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 | 4.67% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.79% |
| Riesgo sistemático de 3M acciones ordinarias | βMMM | 0.96 |
| Tasa de rendimiento esperada de las acciones ordinarias de 3M3 | E(RMMM) | 13.39% |
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]
= 4.67% + 0.96 [13.79% – 4.67%]
= 13.39%