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 3M.
Tasas de retorno
3M Co., tasas mensuales de retorno
| 3M Co. (MMM) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioMMM,t1 | DividendoMMM,t1 | RMMM,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2017 | $174.82 | 2,278.87 | ||||
| 1. | 28 feb 2017 | $186.35 | $1.175 | 7.27% | 2,363.64 | 3.72% |
| 2. | 31 mar 2017 | $191.33 | 2.67% | 2,362.72 | -0.04% | |
| 3. | 30 abr 2017 | $195.83 | 2.35% | 2,384.20 | 0.91% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2021 | $170.04 | $1.48 | -4.01% | 4,567.00 | -0.83% |
| 59. | 31 dic 2021 | $177.63 | 4.46% | 4,766.18 | 4.36% | |
| Promedio (R): | 0.46% | 1.36% | ||||
| Desviación estándar: | 6.03% | 4.48% | ||||
| 3M Co. (MMM) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioMMM,t1 | DividendoMMM,t1 | RMMM,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene 2017 | $174.82 | 2,278.87 | ||||
| 1. | 28 feb 2017 | $186.35 | $1.175 | 7.27% | 2,363.64 | 3.72% |
| 2. | 31 mar 2017 | $191.33 | 2.67% | 2,362.72 | -0.04% | |
| 3. | 30 abr 2017 | $195.83 | 2.35% | 2,384.20 | 0.91% | |
| 4. | 31 may 2017 | $204.47 | $1.175 | 5.01% | 2,411.80 | 1.16% |
| 5. | 30 jun 2017 | $208.19 | 1.82% | 2,423.41 | 0.48% | |
| 6. | 31 jul 2017 | $201.17 | -3.37% | 2,470.30 | 1.93% | |
| 7. | 31 ago 2017 | $204.32 | $1.175 | 2.15% | 2,471.65 | 0.05% |
| 8. | 30 sept 2017 | $209.90 | 2.73% | 2,519.36 | 1.93% | |
| 9. | 31 oct 2017 | $230.19 | 9.67% | 2,575.26 | 2.22% | |
| 10. | 30 nov 2017 | $243.14 | $1.175 | 6.14% | 2,647.58 | 2.81% |
| 11. | 31 dic 2017 | $235.37 | -3.20% | 2,673.61 | 0.98% | |
| 12. | 31 ene 2018 | $250.50 | 6.43% | 2,823.81 | 5.62% | |
| 13. | 28 feb 2018 | $235.51 | $1.36 | -5.44% | 2,713.83 | -3.89% |
| 14. | 31 mar 2018 | $219.52 | -6.79% | 2,640.87 | -2.69% | |
| 15. | 30 abr 2018 | $194.39 | -11.45% | 2,648.05 | 0.27% | |
| 16. | 31 may 2018 | $197.23 | $1.36 | 2.16% | 2,705.27 | 2.16% |
| 17. | 30 jun 2018 | $196.72 | -0.26% | 2,718.37 | 0.48% | |
| 18. | 31 jul 2018 | $212.32 | 7.93% | 2,816.29 | 3.60% | |
| 19. | 31 ago 2018 | $210.92 | $1.36 | -0.02% | 2,901.52 | 3.03% |
| 20. | 30 sept 2018 | $210.71 | -0.10% | 2,913.98 | 0.43% | |
| 21. | 31 oct 2018 | $190.26 | -9.71% | 2,711.74 | -6.94% | |
| 22. | 30 nov 2018 | $207.92 | $1.36 | 10.00% | 2,760.17 | 1.79% |
| 23. | 31 dic 2018 | $190.54 | -8.36% | 2,506.85 | -9.18% | |
| 24. | 31 ene 2019 | $200.30 | 5.12% | 2,704.10 | 7.87% | |
| 25. | 28 feb 2019 | $207.39 | $1.44 | 4.26% | 2,784.49 | 2.97% |
| 26. | 31 mar 2019 | $207.78 | 0.19% | 2,834.40 | 1.79% | |
| 27. | 30 abr 2019 | $189.51 | -8.79% | 2,945.83 | 3.93% | |
| 28. | 31 may 2019 | $159.75 | $1.44 | -14.94% | 2,752.06 | -6.58% |
| 29. | 30 jun 2019 | $173.34 | 8.51% | 2,941.76 | 6.89% | |
| 30. | 31 jul 2019 | $174.72 | 0.80% | 2,980.38 | 1.31% | |
| 31. | 31 ago 2019 | $161.72 | $1.44 | -6.62% | 2,926.46 | -1.81% |
| 32. | 30 sept 2019 | $164.40 | 1.66% | 2,976.74 | 1.72% | |
| 33. | 31 oct 2019 | $164.99 | 0.36% | 3,037.56 | 2.04% | |
| 34. | 30 nov 2019 | $169.77 | $1.44 | 3.77% | 3,140.98 | 3.40% |
| 35. | 31 dic 2019 | $176.42 | 3.92% | 3,230.78 | 2.86% | |
| 36. | 31 ene 2020 | $158.66 | -10.07% | 3,225.52 | -0.16% | |
| 37. | 29 feb 2020 | $149.24 | $1.47 | -5.01% | 2,954.22 | -8.41% |
| 38. | 31 mar 2020 | $136.51 | -8.53% | 2,584.59 | -12.51% | |
| 39. | 30 abr 2020 | $151.92 | 11.29% | 2,912.43 | 12.68% | |
| 40. | 31 may 2020 | $156.44 | $1.47 | 3.94% | 3,044.31 | 4.53% |
| 41. | 30 jun 2020 | $155.99 | -0.29% | 3,100.29 | 1.84% | |
| 42. | 31 jul 2020 | $150.47 | -3.54% | 3,271.12 | 5.51% | |
| 43. | 31 ago 2020 | $163.02 | $1.47 | 9.32% | 3,500.31 | 7.01% |
| 44. | 30 sept 2020 | $160.18 | -1.74% | 3,363.00 | -3.92% | |
| 45. | 31 oct 2020 | $159.96 | -0.14% | 3,269.96 | -2.77% | |
| 46. | 30 nov 2020 | $172.73 | $1.47 | 8.90% | 3,621.63 | 10.75% |
| 47. | 31 dic 2020 | $174.79 | 1.19% | 3,756.07 | 3.71% | |
| 48. | 31 ene 2021 | $175.66 | 0.50% | 3,714.24 | -1.11% | |
| 49. | 28 feb 2021 | $175.06 | $1.48 | 0.50% | 3,811.15 | 2.61% |
| 50. | 31 mar 2021 | $192.68 | 10.07% | 3,972.89 | 4.24% | |
| 51. | 30 abr 2021 | $197.14 | 2.31% | 4,181.17 | 5.24% | |
| 52. | 31 may 2021 | $203.04 | $1.48 | 3.74% | 4,204.11 | 0.55% |
| 53. | 30 jun 2021 | $198.63 | -2.17% | 4,297.50 | 2.22% | |
| 54. | 31 jul 2021 | $197.94 | -0.35% | 4,395.26 | 2.27% | |
| 55. | 31 ago 2021 | $194.74 | $1.48 | -0.87% | 4,522.68 | 2.90% |
| 56. | 30 sept 2021 | $175.42 | -9.92% | 4,307.54 | -4.76% | |
| 57. | 31 oct 2021 | $178.68 | 1.86% | 4,605.38 | 6.91% | |
| 58. | 30 nov 2021 | $170.04 | $1.48 | -4.01% | 4,567.00 | -0.83% |
| 59. | 31 dic 2021 | $177.63 | 4.46% | 4,766.18 | 4.36% | |
| Promedio (R): | 0.46% | 1.36% | ||||
| Desviación estándar: | 6.03% | 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 MMM 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
3M Co., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RMMM,t | RS&P 500,t | (RMMM,t–RMMM)2 | (RS&P 500,t–RS&P 500)2 | (RMMM,t–RMMM)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb 2017 | 7.27% | 3.72% | 46.30 | 5.58 | 16.07 |
| 2. | 31 mar 2017 | 2.67% | -0.04% | 4.88 | 1.95 | -3.09 |
| 3. | 30 abr 2017 | 2.35% | 0.91% | 3.57 | 0.20 | -0.85 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov 2021 | -4.01% | -0.83% | 19.98 | 4.80 | 9.80 |
| 59. | 31 dic 2021 | 4.46% | 4.36% | 16.01 | 9.02 | 12.02 |
| Total (Σ): | 2,109.95 | 1,164.17 | 1,113.30 | |||
| t | Fecha | RMMM,t | RS&P 500,t | (RMMM,t–RMMM)2 | (RS&P 500,t–RS&P 500)2 | (RMMM,t–RMMM)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb 2017 | 7.27% | 3.72% | 46.30 | 5.58 | 16.07 |
| 2. | 31 mar 2017 | 2.67% | -0.04% | 4.88 | 1.95 | -3.09 |
| 3. | 30 abr 2017 | 2.35% | 0.91% | 3.57 | 0.20 | -0.85 |
| 4. | 31 may 2017 | 5.01% | 1.16% | 20.69 | 0.04 | -0.91 |
| 5. | 30 jun 2017 | 1.82% | 0.48% | 1.84 | 0.77 | -1.19 |
| 6. | 31 jul 2017 | -3.37% | 1.93% | 14.71 | 0.33 | -2.21 |
| 7. | 31 ago 2017 | 2.15% | 0.05% | 2.85 | 1.70 | -2.20 |
| 8. | 30 sept 2017 | 2.73% | 1.93% | 5.14 | 0.33 | 1.30 |
| 9. | 31 oct 2017 | 9.67% | 2.22% | 84.70 | 0.74 | 7.92 |
| 10. | 30 nov 2017 | 6.14% | 2.81% | 32.19 | 2.10 | 8.23 |
| 11. | 31 dic 2017 | -3.20% | 0.98% | 13.39 | 0.14 | 1.37 |
| 12. | 31 ene 2018 | 6.43% | 5.62% | 35.58 | 18.15 | 25.41 |
| 13. | 28 feb 2018 | -5.44% | -3.89% | 34.86 | 27.59 | 31.01 |
| 14. | 31 mar 2018 | -6.79% | -2.69% | 52.60 | 16.37 | 29.35 |
| 15. | 30 abr 2018 | -11.45% | 0.27% | 141.86 | 1.18 | 12.94 |
| 16. | 31 may 2018 | 2.16% | 2.16% | 2.88 | 0.64 | 1.36 |
| 17. | 30 jun 2018 | -0.26% | 0.48% | 0.52 | 0.76 | 0.63 |
| 18. | 31 jul 2018 | 7.93% | 3.60% | 55.76 | 5.04 | 16.76 |
| 19. | 31 ago 2018 | -0.02% | 3.03% | 0.23 | 2.78 | -0.80 |
| 20. | 30 sept 2018 | -0.10% | 0.43% | 0.32 | 0.86 | 0.52 |
| 21. | 31 oct 2018 | -9.71% | -6.94% | 103.39 | 68.86 | 84.38 |
| 22. | 30 nov 2018 | 10.00% | 1.79% | 90.89 | 0.18 | 4.08 |
| 23. | 31 dic 2018 | -8.36% | -9.18% | 77.83 | 111.00 | 92.94 |
| 24. | 31 ene 2019 | 5.12% | 7.87% | 21.71 | 42.39 | 30.33 |
| 25. | 28 feb 2019 | 4.26% | 2.97% | 14.41 | 2.61 | 6.13 |
| 26. | 31 mar 2019 | 0.19% | 1.79% | 0.08 | 0.19 | -0.12 |
| 27. | 30 abr 2019 | -8.79% | 3.93% | 85.67 | 6.62 | -23.82 |
| 28. | 31 may 2019 | -14.94% | -6.58% | 237.37 | 62.97 | 122.26 |
| 29. | 30 jun 2019 | 8.51% | 6.89% | 64.71 | 30.64 | 44.52 |
| 30. | 31 jul 2019 | 0.80% | 1.31% | 0.11 | 0.00 | -0.02 |
| 31. | 31 ago 2019 | -6.62% | -1.81% | 50.12 | 10.03 | 22.42 |
| 32. | 30 sept 2019 | 1.66% | 1.72% | 1.43 | 0.13 | 0.43 |
| 33. | 31 oct 2019 | 0.36% | 2.04% | 0.01 | 0.47 | -0.07 |
| 34. | 30 nov 2019 | 3.77% | 3.40% | 10.94 | 4.19 | 6.77 |
| 35. | 31 dic 2019 | 3.92% | 2.86% | 11.93 | 2.25 | 5.18 |
| 36. | 31 ene 2020 | -10.07% | -0.16% | 110.88 | 2.31 | 16.01 |
| 37. | 29 feb 2020 | -5.01% | -8.41% | 29.96 | 95.43 | 53.47 |
| 38. | 31 mar 2020 | -8.53% | -12.51% | 80.87 | 192.37 | 124.73 |
| 39. | 30 abr 2020 | 11.29% | 12.68% | 117.19 | 128.29 | 122.62 |
| 40. | 31 may 2020 | 3.94% | 4.53% | 12.11 | 10.05 | 11.03 |
| 41. | 30 jun 2020 | -0.29% | 1.84% | 0.56 | 0.23 | -0.36 |
| 42. | 31 jul 2020 | -3.54% | 5.51% | 16.01 | 17.24 | -16.62 |
| 43. | 31 ago 2020 | 9.32% | 7.01% | 78.40 | 31.91 | 50.02 |
| 44. | 30 sept 2020 | -1.74% | -3.92% | 4.86 | 27.89 | 11.64 |
| 45. | 31 oct 2020 | -0.14% | -2.77% | 0.36 | 17.01 | 2.48 |
| 46. | 30 nov 2020 | 8.90% | 10.75% | 71.22 | 88.30 | 79.30 |
| 47. | 31 dic 2020 | 1.19% | 3.71% | 0.53 | 5.54 | 1.72 |
| 48. | 31 ene 2021 | 0.50% | -1.11% | 0.00 | 6.11 | -0.09 |
| 49. | 28 feb 2021 | 0.50% | 2.61% | 0.00 | 1.57 | 0.05 |
| 50. | 31 mar 2021 | 10.07% | 4.24% | 92.20 | 8.33 | 27.71 |
| 51. | 30 abr 2021 | 2.31% | 5.24% | 3.43 | 15.09 | 7.19 |
| 52. | 31 may 2021 | 3.74% | 0.55% | 10.76 | 0.65 | -2.65 |
| 53. | 30 jun 2021 | -2.17% | 2.22% | 6.94 | 0.75 | -2.28 |
| 54. | 31 jul 2021 | -0.35% | 2.27% | 0.66 | 0.84 | -0.74 |
| 55. | 31 ago 2021 | -0.87% | 2.90% | 1.77 | 2.38 | -2.05 |
| 56. | 30 sept 2021 | -9.92% | -4.76% | 107.83 | 37.39 | 63.50 |
| 57. | 31 oct 2021 | 1.86% | 6.91% | 1.95 | 30.87 | 7.75 |
| 58. | 30 nov 2021 | -4.01% | -0.83% | 19.98 | 4.80 | 9.80 |
| 59. | 31 dic 2021 | 4.46% | 4.36% | 16.01 | 9.02 | 12.02 |
| Total (Σ): | 2,109.95 | 1,164.17 | 1,113.30 | |||
Mostrar todo
VarianzaMMM = Σ(RMMM,t–RMMM)2 ÷ (59 – 1)
= 2,109.95 ÷ (59 – 1)
= 36.38
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,164.17 ÷ (59 – 1)
= 20.07
CovarianzaMMM, S&P 500 = Σ(RMMM,t–RMMM)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,113.30 ÷ (59 – 1)
= 19.19
Estimación sistemática del riesgo (β)
| VarianzaMMM | 36.38 |
| VarianzaS&P 500 | 20.07 |
| CovarianzaMMM, S&P 500 | 19.19 |
| Coeficiente de correlaciónMMM, S&P 5001 | 0.71 |
| βMMM2 | 0.96 |
| αMMM3 | -0.84% |
Cálculos
1 Coeficiente de correlaciónMMM, S&P 500
= CovarianzaMMM, S&P 500 ÷ (Desviación estándarMMM × Desviación estándarS&P 500)
= 19.19 ÷ (6.03% × 4.48%)
= 0.71
2 βMMM
= CovarianzaMMM, S&P 500 ÷ VarianzaS&P 500
= 19.19 ÷ 20.07
= 0.96
3 αMMM
= PromedioMMM – βMMM × PromedioS&P 500
= 0.46% – 0.96 × 1.36%
= -0.84%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.85% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 14.71% |
| Riesgo sistemático de 3M acciones ordinarias | βMMM | 0.96 |
| Tasa de rendimiento esperada de las acciones ordinarias de 3M3 | E(RMMM) | 14.28% |
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(RMMM) = RF + βMMM [E(RM) – RF]
= 4.85% + 0.96 [14.71% – 4.85%]
= 14.28%