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 Time Warner.
Tasas de retorno
Time Warner Inc., tasas mensuales de retorno
| Time Warner Inc. (TWX) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioTWX,t1 | DividendoTWX,t1 | RTWX,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2013 | $50.52 | 1,498.11 | ||||
| 1. | 28 feb. 2013 | $53.17 | $0.2875 | 5.81% | 1,514.68 | 1.11% |
| 2. | 31 mar. 2013 | $57.62 | 8.37% | 1,569.19 | 3.60% | |
| 3. | 30 abr. 2013 | $59.78 | 3.75% | 1,597.57 | 1.81% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2017 | $91.51 | -6.90% | 2,647.58 | 2.81% | |
| 59. | 31 dic. 2017 | $91.47 | -0.04% | 2,673.61 | 0.98% | |
| Promedio (R): | 1.35% | 1.02% | ||||
| Desviación estándar: | 6.00% | 2.71% | ||||
| Time Warner Inc. (TWX) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioTWX,t1 | DividendoTWX,t1 | RTWX,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2013 | $50.52 | 1,498.11 | ||||
| 1. | 28 feb. 2013 | $53.17 | $0.2875 | 5.81% | 1,514.68 | 1.11% |
| 2. | 31 mar. 2013 | $57.62 | 8.37% | 1,569.19 | 3.60% | |
| 3. | 30 abr. 2013 | $59.78 | 3.75% | 1,597.57 | 1.81% | |
| 4. | 31 may. 2013 | $58.37 | $0.2875 | -1.88% | 1,630.74 | 2.08% |
| 5. | 30 jun. 2013 | $57.82 | -0.94% | 1,606.28 | -1.50% | |
| 6. | 31 jul. 2013 | $62.26 | 7.68% | 1,685.73 | 4.95% | |
| 7. | 31 ago. 2013 | $60.53 | $0.2875 | -2.32% | 1,632.97 | -3.13% |
| 8. | 30 sept. 2013 | $65.81 | 8.72% | 1,681.55 | 2.97% | |
| 9. | 31 oct. 2013 | $68.74 | 4.45% | 1,756.54 | 4.46% | |
| 10. | 30 nov. 2013 | $65.71 | $0.2875 | -3.99% | 1,805.81 | 2.80% |
| 11. | 31 dic. 2013 | $69.72 | 6.10% | 1,848.36 | 2.36% | |
| 12. | 31 ene. 2014 | $62.83 | -9.88% | 1,782.59 | -3.56% | |
| 13. | 28 feb. 2014 | $67.13 | $0.3175 | 7.35% | 1,859.45 | 4.31% |
| 14. | 31 mar. 2014 | $65.33 | -2.68% | 1,872.34 | 0.69% | |
| 15. | 30 abr. 2014 | $66.46 | 1.73% | 1,883.95 | 0.62% | |
| 16. | 31 may. 2014 | $69.83 | $0.3175 | 5.55% | 1,923.57 | 2.10% |
| 17. | 30 jun. 2014 | $70.25 | 0.60% | 1,960.23 | 1.91% | |
| 18. | 31 jul. 2014 | $83.02 | 18.18% | 1,930.67 | -1.51% | |
| 19. | 31 ago. 2014 | $77.03 | $0.3175 | -6.83% | 2,003.37 | 3.77% |
| 20. | 30 sept. 2014 | $75.21 | -2.36% | 1,972.29 | -1.55% | |
| 21. | 31 oct. 2014 | $79.47 | 5.66% | 2,018.05 | 2.32% | |
| 22. | 30 nov. 2014 | $85.12 | $0.3175 | 7.51% | 2,067.56 | 2.45% |
| 23. | 31 dic. 2014 | $85.42 | 0.35% | 2,058.90 | -0.42% | |
| 24. | 31 ene. 2015 | $77.93 | -8.77% | 1,994.99 | -3.10% | |
| 25. | 28 feb. 2015 | $81.86 | $0.35 | 5.49% | 2,104.50 | 5.49% |
| 26. | 31 mar. 2015 | $84.44 | 3.15% | 2,067.89 | -1.74% | |
| 27. | 30 abr. 2015 | $84.41 | -0.04% | 2,085.51 | 0.85% | |
| 28. | 31 may. 2015 | $84.48 | $0.35 | 0.50% | 2,107.39 | 1.05% |
| 29. | 30 jun. 2015 | $87.41 | 3.47% | 2,063.11 | -2.10% | |
| 30. | 31 jul. 2015 | $88.04 | 0.72% | 2,103.84 | 1.97% | |
| 31. | 31 ago. 2015 | $71.10 | $0.35 | -18.84% | 1,972.18 | -6.26% |
| 32. | 30 sept. 2015 | $68.75 | -3.31% | 1,920.03 | -2.64% | |
| 33. | 31 oct. 2015 | $75.34 | 9.59% | 2,079.36 | 8.30% | |
| 34. | 30 nov. 2015 | $69.98 | $0.35 | -6.65% | 2,080.41 | 0.05% |
| 35. | 31 dic. 2015 | $64.67 | -7.59% | 2,043.94 | -1.75% | |
| 36. | 31 ene. 2016 | $70.44 | 8.92% | 1,940.24 | -5.07% | |
| 37. | 29 feb. 2016 | $66.20 | $0.4025 | -5.45% | 1,932.23 | -0.41% |
| 38. | 31 mar. 2016 | $72.55 | 9.59% | 2,059.74 | 6.60% | |
| 39. | 30 abr. 2016 | $75.14 | 3.57% | 2,065.30 | 0.27% | |
| 40. | 31 may. 2016 | $75.66 | $0.4025 | 1.23% | 2,096.95 | 1.53% |
| 41. | 30 jun. 2016 | $73.54 | -2.80% | 2,098.86 | 0.09% | |
| 42. | 31 jul. 2016 | $76.65 | 4.23% | 2,173.60 | 3.56% | |
| 43. | 31 ago. 2016 | $78.41 | $0.4025 | 2.82% | 2,170.95 | -0.12% |
| 44. | 30 sept. 2016 | $79.61 | 1.53% | 2,168.27 | -0.12% | |
| 45. | 31 oct. 2016 | $88.99 | 11.78% | 2,126.15 | -1.94% | |
| 46. | 30 nov. 2016 | $91.82 | $0.4025 | 3.63% | 2,198.81 | 3.42% |
| 47. | 31 dic. 2016 | $96.53 | 5.13% | 2,238.83 | 1.82% | |
| 48. | 31 ene. 2017 | $96.85 | 0.33% | 2,278.87 | 1.79% | |
| 49. | 28 feb. 2017 | $98.21 | $0.4025 | 1.82% | 2,363.64 | 3.72% |
| 50. | 31 mar. 2017 | $97.71 | -0.51% | 2,362.72 | -0.04% | |
| 51. | 30 abr. 2017 | $99.27 | 1.60% | 2,384.20 | 0.91% | |
| 52. | 31 may. 2017 | $99.49 | $0.4025 | 0.63% | 2,411.80 | 1.16% |
| 53. | 30 jun. 2017 | $100.41 | 0.92% | 2,423.41 | 0.48% | |
| 54. | 31 jul. 2017 | $102.42 | $0.4025 | 2.40% | 2,470.30 | 1.93% |
| 55. | 31 ago. 2017 | $101.10 | -1.29% | 2,471.65 | 0.05% | |
| 56. | 30 sept. 2017 | $102.45 | 1.34% | 2,519.36 | 1.93% | |
| 57. | 31 oct. 2017 | $98.29 | $0.4025 | -3.67% | 2,575.26 | 2.22% |
| 58. | 30 nov. 2017 | $91.51 | -6.90% | 2,647.58 | 2.81% | |
| 59. | 31 dic. 2017 | $91.47 | -0.04% | 2,673.61 | 0.98% | |
| Promedio (R): | 1.35% | 1.02% | ||||
| Desviación estándar: | 6.00% | 2.71% | ||||
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 TWX 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
Time Warner Inc., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RTWX,t | RS&P 500,t | (RTWX,t–RTWX)2 | (RS&P 500,t–RS&P 500)2 | (RTWX,t–RTWX)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2013 | 5.81% | 1.11% | 19.96 | 0.01 | 0.37 |
| 2. | 31 mar. 2013 | 8.37% | 3.60% | 49.31 | 6.64 | 18.09 |
| 3. | 30 abr. 2013 | 3.75% | 1.81% | 5.77 | 0.62 | 1.89 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2017 | -6.90% | 2.81% | 67.98 | 3.19 | -14.73 |
| 59. | 31 dic. 2017 | -0.04% | 0.98% | 1.93 | 0.00 | 0.05 |
| Total (Σ): | 2,090.08 | 424.45 | 406.20 | |||
| t | Fecha | RTWX,t | RS&P 500,t | (RTWX,t–RTWX)2 | (RS&P 500,t–RS&P 500)2 | (RTWX,t–RTWX)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2013 | 5.81% | 1.11% | 19.96 | 0.01 | 0.37 |
| 2. | 31 mar. 2013 | 8.37% | 3.60% | 49.31 | 6.64 | 18.09 |
| 3. | 30 abr. 2013 | 3.75% | 1.81% | 5.77 | 0.62 | 1.89 |
| 4. | 31 may. 2013 | -1.88% | 2.08% | 10.40 | 1.11 | -3.40 |
| 5. | 30 jun. 2013 | -0.94% | -1.50% | 5.24 | 6.36 | 5.77 |
| 6. | 31 jul. 2013 | 7.68% | 4.95% | 40.09 | 15.40 | 24.85 |
| 7. | 31 ago. 2013 | -2.32% | -3.13% | 13.42 | 17.24 | 15.21 |
| 8. | 30 sept. 2013 | 8.72% | 2.97% | 54.40 | 3.81 | 14.40 |
| 9. | 31 oct. 2013 | 4.45% | 4.46% | 9.64 | 11.81 | 10.67 |
| 10. | 30 nov. 2013 | -3.99% | 2.80% | 28.48 | 3.18 | -9.51 |
| 11. | 31 dic. 2013 | 6.10% | 2.36% | 22.61 | 1.78 | 6.34 |
| 12. | 31 ene. 2014 | -9.88% | -3.56% | 126.10 | 20.98 | 51.44 |
| 13. | 28 feb. 2014 | 7.35% | 4.31% | 36.03 | 10.82 | 19.74 |
| 14. | 31 mar. 2014 | -2.68% | 0.69% | 16.23 | 0.11 | 1.33 |
| 15. | 30 abr. 2014 | 1.73% | 0.62% | 0.15 | 0.16 | -0.15 |
| 16. | 31 may. 2014 | 5.55% | 2.10% | 17.65 | 1.17 | 4.54 |
| 17. | 30 jun. 2014 | 0.60% | 1.91% | 0.56 | 0.78 | -0.66 |
| 18. | 31 jul. 2014 | 18.18% | -1.51% | 283.28 | 6.40 | -42.59 |
| 19. | 31 ago. 2014 | -6.83% | 3.77% | 66.91 | 7.53 | -22.44 |
| 20. | 30 sept. 2014 | -2.36% | -1.55% | 13.76 | 6.62 | 9.55 |
| 21. | 31 oct. 2014 | 5.66% | 2.32% | 18.64 | 1.68 | 5.60 |
| 22. | 30 nov. 2014 | 7.51% | 2.45% | 37.97 | 2.05 | 8.82 |
| 23. | 31 dic. 2014 | 0.35% | -0.42% | 0.99 | 2.08 | 1.43 |
| 24. | 31 ene. 2015 | -8.77% | -3.10% | 102.32 | 17.03 | 41.74 |
| 25. | 28 feb. 2015 | 5.49% | 5.49% | 17.18 | 19.95 | 18.52 |
| 26. | 31 mar. 2015 | 3.15% | -1.74% | 3.26 | 7.63 | -4.98 |
| 27. | 30 abr. 2015 | -0.04% | 0.85% | 1.91 | 0.03 | 0.24 |
| 28. | 31 may. 2015 | 0.50% | 1.05% | 0.72 | 0.00 | -0.02 |
| 29. | 30 jun. 2015 | 3.47% | -2.10% | 4.50 | 9.76 | -6.63 |
| 30. | 31 jul. 2015 | 0.72% | 1.97% | 0.39 | 0.91 | -0.60 |
| 31. | 31 ago. 2015 | -18.84% | -6.26% | 407.67 | 53.00 | 147.00 |
| 32. | 30 sept. 2015 | -3.31% | -2.64% | 21.64 | 13.44 | 17.06 |
| 33. | 31 oct. 2015 | 9.59% | 8.30% | 67.87 | 52.94 | 59.94 |
| 34. | 30 nov. 2015 | -6.65% | 0.05% | 63.95 | 0.94 | 7.77 |
| 35. | 31 dic. 2015 | -7.59% | -1.75% | 79.83 | 7.70 | 24.80 |
| 36. | 31 ene. 2016 | 8.92% | -5.07% | 57.38 | 37.16 | -46.18 |
| 37. | 29 feb. 2016 | -5.45% | -0.41% | 46.17 | 2.06 | 9.75 |
| 38. | 31 mar. 2016 | 9.59% | 6.60% | 67.98 | 31.10 | 45.98 |
| 39. | 30 abr. 2016 | 3.57% | 0.27% | 4.94 | 0.57 | -1.67 |
| 40. | 31 may. 2016 | 1.23% | 1.53% | 0.01 | 0.26 | -0.06 |
| 41. | 30 jun. 2016 | -2.80% | 0.09% | 17.22 | 0.87 | 3.86 |
| 42. | 31 jul. 2016 | 4.23% | 3.56% | 8.31 | 6.44 | 7.32 |
| 43. | 31 ago. 2016 | 2.82% | -0.12% | 2.17 | 1.31 | -1.69 |
| 44. | 30 sept. 2016 | 1.53% | -0.12% | 0.03 | 1.31 | -0.21 |
| 45. | 31 oct. 2016 | 11.78% | -1.94% | 108.90 | 8.79 | -30.94 |
| 46. | 30 nov. 2016 | 3.63% | 3.42% | 5.22 | 5.74 | 5.47 |
| 47. | 31 dic. 2016 | 5.13% | 1.82% | 14.31 | 0.64 | 3.02 |
| 48. | 31 ene. 2017 | 0.33% | 1.79% | 1.03 | 0.59 | -0.78 |
| 49. | 28 feb. 2017 | 1.82% | 3.72% | 0.22 | 7.28 | 1.28 |
| 50. | 31 mar. 2017 | -0.51% | -0.04% | 3.45 | 1.13 | 1.97 |
| 51. | 30 abr. 2017 | 1.60% | 0.91% | 0.06 | 0.01 | -0.03 |
| 52. | 31 may. 2017 | 0.63% | 1.16% | 0.52 | 0.02 | -0.10 |
| 53. | 30 jun. 2017 | 0.92% | 0.48% | 0.18 | 0.29 | 0.23 |
| 54. | 31 jul. 2017 | 2.40% | 1.93% | 1.11 | 0.83 | 0.96 |
| 55. | 31 ago. 2017 | -1.29% | 0.05% | 6.95 | 0.94 | 2.55 |
| 56. | 30 sept. 2017 | 1.34% | 1.93% | 0.00 | 0.82 | -0.01 |
| 57. | 31 oct. 2017 | -3.67% | 2.22% | 25.15 | 1.43 | -6.00 |
| 58. | 30 nov. 2017 | -6.90% | 2.81% | 67.98 | 3.19 | -14.73 |
| 59. | 31 dic. 2017 | -0.04% | 0.98% | 1.93 | 0.00 | 0.05 |
| Total (Σ): | 2,090.08 | 424.45 | 406.20 | |||
Mostrar todo
VarianzaTWX = Σ(RTWX,t–RTWX)2 ÷ (59 – 1)
= 2,090.08 ÷ (59 – 1)
= 36.04
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 424.45 ÷ (59 – 1)
= 7.32
CovarianzaTWX, S&P 500 = Σ(RTWX,t–RTWX)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 406.20 ÷ (59 – 1)
= 7.00
Estimación sistemática del riesgo (β)
| VarianzaTWX | 36.04 |
| VarianzaS&P 500 | 7.32 |
| CovarianzaTWX, S&P 500 | 7.00 |
| Coeficiente de correlaciónTWX, S&P 5001 | 0.43 |
| βTWX2 | 0.96 |
| αTWX3 | 0.37% |
Cálculos
1 Coeficiente de correlaciónTWX, S&P 500
= CovarianzaTWX, S&P 500 ÷ (Desviación estándarTWX × Desviación estándarS&P 500)
= 7.00 ÷ (6.00% × 2.71%)
= 0.43
2 βTWX
= CovarianzaTWX, S&P 500 ÷ VarianzaS&P 500
= 7.00 ÷ 7.32
= 0.96
3 αTWX
= PromedioTWX – βTWX × PromedioS&P 500
= 1.35% – 0.96 × 1.02%
= 0.37%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.69% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.79% |
| Riesgo sistemático de Time Warner acciones ordinarias | βTWX | 0.96 |
| Tasa de rendimiento esperada de las acciones ordinarias de Time Warner3 | E(RTWX) | 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(RTWX) = RF + βTWX [E(RM) – RF]
= 4.69% + 0.96 [13.79% – 4.69%]
= 13.39%