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 Apple.
Tasas de retorno
Apple Inc. (AAPL) | Standard & Poor’s 500 (S&P 500) | |||||
---|---|---|---|---|---|---|
t | Fecha | PrecioAAPL,t1 | DividendoAAPL,t1 | RAAPL,t2 | PrecioS&P 500,t | RS&P 500,t3 |
31 oct. 2015 | $29.88 | 2,079.36 | ||||
1. | 30 nov. 2015 | $29.58 | $0.13 | -0.57% | 2,080.41 | 0.05% |
2. | 31 dic. 2015 | $26.32 | -11.02% | 2,043.94 | -1.75% | |
3. | 31 ene. 2016 | $24.34 | -7.52% | 1,940.24 | -5.07% | |
. | . | . | . | . | . | . |
. | . | . | . | . | . | . |
. | . | . | . | . | . | . |
70. | 31 ago. 2021 | $151.83 | $0.22 | 4.24% | 4,522.68 | 2.90% |
71. | 30 sept. 2021 | $141.50 | -6.80% | 4,307.54 | -4.76% | |
Promedio (R): | 2.69% | 1.12% | ||||
Desviación estándar: | 8.49% | 4.19% |
Apple Inc. (AAPL) | Standard & Poor’s 500 (S&P 500) | |||||
---|---|---|---|---|---|---|
t | Fecha | PrecioAAPL,t1 | DividendoAAPL,t1 | RAAPL,t2 | PrecioS&P 500,t | RS&P 500,t3 |
31 oct. 2015 | $29.88 | 2,079.36 | ||||
1. | 30 nov. 2015 | $29.58 | $0.13 | -0.57% | 2,080.41 | 0.05% |
2. | 31 dic. 2015 | $26.32 | -11.02% | 2,043.94 | -1.75% | |
3. | 31 ene. 2016 | $24.34 | -7.52% | 1,940.24 | -5.07% | |
4. | 29 feb. 2016 | $24.17 | $0.13 | -0.16% | 1,932.23 | -0.41% |
5. | 31 mar. 2016 | $27.25 | 12.74% | 2,059.74 | 6.60% | |
6. | 30 abr. 2016 | $23.44 | -13.98% | 2,065.30 | 0.27% | |
7. | 31 may. 2016 | $24.97 | $0.1425 | 7.14% | 2,096.95 | 1.53% |
8. | 30 jun. 2016 | $23.90 | -4.29% | 2,098.86 | 0.09% | |
9. | 31 jul. 2016 | $26.05 | 9.00% | 2,173.60 | 3.56% | |
10. | 31 ago. 2016 | $26.53 | $0.1425 | 2.39% | 2,170.95 | -0.12% |
11. | 30 sept. 2016 | $28.26 | 6.52% | 2,168.27 | -0.12% | |
12. | 31 oct. 2016 | $28.39 | 0.46% | 2,126.15 | -1.94% | |
13. | 30 nov. 2016 | $27.63 | $0.1425 | -2.18% | 2,198.81 | 3.42% |
14. | 31 dic. 2016 | $28.96 | 4.81% | 2,238.83 | 1.82% | |
15. | 31 ene. 2017 | $30.34 | 4.77% | 2,278.87 | 1.79% | |
16. | 28 feb. 2017 | $34.25 | $0.1425 | 13.36% | 2,363.64 | 3.72% |
17. | 31 mar. 2017 | $35.92 | 4.88% | 2,362.72 | -0.04% | |
18. | 30 abr. 2017 | $35.91 | -0.03% | 2,384.20 | 0.91% | |
19. | 31 may. 2017 | $38.19 | $0.1575 | 6.79% | 2,411.80 | 1.16% |
20. | 30 jun. 2017 | $36.01 | -5.71% | 2,423.41 | 0.48% | |
21. | 31 jul. 2017 | $37.18 | 3.25% | 2,470.30 | 1.93% | |
22. | 31 ago. 2017 | $41.00 | $0.1575 | 10.70% | 2,471.65 | 0.05% |
23. | 30 sept. 2017 | $38.53 | -6.02% | 2,519.36 | 1.93% | |
24. | 31 oct. 2017 | $42.26 | 9.68% | 2,575.26 | 2.22% | |
25. | 30 nov. 2017 | $42.96 | $0.1575 | 2.03% | 2,647.58 | 2.81% |
26. | 31 dic. 2017 | $42.31 | -1.51% | 2,673.61 | 0.98% | |
27. | 31 ene. 2018 | $41.86 | -1.06% | 2,823.81 | 5.62% | |
28. | 28 feb. 2018 | $44.53 | $0.1575 | 6.75% | 2,713.83 | -3.89% |
29. | 31 mar. 2018 | $41.95 | -5.79% | 2,640.87 | -2.69% | |
30. | 30 abr. 2018 | $41.32 | -1.50% | 2,648.05 | 0.27% | |
31. | 31 may. 2018 | $46.72 | $0.1825 | 13.51% | 2,705.27 | 2.16% |
32. | 30 jun. 2018 | $46.28 | -0.94% | 2,718.37 | 0.48% | |
33. | 31 jul. 2018 | $47.57 | 2.79% | 2,816.29 | 3.60% | |
34. | 31 ago. 2018 | $56.91 | $0.1825 | 20.02% | 2,901.52 | 3.03% |
35. | 30 sept. 2018 | $56.44 | -0.83% | 2,913.98 | 0.43% | |
36. | 31 oct. 2018 | $54.72 | -3.05% | 2,711.74 | -6.94% | |
37. | 30 nov. 2018 | $44.65 | $0.1825 | -18.07% | 2,760.17 | 1.79% |
38. | 31 dic. 2018 | $39.44 | -11.67% | 2,506.85 | -9.18% | |
39. | 31 ene. 2019 | $41.61 | 5.50% | 2,704.10 | 7.87% | |
40. | 28 feb. 2019 | $43.29 | $0.1825 | 4.48% | 2,784.49 | 2.97% |
41. | 31 mar. 2019 | $47.49 | 9.70% | 2,834.40 | 1.79% | |
42. | 30 abr. 2019 | $50.17 | 5.64% | 2,945.83 | 3.93% | |
43. | 31 may. 2019 | $43.77 | $0.1925 | -12.37% | 2,752.06 | -6.58% |
44. | 30 jun. 2019 | $49.48 | 13.05% | 2,941.76 | 6.89% | |
45. | 31 jul. 2019 | $53.26 | 7.64% | 2,980.38 | 1.31% | |
46. | 31 ago. 2019 | $52.19 | $0.1925 | -1.65% | 2,926.46 | -1.81% |
47. | 30 sept. 2019 | $55.99 | 7.28% | 2,976.74 | 1.72% | |
48. | 31 oct. 2019 | $62.19 | 11.07% | 3,037.56 | 2.04% | |
49. | 30 nov. 2019 | $66.81 | $0.1925 | 7.74% | 3,140.98 | 3.40% |
50. | 31 dic. 2019 | $73.41 | 9.88% | 3,230.78 | 2.86% | |
51. | 31 ene. 2020 | $77.38 | 5.41% | 3,225.52 | -0.16% | |
52. | 29 feb. 2020 | $68.34 | $0.1925 | -11.43% | 2,954.22 | -8.41% |
53. | 31 mar. 2020 | $63.57 | -6.98% | 2,584.59 | -12.51% | |
54. | 30 abr. 2020 | $73.45 | 15.54% | 2,912.43 | 12.68% | |
55. | 31 may. 2020 | $79.49 | $0.205 | 8.50% | 3,044.31 | 4.53% |
56. | 30 jun. 2020 | $91.20 | 14.73% | 3,100.29 | 1.84% | |
57. | 31 jul. 2020 | $106.26 | 16.51% | 3,271.12 | 5.51% | |
58. | 31 ago. 2020 | $129.04 | $0.205 | 21.63% | 3,500.31 | 7.01% |
59. | 30 sept. 2020 | $115.81 | -10.25% | 3,363.00 | -3.92% | |
60. | 31 oct. 2020 | $108.86 | -6.00% | 3,269.96 | -2.77% | |
61. | 30 nov. 2020 | $119.05 | $0.205 | 9.55% | 3,621.63 | 10.75% |
62. | 31 dic. 2020 | $132.69 | 11.46% | 3,756.07 | 3.71% | |
63. | 31 ene. 2021 | $131.96 | -0.55% | 3,714.24 | -1.11% | |
64. | 28 feb. 2021 | $121.26 | $0.205 | -7.95% | 3,811.15 | 2.61% |
65. | 31 mar. 2021 | $122.15 | 0.73% | 3,972.89 | 4.24% | |
66. | 30 abr. 2021 | $131.46 | 7.62% | 4,181.17 | 5.24% | |
67. | 31 may. 2021 | $124.61 | $0.22 | -5.04% | 4,204.11 | 0.55% |
68. | 30 jun. 2021 | $136.96 | 9.91% | 4,297.50 | 2.22% | |
69. | 31 jul. 2021 | $145.86 | 6.50% | 4,395.26 | 2.27% | |
70. | 31 ago. 2021 | $151.83 | $0.22 | 4.24% | 4,522.68 | 2.90% |
71. | 30 sept. 2021 | $141.50 | -6.80% | 4,307.54 | -4.76% | |
Promedio (R): | 2.69% | 1.12% | ||||
Desviación estándar: | 8.49% | 4.19% |
Mostrar todo
1 Datos en US$ por acción de acciones ordinarias, ajustados por escisiones y dividendos de acciones.
2 Tasa de rendimiento de las acciones ordinarias de AAPL 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
t | Fecha | RAAPL,t | RS&P 500,t | (RAAPL,t–RAAPL)2 | (RS&P 500,t–RS&P 500)2 | (RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) |
---|---|---|---|---|---|---|
1. | 30 nov. 2015 | -0.57% | 0.05% | 10.62 | 1.14 | 3.48 |
2. | 31 dic. 2015 | -11.02% | -1.75% | 187.98 | 8.24 | 39.36 |
3. | 31 ene. 2016 | -7.52% | -5.07% | 104.29 | 38.33 | 63.23 |
. | . | . | . | . | . | . |
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70. | 31 ago. 2021 | 4.24% | 2.90% | 2.42 | 3.17 | 2.77 |
71. | 30 sept. 2021 | -6.80% | -4.76% | 90.12 | 34.51 | 55.77 |
Total (Σ): | 5,051.17 | 1,226.33 | 1,589.97 |
t | Fecha | RAAPL,t | RS&P 500,t | (RAAPL,t–RAAPL)2 | (RS&P 500,t–RS&P 500)2 | (RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) |
---|---|---|---|---|---|---|
1. | 30 nov. 2015 | -0.57% | 0.05% | 10.62 | 1.14 | 3.48 |
2. | 31 dic. 2015 | -11.02% | -1.75% | 187.98 | 8.24 | 39.36 |
3. | 31 ene. 2016 | -7.52% | -5.07% | 104.29 | 38.33 | 63.23 |
4. | 29 feb. 2016 | -0.16% | -0.41% | 8.14 | 2.34 | 4.37 |
5. | 31 mar. 2016 | 12.74% | 6.60% | 101.08 | 30.04 | 55.11 |
6. | 30 abr. 2016 | -13.98% | 0.27% | 277.93 | 0.72 | 14.14 |
7. | 31 may. 2016 | 7.14% | 1.53% | 19.77 | 0.17 | 1.84 |
8. | 30 jun. 2016 | -4.29% | 0.09% | 48.64 | 1.05 | 7.16 |
9. | 31 jul. 2016 | 9.00% | 3.56% | 39.77 | 5.97 | 15.41 |
10. | 31 ago. 2016 | 2.39% | -0.12% | 0.09 | 1.54 | 0.37 |
11. | 30 sept. 2016 | 6.52% | -0.12% | 14.68 | 1.54 | -4.76 |
12. | 31 oct. 2016 | 0.46% | -1.94% | 4.97 | 9.37 | 6.82 |
13. | 30 nov. 2016 | -2.18% | 3.42% | 23.66 | 5.29 | -11.19 |
14. | 31 dic. 2016 | 4.81% | 1.82% | 4.51 | 0.49 | 1.49 |
15. | 31 ene. 2017 | 4.77% | 1.79% | 4.31 | 0.45 | 1.39 |
16. | 28 feb. 2017 | 13.36% | 3.72% | 113.80 | 6.77 | 27.76 |
17. | 31 mar. 2017 | 4.88% | -0.04% | 4.78 | 1.34 | -2.53 |
18. | 30 abr. 2017 | -0.03% | 0.91% | 7.38 | 0.04 | 0.57 |
19. | 31 may. 2017 | 6.79% | 1.16% | 16.80 | 0.00 | 0.16 |
20. | 30 jun. 2017 | -5.71% | 0.48% | 70.52 | 0.41 | 5.35 |
21. | 31 jul. 2017 | 3.25% | 1.93% | 0.31 | 0.67 | 0.46 |
22. | 31 ago. 2017 | 10.70% | 0.05% | 64.14 | 1.13 | -8.52 |
23. | 30 sept. 2017 | -6.02% | 1.93% | 75.93 | 0.66 | -7.08 |
24. | 31 oct. 2017 | 9.68% | 2.22% | 48.88 | 1.21 | 7.70 |
25. | 30 nov. 2017 | 2.03% | 2.81% | 0.44 | 2.86 | -1.12 |
26. | 31 dic. 2017 | -1.51% | 0.98% | 17.66 | 0.02 | 0.57 |
27. | 31 ene. 2018 | -1.06% | 5.62% | 14.09 | 20.25 | -16.89 |
28. | 28 feb. 2018 | 6.75% | -3.89% | 16.53 | 25.13 | -20.38 |
29. | 31 mar. 2018 | -5.79% | -2.69% | 71.97 | 14.49 | 32.29 |
30. | 30 abr. 2018 | -1.50% | 0.27% | 17.57 | 0.72 | 3.55 |
31. | 31 may. 2018 | 13.51% | 2.16% | 117.09 | 1.09 | 11.28 |
32. | 30 jun. 2018 | -0.94% | 0.48% | 13.19 | 0.40 | 2.30 |
33. | 31 jul. 2018 | 2.79% | 3.60% | 0.01 | 6.17 | 0.24 |
34. | 31 ago. 2018 | 20.02% | 3.03% | 300.27 | 3.64 | 33.07 |
35. | 30 sept. 2018 | -0.83% | 0.43% | 12.36 | 0.47 | 2.42 |
36. | 31 oct. 2018 | -3.05% | -6.94% | 32.91 | 64.94 | 46.23 |
37. | 30 nov. 2018 | -18.07% | 1.79% | 430.92 | 0.45 | -13.87 |
38. | 31 dic. 2018 | -11.67% | -9.18% | 206.15 | 106.00 | 147.83 |
39. | 31 ene. 2019 | 5.50% | 7.87% | 7.91 | 45.57 | 18.99 |
40. | 28 feb. 2019 | 4.48% | 2.97% | 3.19 | 3.44 | 3.31 |
41. | 31 mar. 2019 | 9.70% | 1.79% | 49.18 | 0.45 | 4.73 |
42. | 30 abr. 2019 | 5.64% | 3.93% | 8.73 | 7.92 | 8.31 |
43. | 31 may. 2019 | -12.37% | -6.58% | 226.88 | 59.22 | 115.92 |
44. | 30 jun. 2019 | 13.05% | 6.89% | 107.25 | 33.35 | 59.81 |
45. | 31 jul. 2019 | 7.64% | 1.31% | 24.50 | 0.04 | 0.96 |
46. | 31 ago. 2019 | -1.65% | -1.81% | 18.81 | 8.57 | 12.70 |
47. | 30 sept. 2019 | 7.28% | 1.72% | 21.08 | 0.36 | 2.76 |
48. | 31 oct. 2019 | 11.07% | 2.04% | 70.29 | 0.86 | 7.76 |
49. | 30 nov. 2019 | 7.74% | 3.40% | 25.49 | 5.23 | 11.55 |
50. | 31 dic. 2019 | 9.88% | 2.86% | 51.69 | 3.03 | 12.52 |
51. | 31 ene. 2020 | 5.41% | -0.16% | 7.39 | 1.64 | -3.48 |
52. | 29 feb. 2020 | -11.43% | -8.41% | 199.47 | 90.80 | 134.58 |
53. | 31 mar. 2020 | -6.98% | -12.51% | 93.49 | 185.77 | 131.79 |
54. | 30 abr. 2020 | 15.54% | 12.68% | 165.19 | 133.78 | 148.66 |
55. | 31 may. 2020 | 8.50% | 4.53% | 33.79 | 11.63 | 19.82 |
56. | 30 jun. 2020 | 14.73% | 1.84% | 145.01 | 0.52 | 8.68 |
57. | 31 jul. 2020 | 16.51% | 5.51% | 191.09 | 19.29 | 60.72 |
58. | 31 ago. 2020 | 21.63% | 7.01% | 358.78 | 34.67 | 111.54 |
59. | 30 sept. 2020 | -10.25% | -3.92% | 167.50 | 25.41 | 65.24 |
60. | 31 oct. 2020 | -6.00% | -2.77% | 75.53 | 15.09 | 33.76 |
61. | 30 nov. 2020 | 9.55% | 10.75% | 47.05 | 92.86 | 66.10 |
62. | 31 dic. 2020 | 11.46% | 3.71% | 76.88 | 6.73 | 22.75 |
63. | 31 ene. 2021 | -0.55% | -1.11% | 10.50 | 4.98 | 7.23 |
64. | 28 feb. 2021 | -7.95% | 2.61% | 113.27 | 2.22 | -15.87 |
65. | 31 mar. 2021 | 0.73% | 4.24% | 3.82 | 9.77 | -6.11 |
66. | 30 abr. 2021 | 7.62% | 5.24% | 24.33 | 17.01 | 20.34 |
67. | 31 may. 2021 | -5.04% | 0.55% | 59.80 | 0.32 | 4.40 |
68. | 30 jun. 2021 | 9.91% | 2.22% | 52.15 | 1.22 | 7.97 |
69. | 31 jul. 2021 | 6.50% | 2.27% | 14.51 | 1.34 | 4.41 |
70. | 31 ago. 2021 | 4.24% | 2.90% | 2.42 | 3.17 | 2.77 |
71. | 30 sept. 2021 | -6.80% | -4.76% | 90.12 | 34.51 | 55.77 |
Total (Σ): | 5,051.17 | 1,226.33 | 1,589.97 |
Mostrar todo
VarianzaAAPL = Σ(RAAPL,t–RAAPL)2 ÷ (71 – 1)
= 5,051.17 ÷ (71 – 1)
= 72.16
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,226.33 ÷ (71 – 1)
= 17.52
CovarianzaAAPL, S&P 500 = Σ(RAAPL,t–RAAPL)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 1,589.97 ÷ (71 – 1)
= 22.71
Estimación sistemática del riesgo (β)
VarianzaAAPL | 72.16 |
VarianzaS&P 500 | 17.52 |
CovarianzaAAPL, S&P 500 | 22.71 |
Coeficiente de correlaciónAAPL, S&P 5001 | 0.64 |
βAAPL2 | 1.30 |
αAAPL3 | 1.24% |
Cálculos
1 Coeficiente de correlaciónAAPL, S&P 500
= CovarianzaAAPL, S&P 500 ÷ (Desviación estándarAAPL × Desviación estándarS&P 500)
= 22.71 ÷ (8.49% × 4.19%)
= 0.64
2 βAAPL
= CovarianzaAAPL, S&P 500 ÷ VarianzaS&P 500
= 22.71 ÷ 17.52
= 1.30
3 αAAPL
= PromedioAAPL – βAAPL × PromedioS&P 500
= 2.69% – 1.30 × 1.12%
= 1.24%
Tasa de rendimiento esperada
Suposiciones | ||
Tasa de rendimiento de LT Treasury Composite1 | RF | 3.26% |
Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.05% |
Riesgo sistemático de acciones ordinarias Apple | βAAPL | 1.30 |
Tasa de rendimiento esperada de las acciones ordinarias de Apple3 | E(RAAPL) | 15.96% |
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(RAAPL) = RF + βAAPL [E(RM) – RF]
= 3.26% + 1.30 [13.05% – 3.26%]
= 15.96%