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 Trane Technologies plc common stock.
Rates of Return
Trane Technologies plc, monthly rates of return
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceTT,t1 | DividendTT,t1 | RTT,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Jan 31, 2018 | $94.63 | 2,823.81 | ||||
| 1. | Feb 28, 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | Mar 31, 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | Apr 30, 2018 | $83.89 | -1.89% | 2,648.05 | 0.27% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2022 | $178.42 | 11.77% | 4,080.11 | 5.38% | |
| 59. | Dec 31, 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Average (R): | 1.92% | 0.67% | ||||
| Standard deviation: | 7.50% | 5.40% | ||||
| Trane Technologies plc (TT) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceTT,t1 | DividendTT,t1 | RTT,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Jan 31, 2018 | $94.63 | 2,823.81 | ||||
| 1. | Feb 28, 2018 | $88.80 | -6.16% | 2,713.83 | -3.89% | |
| 2. | Mar 31, 2018 | $85.51 | $0.45 | -3.20% | 2,640.87 | -2.69% |
| 3. | Apr 30, 2018 | $83.89 | -1.89% | 2,648.05 | 0.27% | |
| 4. | May 31, 2018 | $87.54 | 4.35% | 2,705.27 | 2.16% | |
| 5. | Jun 30, 2018 | $89.73 | $0.45 | 3.02% | 2,718.37 | 0.48% |
| 6. | Jul 31, 2018 | $98.51 | 9.78% | 2,816.29 | 3.60% | |
| 7. | Aug 31, 2018 | $101.29 | 2.82% | 2,901.52 | 3.03% | |
| 8. | Sep 30, 2018 | $102.30 | $0.53 | 1.52% | 2,913.98 | 0.43% |
| 9. | Oct 31, 2018 | $95.94 | -6.22% | 2,711.74 | -6.94% | |
| 10. | Nov 30, 2018 | $103.52 | 7.90% | 2,760.17 | 1.79% | |
| 11. | Dec 31, 2018 | $91.23 | $0.53 | -11.36% | 2,506.85 | -9.18% |
| 12. | Jan 31, 2019 | $100.04 | 9.66% | 2,704.10 | 7.87% | |
| 13. | Feb 28, 2019 | $105.56 | 5.52% | 2,784.49 | 2.97% | |
| 14. | Mar 31, 2019 | $107.95 | $0.53 | 2.77% | 2,834.40 | 1.79% |
| 15. | Apr 30, 2019 | $122.61 | 13.58% | 2,945.83 | 3.93% | |
| 16. | May 31, 2019 | $118.34 | -3.48% | 2,752.06 | -6.58% | |
| 17. | Jun 30, 2019 | $126.67 | $0.53 | 7.49% | 2,941.76 | 6.89% |
| 18. | Jul 31, 2019 | $123.66 | -2.38% | 2,980.38 | 1.31% | |
| 19. | Aug 31, 2019 | $121.09 | -2.08% | 2,926.46 | -1.81% | |
| 20. | Sep 30, 2019 | $123.21 | $0.53 | 2.19% | 2,976.74 | 1.72% |
| 21. | Oct 31, 2019 | $126.89 | 2.99% | 3,037.56 | 2.04% | |
| 22. | Nov 30, 2019 | $131.11 | 3.33% | 3,140.98 | 3.40% | |
| 23. | Dec 31, 2019 | $132.92 | $0.53 | 1.78% | 3,230.78 | 2.86% |
| 24. | Jan 31, 2020 | $133.23 | 0.23% | 3,225.52 | -0.16% | |
| 25. | Feb 29, 2020 | $129.04 | -3.14% | 2,954.22 | -8.41% | |
| 26. | Mar 31, 2020 | $82.59 | $29.46 | -13.17% | 2,584.59 | -12.51% |
| 27. | Apr 30, 2020 | $87.42 | 5.85% | 2,912.43 | 12.68% | |
| 28. | May 31, 2020 | $90.21 | 3.19% | 3,044.31 | 4.53% | |
| 29. | Jun 30, 2020 | $88.98 | $0.53 | -0.78% | 3,100.29 | 1.84% |
| 30. | Jul 31, 2020 | $111.87 | 25.72% | 3,271.12 | 5.51% | |
| 31. | Aug 31, 2020 | $118.39 | 5.83% | 3,500.31 | 7.01% | |
| 32. | Sep 30, 2020 | $121.25 | $0.53 | 2.86% | 3,363.00 | -3.92% |
| 33. | Oct 31, 2020 | $132.75 | 9.48% | 3,269.96 | -2.77% | |
| 34. | Nov 30, 2020 | $146.24 | 10.16% | 3,621.63 | 10.75% | |
| 35. | Dec 31, 2020 | $145.16 | $0.53 | -0.38% | 3,756.07 | 3.71% |
| 36. | Jan 31, 2021 | $143.35 | -1.25% | 3,714.24 | -1.11% | |
| 37. | Feb 28, 2021 | $153.24 | 6.90% | 3,811.15 | 2.61% | |
| 38. | Mar 31, 2021 | $165.56 | $0.59 | 8.42% | 3,972.89 | 4.24% |
| 39. | Apr 30, 2021 | $173.83 | 5.00% | 4,181.17 | 5.24% | |
| 40. | May 31, 2021 | $186.40 | 7.23% | 4,204.11 | 0.55% | |
| 41. | Jun 30, 2021 | $184.14 | $0.59 | -0.90% | 4,297.50 | 2.22% |
| 42. | Jul 31, 2021 | $203.61 | 10.57% | 4,395.26 | 2.27% | |
| 43. | Aug 31, 2021 | $198.50 | -2.51% | 4,522.68 | 2.90% | |
| 44. | Sep 30, 2021 | $172.65 | $0.59 | -12.73% | 4,307.54 | -4.76% |
| 45. | Oct 31, 2021 | $180.93 | 4.80% | 4,605.38 | 6.91% | |
| 46. | Nov 30, 2021 | $186.65 | 3.16% | 4,567.00 | -0.83% | |
| 47. | Dec 31, 2021 | $202.03 | $0.59 | 8.56% | 4,766.18 | 4.36% |
| 48. | Jan 31, 2022 | $173.10 | -14.32% | 4,515.55 | -5.26% | |
| 49. | Feb 28, 2022 | $153.93 | -11.07% | 4,373.94 | -3.14% | |
| 50. | Mar 31, 2022 | $152.70 | $0.67 | -0.36% | 4,530.41 | 3.58% |
| 51. | Apr 30, 2022 | $139.89 | -8.39% | 4,131.93 | -8.80% | |
| 52. | May 31, 2022 | $138.06 | -1.31% | 4,132.15 | 0.01% | |
| 53. | Jun 30, 2022 | $129.87 | $0.67 | -5.45% | 3,785.38 | -8.39% |
| 54. | Jul 31, 2022 | $146.99 | 13.18% | 4,130.29 | 9.11% | |
| 55. | Aug 31, 2022 | $154.07 | 4.82% | 3,955.00 | -4.24% | |
| 56. | Sep 30, 2022 | $144.81 | $0.67 | -5.58% | 3,585.62 | -9.34% |
| 57. | Oct 31, 2022 | $159.63 | 10.23% | 3,871.98 | 7.99% | |
| 58. | Nov 30, 2022 | $178.42 | 11.77% | 4,080.11 | 5.38% | |
| 59. | Dec 31, 2022 | $168.09 | $0.67 | -5.41% | 3,839.50 | -5.90% |
| Average (R): | 1.92% | 0.67% | ||||
| Standard deviation: | 7.50% | 5.40% | ||||
Show all
1 Data in US$ per share of common stock, adjusted for splits and stock dividends.
2 Rate of return on common stock of TT during period t
3 Rate of return on S&P 500 (the market portfolio proxy) during period t
Variance and Covariance
Trane Technologies plc, calculation of variance and covariance of returns
| t | Date | RTT,t | RS&P 500,t | (RTT,t–RTT)2 | (RS&P 500,t–RS&P 500)2 | (RTT,t–RTT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Feb 28, 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | Mar 31, 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | Apr 30, 2018 | -1.89% | 0.27% | 14.54 | 0.16 | 1.51 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2022 | 11.77% | 5.38% | 97.08 | 22.17 | 46.39 |
| 59. | Dec 31, 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
| t | Date | RTT,t | RS&P 500,t | (RTT,t–RTT)2 | (RS&P 500,t–RS&P 500)2 | (RTT,t–RTT)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Feb 28, 2018 | -6.16% | -3.89% | 65.27 | 20.81 | 36.85 |
| 2. | Mar 31, 2018 | -3.20% | -2.69% | 26.18 | 11.26 | 17.17 |
| 3. | Apr 30, 2018 | -1.89% | 0.27% | 14.54 | 0.16 | 1.51 |
| 4. | May 31, 2018 | 4.35% | 2.16% | 5.92 | 2.23 | 3.64 |
| 5. | Jun 30, 2018 | 3.02% | 0.48% | 1.20 | 0.03 | -0.20 |
| 6. | Jul 31, 2018 | 9.78% | 3.60% | 61.89 | 8.62 | 23.09 |
| 7. | Aug 31, 2018 | 2.82% | 3.03% | 0.82 | 5.57 | 2.13 |
| 8. | Sep 30, 2018 | 1.52% | 0.43% | 0.16 | 0.06 | 0.09 |
| 9. | Oct 31, 2018 | -6.22% | -6.94% | 66.18 | 57.87 | 61.88 |
| 10. | Nov 30, 2018 | 7.90% | 1.79% | 35.79 | 1.25 | 6.70 |
| 11. | Dec 31, 2018 | -11.36% | -9.18% | 176.31 | 96.91 | 130.71 |
| 12. | Jan 31, 2019 | 9.66% | 7.87% | 59.89 | 51.87 | 55.73 |
| 13. | Feb 28, 2019 | 5.52% | 2.97% | 12.96 | 5.32 | 8.30 |
| 14. | Mar 31, 2019 | 2.77% | 1.79% | 0.72 | 1.27 | 0.95 |
| 15. | Apr 30, 2019 | 13.58% | 3.93% | 136.01 | 10.66 | 38.07 |
| 16. | May 31, 2019 | -3.48% | -6.58% | 29.17 | 52.48 | 39.12 |
| 17. | Jun 30, 2019 | 7.49% | 6.89% | 31.01 | 38.77 | 34.67 |
| 18. | Jul 31, 2019 | -2.38% | 1.31% | 18.44 | 0.42 | -2.77 |
| 19. | Aug 31, 2019 | -2.08% | -1.81% | 15.97 | 6.13 | 9.89 |
| 20. | Sep 30, 2019 | 2.19% | 1.72% | 0.07 | 1.11 | 0.28 |
| 21. | Oct 31, 2019 | 2.99% | 2.04% | 1.14 | 1.89 | 1.47 |
| 22. | Nov 30, 2019 | 3.33% | 3.40% | 1.98 | 7.50 | 3.85 |
| 23. | Dec 31, 2019 | 1.78% | 2.86% | 0.02 | 4.81 | -0.29 |
| 24. | Jan 31, 2020 | 0.23% | -0.16% | 2.84 | 0.69 | 1.40 |
| 25. | Feb 29, 2020 | -3.14% | -8.41% | 25.63 | 82.40 | 45.96 |
| 26. | Mar 31, 2020 | -13.17% | -12.51% | 227.54 | 173.67 | 198.79 |
| 27. | Apr 30, 2020 | 5.85% | 12.68% | 15.45 | 144.43 | 47.23 |
| 28. | May 31, 2020 | 3.19% | 4.53% | 1.62 | 14.91 | 4.92 |
| 29. | Jun 30, 2020 | -0.78% | 1.84% | 7.26 | 1.37 | -3.16 |
| 30. | Jul 31, 2020 | 25.72% | 5.51% | 566.77 | 23.46 | 115.31 |
| 31. | Aug 31, 2020 | 5.83% | 7.01% | 15.29 | 40.19 | 24.79 |
| 32. | Sep 30, 2020 | 2.86% | -3.92% | 0.89 | 21.06 | -4.34 |
| 33. | Oct 31, 2020 | 9.48% | -2.77% | 57.25 | 11.79 | -25.98 |
| 34. | Nov 30, 2020 | 10.16% | 10.75% | 67.96 | 101.77 | 83.16 |
| 35. | Dec 31, 2020 | -0.38% | 3.71% | 5.26 | 9.28 | -6.99 |
| 36. | Jan 31, 2021 | -1.25% | -1.11% | 10.02 | 3.17 | 5.63 |
| 37. | Feb 28, 2021 | 6.90% | 2.61% | 24.81 | 3.77 | 9.68 |
| 38. | Mar 31, 2021 | 8.42% | 4.24% | 42.34 | 12.80 | 23.28 |
| 39. | Apr 30, 2021 | 5.00% | 5.24% | 9.47 | 20.94 | 14.08 |
| 40. | May 31, 2021 | 7.23% | 0.55% | 28.23 | 0.01 | -0.63 |
| 41. | Jun 30, 2021 | -0.90% | 2.22% | 7.92 | 2.42 | -4.38 |
| 42. | Jul 31, 2021 | 10.57% | 2.27% | 74.92 | 2.59 | 13.92 |
| 43. | Aug 31, 2021 | -2.51% | 2.90% | 19.60 | 4.98 | -9.88 |
| 44. | Sep 30, 2021 | -12.73% | -4.76% | 214.43 | 29.42 | 79.42 |
| 45. | Oct 31, 2021 | 4.80% | 6.91% | 8.28 | 39.03 | 17.98 |
| 46. | Nov 30, 2021 | 3.16% | -0.83% | 1.55 | 2.25 | -1.87 |
| 47. | Dec 31, 2021 | 8.56% | 4.36% | 44.06 | 13.65 | 24.53 |
| 48. | Jan 31, 2022 | -14.32% | -5.26% | 263.66 | 35.11 | 96.21 |
| 49. | Feb 28, 2022 | -11.07% | -3.14% | 168.81 | 14.46 | 49.41 |
| 50. | Mar 31, 2022 | -0.36% | 3.58% | 5.21 | 8.47 | -6.64 |
| 51. | Apr 30, 2022 | -8.39% | -8.80% | 106.24 | 89.54 | 97.53 |
| 52. | May 31, 2022 | -1.31% | 0.01% | 10.41 | 0.44 | 2.13 |
| 53. | Jun 30, 2022 | -5.45% | -8.39% | 54.24 | 82.06 | 66.72 |
| 54. | Jul 31, 2022 | 13.18% | 9.11% | 126.89 | 71.32 | 95.13 |
| 55. | Aug 31, 2022 | 4.82% | -4.24% | 8.40 | 24.11 | -14.23 |
| 56. | Sep 30, 2022 | -5.58% | -9.34% | 56.15 | 100.12 | 74.98 |
| 57. | Oct 31, 2022 | 10.23% | 7.99% | 69.16 | 53.58 | 60.87 |
| 58. | Nov 30, 2022 | 11.77% | 5.38% | 97.08 | 22.17 | 46.39 |
| 59. | Dec 31, 2022 | -5.41% | -5.90% | 53.76 | 43.08 | 48.13 |
| Total (Σ): | 3,261.02 | 1,691.48 | 1,742.35 | |||
Show all
VarianceTT = Σ(RTT,t–RTT)2 ÷ (59 – 1)
= 3,261.02 ÷ (59 – 1)
= 56.22
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,691.48 ÷ (59 – 1)
= 29.16
CovarianceTT, S&P 500 = Σ(RTT,t–RTT)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,742.35 ÷ (59 – 1)
= 30.04
Systematic Risk (β) Estimation
| VarianceTT | 56.22 |
| VarianceS&P 500 | 29.16 |
| CovarianceTT, S&P 500 | 30.04 |
| Correlation coefficientTT, S&P 5001 | 0.74 |
| βTT2 | 1.03 |
| αTT3 | 1.23% |
Calculations
1 Correlation coefficientTT, S&P 500
= CovarianceTT, S&P 500 ÷ (Standard deviationTT × Standard deviationS&P 500)
= 30.04 ÷ (7.50% × 5.40%)
= 0.74
2 βTT
= CovarianceTT, S&P 500 ÷ VarianceS&P 500
= 30.04 ÷ 29.16
= 1.03
3 αTT
= AverageTT – βTT × AverageS&P 500
= 1.92% – 1.03 × 0.67%
= 1.23%
Expected Rate of Return
| Assumptions | ||
| Rate of return on LT Treasury Composite1 | RF | 4.62% |
| Expected rate of return on market portfolio2 | E(RM) | 13.69% |
| Systematic risk (β) of Trane Technologies plc common stock | βTT | 1.03 |
| Expected rate of return on Trane Technologies plc common stock3 | E(RTT) | 13.96% |
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(RTT) = RF + βTT [E(RM) – RF]
= 4.62% + 1.03 [13.69% – 4.62%]
= 13.96%