Modèle d’évaluation des immobilisations (CAPM)
Le modèle d’évaluation des actifs financiers (MEDAF) indique quel devrait être le taux de rendement attendu ou requis sur les actifs risqués comme les actions ordinaires de Trane Technologies.
Taux de rendement
Trane Technologies plc, taux de rendement mensuels
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PrixTT,t1 | DividendeTT,t1 | RTT,t2 | PrixS&P 500,t | RS&P 500,t3 |
| 31 janv. 2018 | $94.63 | 2,823.81 | ||||
| 1. | 28 févr. 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | 31 mars 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | 30 avr. 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 déc. 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Moyenne (R): | 1.92% | 0.67% | ||||
| Écart type: | 7.50% | 5.40% | ||||
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PrixTT,t1 | DividendeTT,t1 | RTT,t2 | PrixS&P 500,t | RS&P 500,t3 |
| 31 janv. 2018 | $94.63 | 2,823.81 | ||||
| 1. | 28 févr. 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | 31 mars 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | 30 avr. 2018 | $83.89 | -1.89% | 2,648.05 | 0.27% | |
| 4. | 31 mai 2018 | $87.54 | 4.35% | 2,705.27 | 2.16% | |
| 5. | 30 juin 2018 | $89.73 | $0.45 | 3.02% | 2,718.37 | 0.48% |
| 6. | 31 juil. 2018 | $98.51 | 9.78% | 2,816.29 | 3.60% | |
| 7. | 31 août 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 déc. 2018 | $91.23 | $0.53 | -11.36% | 2,506.85 | -9.18% |
| 12. | 31 janv. 2019 | $100.04 | 9.66% | 2,704.10 | 7.87% | |
| 13. | 28 févr. 2019 | $105.56 | 5.52% | 2,784.49 | 2.97% | |
| 14. | 31 mars 2019 | $107.95 | $0.53 | 2.77% | 2,834.40 | 1.79% |
| 15. | 30 avr. 2019 | $122.61 | 13.58% | 2,945.83 | 3.93% | |
| 16. | 31 mai 2019 | $118.34 | -3.48% | 2,752.06 | -6.58% | |
| 17. | 30 juin 2019 | $126.67 | $0.53 | 7.49% | 2,941.76 | 6.89% |
| 18. | 31 juil. 2019 | $123.66 | -2.38% | 2,980.38 | 1.31% | |
| 19. | 31 août 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 déc. 2019 | $132.92 | $0.53 | 1.78% | 3,230.78 | 2.86% |
| 24. | 31 janv. 2020 | $133.23 | 0.23% | 3,225.52 | -0.16% | |
| 25. | 29 févr. 2020 | $129.04 | -3.14% | 2,954.22 | -8.41% | |
| 26. | 31 mars 2020 | $82.59 | $29.46 | -13.17% | 2,584.59 | -12.51% |
| 27. | 30 avr. 2020 | $87.42 | 5.85% | 2,912.43 | 12.68% | |
| 28. | 31 mai 2020 | $90.21 | 3.19% | 3,044.31 | 4.53% | |
| 29. | 30 juin 2020 | $88.98 | $0.53 | -0.78% | 3,100.29 | 1.84% |
| 30. | 31 juil. 2020 | $111.87 | 25.72% | 3,271.12 | 5.51% | |
| 31. | 31 août 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 déc. 2020 | $145.16 | $0.53 | -0.38% | 3,756.07 | 3.71% |
| 36. | 31 janv. 2021 | $143.35 | -1.25% | 3,714.24 | -1.11% | |
| 37. | 28 févr. 2021 | $153.24 | 6.90% | 3,811.15 | 2.61% | |
| 38. | 31 mars 2021 | $165.56 | $0.59 | 8.42% | 3,972.89 | 4.24% |
| 39. | 30 avr. 2021 | $173.83 | 5.00% | 4,181.17 | 5.24% | |
| 40. | 31 mai 2021 | $186.40 | 7.23% | 4,204.11 | 0.55% | |
| 41. | 30 juin 2021 | $184.14 | $0.59 | -0.90% | 4,297.50 | 2.22% |
| 42. | 31 juil. 2021 | $203.61 | 10.57% | 4,395.26 | 2.27% | |
| 43. | 31 août 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 déc. 2021 | $202.03 | $0.59 | 8.56% | 4,766.18 | 4.36% |
| 48. | 31 janv. 2022 | $173.10 | -14.32% | 4,515.55 | -5.26% | |
| 49. | 28 févr. 2022 | $153.93 | -11.07% | 4,373.94 | -3.14% | |
| 50. | 31 mars 2022 | $152.70 | $0.67 | -0.36% | 4,530.41 | 3.58% |
| 51. | 30 avr. 2022 | $139.89 | -8.39% | 4,131.93 | -8.80% | |
| 52. | 31 mai 2022 | $138.06 | -1.31% | 4,132.15 | 0.01% | |
| 53. | 30 juin 2022 | $129.87 | $0.67 | -5.45% | 3,785.38 | -8.39% |
| 54. | 31 juil. 2022 | $146.99 | 13.18% | 4,130.29 | 9.11% | |
| 55. | 31 août 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 déc. 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Moyenne (R): | 1.92% | 0.67% | ||||
| Écart type: | 7.50% | 5.40% | ||||
Tout afficher
1 Données en dollars américains par action ordinaire, ajustées pour tenir compte des fractionnements et des dividendes en actions.
2 Taux de rendement des actions ordinaires de TT au cours de la période t.
3 Taux de rendement de l’indice S&P 500 (l’indicateur du portefeuille de marché) au cours de la période t.
Variance et covariance
Trane Technologies plc, calcul de la variance et de la covariance des rendements
| t | Date | 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 févr. 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | 31 mars 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | 30 avr. 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 déc. 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
| t | Date | 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 févr. 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | 31 mars 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | 30 avr. 2018 | -1.89% | 0.27% | 14.54 | 0.16 | 1.51 |
| 4. | 31 mai 2018 | 4.35% | 2.16% | 5.92 | 2.23 | 3.64 |
| 5. | 30 juin 2018 | 3.02% | 0.48% | 1.20 | 0.03 | -0.20 |
| 6. | 31 juil. 2018 | 9.78% | 3.60% | 61.89 | 8.62 | 23.09 |
| 7. | 31 août 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 déc. 2018 | -11.36% | -9.18% | 176.31 | 96.91 | 130.71 |
| 12. | 31 janv. 2019 | 9.66% | 7.87% | 59.89 | 51.87 | 55.73 |
| 13. | 28 févr. 2019 | 5.52% | 2.97% | 12.96 | 5.32 | 8.30 |
| 14. | 31 mars 2019 | 2.77% | 1.79% | 0.72 | 1.27 | 0.95 |
| 15. | 30 avr. 2019 | 13.58% | 3.93% | 136.01 | 10.66 | 38.07 |
| 16. | 31 mai 2019 | -3.48% | -6.58% | 29.17 | 52.48 | 39.12 |
| 17. | 30 juin 2019 | 7.49% | 6.89% | 31.01 | 38.77 | 34.67 |
| 18. | 31 juil. 2019 | -2.38% | 1.31% | 18.44 | 0.42 | -2.77 |
| 19. | 31 août 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 déc. 2019 | 1.78% | 2.86% | 0.02 | 4.81 | -0.29 |
| 24. | 31 janv. 2020 | 0.23% | -0.16% | 2.84 | 0.69 | 1.40 |
| 25. | 29 févr. 2020 | -3.14% | -8.41% | 25.63 | 82.40 | 45.96 |
| 26. | 31 mars 2020 | -13.17% | -12.51% | 227.54 | 173.67 | 198.79 |
| 27. | 30 avr. 2020 | 5.85% | 12.68% | 15.45 | 144.43 | 47.23 |
| 28. | 31 mai 2020 | 3.19% | 4.53% | 1.62 | 14.91 | 4.92 |
| 29. | 30 juin 2020 | -0.78% | 1.84% | 7.26 | 1.37 | -3.16 |
| 30. | 31 juil. 2020 | 25.72% | 5.51% | 566.77 | 23.46 | 115.31 |
| 31. | 31 août 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 déc. 2020 | -0.38% | 3.71% | 5.26 | 9.28 | -6.99 |
| 36. | 31 janv. 2021 | -1.25% | -1.11% | 10.02 | 3.17 | 5.63 |
| 37. | 28 févr. 2021 | 6.90% | 2.61% | 24.81 | 3.77 | 9.68 |
| 38. | 31 mars 2021 | 8.42% | 4.24% | 42.34 | 12.80 | 23.28 |
| 39. | 30 avr. 2021 | 5.00% | 5.24% | 9.47 | 20.94 | 14.08 |
| 40. | 31 mai 2021 | 7.23% | 0.55% | 28.23 | 0.01 | -0.63 |
| 41. | 30 juin 2021 | -0.90% | 2.22% | 7.92 | 2.42 | -4.38 |
| 42. | 31 juil. 2021 | 10.57% | 2.27% | 74.92 | 2.59 | 13.92 |
| 43. | 31 août 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 déc. 2021 | 8.56% | 4.36% | 44.06 | 13.65 | 24.53 |
| 48. | 31 janv. 2022 | -14.32% | -5.26% | 263.66 | 35.11 | 96.21 |
| 49. | 28 févr. 2022 | -11.07% | -3.14% | 168.81 | 14.46 | 49.41 |
| 50. | 31 mars 2022 | -0.36% | 3.58% | 5.21 | 8.47 | -6.64 |
| 51. | 30 avr. 2022 | -8.39% | -8.80% | 106.24 | 89.54 | 97.53 |
| 52. | 31 mai 2022 | -1.31% | 0.01% | 10.41 | 0.44 | 2.13 |
| 53. | 30 juin 2022 | -5.45% | -8.39% | 54.24 | 82.06 | 66.72 |
| 54. | 31 juil. 2022 | 13.18% | 9.11% | 126.89 | 71.32 | 95.13 |
| 55. | 31 août 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 déc. 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
Tout afficher
VarianceTT = Σ(RTT,t–RTT)2 ÷ (59 – 1)
= 3,261.02 ÷ (59 – 1)
= 56.22
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,691.48 ÷ (59 – 1)
= 29.16
CovarianceTT, S&P 500 = Σ(RTT,t–RTT)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,742.35 ÷ (59 – 1)
= 30.04
Estimation systématique des risques (β)
| VarianceTT | 56.22 |
| VarianceS&P 500 | 29.16 |
| CovarianceTT, S&P 500 | 30.04 |
| Coefficient de corrélationTT, S&P 5001 | 0.74 |
| βTT2 | 1.03 |
| αTT3 | 1.23% |
Calculs
1 Coefficient de corrélationTT, S&P 500
= CovarianceTT, S&P 500 ÷ (Écart typeTT × Écart typeS&P 500)
= 30.04 ÷ (7.50% × 5.40%)
= 0.74
2 βTT
= CovarianceTT, S&P 500 ÷ VarianceS&P 500
= 30.04 ÷ 29.16
= 1.03
3 αTT
= MoyenneTT – βTT × MoyenneS&P 500
= 1.92% – 1.03 × 0.67%
= 1.23%
Taux de rendement attendu
| Hypothèses | ||
| Taux de rendement de LT Treasury Composite1 | RF | 4.53% |
| Taux de rendement attendu du portefeuille de marché2 | E(RM) | 14.88% |
| Risque systématique de Trane Technologies actions ordinaires | βTT | 1.03 |
| Taux de rendement attendu sur les actions ordinaires de Trane Technologies3 | E(RTT) | 15.19% |
1 Moyenne non pondérée des rendements des offres sur tous les bons du Trésor américain à coupon fixe en circulation qui ne sont ni échus ni remboursables par anticipation dans moins de 10 ans (indicateur du taux de rendement sans risque).
3 E(RTT) = RF + βTT [E(RM) – RF]
= 4.53% + 1.03 [14.88% – 4.53%]
= 15.19%