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 Trane Technologies.
Tasas de retorno
Trane Technologies plc, tasas mensuales de retorno
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioTT,t1 | DividendoTT,t1 | RTT,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2018 | $94.63 | 2,823.81 | ||||
| 1. | 28 feb. 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | 31 mar. 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | 30 abr. 2018 | $83.89 | -1.89% | 2,648.05 | 0.27% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2022 | $178.42 | 11.77% | 4,080.11 | 5.38% | |
| 59. | 31 dic. 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Promedio (R): | 1.92% | 0.67% | ||||
| Desviación estándar: | 7.50% | 5.40% | ||||
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioTT,t1 | DividendoTT,t1 | RTT,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2018 | $94.63 | 2,823.81 | ||||
| 1. | 28 feb. 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | 31 mar. 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | 30 abr. 2018 | $83.89 | -1.89% | 2,648.05 | 0.27% | |
| 4. | 31 may. 2018 | $87.54 | 4.35% | 2,705.27 | 2.16% | |
| 5. | 30 jun. 2018 | $89.73 | $0.45 | 3.02% | 2,718.37 | 0.48% |
| 6. | 31 jul. 2018 | $98.51 | 9.78% | 2,816.29 | 3.60% | |
| 7. | 31 ago. 2018 | $101.29 | 2.82% | 2,901.52 | 3.03% | |
| 8. | 30 sept. 2018 | $102.30 | $0.53 | 1.52% | 2,913.98 | 0.43% |
| 9. | 31 oct. 2018 | $95.94 | -6.22% | 2,711.74 | -6.94% | |
| 10. | 30 nov. 2018 | $103.52 | 7.90% | 2,760.17 | 1.79% | |
| 11. | 31 dic. 2018 | $91.23 | $0.53 | -11.36% | 2,506.85 | -9.18% |
| 12. | 31 ene. 2019 | $100.04 | 9.66% | 2,704.10 | 7.87% | |
| 13. | 28 feb. 2019 | $105.56 | 5.52% | 2,784.49 | 2.97% | |
| 14. | 31 mar. 2019 | $107.95 | $0.53 | 2.77% | 2,834.40 | 1.79% |
| 15. | 30 abr. 2019 | $122.61 | 13.58% | 2,945.83 | 3.93% | |
| 16. | 31 may. 2019 | $118.34 | -3.48% | 2,752.06 | -6.58% | |
| 17. | 30 jun. 2019 | $126.67 | $0.53 | 7.49% | 2,941.76 | 6.89% |
| 18. | 31 jul. 2019 | $123.66 | -2.38% | 2,980.38 | 1.31% | |
| 19. | 31 ago. 2019 | $121.09 | -2.08% | 2,926.46 | -1.81% | |
| 20. | 30 sept. 2019 | $123.21 | $0.53 | 2.19% | 2,976.74 | 1.72% |
| 21. | 31 oct. 2019 | $126.89 | 2.99% | 3,037.56 | 2.04% | |
| 22. | 30 nov. 2019 | $131.11 | 3.33% | 3,140.98 | 3.40% | |
| 23. | 31 dic. 2019 | $132.92 | $0.53 | 1.78% | 3,230.78 | 2.86% |
| 24. | 31 ene. 2020 | $133.23 | 0.23% | 3,225.52 | -0.16% | |
| 25. | 29 feb. 2020 | $129.04 | -3.14% | 2,954.22 | -8.41% | |
| 26. | 31 mar. 2020 | $82.59 | $29.46 | -13.17% | 2,584.59 | -12.51% |
| 27. | 30 abr. 2020 | $87.42 | 5.85% | 2,912.43 | 12.68% | |
| 28. | 31 may. 2020 | $90.21 | 3.19% | 3,044.31 | 4.53% | |
| 29. | 30 jun. 2020 | $88.98 | $0.53 | -0.78% | 3,100.29 | 1.84% |
| 30. | 31 jul. 2020 | $111.87 | 25.72% | 3,271.12 | 5.51% | |
| 31. | 31 ago. 2020 | $118.39 | 5.83% | 3,500.31 | 7.01% | |
| 32. | 30 sept. 2020 | $121.25 | $0.53 | 2.86% | 3,363.00 | -3.92% |
| 33. | 31 oct. 2020 | $132.75 | 9.48% | 3,269.96 | -2.77% | |
| 34. | 30 nov. 2020 | $146.24 | 10.16% | 3,621.63 | 10.75% | |
| 35. | 31 dic. 2020 | $145.16 | $0.53 | -0.38% | 3,756.07 | 3.71% |
| 36. | 31 ene. 2021 | $143.35 | -1.25% | 3,714.24 | -1.11% | |
| 37. | 28 feb. 2021 | $153.24 | 6.90% | 3,811.15 | 2.61% | |
| 38. | 31 mar. 2021 | $165.56 | $0.59 | 8.42% | 3,972.89 | 4.24% |
| 39. | 30 abr. 2021 | $173.83 | 5.00% | 4,181.17 | 5.24% | |
| 40. | 31 may. 2021 | $186.40 | 7.23% | 4,204.11 | 0.55% | |
| 41. | 30 jun. 2021 | $184.14 | $0.59 | -0.90% | 4,297.50 | 2.22% |
| 42. | 31 jul. 2021 | $203.61 | 10.57% | 4,395.26 | 2.27% | |
| 43. | 31 ago. 2021 | $198.50 | -2.51% | 4,522.68 | 2.90% | |
| 44. | 30 sept. 2021 | $172.65 | $0.59 | -12.73% | 4,307.54 | -4.76% |
| 45. | 31 oct. 2021 | $180.93 | 4.80% | 4,605.38 | 6.91% | |
| 46. | 30 nov. 2021 | $186.65 | 3.16% | 4,567.00 | -0.83% | |
| 47. | 31 dic. 2021 | $202.03 | $0.59 | 8.56% | 4,766.18 | 4.36% |
| 48. | 31 ene. 2022 | $173.10 | -14.32% | 4,515.55 | -5.26% | |
| 49. | 28 feb. 2022 | $153.93 | -11.07% | 4,373.94 | -3.14% | |
| 50. | 31 mar. 2022 | $152.70 | $0.67 | -0.36% | 4,530.41 | 3.58% |
| 51. | 30 abr. 2022 | $139.89 | -8.39% | 4,131.93 | -8.80% | |
| 52. | 31 may. 2022 | $138.06 | -1.31% | 4,132.15 | 0.01% | |
| 53. | 30 jun. 2022 | $129.87 | $0.67 | -5.45% | 3,785.38 | -8.39% |
| 54. | 31 jul. 2022 | $146.99 | 13.18% | 4,130.29 | 9.11% | |
| 55. | 31 ago. 2022 | $154.07 | 4.82% | 3,955.00 | -4.24% | |
| 56. | 30 sept. 2022 | $144.81 | $0.67 | -5.58% | 3,585.62 | -9.34% |
| 57. | 31 oct. 2022 | $159.63 | 10.23% | 3,871.98 | 7.99% | |
| 58. | 30 nov. 2022 | $178.42 | 11.77% | 4,080.11 | 5.38% | |
| 59. | 31 dic. 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Promedio (R): | 1.92% | 0.67% | ||||
| Desviación estándar: | 7.50% | 5.40% | ||||
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 TT 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
Trane Technologies plc, cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RTT,t | RS&P 500,t | (RTT,t–RTT)2 | (RS&P 500,t–RS&P 500)2 | (RTT,t–RTT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | 31 mar. 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | 30 abr. 2018 | -1.89% | 0.27% | 14.54 | 0.16 | 1.51 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2022 | 11.77% | 5.38% | 97.08 | 22.17 | 46.39 |
| 59. | 31 dic. 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
| t | Fecha | RTT,t | RS&P 500,t | (RTT,t–RTT)2 | (RS&P 500,t–RS&P 500)2 | (RTT,t–RTT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | 31 mar. 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | 30 abr. 2018 | -1.89% | 0.27% | 14.54 | 0.16 | 1.51 |
| 4. | 31 may. 2018 | 4.35% | 2.16% | 5.92 | 2.23 | 3.64 |
| 5. | 30 jun. 2018 | 3.02% | 0.48% | 1.20 | 0.03 | -0.20 |
| 6. | 31 jul. 2018 | 9.78% | 3.60% | 61.89 | 8.62 | 23.09 |
| 7. | 31 ago. 2018 | 2.82% | 3.03% | 0.82 | 5.57 | 2.13 |
| 8. | 30 sept. 2018 | 1.52% | 0.43% | 0.16 | 0.06 | 0.09 |
| 9. | 31 oct. 2018 | -6.22% | -6.94% | 66.18 | 57.87 | 61.88 |
| 10. | 30 nov. 2018 | 7.90% | 1.79% | 35.79 | 1.25 | 6.70 |
| 11. | 31 dic. 2018 | -11.36% | -9.18% | 176.31 | 96.91 | 130.71 |
| 12. | 31 ene. 2019 | 9.66% | 7.87% | 59.89 | 51.87 | 55.73 |
| 13. | 28 feb. 2019 | 5.52% | 2.97% | 12.96 | 5.32 | 8.30 |
| 14. | 31 mar. 2019 | 2.77% | 1.79% | 0.72 | 1.27 | 0.95 |
| 15. | 30 abr. 2019 | 13.58% | 3.93% | 136.01 | 10.66 | 38.07 |
| 16. | 31 may. 2019 | -3.48% | -6.58% | 29.17 | 52.48 | 39.12 |
| 17. | 30 jun. 2019 | 7.49% | 6.89% | 31.01 | 38.77 | 34.67 |
| 18. | 31 jul. 2019 | -2.38% | 1.31% | 18.44 | 0.42 | -2.77 |
| 19. | 31 ago. 2019 | -2.08% | -1.81% | 15.97 | 6.13 | 9.89 |
| 20. | 30 sept. 2019 | 2.19% | 1.72% | 0.07 | 1.11 | 0.28 |
| 21. | 31 oct. 2019 | 2.99% | 2.04% | 1.14 | 1.89 | 1.47 |
| 22. | 30 nov. 2019 | 3.33% | 3.40% | 1.98 | 7.50 | 3.85 |
| 23. | 31 dic. 2019 | 1.78% | 2.86% | 0.02 | 4.81 | -0.29 |
| 24. | 31 ene. 2020 | 0.23% | -0.16% | 2.84 | 0.69 | 1.40 |
| 25. | 29 feb. 2020 | -3.14% | -8.41% | 25.63 | 82.40 | 45.96 |
| 26. | 31 mar. 2020 | -13.17% | -12.51% | 227.54 | 173.67 | 198.79 |
| 27. | 30 abr. 2020 | 5.85% | 12.68% | 15.45 | 144.43 | 47.23 |
| 28. | 31 may. 2020 | 3.19% | 4.53% | 1.62 | 14.91 | 4.92 |
| 29. | 30 jun. 2020 | -0.78% | 1.84% | 7.26 | 1.37 | -3.16 |
| 30. | 31 jul. 2020 | 25.72% | 5.51% | 566.77 | 23.46 | 115.31 |
| 31. | 31 ago. 2020 | 5.83% | 7.01% | 15.29 | 40.19 | 24.79 |
| 32. | 30 sept. 2020 | 2.86% | -3.92% | 0.89 | 21.06 | -4.34 |
| 33. | 31 oct. 2020 | 9.48% | -2.77% | 57.25 | 11.79 | -25.98 |
| 34. | 30 nov. 2020 | 10.16% | 10.75% | 67.96 | 101.77 | 83.16 |
| 35. | 31 dic. 2020 | -0.38% | 3.71% | 5.26 | 9.28 | -6.99 |
| 36. | 31 ene. 2021 | -1.25% | -1.11% | 10.02 | 3.17 | 5.63 |
| 37. | 28 feb. 2021 | 6.90% | 2.61% | 24.81 | 3.77 | 9.68 |
| 38. | 31 mar. 2021 | 8.42% | 4.24% | 42.34 | 12.80 | 23.28 |
| 39. | 30 abr. 2021 | 5.00% | 5.24% | 9.47 | 20.94 | 14.08 |
| 40. | 31 may. 2021 | 7.23% | 0.55% | 28.23 | 0.01 | -0.63 |
| 41. | 30 jun. 2021 | -0.90% | 2.22% | 7.92 | 2.42 | -4.38 |
| 42. | 31 jul. 2021 | 10.57% | 2.27% | 74.92 | 2.59 | 13.92 |
| 43. | 31 ago. 2021 | -2.51% | 2.90% | 19.60 | 4.98 | -9.88 |
| 44. | 30 sept. 2021 | -12.73% | -4.76% | 214.43 | 29.42 | 79.42 |
| 45. | 31 oct. 2021 | 4.80% | 6.91% | 8.28 | 39.03 | 17.98 |
| 46. | 30 nov. 2021 | 3.16% | -0.83% | 1.55 | 2.25 | -1.87 |
| 47. | 31 dic. 2021 | 8.56% | 4.36% | 44.06 | 13.65 | 24.53 |
| 48. | 31 ene. 2022 | -14.32% | -5.26% | 263.66 | 35.11 | 96.21 |
| 49. | 28 feb. 2022 | -11.07% | -3.14% | 168.81 | 14.46 | 49.41 |
| 50. | 31 mar. 2022 | -0.36% | 3.58% | 5.21 | 8.47 | -6.64 |
| 51. | 30 abr. 2022 | -8.39% | -8.80% | 106.24 | 89.54 | 97.53 |
| 52. | 31 may. 2022 | -1.31% | 0.01% | 10.41 | 0.44 | 2.13 |
| 53. | 30 jun. 2022 | -5.45% | -8.39% | 54.24 | 82.06 | 66.72 |
| 54. | 31 jul. 2022 | 13.18% | 9.11% | 126.89 | 71.32 | 95.13 |
| 55. | 31 ago. 2022 | 4.82% | -4.24% | 8.40 | 24.11 | -14.23 |
| 56. | 30 sept. 2022 | -5.58% | -9.34% | 56.15 | 100.12 | 74.98 |
| 57. | 31 oct. 2022 | 10.23% | 7.99% | 69.16 | 53.58 | 60.87 |
| 58. | 30 nov. 2022 | 11.77% | 5.38% | 97.08 | 22.17 | 46.39 |
| 59. | 31 dic. 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
Mostrar todo
VarianzaTT = Σ(RTT,t–RTT)2 ÷ (59 – 1)
= 3,261.02 ÷ (59 – 1)
= 56.22
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,691.48 ÷ (59 – 1)
= 29.16
CovarianzaTT, S&P 500 = Σ(RTT,t–RTT)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,742.35 ÷ (59 – 1)
= 30.04
Estimación sistemática del riesgo (β)
| VarianzaTT | 56.22 |
| VarianzaS&P 500 | 29.16 |
| CovarianzaTT, S&P 500 | 30.04 |
| Coeficiente de correlaciónTT, S&P 5001 | 0.74 |
| βTT2 | 1.03 |
| αTT3 | 1.23% |
Cálculos
1 Coeficiente de correlaciónTT, S&P 500
= CovarianzaTT, S&P 500 ÷ (Desviación estándarTT × Desviación estándarS&P 500)
= 30.04 ÷ (7.50% × 5.40%)
= 0.74
2 βTT
= CovarianzaTT, S&P 500 ÷ VarianzaS&P 500
= 30.04 ÷ 29.16
= 1.03
3 αTT
= PromedioTT – βTT × PromedioS&P 500
= 1.92% – 1.03 × 0.67%
= 1.23%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.62% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.69% |
| Riesgo sistemático de Trane Technologies acciones ordinarias | βTT | 1.03 |
| Tasa de rendimiento esperada de las acciones ordinarias de Trane Technologies3 | E(RTT) | 13.96% |
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(RTT) = RF + βTT [E(RM) – RF]
= 4.62% + 1.03 [13.69% – 4.62%]
= 13.96%