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 Air Products & Chemicals Inc. common stock.
Rates of Return
Air Products & Chemicals Inc., monthly rates of return
| Air Products & Chemicals Inc. (APD) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceAPD,t1 | DividendAPD,t1 | RAPD,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Oct 31, 2014 | $134.66 | 2,018.05 | ||||
| 1. | Nov 30, 2014 | $143.83 | 6.81% | 2,067.56 | 2.45% | |
| 2. | Dec 31, 2014 | $144.23 | $0.77 | 0.81% | 2,058.90 | -0.42% |
| 3. | Jan 31, 2015 | $145.61 | 0.96% | 1,994.99 | -3.10% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | Aug 31, 2020 | $292.26 | 1.96% | 3,500.31 | 7.01% | |
| 71. | Sep 30, 2020 | $297.86 | $1.34 | 2.37% | 3,363.00 | -3.92% |
| Average (R): | 1.58% | 0.81% | ||||
| Standard deviation: | 5.21% | 4.16% | ||||
| Air Products & Chemicals Inc. (APD) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceAPD,t1 | DividendAPD,t1 | RAPD,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Oct 31, 2014 | $134.66 | 2,018.05 | ||||
| 1. | Nov 30, 2014 | $143.83 | 6.81% | 2,067.56 | 2.45% | |
| 2. | Dec 31, 2014 | $144.23 | $0.77 | 0.81% | 2,058.90 | -0.42% |
| 3. | Jan 31, 2015 | $145.61 | 0.96% | 1,994.99 | -3.10% | |
| 4. | Feb 28, 2015 | $156.14 | 7.23% | 2,104.50 | 5.49% | |
| 5. | Mar 31, 2015 | $151.28 | $0.81 | -2.59% | 2,067.89 | -1.74% |
| 6. | Apr 30, 2015 | $143.43 | -5.19% | 2,085.51 | 0.85% | |
| 7. | May 31, 2015 | $146.76 | 2.32% | 2,107.39 | 1.05% | |
| 8. | Jun 30, 2015 | $136.83 | $0.81 | -6.21% | 2,063.11 | -2.10% |
| 9. | Jul 31, 2015 | $142.51 | 4.15% | 2,103.84 | 1.97% | |
| 10. | Aug 31, 2015 | $139.53 | -2.09% | 1,972.18 | -6.26% | |
| 11. | Sep 30, 2015 | $127.58 | $0.81 | -7.98% | 1,920.03 | -2.64% |
| 12. | Oct 31, 2015 | $138.98 | 8.94% | 2,079.36 | 8.30% | |
| 13. | Nov 30, 2015 | $136.89 | -1.50% | 2,080.41 | 0.05% | |
| 14. | Dec 31, 2015 | $130.11 | $0.81 | -4.36% | 2,043.94 | -1.75% |
| 15. | Jan 31, 2016 | $126.71 | -2.61% | 1,940.24 | -5.07% | |
| 16. | Feb 29, 2016 | $132.47 | 4.55% | 1,932.23 | -0.41% | |
| 17. | Mar 31, 2016 | $144.05 | $0.86 | 9.39% | 2,059.74 | 6.60% |
| 18. | Apr 30, 2016 | $145.89 | 1.28% | 2,065.30 | 0.27% | |
| 19. | May 31, 2016 | $142.64 | -2.23% | 2,096.95 | 1.53% | |
| 20. | Jun 30, 2016 | $142.04 | $0.86 | 0.18% | 2,098.86 | 0.09% |
| 21. | Jul 31, 2016 | $149.42 | 5.20% | 2,173.60 | 3.56% | |
| 22. | Aug 31, 2016 | $155.62 | 4.15% | 2,170.95 | -0.12% | |
| 23. | Sep 30, 2016 | $150.34 | $0.86 | -2.84% | 2,168.27 | -0.12% |
| 24. | Oct 31, 2016 | $133.42 | $11.35 | -3.70% | 2,126.15 | -1.94% |
| 25. | Nov 30, 2016 | $144.46 | 8.27% | 2,198.81 | 3.42% | |
| 26. | Dec 31, 2016 | $143.82 | $0.86 | 0.15% | 2,238.83 | 1.82% |
| 27. | Jan 31, 2017 | $139.76 | -2.82% | 2,278.87 | 1.79% | |
| 28. | Feb 28, 2017 | $140.47 | 0.51% | 2,363.64 | 3.72% | |
| 29. | Mar 31, 2017 | $135.29 | $0.95 | -3.01% | 2,362.72 | -0.04% |
| 30. | Apr 30, 2017 | $140.50 | 3.85% | 2,384.20 | 0.91% | |
| 31. | May 31, 2017 | $144.06 | 2.53% | 2,411.80 | 1.16% | |
| 32. | Jun 30, 2017 | $143.06 | $0.95 | -0.03% | 2,423.41 | 0.48% |
| 33. | Jul 31, 2017 | $142.15 | -0.64% | 2,470.30 | 1.93% | |
| 34. | Aug 31, 2017 | $145.37 | 2.27% | 2,471.65 | 0.05% | |
| 35. | Sep 30, 2017 | $151.22 | 4.02% | 2,519.36 | 1.93% | |
| 36. | Oct 31, 2017 | $159.43 | $0.95 | 6.06% | 2,575.26 | 2.22% |
| 37. | Nov 30, 2017 | $163.04 | 2.26% | 2,647.58 | 2.81% | |
| 38. | Dec 31, 2017 | $164.08 | $0.95 | 1.22% | 2,673.61 | 0.98% |
| 39. | Jan 31, 2018 | $168.37 | 2.61% | 2,823.81 | 5.62% | |
| 40. | Feb 28, 2018 | $160.79 | -4.50% | 2,713.83 | -3.89% | |
| 41. | Mar 31, 2018 | $159.03 | $1.10 | -0.41% | 2,640.87 | -2.69% |
| 42. | Apr 30, 2018 | $162.29 | 2.05% | 2,648.05 | 0.27% | |
| 43. | May 31, 2018 | $161.41 | -0.54% | 2,705.27 | 2.16% | |
| 44. | Jun 30, 2018 | $155.73 | $1.10 | -2.84% | 2,718.37 | 0.48% |
| 45. | Jul 31, 2018 | $164.17 | 5.42% | 2,816.29 | 3.60% | |
| 46. | Aug 31, 2018 | $166.29 | 1.29% | 2,901.52 | 3.03% | |
| 47. | Sep 30, 2018 | $167.05 | $1.10 | 1.12% | 2,913.98 | 0.43% |
| 48. | Oct 31, 2018 | $154.35 | -7.60% | 2,711.74 | -6.94% | |
| 49. | Nov 30, 2018 | $160.87 | 4.22% | 2,760.17 | 1.79% | |
| 50. | Dec 31, 2018 | $160.05 | $1.10 | 0.17% | 2,506.85 | -9.18% |
| 51. | Jan 31, 2019 | $164.39 | 2.71% | 2,704.10 | 7.87% | |
| 52. | Feb 28, 2019 | $181.18 | 10.21% | 2,784.49 | 2.97% | |
| 53. | Mar 31, 2019 | $190.96 | $1.16 | 6.04% | 2,834.40 | 1.79% |
| 54. | Apr 30, 2019 | $205.79 | 7.77% | 2,945.83 | 3.93% | |
| 55. | May 31, 2019 | $203.59 | -1.07% | 2,752.06 | -6.58% | |
| 56. | Jun 30, 2019 | $226.37 | $1.16 | 11.76% | 2,941.76 | 6.89% |
| 57. | Jul 31, 2019 | $228.27 | 0.84% | 2,980.38 | 1.31% | |
| 58. | Aug 31, 2019 | $225.92 | -1.03% | 2,926.46 | -1.81% | |
| 59. | Sep 30, 2019 | $221.86 | $1.16 | -1.28% | 2,976.74 | 1.72% |
| 60. | Oct 31, 2019 | $213.26 | -3.88% | 3,037.56 | 2.04% | |
| 61. | Nov 30, 2019 | $236.33 | 10.82% | 3,140.98 | 3.40% | |
| 62. | Dec 31, 2019 | $234.99 | $1.16 | -0.08% | 3,230.78 | 2.86% |
| 63. | Jan 31, 2020 | $238.71 | 1.58% | 3,225.52 | -0.16% | |
| 64. | Feb 29, 2020 | $219.61 | -8.00% | 2,954.22 | -8.41% | |
| 65. | Mar 31, 2020 | $199.61 | $1.34 | -8.50% | 2,584.59 | -12.51% |
| 66. | Apr 30, 2020 | $225.58 | 13.01% | 2,912.43 | 12.68% | |
| 67. | May 31, 2020 | $241.65 | 7.12% | 3,044.31 | 4.53% | |
| 68. | Jun 30, 2020 | $241.46 | $1.34 | 0.48% | 3,100.29 | 1.84% |
| 69. | Jul 31, 2020 | $286.63 | 18.71% | 3,271.12 | 5.51% | |
| 70. | Aug 31, 2020 | $292.26 | 1.96% | 3,500.31 | 7.01% | |
| 71. | Sep 30, 2020 | $297.86 | $1.34 | 2.37% | 3,363.00 | -3.92% |
| Average (R): | 1.58% | 0.81% | ||||
| Standard deviation: | 5.21% | 4.16% | ||||
Show all
1 Data in US$ per share of common stock, adjusted for splits and stock dividends.
2 Rate of return on common stock of APD during period t
3 Rate of return on S&P 500 (the market portfolio proxy) during period t
Variance and Covariance
Air Products & Chemicals Inc., calculation of variance and covariance of returns
| t | Date | RAPD,t | RS&P 500,t | (RAPD,t–RAPD)2 | (RS&P 500,t–RS&P 500)2 | (RAPD,t–RAPD)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Nov 30, 2014 | 6.81% | 2.45% | 27.40 | 2.71 | 8.61 |
| 2. | Dec 31, 2014 | 0.81% | -0.42% | 0.58 | 1.50 | 0.93 |
| 3. | Jan 31, 2015 | 0.96% | -3.10% | 0.38 | 15.30 | 2.42 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 70. | Aug 31, 2020 | 1.96% | 7.01% | 0.15 | 38.42 | 2.41 |
| 71. | Sep 30, 2020 | 2.37% | -3.92% | 0.64 | 22.38 | -3.78 |
| Total (Σ): | 1,898.15 | 1,210.80 | 1,061.47 | |||
| t | Date | RAPD,t | RS&P 500,t | (RAPD,t–RAPD)2 | (RS&P 500,t–RS&P 500)2 | (RAPD,t–RAPD)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Nov 30, 2014 | 6.81% | 2.45% | 27.40 | 2.71 | 8.61 |
| 2. | Dec 31, 2014 | 0.81% | -0.42% | 0.58 | 1.50 | 0.93 |
| 3. | Jan 31, 2015 | 0.96% | -3.10% | 0.38 | 15.30 | 2.42 |
| 4. | Feb 28, 2015 | 7.23% | 5.49% | 32.00 | 21.92 | 26.48 |
| 5. | Mar 31, 2015 | -2.59% | -1.74% | 17.38 | 6.49 | 10.62 |
| 6. | Apr 30, 2015 | -5.19% | 0.85% | 45.75 | 0.00 | -0.30 |
| 7. | May 31, 2015 | 2.32% | 1.05% | 0.56 | 0.06 | 0.18 |
| 8. | Jun 30, 2015 | -6.21% | -2.10% | 60.67 | 8.46 | 22.66 |
| 9. | Jul 31, 2015 | 4.15% | 1.97% | 6.64 | 1.36 | 3.00 |
| 10. | Aug 31, 2015 | -2.09% | -6.26% | 13.44 | 49.93 | 25.91 |
| 11. | Sep 30, 2015 | -7.98% | -2.64% | 91.38 | 11.92 | 33.00 |
| 12. | Oct 31, 2015 | 8.94% | 8.30% | 54.18 | 56.11 | 55.13 |
| 13. | Nov 30, 2015 | -1.50% | 0.05% | 9.48 | 0.57 | 2.33 |
| 14. | Dec 31, 2015 | -4.36% | -1.75% | 35.24 | 6.56 | 15.20 |
| 15. | Jan 31, 2016 | -2.61% | -5.07% | 17.54 | 34.59 | 24.63 |
| 16. | Feb 29, 2016 | 4.55% | -0.41% | 8.83 | 1.49 | -3.63 |
| 17. | Mar 31, 2016 | 9.39% | 6.60% | 61.08 | 33.54 | 45.26 |
| 18. | Apr 30, 2016 | 1.28% | 0.27% | 0.09 | 0.29 | 0.16 |
| 19. | May 31, 2016 | -2.23% | 1.53% | 14.46 | 0.53 | -2.76 |
| 20. | Jun 30, 2016 | 0.18% | 0.09% | 1.94 | 0.51 | 1.00 |
| 21. | Jul 31, 2016 | 5.20% | 3.56% | 13.11 | 7.58 | 9.97 |
| 22. | Aug 31, 2016 | 4.15% | -0.12% | 6.63 | 0.86 | -2.39 |
| 23. | Sep 30, 2016 | -2.84% | -0.12% | 19.50 | 0.87 | 4.11 |
| 24. | Oct 31, 2016 | -3.70% | -1.94% | 27.88 | 7.56 | 14.52 |
| 25. | Nov 30, 2016 | 8.27% | 3.42% | 44.88 | 6.81 | 17.48 |
| 26. | Dec 31, 2016 | 0.15% | 1.82% | 2.02 | 1.02 | -1.44 |
| 27. | Jan 31, 2017 | -2.82% | 1.79% | 19.34 | 0.96 | -4.31 |
| 28. | Feb 28, 2017 | 0.51% | 3.72% | 1.14 | 8.48 | -3.11 |
| 29. | Mar 31, 2017 | -3.01% | -0.04% | 21.04 | 0.72 | 3.88 |
| 30. | Apr 30, 2017 | 3.85% | 0.91% | 5.18 | 0.01 | 0.23 |
| 31. | May 31, 2017 | 2.53% | 1.16% | 0.92 | 0.12 | 0.34 |
| 32. | Jun 30, 2017 | -0.03% | 0.48% | 2.59 | 0.11 | 0.53 |
| 33. | Jul 31, 2017 | -0.64% | 1.93% | 4.89 | 1.27 | -2.49 |
| 34. | Aug 31, 2017 | 2.27% | 0.05% | 0.48 | 0.57 | -0.52 |
| 35. | Sep 30, 2017 | 4.02% | 1.93% | 6.00 | 1.26 | 2.75 |
| 36. | Oct 31, 2017 | 6.06% | 2.22% | 20.09 | 1.99 | 6.32 |
| 37. | Nov 30, 2017 | 2.26% | 2.81% | 0.47 | 4.00 | 1.38 |
| 38. | Dec 31, 2017 | 1.22% | 0.98% | 0.13 | 0.03 | -0.06 |
| 39. | Jan 31, 2018 | 2.61% | 5.62% | 1.08 | 23.14 | 5.00 |
| 40. | Feb 28, 2018 | -4.50% | -3.89% | 36.93 | 22.11 | 28.58 |
| 41. | Mar 31, 2018 | -0.41% | -2.69% | 3.94 | 12.22 | 6.94 |
| 42. | Apr 30, 2018 | 2.05% | 0.27% | 0.23 | 0.29 | -0.25 |
| 43. | May 31, 2018 | -0.54% | 2.16% | 4.48 | 1.83 | -2.86 |
| 44. | Jun 30, 2018 | -2.84% | 0.48% | 19.47 | 0.10 | 1.43 |
| 45. | Jul 31, 2018 | 5.42% | 3.60% | 14.78 | 7.81 | 10.74 |
| 46. | Aug 31, 2018 | 1.29% | 3.03% | 0.08 | 4.92 | -0.63 |
| 47. | Sep 30, 2018 | 1.12% | 0.43% | 0.21 | 0.14 | 0.17 |
| 48. | Oct 31, 2018 | -7.60% | -6.94% | 84.23 | 60.03 | 71.11 |
| 49. | Nov 30, 2018 | 4.22% | 1.79% | 7.02 | 0.96 | 2.59 |
| 50. | Dec 31, 2018 | 0.17% | -9.18% | 1.96 | 99.71 | 13.99 |
| 51. | Jan 31, 2019 | 2.71% | 7.87% | 1.29 | 49.85 | 8.02 |
| 52. | Feb 28, 2019 | 10.21% | 2.97% | 74.62 | 4.69 | 18.70 |
| 53. | Mar 31, 2019 | 6.04% | 1.79% | 19.92 | 0.97 | 4.39 |
| 54. | Apr 30, 2019 | 7.77% | 3.93% | 38.33 | 9.76 | 19.34 |
| 55. | May 31, 2019 | -1.07% | -6.58% | 6.99 | 54.55 | 19.53 |
| 56. | Jun 30, 2019 | 11.76% | 6.89% | 103.71 | 37.03 | 61.97 |
| 57. | Jul 31, 2019 | 0.84% | 1.31% | 0.54 | 0.25 | -0.37 |
| 58. | Aug 31, 2019 | -1.03% | -1.81% | 6.78 | 6.85 | 6.82 |
| 59. | Sep 30, 2019 | -1.28% | 1.72% | 8.17 | 0.83 | -2.60 |
| 60. | Oct 31, 2019 | -3.88% | 2.04% | 29.72 | 1.53 | -6.73 |
| 61. | Nov 30, 2019 | 10.82% | 3.40% | 85.43 | 6.74 | 24.00 |
| 62. | Dec 31, 2019 | -0.08% | 2.86% | 2.73 | 4.21 | -3.39 |
| 63. | Jan 31, 2020 | 1.58% | -0.16% | 0.00 | 0.94 | -0.01 |
| 64. | Feb 29, 2020 | -8.00% | -8.41% | 91.71 | 84.99 | 88.28 |
| 65. | Mar 31, 2020 | -8.50% | -12.51% | 101.45 | 177.42 | 134.16 |
| 66. | Apr 30, 2020 | 13.01% | 12.68% | 130.76 | 141.05 | 135.81 |
| 67. | May 31, 2020 | 7.12% | 4.53% | 30.79 | 13.84 | 20.64 |
| 68. | Jun 30, 2020 | 0.48% | 1.84% | 1.21 | 1.06 | -1.13 |
| 69. | Jul 31, 2020 | 18.71% | 5.51% | 293.50 | 22.11 | 80.56 |
| 70. | Aug 31, 2020 | 1.96% | 7.01% | 0.15 | 38.42 | 2.41 |
| 71. | Sep 30, 2020 | 2.37% | -3.92% | 0.64 | 22.38 | -3.78 |
| Total (Σ): | 1,898.15 | 1,210.80 | 1,061.47 | |||
Show all
VarianceAPD = Σ(RAPD,t–RAPD)2 ÷ (71 – 1)
= 1,898.15 ÷ (71 – 1)
= 27.12
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (71 – 1)
= 1,210.80 ÷ (71 – 1)
= 17.30
CovarianceAPD, S&P 500 = Σ(RAPD,t–RAPD)×(RS&P 500,t–RS&P 500) ÷ (71 – 1)
= 1,061.47 ÷ (71 – 1)
= 15.16
Systematic Risk (β) Estimation
| VarianceAPD | 27.12 |
| VarianceS&P 500 | 17.30 |
| CovarianceAPD, S&P 500 | 15.16 |
| Correlation coefficientAPD, S&P 5001 | 0.70 |
| βAPD2 | 0.88 |
| αAPD3 | 0.87% |
Calculations
1 Correlation coefficientAPD, S&P 500
= CovarianceAPD, S&P 500 ÷ (Standard deviationAPD × Standard deviationS&P 500)
= 15.16 ÷ (5.21% × 4.16%)
= 0.70
2 βAPD
= CovarianceAPD, S&P 500 ÷ VarianceS&P 500
= 15.16 ÷ 17.30
= 0.88
3 αAPD
= AverageAPD – βAPD × AverageS&P 500
= 1.58% – 0.88 × 0.81%
= 0.87%
Expected Rate of Return
| Assumptions | ||
| Rate of return on LT Treasury Composite1 | RF | 4.86% |
| Expected rate of return on market portfolio2 | E(RM) | 13.54% |
| Systematic risk (β) of Air Products & Chemicals Inc. common stock | βAPD | 0.88 |
| Expected rate of return on Air Products & Chemicals Inc. common stock3 | E(RAPD) | 12.47% |
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(RAPD) = RF + βAPD [E(RM) – RF]
= 4.86% + 0.88 [13.54% – 4.86%]
= 12.47%