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 Lilly.
Tasas de retorno
Eli Lilly & Co., tasas mensuales de retorno
| Eli Lilly & Co. (LLY) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioLLY,t1 | DividendoLLY,t1 | RLLY,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2019 | $119.86 | 2,704.10 | ||||
| 1. | 28 feb. 2019 | $126.29 | $0.645 | 5.90% | 2,784.49 | 2.97% |
| 2. | 31 mar. 2019 | $129.76 | 2.75% | 2,834.40 | 1.79% | |
| 3. | 30 abr. 2019 | $117.04 | -9.80% | 2,945.83 | 3.93% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2023 | $591.04 | $1.13 | 6.90% | 4,567.80 | 8.92% |
| 59. | 31 dic. 2023 | $582.92 | -1.37% | 4,769.83 | 4.42% | |
| Promedio (R): | 3.19% | 1.11% | ||||
| Desviación estándar: | 8.33% | 5.31% | ||||
| Eli Lilly & Co. (LLY) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Fecha | PrecioLLY,t1 | DividendoLLY,t1 | RLLY,t2 | PrecioS&P 500,t | RS&P 500,t3 |
| 31 ene. 2019 | $119.86 | 2,704.10 | ||||
| 1. | 28 feb. 2019 | $126.29 | $0.645 | 5.90% | 2,784.49 | 2.97% |
| 2. | 31 mar. 2019 | $129.76 | 2.75% | 2,834.40 | 1.79% | |
| 3. | 30 abr. 2019 | $117.04 | -9.80% | 2,945.83 | 3.93% | |
| 4. | 31 may. 2019 | $115.94 | $0.645 | -0.39% | 2,752.06 | -6.58% |
| 5. | 30 jun. 2019 | $110.79 | -4.44% | 2,941.76 | 6.89% | |
| 6. | 31 jul. 2019 | $108.95 | -1.66% | 2,980.38 | 1.31% | |
| 7. | 31 ago. 2019 | $112.97 | $0.645 | 4.28% | 2,926.46 | -1.81% |
| 8. | 30 sept. 2019 | $111.83 | -1.01% | 2,976.74 | 1.72% | |
| 9. | 31 oct. 2019 | $113.95 | 1.90% | 3,037.56 | 2.04% | |
| 10. | 30 nov. 2019 | $117.35 | $0.645 | 3.55% | 3,140.98 | 3.40% |
| 11. | 31 dic. 2019 | $131.43 | 12.00% | 3,230.78 | 2.86% | |
| 12. | 31 ene. 2020 | $139.64 | 6.25% | 3,225.52 | -0.16% | |
| 13. | 29 feb. 2020 | $126.13 | $0.74 | -9.14% | 2,954.22 | -8.41% |
| 14. | 31 mar. 2020 | $138.72 | 9.98% | 2,584.59 | -12.51% | |
| 15. | 30 abr. 2020 | $154.64 | 11.48% | 2,912.43 | 12.68% | |
| 16. | 31 may. 2020 | $152.95 | $0.74 | -0.61% | 3,044.31 | 4.53% |
| 17. | 30 jun. 2020 | $164.18 | 7.34% | 3,100.29 | 1.84% | |
| 18. | 31 jul. 2020 | $150.29 | -8.46% | 3,271.12 | 5.51% | |
| 19. | 31 ago. 2020 | $148.39 | $0.74 | -0.77% | 3,500.31 | 7.01% |
| 20. | 30 sept. 2020 | $148.02 | -0.25% | 3,363.00 | -3.92% | |
| 21. | 31 oct. 2020 | $130.46 | -11.86% | 3,269.96 | -2.77% | |
| 22. | 30 nov. 2020 | $145.65 | $0.74 | 12.21% | 3,621.63 | 10.75% |
| 23. | 31 dic. 2020 | $168.84 | 15.92% | 3,756.07 | 3.71% | |
| 24. | 31 ene. 2021 | $207.97 | 23.18% | 3,714.24 | -1.11% | |
| 25. | 28 feb. 2021 | $204.89 | $0.85 | -1.07% | 3,811.15 | 2.61% |
| 26. | 31 mar. 2021 | $186.82 | -8.82% | 3,972.89 | 4.24% | |
| 27. | 30 abr. 2021 | $182.77 | -2.17% | 4,181.17 | 5.24% | |
| 28. | 31 may. 2021 | $199.74 | $0.85 | 9.75% | 4,204.11 | 0.55% |
| 29. | 30 jun. 2021 | $229.52 | 14.91% | 4,297.50 | 2.22% | |
| 30. | 31 jul. 2021 | $243.50 | 6.09% | 4,395.26 | 2.27% | |
| 31. | 31 ago. 2021 | $258.29 | $0.85 | 6.42% | 4,522.68 | 2.90% |
| 32. | 30 sept. 2021 | $231.05 | -10.55% | 4,307.54 | -4.76% | |
| 33. | 31 oct. 2021 | $254.76 | 10.26% | 4,605.38 | 6.91% | |
| 34. | 30 nov. 2021 | $248.04 | $0.85 | -2.30% | 4,567.00 | -0.83% |
| 35. | 31 dic. 2021 | $276.22 | 11.36% | 4,766.18 | 4.36% | |
| 36. | 31 ene. 2022 | $245.39 | -11.16% | 4,515.55 | -5.26% | |
| 37. | 28 feb. 2022 | $249.95 | $0.98 | 2.26% | 4,373.94 | -3.14% |
| 38. | 31 mar. 2022 | $286.37 | 14.57% | 4,530.41 | 3.58% | |
| 39. | 30 abr. 2022 | $292.13 | 2.01% | 4,131.93 | -8.80% | |
| 40. | 31 may. 2022 | $313.44 | $0.98 | 7.63% | 4,132.15 | 0.01% |
| 41. | 30 jun. 2022 | $324.23 | 3.44% | 3,785.38 | -8.39% | |
| 42. | 31 jul. 2022 | $329.69 | 1.68% | 4,130.29 | 9.11% | |
| 43. | 31 ago. 2022 | $301.23 | $0.98 | -8.34% | 3,955.00 | -4.24% |
| 44. | 30 sept. 2022 | $323.35 | 7.34% | 3,585.62 | -9.34% | |
| 45. | 31 oct. 2022 | $362.09 | 11.98% | 3,871.98 | 7.99% | |
| 46. | 30 nov. 2022 | $371.08 | $0.98 | 2.75% | 4,080.11 | 5.38% |
| 47. | 31 dic. 2022 | $365.84 | -1.41% | 3,839.50 | -5.90% | |
| 48. | 31 ene. 2023 | $344.15 | -5.93% | 4,076.60 | 6.18% | |
| 49. | 28 feb. 2023 | $311.22 | $1.13 | -9.24% | 3,970.15 | -2.61% |
| 50. | 31 mar. 2023 | $343.42 | 10.35% | 4,109.31 | 3.51% | |
| 51. | 30 abr. 2023 | $395.86 | 15.27% | 4,169.48 | 1.46% | |
| 52. | 31 may. 2023 | $429.46 | $1.13 | 8.77% | 4,179.83 | 0.25% |
| 53. | 30 jun. 2023 | $468.98 | 9.20% | 4,376.86 | 4.71% | |
| 54. | 31 jul. 2023 | $454.55 | -3.08% | 4,588.96 | 4.85% | |
| 55. | 31 ago. 2023 | $554.20 | $1.13 | 22.17% | 4,507.66 | -1.77% |
| 56. | 30 sept. 2023 | $537.13 | -3.08% | 4,288.05 | -4.87% | |
| 57. | 31 oct. 2023 | $553.93 | 3.13% | 4,193.80 | -2.20% | |
| 58. | 30 nov. 2023 | $591.04 | $1.13 | 6.90% | 4,567.80 | 8.92% |
| 59. | 31 dic. 2023 | $582.92 | -1.37% | 4,769.83 | 4.42% | |
| Promedio (R): | 3.19% | 1.11% | ||||
| Desviación estándar: | 8.33% | 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 LLY 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
Eli Lilly & Co., cálculo de la varianza y covarianza de los rendimientos
| t | Fecha | RLLY,t | RS&P 500,t | (RLLY,t–RLLY)2 | (RS&P 500,t–RS&P 500)2 | (RLLY,t–RLLY)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2019 | 5.90% | 2.97% | 7.37 | 3.49 | 5.07 |
| 2. | 31 mar. 2019 | 2.75% | 1.79% | 0.19 | 0.47 | -0.30 |
| 3. | 30 abr. 2019 | -9.80% | 3.93% | 168.75 | 7.98 | -36.70 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | 30 nov. 2023 | 6.90% | 8.92% | 13.81 | 61.03 | 29.03 |
| 59. | 31 dic. 2023 | -1.37% | 4.42% | 20.81 | 11.00 | -15.13 |
| Total (Σ): | 4,020.05 | 1,634.30 | 491.40 | |||
| t | Fecha | RLLY,t | RS&P 500,t | (RLLY,t–RLLY)2 | (RS&P 500,t–RS&P 500)2 | (RLLY,t–RLLY)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | 28 feb. 2019 | 5.90% | 2.97% | 7.37 | 3.49 | 5.07 |
| 2. | 31 mar. 2019 | 2.75% | 1.79% | 0.19 | 0.47 | -0.30 |
| 3. | 30 abr. 2019 | -9.80% | 3.93% | 168.75 | 7.98 | -36.70 |
| 4. | 31 may. 2019 | -0.39% | -6.58% | 12.79 | 59.04 | 27.48 |
| 5. | 30 jun. 2019 | -4.44% | 6.89% | 58.21 | 33.49 | -44.15 |
| 6. | 31 jul. 2019 | -1.66% | 1.31% | 23.51 | 0.04 | -1.00 |
| 7. | 31 ago. 2019 | 4.28% | -1.81% | 1.20 | 8.50 | -3.19 |
| 8. | 30 sept. 2019 | -1.01% | 1.72% | 17.61 | 0.37 | -2.57 |
| 9. | 31 oct. 2019 | 1.90% | 2.04% | 1.67 | 0.88 | -1.21 |
| 10. | 30 nov. 2019 | 3.55% | 3.40% | 0.13 | 5.28 | 0.83 |
| 11. | 31 dic. 2019 | 12.00% | 2.86% | 77.63 | 3.07 | 15.45 |
| 12. | 31 ene. 2020 | 6.25% | -0.16% | 9.36 | 1.61 | -3.88 |
| 13. | 29 feb. 2020 | -9.14% | -8.41% | 152.09 | 90.57 | 117.37 |
| 14. | 31 mar. 2020 | 9.98% | -12.51% | 46.16 | 185.44 | -92.52 |
| 15. | 30 abr. 2020 | 11.48% | 12.68% | 68.70 | 134.06 | 95.97 |
| 16. | 31 may. 2020 | -0.61% | 4.53% | 14.45 | 11.71 | -13.01 |
| 17. | 30 jun. 2020 | 7.34% | 1.84% | 17.26 | 0.54 | 3.05 |
| 18. | 31 jul. 2020 | -8.46% | 5.51% | 135.67 | 19.40 | -51.30 |
| 19. | 31 ago. 2020 | -0.77% | 7.01% | 15.68 | 34.82 | -23.36 |
| 20. | 30 sept. 2020 | -0.25% | -3.92% | 11.81 | 25.29 | 17.28 |
| 21. | 31 oct. 2020 | -11.86% | -2.77% | 226.53 | 15.00 | 58.28 |
| 22. | 30 nov. 2020 | 12.21% | 10.75% | 81.41 | 93.10 | 87.06 |
| 23. | 31 dic. 2020 | 15.92% | 3.71% | 162.16 | 6.79 | 33.19 |
| 24. | 31 ene. 2021 | 23.18% | -1.11% | 399.53 | 4.93 | -44.36 |
| 25. | 28 feb. 2021 | -1.07% | 2.61% | 18.15 | 2.26 | -6.40 |
| 26. | 31 mar. 2021 | -8.82% | 4.24% | 144.17 | 9.85 | -37.68 |
| 27. | 30 abr. 2021 | -2.17% | 5.24% | 28.68 | 17.11 | -22.15 |
| 28. | 31 may. 2021 | 9.75% | 0.55% | 43.06 | 0.31 | -3.66 |
| 29. | 30 jun. 2021 | 14.91% | 2.22% | 137.40 | 1.24 | 13.08 |
| 30. | 31 jul. 2021 | 6.09% | 2.27% | 8.43 | 1.37 | 3.39 |
| 31. | 31 ago. 2021 | 6.42% | 2.90% | 10.47 | 3.22 | 5.80 |
| 32. | 30 sept. 2021 | -10.55% | -4.76% | 188.62 | 34.37 | 80.52 |
| 33. | 31 oct. 2021 | 10.26% | 6.91% | 50.04 | 33.74 | 41.09 |
| 34. | 30 nov. 2021 | -2.30% | -0.83% | 30.16 | 3.76 | 10.65 |
| 35. | 31 dic. 2021 | 11.36% | 4.36% | 66.81 | 10.60 | 26.61 |
| 36. | 31 ene. 2022 | -11.16% | -5.26% | 205.89 | 40.51 | 91.32 |
| 37. | 28 feb. 2022 | 2.26% | -3.14% | 0.86 | 17.99 | 3.94 |
| 38. | 31 mar. 2022 | 14.57% | 3.58% | 129.58 | 6.11 | 28.13 |
| 39. | 30 abr. 2022 | 2.01% | -8.80% | 1.38 | 98.04 | 11.65 |
| 40. | 31 may. 2022 | 7.63% | 0.01% | 19.74 | 1.21 | -4.89 |
| 41. | 30 jun. 2022 | 3.44% | -8.39% | 0.06 | 90.21 | -2.42 |
| 42. | 31 jul. 2022 | 1.68% | 9.11% | 2.26 | 64.09 | -12.04 |
| 43. | 31 ago. 2022 | -8.34% | -4.24% | 132.77 | 28.62 | 61.64 |
| 44. | 30 sept. 2022 | 7.34% | -9.34% | 17.27 | 109.11 | -43.41 |
| 45. | 31 oct. 2022 | 11.98% | 7.99% | 77.32 | 47.34 | 60.50 |
| 46. | 30 nov. 2022 | 2.75% | 5.38% | 0.19 | 18.23 | -1.85 |
| 47. | 31 dic. 2022 | -1.41% | -5.90% | 21.16 | 49.04 | 32.21 |
| 48. | 31 ene. 2023 | -5.93% | 6.18% | 83.11 | 25.70 | -46.22 |
| 49. | 28 feb. 2023 | -9.24% | -2.61% | 154.45 | 13.82 | 46.20 |
| 50. | 31 mar. 2023 | 10.35% | 3.51% | 51.25 | 5.76 | 17.18 |
| 51. | 30 abr. 2023 | 15.27% | 1.46% | 145.98 | 0.13 | 4.33 |
| 52. | 31 may. 2023 | 8.77% | 0.25% | 31.20 | 0.74 | -4.79 |
| 53. | 30 jun. 2023 | 9.20% | 4.71% | 36.18 | 13.02 | 21.70 |
| 54. | 31 jul. 2023 | -3.08% | 4.85% | 39.24 | 13.99 | -23.43 |
| 55. | 31 ago. 2023 | 22.17% | -1.77% | 360.38 | 8.28 | -54.63 |
| 56. | 30 sept. 2023 | -3.08% | -4.87% | 39.28 | 35.73 | 37.47 |
| 57. | 31 oct. 2023 | 3.13% | -2.20% | 0.00 | 10.92 | 0.20 |
| 58. | 30 nov. 2023 | 6.90% | 8.92% | 13.81 | 61.03 | 29.03 |
| 59. | 31 dic. 2023 | -1.37% | 4.42% | 20.81 | 11.00 | -15.13 |
| Total (Σ): | 4,020.05 | 1,634.30 | 491.40 | |||
Mostrar todo
VarianzaLLY = Σ(RLLY,t–RLLY)2 ÷ (59 – 1)
= 4,020.05 ÷ (59 – 1)
= 69.31
VarianzaS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,634.30 ÷ (59 – 1)
= 28.18
CovarianzaLLY, S&P 500 = Σ(RLLY,t–RLLY)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 491.40 ÷ (59 – 1)
= 8.47
Estimación sistemática del riesgo (β)
| VarianzaLLY | 69.31 |
| VarianzaS&P 500 | 28.18 |
| CovarianzaLLY, S&P 500 | 8.47 |
| Coeficiente de correlaciónLLY, S&P 5001 | 0.19 |
| βLLY2 | 0.30 |
| αLLY3 | 2.86% |
Cálculos
1 Coeficiente de correlaciónLLY, S&P 500
= CovarianzaLLY, S&P 500 ÷ (Desviación estándarLLY × Desviación estándarS&P 500)
= 8.47 ÷ (8.33% × 5.31%)
= 0.19
2 βLLY
= CovarianzaLLY, S&P 500 ÷ VarianzaS&P 500
= 8.47 ÷ 28.18
= 0.30
3 αLLY
= PromedioLLY – βLLY × PromedioS&P 500
= 3.19% – 0.30 × 1.11%
= 2.86%
Tasa de rendimiento esperada
| Suposiciones | ||
| Tasa de rendimiento del LT Treasury Composite1 | RF | 4.83% |
| Tasa de rendimiento esperada de la cartera de mercado2 | E(RM) | 13.48% |
| Riesgo sistemático de Lilly acciones ordinarias | βLLY | 0.30 |
| Tasa de rendimiento esperada de las acciones ordinarias de Lilly3 | E(RLLY) | 7.43% |
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(RLLY) = RF + βLLY [E(RM) – RF]
= 4.83% + 0.30 [13.48% – 4.83%]
= 7.43%