Modelo de fijación de precios de activos de capital (CAPM)
El modelo de fijación de precios de activos de capital (CAPM) indica cuál debería ser la tasa de rendimiento esperada o requerida de los 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 | $177.63 | 4.46% | 4,766.18 | 4.36% | |
| Promedio (R): | 0.46% | 1.36% | ||||
| Desviación estándar: | 6.03% | 4.48% | ||||
Mostrar todo
1 Datos en US$ por acción de acciones ordinarias, ajustados por escisiones 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 varianza y covarianza de 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 de LT Treasury Composite1 | RF | 3.62% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 12.55% |
| Riesgo sistemático de acciones ordinarias 3M | βMMM | 0.96 |
| Tasa de rendimiento esperada de las acciones ordinarias de 3M3 | E(RMMM) | 12.16% |
1 Promedio no ponderado de los rendimientos de las ofertas en todos los bonos del Tesoro de los Estados Unidos con cupón fijo pendientes, ni vencidos ni exigibles en menos de 10 años (proxy de tasa de rendimiento libre de riesgo).
3 E(RMMM) = RF + βMMM [E(RM) – RF]
= 3.62% + 0.96 [12.55% – 3.62%]
= 12.16%