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 Union Pacific Corp. common stock.
Rates of Return
Union Pacific Corp., monthly rates of return
| Union Pacific Corp. (UNP) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceUNP,t1 | DividendUNP,t1 | RUNP,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Jan 31, 2019 | $159.07 | 2,704.10 | ||||
| 1. | Feb 28, 2019 | $167.70 | $0.88 | 5.98% | 2,784.49 | 2.97% |
| 2. | Mar 31, 2019 | $167.20 | -0.30% | 2,834.40 | 1.79% | |
| 3. | Apr 30, 2019 | $177.04 | 5.89% | 2,945.83 | 3.93% | |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2023 | $225.27 | 8.51% | 4,567.80 | 8.92% | |
| 59. | Dec 31, 2023 | $245.62 | $1.30 | 9.61% | 4,769.83 | 4.42% |
| Average (R): | 1.19% | 1.11% | ||||
| Standard deviation: | 7.41% | 5.31% | ||||
| Union Pacific Corp. (UNP) | Standard & Poor’s 500 (S&P 500) | |||||
|---|---|---|---|---|---|---|
| t | Date | PriceUNP,t1 | DividendUNP,t1 | RUNP,t2 | PriceS&P 500,t | RS&P 500,t3 |
| Jan 31, 2019 | $159.07 | 2,704.10 | ||||
| 1. | Feb 28, 2019 | $167.70 | $0.88 | 5.98% | 2,784.49 | 2.97% |
| 2. | Mar 31, 2019 | $167.20 | -0.30% | 2,834.40 | 1.79% | |
| 3. | Apr 30, 2019 | $177.04 | 5.89% | 2,945.83 | 3.93% | |
| 4. | May 31, 2019 | $166.78 | $0.88 | -5.30% | 2,752.06 | -6.58% |
| 5. | Jun 30, 2019 | $169.11 | 1.40% | 2,941.76 | 6.89% | |
| 6. | Jul 31, 2019 | $179.95 | 6.41% | 2,980.38 | 1.31% | |
| 7. | Aug 31, 2019 | $161.96 | $0.97 | -9.46% | 2,926.46 | -1.81% |
| 8. | Sep 30, 2019 | $161.98 | 0.01% | 2,976.74 | 1.72% | |
| 9. | Oct 31, 2019 | $165.46 | 2.15% | 3,037.56 | 2.04% | |
| 10. | Nov 30, 2019 | $175.99 | $0.97 | 6.95% | 3,140.98 | 3.40% |
| 11. | Dec 31, 2019 | $180.79 | 2.73% | 3,230.78 | 2.86% | |
| 12. | Jan 31, 2020 | $179.42 | -0.76% | 3,225.52 | -0.16% | |
| 13. | Feb 29, 2020 | $159.81 | $0.97 | -10.39% | 2,954.22 | -8.41% |
| 14. | Mar 31, 2020 | $141.04 | -11.75% | 2,584.59 | -12.51% | |
| 15. | Apr 30, 2020 | $159.79 | 13.29% | 2,912.43 | 12.68% | |
| 16. | May 31, 2020 | $169.86 | $0.97 | 6.91% | 3,044.31 | 4.53% |
| 17. | Jun 30, 2020 | $169.07 | -0.47% | 3,100.29 | 1.84% | |
| 18. | Jul 31, 2020 | $173.35 | 2.53% | 3,271.12 | 5.51% | |
| 19. | Aug 31, 2020 | $192.44 | $0.97 | 11.57% | 3,500.31 | 7.01% |
| 20. | Sep 30, 2020 | $196.87 | 2.30% | 3,363.00 | -3.92% | |
| 21. | Oct 31, 2020 | $177.19 | -10.00% | 3,269.96 | -2.77% | |
| 22. | Nov 30, 2020 | $204.08 | $0.97 | 15.72% | 3,621.63 | 10.75% |
| 23. | Dec 31, 2020 | $208.22 | 2.03% | 3,756.07 | 3.71% | |
| 24. | Jan 31, 2021 | $197.47 | -5.16% | 3,714.24 | -1.11% | |
| 25. | Feb 28, 2021 | $205.96 | $0.97 | 4.79% | 3,811.15 | 2.61% |
| 26. | Mar 31, 2021 | $220.41 | 7.02% | 3,972.89 | 4.24% | |
| 27. | Apr 30, 2021 | $222.09 | 0.76% | 4,181.17 | 5.24% | |
| 28. | May 31, 2021 | $224.73 | $1.07 | 1.67% | 4,204.11 | 0.55% |
| 29. | Jun 30, 2021 | $219.93 | -2.14% | 4,297.50 | 2.22% | |
| 30. | Jul 31, 2021 | $218.76 | -0.53% | 4,395.26 | 2.27% | |
| 31. | Aug 31, 2021 | $216.84 | $1.07 | -0.39% | 4,522.68 | 2.90% |
| 32. | Sep 30, 2021 | $196.01 | -9.61% | 4,307.54 | -4.76% | |
| 33. | Oct 31, 2021 | $241.40 | 23.16% | 4,605.38 | 6.91% | |
| 34. | Nov 30, 2021 | $235.64 | -2.39% | 4,567.00 | -0.83% | |
| 35. | Dec 31, 2021 | $251.93 | $1.18 | 7.41% | 4,766.18 | 4.36% |
| 36. | Jan 31, 2022 | $244.55 | -2.93% | 4,515.55 | -5.26% | |
| 37. | Feb 28, 2022 | $245.95 | $1.18 | 1.05% | 4,373.94 | -3.14% |
| 38. | Mar 31, 2022 | $273.21 | 11.08% | 4,530.41 | 3.58% | |
| 39. | Apr 30, 2022 | $234.29 | -14.25% | 4,131.93 | -8.80% | |
| 40. | May 31, 2022 | $219.78 | $1.30 | -5.64% | 4,132.15 | 0.01% |
| 41. | Jun 30, 2022 | $213.28 | -2.96% | 3,785.38 | -8.39% | |
| 42. | Jul 31, 2022 | $227.30 | 6.57% | 4,130.29 | 9.11% | |
| 43. | Aug 31, 2022 | $224.51 | $1.30 | -0.66% | 3,955.00 | -4.24% |
| 44. | Sep 30, 2022 | $194.82 | -13.22% | 3,585.62 | -9.34% | |
| 45. | Oct 31, 2022 | $197.14 | 1.19% | 3,871.98 | 7.99% | |
| 46. | Nov 30, 2022 | $217.43 | 10.29% | 4,080.11 | 5.38% | |
| 47. | Dec 31, 2022 | $207.07 | $1.30 | -4.17% | 3,839.50 | -5.90% |
| 48. | Jan 31, 2023 | $204.19 | -1.39% | 4,076.60 | 6.18% | |
| 49. | Feb 28, 2023 | $207.28 | $1.30 | 2.15% | 3,970.15 | -2.61% |
| 50. | Mar 31, 2023 | $201.26 | -2.90% | 4,109.31 | 3.51% | |
| 51. | Apr 30, 2023 | $195.70 | -2.76% | 4,169.48 | 1.46% | |
| 52. | May 31, 2023 | $192.52 | $1.30 | -0.96% | 4,179.83 | 0.25% |
| 53. | Jun 30, 2023 | $204.62 | 6.29% | 4,376.86 | 4.71% | |
| 54. | Jul 31, 2023 | $232.02 | 13.39% | 4,588.96 | 4.85% | |
| 55. | Aug 31, 2023 | $220.57 | $1.30 | -4.37% | 4,507.66 | -1.77% |
| 56. | Sep 30, 2023 | $203.63 | -7.68% | 4,288.05 | -4.87% | |
| 57. | Oct 31, 2023 | $207.61 | 1.95% | 4,193.80 | -2.20% | |
| 58. | Nov 30, 2023 | $225.27 | 8.51% | 4,567.80 | 8.92% | |
| 59. | Dec 31, 2023 | $245.62 | $1.30 | 9.61% | 4,769.83 | 4.42% |
| Average (R): | 1.19% | 1.11% | ||||
| Standard deviation: | 7.41% | 5.31% | ||||
Show all
1 Data in US$ per share of common stock, adjusted for splits and stock dividends.
2 Rate of return on common stock of UNP during period t
3 Rate of return on S&P 500 (the market portfolio proxy) during period t
Variance and Covariance
Union Pacific Corp., calculation of variance and covariance of returns
| t | Date | RUNP,t | RS&P 500,t | (RUNP,t–RUNP)2 | (RS&P 500,t–RS&P 500)2 | (RUNP,t–RUNP)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Feb 28, 2019 | 5.98% | 2.97% | 22.92 | 3.49 | 8.94 |
| 2. | Mar 31, 2019 | -0.30% | 1.79% | 2.22 | 0.47 | -1.02 |
| 3. | Apr 30, 2019 | 5.89% | 3.93% | 22.04 | 7.98 | 13.26 |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| . | . | . | . | . | . | . |
| 58. | Nov 30, 2023 | 8.51% | 8.92% | 53.52 | 61.03 | 57.15 |
| 59. | Dec 31, 2023 | 9.61% | 4.42% | 70.89 | 11.00 | 27.93 |
| Total (Σ): | 3,180.39 | 1,634.30 | 1,780.86 | |||
| t | Date | RUNP,t | RS&P 500,t | (RUNP,t–RUNP)2 | (RS&P 500,t–RS&P 500)2 | (RUNP,t–RUNP)×(RS&P 500,t–RS&P 500) |
|---|---|---|---|---|---|---|
| 1. | Feb 28, 2019 | 5.98% | 2.97% | 22.92 | 3.49 | 8.94 |
| 2. | Mar 31, 2019 | -0.30% | 1.79% | 2.22 | 0.47 | -1.02 |
| 3. | Apr 30, 2019 | 5.89% | 3.93% | 22.04 | 7.98 | 13.26 |
| 4. | May 31, 2019 | -5.30% | -6.58% | 42.11 | 59.04 | 49.86 |
| 5. | Jun 30, 2019 | 1.40% | 6.89% | 0.04 | 33.49 | 1.19 |
| 6. | Jul 31, 2019 | 6.41% | 1.31% | 27.24 | 0.04 | 1.08 |
| 7. | Aug 31, 2019 | -9.46% | -1.81% | 113.40 | 8.50 | 31.04 |
| 8. | Sep 30, 2019 | 0.01% | 1.72% | 1.39 | 0.37 | -0.72 |
| 9. | Oct 31, 2019 | 2.15% | 2.04% | 0.92 | 0.88 | 0.90 |
| 10. | Nov 30, 2019 | 6.95% | 3.40% | 33.17 | 5.28 | 13.24 |
| 11. | Dec 31, 2019 | 2.73% | 2.86% | 2.36 | 3.07 | 2.69 |
| 12. | Jan 31, 2020 | -0.76% | -0.16% | 3.80 | 1.61 | 2.47 |
| 13. | Feb 29, 2020 | -10.39% | -8.41% | 134.09 | 90.57 | 110.20 |
| 14. | Mar 31, 2020 | -11.75% | -12.51% | 167.34 | 185.44 | 176.16 |
| 15. | Apr 30, 2020 | 13.29% | 12.68% | 146.49 | 134.06 | 140.14 |
| 16. | May 31, 2020 | 6.91% | 4.53% | 32.70 | 11.71 | 19.57 |
| 17. | Jun 30, 2020 | -0.47% | 1.84% | 2.74 | 0.54 | -1.21 |
| 18. | Jul 31, 2020 | 2.53% | 5.51% | 1.80 | 19.40 | 5.90 |
| 19. | Aug 31, 2020 | 11.57% | 7.01% | 107.77 | 34.82 | 61.25 |
| 20. | Sep 30, 2020 | 2.30% | -3.92% | 1.23 | 25.29 | -5.59 |
| 21. | Oct 31, 2020 | -10.00% | -2.77% | 125.16 | 15.00 | 43.32 |
| 22. | Nov 30, 2020 | 15.72% | 10.75% | 211.19 | 93.10 | 140.22 |
| 23. | Dec 31, 2020 | 2.03% | 3.71% | 0.70 | 6.79 | 2.18 |
| 24. | Jan 31, 2021 | -5.16% | -1.11% | 40.37 | 4.93 | 14.10 |
| 25. | Feb 28, 2021 | 4.79% | 2.61% | 12.96 | 2.26 | 5.41 |
| 26. | Mar 31, 2021 | 7.02% | 4.24% | 33.93 | 9.85 | 18.28 |
| 27. | Apr 30, 2021 | 0.76% | 5.24% | 0.18 | 17.11 | -1.77 |
| 28. | May 31, 2021 | 1.67% | 0.55% | 0.23 | 0.31 | -0.27 |
| 29. | Jun 30, 2021 | -2.14% | 2.22% | 11.07 | 1.24 | -3.71 |
| 30. | Jul 31, 2021 | -0.53% | 2.27% | 2.97 | 1.37 | -2.01 |
| 31. | Aug 31, 2021 | -0.39% | 2.90% | 2.49 | 3.22 | -2.83 |
| 32. | Sep 30, 2021 | -9.61% | -4.76% | 116.58 | 34.37 | 63.30 |
| 33. | Oct 31, 2021 | 23.16% | 6.91% | 482.51 | 33.74 | 127.59 |
| 34. | Nov 30, 2021 | -2.39% | -0.83% | 12.79 | 3.76 | 6.94 |
| 35. | Dec 31, 2021 | 7.41% | 4.36% | 38.73 | 10.60 | 20.26 |
| 36. | Jan 31, 2022 | -2.93% | -5.26% | 16.98 | 40.51 | 26.22 |
| 37. | Feb 28, 2022 | 1.05% | -3.14% | 0.02 | 17.99 | 0.58 |
| 38. | Mar 31, 2022 | 11.08% | 3.58% | 97.86 | 6.11 | 24.45 |
| 39. | Apr 30, 2022 | -14.25% | -8.80% | 238.28 | 98.04 | 152.84 |
| 40. | May 31, 2022 | -5.64% | 0.01% | 46.64 | 1.21 | 7.52 |
| 41. | Jun 30, 2022 | -2.96% | -8.39% | 17.21 | 90.21 | 39.40 |
| 42. | Jul 31, 2022 | 6.57% | 9.11% | 28.97 | 64.09 | 43.09 |
| 43. | Aug 31, 2022 | -0.66% | -4.24% | 3.41 | 28.62 | 9.88 |
| 44. | Sep 30, 2022 | -13.22% | -9.34% | 207.80 | 109.11 | 150.57 |
| 45. | Oct 31, 2022 | 1.19% | 7.99% | 0.00 | 47.34 | 0.00 |
| 46. | Nov 30, 2022 | 10.29% | 5.38% | 82.83 | 18.23 | 38.86 |
| 47. | Dec 31, 2022 | -4.17% | -5.90% | 28.71 | 49.04 | 37.52 |
| 48. | Jan 31, 2023 | -1.39% | 6.18% | 6.67 | 25.70 | -13.09 |
| 49. | Feb 28, 2023 | 2.15% | -2.61% | 0.92 | 13.82 | -3.56 |
| 50. | Mar 31, 2023 | -2.90% | 3.51% | 16.77 | 5.76 | -9.83 |
| 51. | Apr 30, 2023 | -2.76% | 1.46% | 15.63 | 0.13 | -1.42 |
| 52. | May 31, 2023 | -0.96% | 0.25% | 4.63 | 0.74 | 1.85 |
| 53. | Jun 30, 2023 | 6.29% | 4.71% | 25.95 | 13.02 | 18.38 |
| 54. | Jul 31, 2023 | 13.39% | 4.85% | 148.83 | 13.99 | 45.63 |
| 55. | Aug 31, 2023 | -4.37% | -1.77% | 30.97 | 8.28 | 16.01 |
| 56. | Sep 30, 2023 | -7.68% | -4.87% | 78.69 | 35.73 | 53.03 |
| 57. | Oct 31, 2023 | 1.95% | -2.20% | 0.58 | 10.92 | -2.52 |
| 58. | Nov 30, 2023 | 8.51% | 8.92% | 53.52 | 61.03 | 57.15 |
| 59. | Dec 31, 2023 | 9.61% | 4.42% | 70.89 | 11.00 | 27.93 |
| Total (Σ): | 3,180.39 | 1,634.30 | 1,780.86 | |||
Show all
VarianceUNP = Σ(RUNP,t–RUNP)2 ÷ (59 – 1)
= 3,180.39 ÷ (59 – 1)
= 54.83
VarianceS&P 500 = Σ(RS&P 500,t–RS&P 500)2 ÷ (59 – 1)
= 1,634.30 ÷ (59 – 1)
= 28.18
CovarianceUNP, S&P 500 = Σ(RUNP,t–RUNP)×(RS&P 500,t–RS&P 500) ÷ (59 – 1)
= 1,780.86 ÷ (59 – 1)
= 30.70
Systematic Risk (β) Estimation
| VarianceUNP | 54.83 |
| VarianceS&P 500 | 28.18 |
| CovarianceUNP, S&P 500 | 30.70 |
| Correlation coefficientUNP, S&P 5001 | 0.78 |
| βUNP2 | 1.09 |
| αUNP3 | -0.01% |
Calculations
1 Correlation coefficientUNP, S&P 500
= CovarianceUNP, S&P 500 ÷ (Standard deviationUNP × Standard deviationS&P 500)
= 30.70 ÷ (7.41% × 5.31%)
= 0.78
2 βUNP
= CovarianceUNP, S&P 500 ÷ VarianceS&P 500
= 30.70 ÷ 28.18
= 1.09
3 αUNP
= AverageUNP – βUNP × AverageS&P 500
= 1.19% – 1.09 × 1.11%
= -0.01%
Expected Rate of Return
| Assumptions | ||
| Rate of return on LT Treasury Composite1 | RF | 4.65% |
| Expected rate of return on market portfolio2 | E(RM) | 13.79% |
| Systematic risk (β) of Union Pacific Corp. common stock | βUNP | 1.09 |
| Expected rate of return on Union Pacific Corp. common stock3 | E(RUNP) | 14.61% |
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(RUNP) = RF + βUNP [E(RM) – RF]
= 4.65% + 1.09 [13.79% – 4.65%]
= 14.61%