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.
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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 | ||||||
| 1. | Feb 28, 2018 | |||||
| 2. | Mar 31, 2018 | |||||
| 3. | Apr 30, 2018 | |||||
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2022 | |||||
| 59. | Dec 31, 2022 | |||||
| Average (R): | ||||||
| Standard deviation: | ||||||
| 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 | ||||||
| 1. | Feb 28, 2018 | |||||
| 2. | Mar 31, 2018 | |||||
| 3. | Apr 30, 2018 | |||||
| 4. | May 31, 2018 | |||||
| 5. | Jun 30, 2018 | |||||
| 6. | Jul 31, 2018 | |||||
| 7. | Aug 31, 2018 | |||||
| 8. | Sep 30, 2018 | |||||
| 9. | Oct 31, 2018 | |||||
| 10. | Nov 30, 2018 | |||||
| 11. | Dec 31, 2018 | |||||
| 12. | Jan 31, 2019 | |||||
| 13. | Feb 28, 2019 | |||||
| 14. | Mar 31, 2019 | |||||
| 15. | Apr 30, 2019 | |||||
| 16. | May 31, 2019 | |||||
| 17. | Jun 30, 2019 | |||||
| 18. | Jul 31, 2019 | |||||
| 19. | Aug 31, 2019 | |||||
| 20. | Sep 30, 2019 | |||||
| 21. | Oct 31, 2019 | |||||
| 22. | Nov 30, 2019 | |||||
| 23. | Dec 31, 2019 | |||||
| 24. | Jan 31, 2020 | |||||
| 25. | Feb 29, 2020 | |||||
| 26. | Mar 31, 2020 | |||||
| 27. | Apr 30, 2020 | |||||
| 28. | May 31, 2020 | |||||
| 29. | Jun 30, 2020 | |||||
| 30. | Jul 31, 2020 | |||||
| 31. | Aug 31, 2020 | |||||
| 32. | Sep 30, 2020 | |||||
| 33. | Oct 31, 2020 | |||||
| 34. | Nov 30, 2020 | |||||
| 35. | Dec 31, 2020 | |||||
| 36. | Jan 31, 2021 | |||||
| 37. | Feb 28, 2021 | |||||
| 38. | Mar 31, 2021 | |||||
| 39. | Apr 30, 2021 | |||||
| 40. | May 31, 2021 | |||||
| 41. | Jun 30, 2021 | |||||
| 42. | Jul 31, 2021 | |||||
| 43. | Aug 31, 2021 | |||||
| 44. | Sep 30, 2021 | |||||
| 45. | Oct 31, 2021 | |||||
| 46. | Nov 30, 2021 | |||||
| 47. | Dec 31, 2021 | |||||
| 48. | Jan 31, 2022 | |||||
| 49. | Feb 28, 2022 | |||||
| 50. | Mar 31, 2022 | |||||
| 51. | Apr 30, 2022 | |||||
| 52. | May 31, 2022 | |||||
| 53. | Jun 30, 2022 | |||||
| 54. | Jul 31, 2022 | |||||
| 55. | Aug 31, 2022 | |||||
| 56. | Sep 30, 2022 | |||||
| 57. | Oct 31, 2022 | |||||
| 58. | Nov 30, 2022 | |||||
| 59. | Dec 31, 2022 | |||||
| Average (R): | ||||||
| Standard deviation: | ||||||
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 | |||||
| 2. | Mar 31, 2018 | |||||
| 3. | Apr 30, 2018 | |||||
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2022 | |||||
| 59. | Dec 31, 2022 | |||||
| Total (Σ): | ||||||
| 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 | |||||
| 2. | Mar 31, 2018 | |||||
| 3. | Apr 30, 2018 | |||||
| 4. | May 31, 2018 | |||||
| 5. | Jun 30, 2018 | |||||
| 6. | Jul 31, 2018 | |||||
| 7. | Aug 31, 2018 | |||||
| 8. | Sep 30, 2018 | |||||
| 9. | Oct 31, 2018 | |||||
| 10. | Nov 30, 2018 | |||||
| 11. | Dec 31, 2018 | |||||
| 12. | Jan 31, 2019 | |||||
| 13. | Feb 28, 2019 | |||||
| 14. | Mar 31, 2019 | |||||
| 15. | Apr 30, 2019 | |||||
| 16. | May 31, 2019 | |||||
| 17. | Jun 30, 2019 | |||||
| 18. | Jul 31, 2019 | |||||
| 19. | Aug 31, 2019 | |||||
| 20. | Sep 30, 2019 | |||||
| 21. | Oct 31, 2019 | |||||
| 22. | Nov 30, 2019 | |||||
| 23. | Dec 31, 2019 | |||||
| 24. | Jan 31, 2020 | |||||
| 25. | Feb 29, 2020 | |||||
| 26. | Mar 31, 2020 | |||||
| 27. | Apr 30, 2020 | |||||
| 28. | May 31, 2020 | |||||
| 29. | Jun 30, 2020 | |||||
| 30. | Jul 31, 2020 | |||||
| 31. | Aug 31, 2020 | |||||
| 32. | Sep 30, 2020 | |||||
| 33. | Oct 31, 2020 | |||||
| 34. | Nov 30, 2020 | |||||
| 35. | Dec 31, 2020 | |||||
| 36. | Jan 31, 2021 | |||||
| 37. | Feb 28, 2021 | |||||
| 38. | Mar 31, 2021 | |||||
| 39. | Apr 30, 2021 | |||||
| 40. | May 31, 2021 | |||||
| 41. | Jun 30, 2021 | |||||
| 42. | Jul 31, 2021 | |||||
| 43. | Aug 31, 2021 | |||||
| 44. | Sep 30, 2021 | |||||
| 45. | Oct 31, 2021 | |||||
| 46. | Nov 30, 2021 | |||||
| 47. | Dec 31, 2021 | |||||
| 48. | Jan 31, 2022 | |||||
| 49. | Feb 28, 2022 | |||||
| 50. | Mar 31, 2022 | |||||
| 51. | Apr 30, 2022 | |||||
| 52. | May 31, 2022 | |||||
| 53. | Jun 30, 2022 | |||||
| 54. | Jul 31, 2022 | |||||
| 55. | Aug 31, 2022 | |||||
| 56. | Sep 30, 2022 | |||||
| 57. | Oct 31, 2022 | |||||
| 58. | Nov 30, 2022 | |||||
| 59. | Dec 31, 2022 | |||||
| Total (Σ): | ||||||
Show all
VarianceTT = Σ(RTT,t–RTT)2 ÷ (59 – 1)
= ÷ (59 – 1)
=
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= ÷ (59 – 1)
=
CovarianceTT, S&P 500 = Σ(RTT,t–RTT)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= ÷ (59 – 1)
=
Systematic Risk (β) Estimation
| VarianceTT | |
| VarianceS&P 500 | |
| CovarianceTT, S&P 500 | |
| Correlation coefficientTT, S&P 5001 | |
| βTT2 | |
| αTT3 |
Calculations
1 Correlation coefficientTT, S&P 500
= CovarianceTT, S&P 500 ÷ (Standard deviationTT × Standard deviationS&P 500)
= ÷ ( × )
=
2 βTT
= CovarianceTT, S&P 500 ÷ VarianceS&P 500
= ÷
=
3 αTT
= AverageTT – βTT × AverageS&P 500
= – ×
=
Expected Rate of Return
| Assumptions | ||
| Rate of return on LT Treasury Composite1 | RF | |
| Expected rate of return on market portfolio2 | E(RM) | |
| Systematic risk (β) of Trane Technologies plc common stock | βTT | |
| Expected rate of return on Trane Technologies plc common stock3 | E(RTT) | |
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]
= + [ – ]
=