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 Linde.
Tasas de retorno
Linde plc (LIN) | Standard & Poor’s 500 (S&P 500) | |||||
---|---|---|---|---|---|---|
t | Fecha | PrecioLIN,t1 | DividendoLIN,t1 | RLIN,t2 | PrecioS&P 500,t | RS&P 500,t3 |
31 ene. 2019 | $163.01 | 2,704.10 | ||||
1. | 28 feb. 2019 | $173.24 | 6.28% | 2,784.49 | 2.97% | |
2. | 31 mar. 2019 | $175.93 | $0.875 | 2.06% | 2,834.40 | 1.79% |
3. | 30 abr. 2019 | $180.26 | 2.46% | 2,945.83 | 3.93% | |
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58. | 30 nov. 2023 | $413.77 | 8.27% | 4,567.80 | 8.92% | |
59. | 31 dic. 2023 | $410.71 | $1.275 | -0.43% | 4,769.83 | 4.42% |
Promedio (R): | 1.92% | 1.11% | ||||
Desviación estándar: | 6.50% | 5.31% |
Linde plc (LIN) | Standard & Poor’s 500 (S&P 500) | |||||
---|---|---|---|---|---|---|
t | Fecha | PrecioLIN,t1 | DividendoLIN,t1 | RLIN,t2 | PrecioS&P 500,t | RS&P 500,t3 |
31 ene. 2019 | $163.01 | 2,704.10 | ||||
1. | 28 feb. 2019 | $173.24 | 6.28% | 2,784.49 | 2.97% | |
2. | 31 mar. 2019 | $175.93 | $0.875 | 2.06% | 2,834.40 | 1.79% |
3. | 30 abr. 2019 | $180.26 | 2.46% | 2,945.83 | 3.93% | |
4. | 31 may. 2019 | $180.55 | $0.875 | 0.65% | 2,752.06 | -6.58% |
5. | 30 jun. 2019 | $200.80 | 11.22% | 2,941.76 | 6.89% | |
6. | 31 jul. 2019 | $191.28 | -4.74% | 2,980.38 | 1.31% | |
7. | 31 ago. 2019 | $188.91 | $0.875 | -0.78% | 2,926.46 | -1.81% |
8. | 30 sept. 2019 | $193.72 | 2.55% | 2,976.74 | 1.72% | |
9. | 31 oct. 2019 | $198.35 | 2.39% | 3,037.56 | 2.04% | |
10. | 30 nov. 2019 | $206.21 | 3.96% | 3,140.98 | 3.40% | |
11. | 31 dic. 2019 | $212.90 | $0.875 | 3.67% | 3,230.78 | 2.86% |
12. | 31 ene. 2020 | $203.13 | -4.59% | 3,225.52 | -0.16% | |
13. | 29 feb. 2020 | $191.01 | -5.97% | 2,954.22 | -8.41% | |
14. | 31 mar. 2020 | $173.00 | $0.963 | -8.92% | 2,584.59 | -12.51% |
15. | 30 abr. 2020 | $183.99 | 6.35% | 2,912.43 | 12.68% | |
16. | 31 may. 2020 | $202.34 | 9.97% | 3,044.31 | 4.53% | |
17. | 30 jun. 2020 | $212.11 | $0.963 | 5.30% | 3,100.29 | 1.84% |
18. | 31 jul. 2020 | $245.11 | 15.56% | 3,271.12 | 5.51% | |
19. | 31 ago. 2020 | $249.74 | 1.89% | 3,500.31 | 7.01% | |
20. | 30 sept. 2020 | $238.13 | $0.963 | -4.26% | 3,363.00 | -3.92% |
21. | 31 oct. 2020 | $220.34 | -7.47% | 3,269.96 | -2.77% | |
22. | 30 nov. 2020 | $256.42 | 16.37% | 3,621.63 | 10.75% | |
23. | 31 dic. 2020 | $263.51 | $0.963 | 3.14% | 3,756.07 | 3.71% |
24. | 31 ene. 2021 | $245.40 | -6.87% | 3,714.24 | -1.11% | |
25. | 28 feb. 2021 | $244.27 | -0.46% | 3,811.15 | 2.61% | |
26. | 31 mar. 2021 | $280.14 | $1.06 | 15.12% | 3,972.89 | 4.24% |
27. | 30 abr. 2021 | $285.84 | 2.03% | 4,181.17 | 5.24% | |
28. | 31 may. 2021 | $300.60 | 5.16% | 4,204.11 | 0.55% | |
29. | 30 jun. 2021 | $289.10 | $1.06 | -3.47% | 4,297.50 | 2.22% |
30. | 31 jul. 2021 | $307.39 | 6.33% | 4,395.26 | 2.27% | |
31. | 31 ago. 2021 | $314.59 | 2.34% | 4,522.68 | 2.90% | |
32. | 30 sept. 2021 | $293.38 | $1.06 | -6.41% | 4,307.54 | -4.76% |
33. | 31 oct. 2021 | $319.20 | 8.80% | 4,605.38 | 6.91% | |
34. | 30 nov. 2021 | $318.14 | -0.33% | 4,567.00 | -0.83% | |
35. | 31 dic. 2021 | $346.43 | $1.06 | 9.23% | 4,766.18 | 4.36% |
36. | 31 ene. 2022 | $318.68 | -8.01% | 4,515.55 | -5.26% | |
37. | 28 feb. 2022 | $293.24 | -7.98% | 4,373.94 | -3.14% | |
38. | 31 mar. 2022 | $319.43 | $1.17 | 9.33% | 4,530.41 | 3.58% |
39. | 30 abr. 2022 | $311.96 | -2.34% | 4,131.93 | -8.80% | |
40. | 31 may. 2022 | $324.68 | 4.08% | 4,132.15 | 0.01% | |
41. | 30 jun. 2022 | $287.53 | $1.17 | -11.08% | 3,785.38 | -8.39% |
42. | 31 jul. 2022 | $302.00 | 5.03% | 4,130.29 | 9.11% | |
43. | 31 ago. 2022 | $282.86 | -6.34% | 3,955.00 | -4.24% | |
44. | 30 sept. 2022 | $269.59 | $1.17 | -4.28% | 3,585.62 | -9.34% |
45. | 31 oct. 2022 | $297.35 | 10.30% | 3,871.98 | 7.99% | |
46. | 30 nov. 2022 | $336.48 | 13.16% | 4,080.11 | 5.38% | |
47. | 31 dic. 2022 | $326.18 | $1.17 | -2.71% | 3,839.50 | -5.90% |
48. | 31 ene. 2023 | $330.94 | 1.46% | 4,076.60 | 6.18% | |
49. | 28 feb. 2023 | $348.37 | 5.27% | 3,970.15 | -2.61% | |
50. | 31 mar. 2023 | $355.44 | $1.275 | 2.40% | 4,109.31 | 3.51% |
51. | 30 abr. 2023 | $369.45 | 3.94% | 4,169.48 | 1.46% | |
52. | 31 may. 2023 | $353.66 | -4.27% | 4,179.83 | 0.25% | |
53. | 30 jun. 2023 | $381.08 | $1.275 | 8.11% | 4,376.86 | 4.71% |
54. | 31 jul. 2023 | $390.67 | 2.52% | 4,588.96 | 4.85% | |
55. | 31 ago. 2023 | $387.04 | -0.93% | 4,507.66 | -1.77% | |
56. | 30 sept. 2023 | $372.35 | $1.275 | -3.47% | 4,288.05 | -4.87% |
57. | 31 oct. 2023 | $382.16 | 2.63% | 4,193.80 | -2.20% | |
58. | 30 nov. 2023 | $413.77 | 8.27% | 4,567.80 | 8.92% | |
59. | 31 dic. 2023 | $410.71 | $1.275 | -0.43% | 4,769.83 | 4.42% |
Promedio (R): | 1.92% | 1.11% | ||||
Desviación estándar: | 6.50% | 5.31% |
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 LIN 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 | RLIN,t | RS&P 500,t | (RLIN,t–RLIN)2 | (RS&P 500,t–RS&P 500)2 | (RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) |
---|---|---|---|---|---|---|
1. | 28 feb. 2019 | 6.28% | 2.97% | 18.98 | 3.49 | 8.13 |
2. | 31 mar. 2019 | 2.06% | 1.79% | 0.02 | 0.47 | 0.10 |
3. | 30 abr. 2019 | 2.46% | 3.93% | 0.29 | 7.98 | 1.53 |
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58. | 30 nov. 2023 | 8.27% | 8.92% | 40.36 | 61.03 | 49.63 |
59. | 31 dic. 2023 | -0.43% | 4.42% | 5.52 | 11.00 | -7.80 |
Total (Σ): | 2,453.29 | 1,634.30 | 1,499.15 |
t | Fecha | RLIN,t | RS&P 500,t | (RLIN,t–RLIN)2 | (RS&P 500,t–RS&P 500)2 | (RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) |
---|---|---|---|---|---|---|
1. | 28 feb. 2019 | 6.28% | 2.97% | 18.98 | 3.49 | 8.13 |
2. | 31 mar. 2019 | 2.06% | 1.79% | 0.02 | 0.47 | 0.10 |
3. | 30 abr. 2019 | 2.46% | 3.93% | 0.29 | 7.98 | 1.53 |
4. | 31 may. 2019 | 0.65% | -6.58% | 1.62 | 59.04 | 9.78 |
5. | 30 jun. 2019 | 11.22% | 6.89% | 86.44 | 33.49 | 53.80 |
6. | 31 jul. 2019 | -4.74% | 1.31% | 44.35 | 0.04 | -1.38 |
7. | 31 ago. 2019 | -0.78% | -1.81% | 7.29 | 8.50 | 7.87 |
8. | 30 sept. 2019 | 2.55% | 1.72% | 0.39 | 0.37 | 0.38 |
9. | 31 oct. 2019 | 2.39% | 2.04% | 0.22 | 0.88 | 0.44 |
10. | 30 nov. 2019 | 3.96% | 3.40% | 4.18 | 5.28 | 4.70 |
11. | 31 dic. 2019 | 3.67% | 2.86% | 3.06 | 3.07 | 3.07 |
12. | 31 ene. 2020 | -4.59% | -0.16% | 42.35 | 1.61 | 8.26 |
13. | 29 feb. 2020 | -5.97% | -8.41% | 62.18 | 90.57 | 75.04 |
14. | 31 mar. 2020 | -8.92% | -12.51% | 117.58 | 185.44 | 147.66 |
15. | 30 abr. 2020 | 6.35% | 12.68% | 19.66 | 134.06 | 51.34 |
16. | 31 may. 2020 | 9.97% | 4.53% | 64.88 | 11.71 | 27.57 |
17. | 30 jun. 2020 | 5.30% | 1.84% | 11.46 | 0.54 | 2.48 |
18. | 31 jul. 2020 | 15.56% | 5.51% | 186.03 | 19.40 | 60.07 |
19. | 31 ago. 2020 | 1.89% | 7.01% | 0.00 | 34.82 | -0.18 |
20. | 30 sept. 2020 | -4.26% | -3.92% | 38.22 | 25.29 | 31.09 |
21. | 31 oct. 2020 | -7.47% | -2.77% | 88.16 | 15.00 | 36.36 |
22. | 30 nov. 2020 | 16.37% | 10.75% | 208.98 | 93.10 | 139.48 |
23. | 31 dic. 2020 | 3.14% | 3.71% | 1.49 | 6.79 | 3.18 |
24. | 31 ene. 2021 | -6.87% | -1.11% | 77.29 | 4.93 | 19.51 |
25. | 28 feb. 2021 | -0.46% | 2.61% | 5.66 | 2.26 | -3.58 |
26. | 31 mar. 2021 | 15.12% | 4.24% | 174.24 | 9.85 | 41.42 |
27. | 30 abr. 2021 | 2.03% | 5.24% | 0.01 | 17.11 | 0.48 |
28. | 31 may. 2021 | 5.16% | 0.55% | 10.53 | 0.31 | -1.81 |
29. | 30 jun. 2021 | -3.47% | 2.22% | 29.07 | 1.24 | -6.01 |
30. | 31 jul. 2021 | 6.33% | 2.27% | 19.43 | 1.37 | 5.15 |
31. | 31 ago. 2021 | 2.34% | 2.90% | 0.18 | 3.22 | 0.76 |
32. | 30 sept. 2021 | -6.41% | -4.76% | 69.29 | 34.37 | 48.80 |
33. | 31 oct. 2021 | 8.80% | 6.91% | 47.36 | 33.74 | 39.98 |
34. | 30 nov. 2021 | -0.33% | -0.83% | 5.07 | 3.76 | 4.36 |
35. | 31 dic. 2021 | 9.23% | 4.36% | 53.39 | 10.60 | 23.79 |
36. | 31 ene. 2022 | -8.01% | -5.26% | 98.58 | 40.51 | 63.19 |
37. | 28 feb. 2022 | -7.98% | -3.14% | 98.04 | 17.99 | 42.00 |
38. | 31 mar. 2022 | 9.33% | 3.58% | 54.93 | 6.11 | 18.32 |
39. | 30 abr. 2022 | -2.34% | -8.80% | 18.12 | 98.04 | 42.15 |
40. | 31 may. 2022 | 4.08% | 0.01% | 4.66 | 1.21 | -2.38 |
41. | 30 jun. 2022 | -11.08% | -8.39% | 169.01 | 90.21 | 123.48 |
42. | 31 jul. 2022 | 5.03% | 9.11% | 9.70 | 64.09 | 24.93 |
43. | 31 ago. 2022 | -6.34% | -4.24% | 68.17 | 28.62 | 44.17 |
44. | 30 sept. 2022 | -4.28% | -9.34% | 38.40 | 109.11 | 64.72 |
45. | 31 oct. 2022 | 10.30% | 7.99% | 70.20 | 47.34 | 57.65 |
46. | 30 nov. 2022 | 13.16% | 5.38% | 126.36 | 18.23 | 47.99 |
47. | 31 dic. 2022 | -2.71% | -5.90% | 21.46 | 49.04 | 32.44 |
48. | 31 ene. 2023 | 1.46% | 6.18% | 0.21 | 25.70 | -2.33 |
49. | 28 feb. 2023 | 5.27% | -2.61% | 11.21 | 13.82 | -12.45 |
50. | 31 mar. 2023 | 2.40% | 3.51% | 0.23 | 5.76 | 1.14 |
51. | 30 abr. 2023 | 3.94% | 1.46% | 4.09 | 0.13 | 0.72 |
52. | 31 may. 2023 | -4.27% | 0.25% | 38.35 | 0.74 | 5.31 |
53. | 30 jun. 2023 | 8.11% | 4.71% | 38.38 | 13.02 | 22.35 |
54. | 31 jul. 2023 | 2.52% | 4.85% | 0.36 | 13.99 | 2.24 |
55. | 31 ago. 2023 | -0.93% | -1.77% | 8.11 | 8.28 | 8.19 |
56. | 30 sept. 2023 | -3.47% | -4.87% | 29.00 | 35.73 | 32.19 |
57. | 31 oct. 2023 | 2.63% | -2.20% | 0.51 | 10.92 | -2.37 |
58. | 30 nov. 2023 | 8.27% | 8.92% | 40.36 | 61.03 | 49.63 |
59. | 31 dic. 2023 | -0.43% | 4.42% | 5.52 | 11.00 | -7.80 |
Total (Σ): | 2,453.29 | 1,634.30 | 1,499.15 |
Mostrar todo
VarianzaLIN = Σ(RLIN,t–RLIN)2 ÷ (59 – 1)
= 2,453.29 ÷ (59 – 1)
= 42.30
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,634.30 ÷ (59 – 1)
= 28.18
CovarianzaLIN, S&P 500 = Σ(RLIN,t–RLIN)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,499.15 ÷ (59 – 1)
= 25.85
Estimación sistemática del riesgo (β)
VarianzaLIN | 42.30 |
VarianzaS&P 500 | 28.18 |
CovarianzaLIN, S&P 500 | 25.85 |
Coeficiente de correlaciónLIN, S&P 5001 | 0.75 |
βLIN2 | 0.92 |
αLIN3 | 0.90% |
Cálculos
1 Coeficiente de correlaciónLIN, S&P 500
= CovarianzaLIN, S&P 500 ÷ (Desviación estándarLIN × Desviación estándarS&P 500)
= 25.85 ÷ (6.50% × 5.31%)
= 0.75
2 βLIN
= CovarianzaLIN, S&P 500 ÷ VarianzaS&P 500
= 25.85 ÷ 28.18
= 0.92
3 αLIN
= PromedioLIN – βLIN × PromedioS&P 500
= 1.92% – 0.92 × 1.11%
= 0.90%
Tasa de rendimiento esperada
Suposiciones | ||
Tasa de rendimiento del LT Treasury Composite1 | RF | 4.43% |
Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.60% |
Riesgo sistemático de Linde acciones ordinarias | βLIN | 0.92 |
Tasa de rendimiento esperada de las acciones ordinarias de Linde3 | E(RLIN) | 12.85% |
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(RLIN) = RF + βLIN [E(RM) – RF]
= 4.43% + 0.92 [13.60% – 4.43%]
= 12.85%