Modèle d’évaluation des immobilisations (CAPM)
Le modèle d’évaluation des immobilisations (MEDAF) indique quel devrait être le taux de rendement attendu ou requis pour les actifs risqués tels que les actions ordinaires de Caterpillar.
Taux de rendement
Caterpillar Inc., taux de rendement mensuels
| Caterpillar Inc. (CAT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PrixCAT,t1 | DividendeCAT,t1 | RCAT,t2 | PrixS&P 500,t | RS&P 500,t3 |
| 31 janv. 2020 | $131.35 | 3,225.52 | ||||
| 1. | 29 févr. 2020 | $124.24 | -5.41% | 2,954.22 | -8.41% | |
| 2. | 31 mars 2020 | $116.04 | -6.60% | 2,584.59 | -12.51% | |
| 3. | 30 avr. 2020 | $116.38 | $1.03 | 1.18% | 2,912.43 | 12.68% |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2024 | $406.11 | 7.95% | 6,032.38 | 5.73% | |
| 59. | 31 déc. 2024 | $362.76 | -10.67% | 5,881.63 | -2.50% | |
| Moyenne (R): | 2.32% | 1.16% | ||||
| Écart type: | 9.25% | 5.28% | ||||
| Caterpillar Inc. (CAT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PrixCAT,t1 | DividendeCAT,t1 | RCAT,t2 | PrixS&P 500,t | RS&P 500,t3 |
| 31 janv. 2020 | $131.35 | 3,225.52 | ||||
| 1. | 29 févr. 2020 | $124.24 | -5.41% | 2,954.22 | -8.41% | |
| 2. | 31 mars 2020 | $116.04 | -6.60% | 2,584.59 | -12.51% | |
| 3. | 30 avr. 2020 | $116.38 | $1.03 | 1.18% | 2,912.43 | 12.68% |
| 4. | 31 mai 2020 | $120.13 | 3.22% | 3,044.31 | 4.53% | |
| 5. | 30 juin 2020 | $126.50 | 5.30% | 3,100.29 | 1.84% | |
| 6. | 31 juil. 2020 | $132.88 | $1.03 | 5.86% | 3,271.12 | 5.51% |
| 7. | 31 août 2020 | $142.31 | 7.10% | 3,500.31 | 7.01% | |
| 8. | 30 sept. 2020 | $149.15 | 4.81% | 3,363.00 | -3.92% | |
| 9. | 31 oct. 2020 | $157.05 | $1.03 | 5.99% | 3,269.96 | -2.77% |
| 10. | 30 nov. 2020 | $173.59 | 10.53% | 3,621.63 | 10.75% | |
| 11. | 31 déc. 2020 | $182.02 | 4.86% | 3,756.07 | 3.71% | |
| 12. | 31 janv. 2021 | $182.84 | $1.03 | 1.02% | 3,714.24 | -1.11% |
| 13. | 28 févr. 2021 | $215.88 | 18.07% | 3,811.15 | 2.61% | |
| 14. | 31 mars 2021 | $231.87 | 7.41% | 3,972.89 | 4.24% | |
| 15. | 30 avr. 2021 | $228.11 | $1.03 | -1.18% | 4,181.17 | 5.24% |
| 16. | 31 mai 2021 | $241.08 | 5.69% | 4,204.11 | 0.55% | |
| 17. | 30 juin 2021 | $217.63 | -9.73% | 4,297.50 | 2.22% | |
| 18. | 31 juil. 2021 | $206.75 | $1.11 | -4.49% | 4,395.26 | 2.27% |
| 19. | 31 août 2021 | $210.87 | 1.99% | 4,522.68 | 2.90% | |
| 20. | 30 sept. 2021 | $191.97 | -8.96% | 4,307.54 | -4.76% | |
| 21. | 31 oct. 2021 | $204.01 | $1.11 | 6.85% | 4,605.38 | 6.91% |
| 22. | 30 nov. 2021 | $193.35 | -5.23% | 4,567.00 | -0.83% | |
| 23. | 31 déc. 2021 | $206.74 | 6.93% | 4,766.18 | 4.36% | |
| 24. | 31 janv. 2022 | $201.56 | $1.11 | -1.97% | 4,515.55 | -5.26% |
| 25. | 28 févr. 2022 | $187.58 | -6.94% | 4,373.94 | -3.14% | |
| 26. | 31 mars 2022 | $222.82 | 18.79% | 4,530.41 | 3.58% | |
| 27. | 30 avr. 2022 | $210.54 | $1.11 | -5.01% | 4,131.93 | -8.80% |
| 28. | 31 mai 2022 | $215.85 | 2.52% | 4,132.15 | 0.01% | |
| 29. | 30 juin 2022 | $178.76 | -17.18% | 3,785.38 | -8.39% | |
| 30. | 31 juil. 2022 | $198.25 | $1.20 | 11.57% | 4,130.29 | 9.11% |
| 31. | 31 août 2022 | $184.71 | -6.83% | 3,955.00 | -4.24% | |
| 32. | 30 sept. 2022 | $164.08 | -11.17% | 3,585.62 | -9.34% | |
| 33. | 31 oct. 2022 | $216.46 | $1.20 | 32.65% | 3,871.98 | 7.99% |
| 34. | 30 nov. 2022 | $236.41 | 9.22% | 4,080.11 | 5.38% | |
| 35. | 31 déc. 2022 | $239.56 | 1.33% | 3,839.50 | -5.90% | |
| 36. | 31 janv. 2023 | $252.29 | $1.20 | 5.81% | 4,076.60 | 6.18% |
| 37. | 28 févr. 2023 | $239.55 | -5.05% | 3,970.15 | -2.61% | |
| 38. | 31 mars 2023 | $228.84 | -4.47% | 4,109.31 | 3.51% | |
| 39. | 30 avr. 2023 | $218.80 | $1.20 | -3.86% | 4,169.48 | 1.46% |
| 40. | 31 mai 2023 | $205.75 | -5.96% | 4,179.83 | 0.25% | |
| 41. | 30 juin 2023 | $246.05 | 19.59% | 4,376.86 | 4.71% | |
| 42. | 31 juil. 2023 | $265.17 | $1.30 | 8.30% | 4,588.96 | 4.85% |
| 43. | 31 août 2023 | $281.13 | 6.02% | 4,507.66 | -1.77% | |
| 44. | 30 sept. 2023 | $273.00 | -2.89% | 4,288.05 | -4.87% | |
| 45. | 31 oct. 2023 | $226.05 | $1.30 | -16.72% | 4,193.80 | -2.20% |
| 46. | 30 nov. 2023 | $250.72 | 10.91% | 4,567.80 | 8.92% | |
| 47. | 31 déc. 2023 | $295.67 | 17.93% | 4,769.83 | 4.42% | |
| 48. | 31 janv. 2024 | $300.31 | $1.30 | 2.01% | 4,845.65 | 1.59% |
| 49. | 29 févr. 2024 | $333.96 | 11.21% | 5,096.27 | 5.17% | |
| 50. | 31 mars 2024 | $366.43 | 9.72% | 5,254.35 | 3.10% | |
| 51. | 30 avr. 2024 | $334.57 | $1.30 | -8.34% | 5,035.69 | -4.16% |
| 52. | 31 mai 2024 | $338.52 | 1.18% | 5,277.51 | 4.80% | |
| 53. | 30 juin 2024 | $333.10 | -1.60% | 5,460.48 | 3.47% | |
| 54. | 31 juil. 2024 | $346.20 | $1.41 | 4.36% | 5,522.30 | 1.13% |
| 55. | 31 août 2024 | $356.10 | 2.86% | 5,648.40 | 2.28% | |
| 56. | 30 sept. 2024 | $391.12 | 9.83% | 5,762.48 | 2.02% | |
| 57. | 31 oct. 2024 | $376.20 | $1.41 | -3.45% | 5,705.45 | -0.99% |
| 58. | 30 nov. 2024 | $406.11 | 7.95% | 6,032.38 | 5.73% | |
| 59. | 31 déc. 2024 | $362.76 | -10.67% | 5,881.63 | -2.50% | |
| Moyenne (R): | 2.32% | 1.16% | ||||
| Écart type: | 9.25% | 5.28% | ||||
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 CAT 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
Caterpillar Inc., calcul de la variance et de la covariance des rendements
| t | Date | RCAT,t | RS&P 500,t | (RCAT,t–RCAT)2 | (RS&P 500,t–RS&P 500)2 | (RCAT,t–RCAT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 29 févr. 2020 | -5.41% | -8.41% | 59.79 | 91.63 | 74.01 |
| 2. | 31 mars 2020 | -6.60% | -12.51% | 79.55 | 186.96 | 121.96 |
| 3. | 30 avr. 2020 | 1.18% | 12.68% | 1.30 | 132.78 | -13.12 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2024 | 7.95% | 5.73% | 31.71 | 20.87 | 25.73 |
| 59. | 31 déc. 2024 | -10.67% | -2.50% | 168.83 | 13.40 | 47.56 |
| Total (Σ): | 4,958.68 | 1,618.62 | 1,796.90 | |||
| t | Date | RCAT,t | RS&P 500,t | (RCAT,t–RCAT)2 | (RS&P 500,t–RS&P 500)2 | (RCAT,t–RCAT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 29 févr. 2020 | -5.41% | -8.41% | 59.79 | 91.63 | 74.01 |
| 2. | 31 mars 2020 | -6.60% | -12.51% | 79.55 | 186.96 | 121.96 |
| 3. | 30 avr. 2020 | 1.18% | 12.68% | 1.30 | 132.78 | -13.12 |
| 4. | 31 mai 2020 | 3.22% | 4.53% | 0.82 | 11.34 | 3.04 |
| 5. | 30 juin 2020 | 5.30% | 1.84% | 8.90 | 0.46 | 2.02 |
| 6. | 31 juil. 2020 | 5.86% | 5.51% | 12.52 | 18.91 | 15.39 |
| 7. | 31 août 2020 | 7.10% | 7.01% | 22.82 | 34.17 | 27.93 |
| 8. | 30 sept. 2020 | 4.81% | -3.92% | 6.19 | 25.85 | -12.65 |
| 9. | 31 oct. 2020 | 5.99% | -2.77% | 13.46 | 15.43 | -14.41 |
| 10. | 30 nov. 2020 | 10.53% | 10.75% | 67.45 | 92.03 | 78.79 |
| 11. | 31 déc. 2020 | 4.86% | 3.71% | 6.44 | 6.51 | 6.47 |
| 12. | 31 janv. 2021 | 1.02% | -1.11% | 1.70 | 5.18 | 2.96 |
| 13. | 28 févr. 2021 | 18.07% | 2.61% | 248.10 | 2.10 | 22.81 |
| 14. | 31 mars 2021 | 7.41% | 4.24% | 25.89 | 9.50 | 15.68 |
| 15. | 30 avr. 2021 | -1.18% | 5.24% | 12.23 | 16.66 | -14.27 |
| 16. | 31 mai 2021 | 5.69% | 0.55% | 11.33 | 0.38 | -2.06 |
| 17. | 30 juin 2021 | -9.73% | 2.22% | 145.11 | 1.12 | -12.77 |
| 18. | 31 juil. 2021 | -4.49% | 2.27% | 46.35 | 1.24 | -7.58 |
| 19. | 31 août 2021 | 1.99% | 2.90% | 0.11 | 3.02 | -0.57 |
| 20. | 30 sept. 2021 | -8.96% | -4.76% | 127.28 | 35.02 | 66.77 |
| 21. | 31 oct. 2021 | 6.85% | 6.91% | 20.53 | 33.10 | 26.07 |
| 22. | 30 nov. 2021 | -5.23% | -0.83% | 56.92 | 3.98 | 15.05 |
| 23. | 31 déc. 2021 | 6.93% | 4.36% | 21.22 | 10.24 | 14.74 |
| 24. | 31 janv. 2022 | -1.97% | -5.26% | 18.39 | 41.21 | 27.53 |
| 25. | 28 févr. 2022 | -6.94% | -3.14% | 85.66 | 18.47 | 39.77 |
| 26. | 31 mars 2022 | 18.79% | 3.58% | 271.18 | 5.84 | 39.79 |
| 27. | 30 avr. 2022 | -5.01% | -8.80% | 53.76 | 99.14 | 73.01 |
| 28. | 31 mai 2022 | 2.52% | 0.01% | 0.04 | 1.34 | -0.23 |
| 29. | 30 juin 2022 | -17.18% | -8.39% | 380.34 | 91.26 | 186.31 |
| 30. | 31 juil. 2022 | 11.57% | 9.11% | 85.66 | 63.21 | 73.58 |
| 31. | 31 août 2022 | -6.83% | -4.24% | 83.70 | 29.22 | 49.45 |
| 32. | 30 sept. 2022 | -11.17% | -9.34% | 181.93 | 110.27 | 141.64 |
| 33. | 31 oct. 2022 | 32.65% | 7.99% | 920.25 | 46.58 | 207.04 |
| 34. | 30 nov. 2022 | 9.22% | 5.38% | 47.57 | 17.76 | 29.07 |
| 35. | 31 déc. 2022 | 1.33% | -5.90% | 0.97 | 49.82 | 6.96 |
| 36. | 31 janv. 2023 | 5.81% | 6.18% | 12.22 | 25.14 | 17.53 |
| 37. | 28 févr. 2023 | -5.05% | -2.61% | 54.30 | 14.23 | 27.80 |
| 38. | 31 mars 2023 | -4.47% | 3.51% | 46.10 | 5.49 | -15.92 |
| 39. | 30 avr. 2023 | -3.86% | 1.46% | 38.22 | 0.09 | -1.87 |
| 40. | 31 mai 2023 | -5.96% | 0.25% | 68.62 | 0.83 | 7.56 |
| 41. | 30 juin 2023 | 19.59% | 4.71% | 298.17 | 12.62 | 61.34 |
| 42. | 31 juil. 2023 | 8.30% | 4.85% | 35.76 | 13.58 | 22.03 |
| 43. | 31 août 2023 | 6.02% | -1.77% | 13.69 | 8.60 | -10.85 |
| 44. | 30 sept. 2023 | -2.89% | -4.87% | 27.16 | 36.40 | 31.44 |
| 45. | 31 oct. 2023 | -16.72% | -2.20% | 362.55 | 11.28 | 63.96 |
| 46. | 30 nov. 2023 | 10.91% | 8.92% | 73.86 | 60.17 | 66.66 |
| 47. | 31 déc. 2023 | 17.93% | 4.42% | 243.65 | 10.64 | 50.91 |
| 48. | 31 janv. 2024 | 2.01% | 1.59% | 0.10 | 0.18 | -0.13 |
| 49. | 29 févr. 2024 | 11.21% | 5.17% | 78.96 | 16.09 | 35.64 |
| 50. | 31 mars 2024 | 9.72% | 3.10% | 54.81 | 3.77 | 14.37 |
| 51. | 30 avr. 2024 | -8.34% | -4.16% | 113.62 | 28.33 | 56.74 |
| 52. | 31 mai 2024 | 1.18% | 4.80% | 1.30 | 13.26 | -4.15 |
| 53. | 30 juin 2024 | -1.60% | 3.47% | 15.37 | 5.32 | -9.04 |
| 54. | 31 juil. 2024 | 4.36% | 1.13% | 4.15 | 0.00 | -0.06 |
| 55. | 31 août 2024 | 2.86% | 2.28% | 0.29 | 1.26 | 0.61 |
| 56. | 30 sept. 2024 | 9.83% | 2.02% | 56.48 | 0.74 | 6.45 |
| 57. | 31 oct. 2024 | -3.45% | -0.99% | 33.33 | 4.63 | 12.42 |
| 58. | 30 nov. 2024 | 7.95% | 5.73% | 31.71 | 20.87 | 25.73 |
| 59. | 31 déc. 2024 | -10.67% | -2.50% | 168.83 | 13.40 | 47.56 |
| Total (Σ): | 4,958.68 | 1,618.62 | 1,796.90 | |||
Tout afficher
VarianceCAT = Σ(RCAT,t–RCAT)2 ÷ (59 – 1)
= 4,958.68 ÷ (59 – 1)
= 85.49
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,618.62 ÷ (59 – 1)
= 27.91
CovarianceCAT, S&P 500 = Σ(RCAT,t–RCAT)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,796.90 ÷ (59 – 1)
= 30.98
Estimation systématique des risques (β)
| VarianceCAT | 85.49 |
| VarianceS&P 500 | 27.91 |
| CovarianceCAT, S&P 500 | 30.98 |
| Coefficient de corrélationCAT, S&P 5001 | 0.63 |
| βCAT2 | 1.11 |
| αCAT3 | 1.03% |
Calculs
1 Coefficient de corrélationCAT, S&P 500
= CovarianceCAT, S&P 500 ÷ (Écart typeCAT × Écart typeS&P 500)
= 30.98 ÷ (9.25% × 5.28%)
= 0.63
2 βCAT
= CovarianceCAT, S&P 500 ÷ VarianceS&P 500
= 30.98 ÷ 27.91
= 1.11
3 αCAT
= MoyenneCAT – βCAT × MoyenneS&P 500
= 2.32% – 1.11 × 1.16%
= 1.03%
Taux de rendement attendu
| Hypothèses | ||
| Taux de rendement de LT Treasury Composite1 | RF | 4.60% |
| Taux de rendement attendu du portefeuille de marché2 | E(RM) | 14.89% |
| Risque systématique lié à Caterpillar actions ordinaires | βCAT | 1.11 |
| Taux de rendement attendu des actions ordinaires de Caterpillar3 | E(RCAT) | 16.02% |
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(RCAT) = RF + βCAT [E(RM) – RF]
= 4.60% + 1.11 [14.89% – 4.60%]
= 16.02%