47. | 31 dic 2023 | -0.43% | 4.42% | 4.01 | 10.64 | -6.53 |
| 48. | 31 ene 2024 | -1.43% | 1.59% | 9.02 | 0.18 | -1.29 |
| 49. | 29 feb 2024 | 10.87% | 5.17% | 86.39 | 16.09 | 37.28 |
| 50. | 31 mar 2024 | 3.76% | 3.10% | 4.80 | 3.77 | 4.25 |
| 51. | 30 abr 2024 | -5.03% | -4.16% | 43.60 | 28.33 | 35.15 |
| 52. | 31 may 2024 | -1.23% | 4.80% | 7.87 | 13.26 | -10.21 |
| 53. | 30 jun 2024 | 1.07% | 3.47% | 0.25 | 5.32 | -1.15 |
| 54. | 31 jul 2024 | 3.35% | 1.13% | 3.15 | 0.00 | -0.05 |
| 55. | 31 ago 2024 | 5.46% | 2.28% | 15.10 | 1.26 | 4.36 |
| 56. | 30 sept 2024 | 0.00% | 2.02% | 2.47 | 0.74 | -1.35 |
| 57. | 31 oct 2024 | -4.34% | -0.99% | 34.99 | 4.63 | 12.72 |
| 58. | 30 nov 2024 | 1.06% | 5.73% | 0.26 | 20.87 | -2.33 |
| 59. | 31 dic 2024 | -8.88% | -2.50% | 109.22 | 13.40 | 38.25 |
| Total (Σ): | 2,565.41 | 1,618.62 | 1,515.53 | |||
Mostrar todo
VarianzaLIN = Σ(RLIN,t–RLIN)2 ÷ (59 – 1)
= 2,565.41 ÷ (59 – 1)
= 44.23
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,618.62 ÷ (59 – 1)
= 27.91
CovarianzaLIN, S&P 500 = Σ(RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,515.53 ÷ (59 – 1)
= 26.13
Estimación sistemática del riesgo (β)
| VarianzaLIN | 44.23 |
| VarianzaS&P 500 | 27.91 |
| CovarianzaLIN, S&P 500 | 26.13 |
| Coeficiente de correlaciónLIN, S&P 5001 | 0.74 |
| βLIN2 | 0.94 |
| αLIN3 | 0.48% |
Cálculos
1 Coeficiente de correlaciónLIN, S&P 500
= CovarianzaLIN, S&P 500 ÷ (Desviación estándarLIN × Desviación estándarS&P 500)
= 26.13 ÷ (6.65% × 5.28%)
= 0.74
2 βLIN
= CovarianzaLIN, S&P 500 ÷ VarianzaS&P 500
= 26.13 ÷ 27.91
= 0.94
3 αLIN
= PromedioLIN – βLIN × PromedioS&P 500
= 1.57% – 0.94 × 1.16%
= 0.48%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.60% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 14.89% |
| Riesgo sistemático de Linde acciones ordinarias | βLIN | 0.94 |
| Tasa de rendimiento esperada de las acciones ordinarias de Linde3 | E(RLIN) | 14.23% |
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(RLIN) = RF + βLIN [E(RM) – RF]
= 4.60% + 0.94 [14.89% – 4.60%]
= 14.23%