Capital Asset Pricing Model (CAPM)
Capital asset pricing model (CAPM) indicates what should be the expected or required rate of return on risky assets like Johnson Controls International plc common stock.
Rates of Return
Johnson Controls International plc, monthly rates of return
| Johnson Controls International plc (JCI) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceJCI,t1 | DividendJCI,t1 | RJCI,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Oct 31, 2017 | $41.39 | 2,575.26 | ||||
| 1. | Nov 30, 2017 | $37.64 | -9.06% | 2,647.58 | 2.81% | |
| 2. | Dec 31, 2017 | $38.11 | $0.26 | 1.94% | 2,673.61 | 0.98% |
| 3. | Jan 31, 2018 | $39.13 | 2.68% | 2,823.81 | 5.62% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | Aug 31, 2023 | $59.06 | -15.08% | 4,507.66 | -1.77% | |
| 71. | Sep 30, 2023 | $53.21 | $0.37 | -9.28% | 4,288.05 | -4.87% |
| Average (R): | 0.92% | 0.85% | ||||
| Standard deviation: | 8.29% | 5.13% | ||||
| Johnson Controls International plc (JCI) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceJCI,t1 | DividendJCI,t1 | RJCI,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Oct 31, 2017 | $41.39 | 2,575.26 | ||||
| 1. | Nov 30, 2017 | $37.64 | -9.06% | 2,647.58 | 2.81% | |
| 2. | Dec 31, 2017 | $38.11 | $0.26 | 1.94% | 2,673.61 | 0.98% |
| 3. | Jan 31, 2018 | $39.13 | 2.68% | 2,823.81 | 5.62% | |
| 4. | Feb 28, 2018 | $36.87 | -5.78% | 2,713.83 | -3.89% | |
| 5. | Mar 31, 2018 | $35.24 | $0.26 | -3.72% | 2,640.87 | -2.69% |
| 6. | Apr 30, 2018 | $33.87 | -3.89% | 2,648.05 | 0.27% | |
| 7. | May 31, 2018 | $33.56 | -0.92% | 2,705.27 | 2.16% | |
| 8. | Jun 30, 2018 | $33.45 | $0.26 | 0.45% | 2,718.37 | 0.48% |
| 9. | Jul 31, 2018 | $37.51 | 12.14% | 2,816.29 | 3.60% | |
| 10. | Aug 31, 2018 | $37.77 | 0.69% | 2,901.52 | 3.03% | |
| 11. | Sep 30, 2018 | $35.00 | $0.26 | -6.65% | 2,913.98 | 0.43% |
| 12. | Oct 31, 2018 | $31.97 | -8.66% | 2,711.74 | -6.94% | |
| 13. | Nov 30, 2018 | $34.78 | 8.79% | 2,760.17 | 1.79% | |
| 14. | Dec 31, 2018 | $29.65 | $0.26 | -14.00% | 2,506.85 | -9.18% |
| 15. | Jan 31, 2019 | $33.77 | 13.90% | 2,704.10 | 7.87% | |
| 16. | Feb 28, 2019 | $35.27 | 4.44% | 2,784.49 | 2.97% | |
| 17. | Mar 31, 2019 | $36.94 | $0.26 | 5.47% | 2,834.40 | 1.79% |
| 18. | Apr 30, 2019 | $37.50 | 1.52% | 2,945.83 | 3.93% | |
| 19. | May 31, 2019 | $38.52 | 2.72% | 2,752.06 | -6.58% | |
| 20. | Jun 30, 2019 | $41.31 | $0.26 | 7.92% | 2,941.76 | 6.89% |
| 21. | Jul 31, 2019 | $42.44 | 2.74% | 2,980.38 | 1.31% | |
| 22. | Aug 31, 2019 | $42.69 | 0.59% | 2,926.46 | -1.81% | |
| 23. | Sep 30, 2019 | $43.89 | $0.26 | 3.42% | 2,976.74 | 1.72% |
| 24. | Oct 31, 2019 | $43.33 | -1.28% | 3,037.56 | 2.04% | |
| 25. | Nov 30, 2019 | $42.83 | -1.15% | 3,140.98 | 3.40% | |
| 26. | Dec 31, 2019 | $40.71 | $0.26 | -4.34% | 3,230.78 | 2.86% |
| 27. | Jan 31, 2020 | $39.45 | -3.10% | 3,225.52 | -0.16% | |
| 28. | Feb 29, 2020 | $36.57 | -7.30% | 2,954.22 | -8.41% | |
| 29. | Mar 31, 2020 | $26.96 | $0.26 | -25.57% | 2,584.59 | -12.51% |
| 30. | Apr 30, 2020 | $29.11 | 7.97% | 2,912.43 | 12.68% | |
| 31. | May 31, 2020 | $31.41 | 7.90% | 3,044.31 | 4.53% | |
| 32. | Jun 30, 2020 | $34.14 | $0.26 | 9.52% | 3,100.29 | 1.84% |
| 33. | Jul 31, 2020 | $38.48 | 12.71% | 3,271.12 | 5.51% | |
| 34. | Aug 31, 2020 | $40.73 | 5.85% | 3,500.31 | 7.01% | |
| 35. | Sep 30, 2020 | $40.85 | $0.26 | 0.93% | 3,363.00 | -3.92% |
| 36. | Oct 31, 2020 | $42.21 | 3.33% | 3,269.96 | -2.77% | |
| 37. | Nov 30, 2020 | $46.04 | 9.07% | 3,621.63 | 10.75% | |
| 38. | Dec 31, 2020 | $46.59 | $0.26 | 1.76% | 3,756.07 | 3.71% |
| 39. | Jan 31, 2021 | $49.82 | 6.93% | 3,714.24 | -1.11% | |
| 40. | Feb 28, 2021 | $55.79 | 11.98% | 3,811.15 | 2.61% | |
| 41. | Mar 31, 2021 | $59.67 | $0.27 | 7.44% | 3,972.89 | 4.24% |
| 42. | Apr 30, 2021 | $62.34 | 4.47% | 4,181.17 | 5.24% | |
| 43. | May 31, 2021 | $66.54 | 6.74% | 4,204.11 | 0.55% | |
| 44. | Jun 30, 2021 | $68.63 | $0.27 | 3.55% | 4,297.50 | 2.22% |
| 45. | Jul 31, 2021 | $71.42 | 4.07% | 4,395.26 | 2.27% | |
| 46. | Aug 31, 2021 | $74.80 | 4.73% | 4,522.68 | 2.90% | |
| 47. | Sep 30, 2021 | $68.08 | $0.27 | -8.62% | 4,307.54 | -4.76% |
| 48. | Oct 31, 2021 | $73.37 | 7.77% | 4,605.38 | 6.91% | |
| 49. | Nov 30, 2021 | $74.76 | 1.89% | 4,567.00 | -0.83% | |
| 50. | Dec 31, 2021 | $81.31 | $0.34 | 9.22% | 4,766.18 | 4.36% |
| 51. | Jan 31, 2022 | $72.67 | -10.63% | 4,515.55 | -5.26% | |
| 52. | Feb 28, 2022 | $64.96 | -10.61% | 4,373.94 | -3.14% | |
| 53. | Mar 31, 2022 | $65.57 | $0.35 | 1.48% | 4,530.41 | 3.58% |
| 54. | Apr 30, 2022 | $59.87 | -8.69% | 4,131.93 | -8.80% | |
| 55. | May 31, 2022 | $54.51 | -8.95% | 4,132.15 | 0.01% | |
| 56. | Jun 30, 2022 | $47.88 | $0.35 | -11.52% | 3,785.38 | -8.39% |
| 57. | Jul 31, 2022 | $53.91 | 12.59% | 4,130.29 | 9.11% | |
| 58. | Aug 31, 2022 | $54.14 | 0.43% | 3,955.00 | -4.24% | |
| 59. | Sep 30, 2022 | $49.22 | $0.35 | -8.44% | 3,585.62 | -9.34% |
| 60. | Oct 31, 2022 | $57.84 | 17.51% | 3,871.98 | 7.99% | |
| 61. | Nov 30, 2022 | $66.44 | 14.87% | 4,080.11 | 5.38% | |
| 62. | Dec 31, 2022 | $64.00 | $0.35 | -3.15% | 3,839.50 | -5.90% |
| 63. | Jan 31, 2023 | $69.57 | 8.70% | 4,076.60 | 6.18% | |
| 64. | Feb 28, 2023 | $62.72 | -9.85% | 3,970.15 | -2.61% | |
| 65. | Mar 31, 2023 | $60.22 | $0.36 | -3.41% | 4,109.31 | 3.51% |
| 66. | Apr 30, 2023 | $59.84 | -0.63% | 4,169.48 | 1.46% | |
| 67. | May 31, 2023 | $59.70 | -0.23% | 4,179.83 | 0.25% | |
| 68. | Jun 30, 2023 | $68.14 | $0.37 | 14.76% | 4,376.86 | 4.71% |
| 69. | Jul 31, 2023 | $69.55 | 2.07% | 4,588.96 | 4.85% | |
| 70. | Aug 31, 2023 | $59.06 | -15.08% | 4,507.66 | -1.77% | |
| 71. | Sep 30, 2023 | $53.21 | $0.37 | -9.28% | 4,288.05 | -4.87% |
| Average (R): | 0.92% | 0.85% | ||||
| Standard deviation: | 8.29% | 5.13% | ||||
Show all
1 Data in US$ per share of common stock, adjusted for splits and stock dividends.
2 Rate of return on common stock of JCI during period t
3 Rate of return on S&P 500 (the market portfolio proxy) during period t
Variance and Covariance
Johnson Controls International plc, calculation of variance and covariance of returns
| t | Date | RJCI,t | RS&P 500,t | (RJCI,t–RJCI)2 | (RS&P 500,t–RS&P 500)2 | (RJCI,t–RJCI)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Nov 30, 2017 | -9.06% | 2.81% | 99.56 | 3.83 | -19.53 |
| 2. | Dec 31, 2017 | 1.94% | 0.98% | 1.04 | 0.02 | 0.13 |
| 3. | Jan 31, 2018 | 2.68% | 5.62% | 3.09 | 22.72 | 8.38 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | Aug 31, 2023 | -15.08% | -1.77% | 256.01 | 6.88 | 41.97 |
| 71. | Sep 30, 2023 | -9.28% | -4.87% | 103.96 | 32.75 | 58.36 |
| Total (Σ): | 4,811.07 | 1,838.69 | 2,278.06 | |||
| t | Date | RJCI,t | RS&P 500,t | (RJCI,t–RJCI)2 | (RS&P 500,t–RS&P 500)2 | (RJCI,t–RJCI)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Nov 30, 2017 | -9.06% | 2.81% | 99.56 | 3.83 | -19.53 |
| 2. | Dec 31, 2017 | 1.94% | 0.98% | 1.04 | 0.02 | 0.13 |
| 3. | Jan 31, 2018 | 2.68% | 5.62% | 3.09 | 22.72 | 8.38 |
| 4. | Feb 28, 2018 | -5.78% | -3.89% | 44.80 | 22.52 | 31.77 |
| 5. | Mar 31, 2018 | -3.72% | -2.69% | 21.47 | 12.53 | 16.40 |
| 6. | Apr 30, 2018 | -3.89% | 0.27% | 23.09 | 0.34 | 2.78 |
| 7. | May 31, 2018 | -0.92% | 2.16% | 3.36 | 1.71 | -2.40 |
| 8. | Jun 30, 2018 | 0.45% | 0.48% | 0.22 | 0.13 | 0.17 |
| 9. | Jul 31, 2018 | 12.14% | 3.60% | 125.89 | 7.57 | 30.86 |
| 10. | Aug 31, 2018 | 0.69% | 3.03% | 0.05 | 4.73 | -0.49 |
| 11. | Sep 30, 2018 | -6.65% | 0.43% | 57.20 | 0.18 | 3.19 |
| 12. | Oct 31, 2018 | -8.66% | -6.94% | 91.68 | 60.71 | 74.60 |
| 13. | Nov 30, 2018 | 8.79% | 1.79% | 61.97 | 0.87 | 7.36 |
| 14. | Dec 31, 2018 | -14.00% | -9.18% | 222.60 | 100.58 | 149.63 |
| 15. | Jan 31, 2019 | 13.90% | 7.87% | 168.42 | 49.24 | 91.07 |
| 16. | Feb 28, 2019 | 4.44% | 2.97% | 12.42 | 4.50 | 7.48 |
| 17. | Mar 31, 2019 | 5.47% | 1.79% | 20.74 | 0.89 | 4.29 |
| 18. | Apr 30, 2019 | 1.52% | 3.93% | 0.36 | 9.49 | 1.84 |
| 19. | May 31, 2019 | 2.72% | -6.58% | 3.25 | 55.19 | -13.39 |
| 20. | Jun 30, 2019 | 7.92% | 6.89% | 49.00 | 36.50 | 42.29 |
| 21. | Jul 31, 2019 | 2.74% | 1.31% | 3.30 | 0.21 | 0.84 |
| 22. | Aug 31, 2019 | 0.59% | -1.81% | 0.11 | 7.08 | 0.87 |
| 23. | Sep 30, 2019 | 3.42% | 1.72% | 6.26 | 0.75 | 2.17 |
| 24. | Oct 31, 2019 | -1.28% | 2.04% | 4.81 | 1.42 | -2.61 |
| 25. | Nov 30, 2019 | -1.15% | 3.40% | 4.29 | 6.52 | -5.29 |
| 26. | Dec 31, 2019 | -4.34% | 2.86% | 27.67 | 4.03 | -10.56 |
| 27. | Jan 31, 2020 | -3.10% | -0.16% | 16.10 | 1.03 | 4.07 |
| 28. | Feb 29, 2020 | -7.30% | -8.41% | 67.54 | 85.79 | 76.12 |
| 29. | Mar 31, 2020 | -25.57% | -12.51% | 701.46 | 178.57 | 353.92 |
| 30. | Apr 30, 2020 | 7.97% | 12.68% | 49.80 | 140.02 | 83.51 |
| 31. | May 31, 2020 | 7.90% | 4.53% | 48.77 | 13.52 | 25.68 |
| 32. | Jun 30, 2020 | 9.52% | 1.84% | 73.99 | 0.98 | 8.49 |
| 33. | Jul 31, 2020 | 12.71% | 5.51% | 139.12 | 21.71 | 54.95 |
| 34. | Aug 31, 2020 | 5.85% | 7.01% | 24.30 | 37.89 | 30.34 |
| 35. | Sep 30, 2020 | 0.93% | -3.92% | 0.00 | 22.79 | -0.07 |
| 36. | Oct 31, 2020 | 3.33% | -2.77% | 5.82 | 13.09 | -8.72 |
| 37. | Nov 30, 2020 | 9.07% | 10.75% | 66.52 | 98.08 | 80.77 |
| 38. | Dec 31, 2020 | 1.76% | 3.71% | 0.71 | 8.18 | 2.41 |
| 39. | Jan 31, 2021 | 6.93% | -1.11% | 36.18 | 3.86 | -11.82 |
| 40. | Feb 28, 2021 | 11.98% | 2.61% | 122.45 | 3.09 | 19.45 |
| 41. | Mar 31, 2021 | 7.44% | 4.24% | 42.52 | 11.51 | 22.12 |
| 42. | Apr 30, 2021 | 4.47% | 5.24% | 12.65 | 19.28 | 15.62 |
| 43. | May 31, 2021 | 6.74% | 0.55% | 33.87 | 0.09 | -1.76 |
| 44. | Jun 30, 2021 | 3.55% | 2.22% | 6.91 | 1.88 | 3.60 |
| 45. | Jul 31, 2021 | 4.07% | 2.27% | 9.91 | 2.03 | 4.48 |
| 46. | Aug 31, 2021 | 4.73% | 2.90% | 14.55 | 4.19 | 7.81 |
| 47. | Sep 30, 2021 | -8.62% | -4.76% | 91.02 | 31.45 | 53.51 |
| 48. | Oct 31, 2021 | 7.77% | 6.91% | 46.96 | 36.76 | 41.55 |
| 49. | Nov 30, 2021 | 1.89% | -0.83% | 0.95 | 2.84 | -1.65 |
| 50. | Dec 31, 2021 | 9.22% | 4.36% | 68.87 | 12.32 | 29.13 |
| 51. | Jan 31, 2022 | -10.63% | -5.26% | 133.26 | 37.33 | 70.53 |
| 52. | Feb 28, 2022 | -10.61% | -3.14% | 132.88 | 15.90 | 45.96 |
| 53. | Mar 31, 2022 | 1.48% | 3.58% | 0.31 | 7.43 | 1.53 |
| 54. | Apr 30, 2022 | -8.69% | -8.80% | 92.36 | 93.06 | 92.71 |
| 55. | May 31, 2022 | -8.95% | 0.01% | 97.42 | 0.72 | 8.35 |
| 56. | Jun 30, 2022 | -11.52% | -8.39% | 154.71 | 85.44 | 114.97 |
| 57. | Jul 31, 2022 | 12.59% | 9.11% | 136.34 | 68.23 | 96.45 |
| 58. | Aug 31, 2022 | 0.43% | -4.24% | 0.24 | 25.96 | 2.50 |
| 59. | Sep 30, 2022 | -8.44% | -9.34% | 87.59 | 103.85 | 95.37 |
| 60. | Oct 31, 2022 | 17.51% | 7.99% | 275.41 | 50.91 | 118.41 |
| 61. | Nov 30, 2022 | 14.87% | 5.38% | 194.63 | 20.47 | 63.11 |
| 62. | Dec 31, 2022 | -3.15% | -5.90% | 16.51 | 45.54 | 27.42 |
| 63. | Jan 31, 2023 | 8.70% | 6.18% | 60.61 | 28.35 | 41.45 |
| 64. | Feb 28, 2023 | -9.85% | -2.61% | 115.86 | 11.99 | 37.27 |
| 65. | Mar 31, 2023 | -3.41% | 3.51% | 18.75 | 7.04 | -11.49 |
| 66. | Apr 30, 2023 | -0.63% | 1.46% | 2.40 | 0.38 | -0.95 |
| 67. | May 31, 2023 | -0.23% | 0.25% | 1.33 | 0.36 | 0.69 |
| 68. | Jun 30, 2023 | 14.76% | 4.71% | 191.53 | 14.92 | 53.46 |
| 69. | Jul 31, 2023 | 2.07% | 4.85% | 1.33 | 15.96 | 4.60 |
| 70. | Aug 31, 2023 | -15.08% | -1.77% | 256.01 | 6.88 | 41.97 |
| 71. | Sep 30, 2023 | -9.28% | -4.87% | 103.96 | 32.75 | 58.36 |
| Total (Σ): | 4,811.07 | 1,838.69 | 2,278.06 | |||
Show all
VarianceJCI = Σ(RJCI,t–RJCI)2 ÷ (71 – 1)
= 4,811.07 ÷ (71 – 1)
= 68.73
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,838.69 ÷ (71 – 1)
= 26.27
CovarianceJCI, S&P 500 = Σ(RJCI,t–RJCI)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 2,278.06 ÷ (71 – 1)
= 32.54
Systematic Risk (β) Estimation
| VarianceJCI | 68.73 |
| VarianceS&P 500 | 26.27 |
| CovarianceJCI, S&P 500 | 32.54 |
| Correlation coefficientJCI, S&P 5001 | 0.77 |
| βJCI2 | 1.24 |
| αJCI3 | -0.14% |
Calculations
1 Correlation coefficientJCI, S&P 500
= CovarianceJCI, S&P 500 ÷ (Standard deviationJCI × Standard deviationS&P 500)
= 32.54 ÷ (8.29% × 5.13%)
= 0.77
2 βJCI
= CovarianceJCI, S&P 500 ÷ VarianceS&P 500
= 32.54 ÷ 26.27
= 1.24
3 αJCI
= AverageJCI – βJCI × AverageS&P 500
= 0.92% – 1.24 × 0.85%
= -0.14%
Expected Rate of Return
| Assumptions | ||
| Rate of return on LT Treasury Composite1 | RF | 3.99% |
| Expected rate of return on market portfolio2 | E(RM) | 13.81% |
| Systematic risk (β) of Johnson Controls International plc common stock | βJCI | 1.24 |
| Expected rate of return on Johnson Controls International plc common stock3 | E(RJCI) | 16.15% |
1 Unweighted average of bid yields on all outstanding fixed-coupon U.S. Treasury bonds neither due or callable in less than 10 years (risk-free rate of return proxy).
3 E(RJCI) = RF + βJCI [E(RM) – RF]
= 3.99% + 1.24 [13.81% – 3.99%]
= 16.15%