Stock Analysis on Net

International Business Machines Corp. (NYSE:IBM)

Capital Asset Pricing Model (CAPM) 

Microsoft Excel LibreOffice Calc

Rates of Return

International Business Machines Corp., monthly rates of return

Microsoft Excel LibreOffice Calc
International Business Machines Corp. (IBM) Standard & Poor’s 500 (S&P 500)
t Date PriceIBM,t1 DividendIBM,t1 RIBM,t2 PriceS&P 500,t RS&P 500,t3
Jan 31, 2017 $174.52 2,278.87
1. Feb 28, 2017 $179.82 $1.40 3.84% 2,363.64 3.72%
2. Mar 31, 2017 $174.14 -3.16% 2,362.72 -0.04%
3. Apr 30, 2017 $160.29 -7.95% 2,384.20 0.91%
. . . . . . .
. . . . . . .
. . . . . . .
58. Nov 30, 2021 $117.10 $1.64 -5.08% 4,567.00 -0.83%
59. Dec 31, 2021 $133.66 14.14% 4,766.18 4.36%
Average (R): 0.21% 1.36%
Standard deviation: 7.27% 4.48%
International Business Machines Corp. (IBM) Standard & Poor’s 500 (S&P 500)
t Date PriceIBM,t1 DividendIBM,t1 RIBM,t2 PriceS&P 500,t RS&P 500,t3
Jan 31, 2017 $174.52 2,278.87
1. Feb 28, 2017 $179.82 $1.40 3.84% 2,363.64 3.72%
2. Mar 31, 2017 $174.14 -3.16% 2,362.72 -0.04%
3. Apr 30, 2017 $160.29 -7.95% 2,384.20 0.91%
4. May 31, 2017 $152.63 $1.50 -3.84% 2,411.80 1.16%
5. Jun 30, 2017 $153.83 0.79% 2,423.41 0.48%
6. Jul 31, 2017 $144.67 -5.95% 2,470.30 1.93%
7. Aug 31, 2017 $143.03 $1.50 -0.10% 2,471.65 0.05%
8. Sep 30, 2017 $145.08 1.43% 2,519.36 1.93%
9. Oct 31, 2017 $154.06 6.19% 2,575.26 2.22%
10. Nov 30, 2017 $153.97 $1.50 0.92% 2,647.58 2.81%
11. Dec 31, 2017 $153.42 -0.36% 2,673.61 0.98%
12. Jan 31, 2018 $163.70 6.70% 2,823.81 5.62%
13. Feb 28, 2018 $155.83 $1.50 -3.89% 2,713.83 -3.89%
14. Mar 31, 2018 $153.43 -1.54% 2,640.87 -2.69%
15. Apr 30, 2018 $144.96 -5.52% 2,648.05 0.27%
16. May 31, 2018 $141.31 $1.57 -1.43% 2,705.27 2.16%
17. Jun 30, 2018 $139.70 -1.14% 2,718.37 0.48%
18. Jul 31, 2018 $144.93 3.74% 2,816.29 3.60%
19. Aug 31, 2018 $146.48 $1.57 2.15% 2,901.52 3.03%
20. Sep 30, 2018 $151.21 3.23% 2,913.98 0.43%
21. Oct 31, 2018 $115.43 -23.66% 2,711.74 -6.94%
22. Nov 30, 2018 $124.27 $1.57 9.02% 2,760.17 1.79%
23. Dec 31, 2018 $113.67 -8.53% 2,506.85 -9.18%
24. Jan 31, 2019 $134.42 18.25% 2,704.10 7.87%
25. Feb 28, 2019 $138.13 $1.57 3.93% 2,784.49 2.97%
26. Mar 31, 2019 $141.10 2.15% 2,834.40 1.79%
27. Apr 30, 2019 $140.27 -0.59% 2,945.83 3.93%
28. May 31, 2019 $126.99 $1.62 -8.31% 2,752.06 -6.58%
29. Jun 30, 2019 $137.90 8.59% 2,941.76 6.89%
30. Jul 31, 2019 $148.24 7.50% 2,980.38 1.31%
31. Aug 31, 2019 $135.53 $1.62 -7.48% 2,926.46 -1.81%
32. Sep 30, 2019 $145.42 7.30% 2,976.74 1.72%
33. Oct 31, 2019 $133.73 -8.04% 3,037.56 2.04%
34. Nov 30, 2019 $134.45 $1.62 1.75% 3,140.98 3.40%
35. Dec 31, 2019 $134.04 -0.30% 3,230.78 2.86%
36. Jan 31, 2020 $143.73 7.23% 3,225.52 -0.16%
37. Feb 29, 2020 $130.15 $1.62 -8.32% 2,954.22 -8.41%
38. Mar 31, 2020 $110.93 -14.77% 2,584.59 -12.51%
39. Apr 30, 2020 $125.56 13.19% 2,912.43 12.68%
40. May 31, 2020 $124.90 $1.63 0.77% 3,044.31 4.53%
41. Jun 30, 2020 $120.77 -3.31% 3,100.29 1.84%
42. Jul 31, 2020 $122.94 1.80% 3,271.12 5.51%
43. Aug 31, 2020 $123.31 $1.63 1.63% 3,500.31 7.01%
44. Sep 30, 2020 $121.67 -1.33% 3,363.00 -3.92%
45. Oct 31, 2020 $111.66 -8.23% 3,269.96 -2.77%
46. Nov 30, 2020 $123.52 $1.63 12.08% 3,621.63 10.75%
47. Dec 31, 2020 $125.88 1.91% 3,756.07 3.71%
48. Jan 31, 2021 $119.11 -5.38% 3,714.24 -1.11%
49. Feb 28, 2021 $118.93 $1.63 1.22% 3,811.15 2.61%
50. Mar 31, 2021 $133.26 12.05% 3,972.89 4.24%
51. Apr 30, 2021 $141.88 6.47% 4,181.17 5.24%
52. May 31, 2021 $143.74 $1.64 2.47% 4,204.11 0.55%
53. Jun 30, 2021 $146.59 1.98% 4,297.50 2.22%
54. Jul 31, 2021 $140.96 -3.84% 4,395.26 2.27%
55. Aug 31, 2021 $140.34 $1.64 0.72% 4,522.68 2.90%
56. Sep 30, 2021 $138.93 -1.00% 4,307.54 -4.76%
57. Oct 31, 2021 $125.10 -9.95% 4,605.38 6.91%
58. Nov 30, 2021 $117.10 $1.64 -5.08% 4,567.00 -0.83%
59. Dec 31, 2021 $133.66 14.14% 4,766.18 4.36%
Average (R): 0.21% 1.36%
Standard deviation: 7.27% 4.48%

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 IBM during period t

3 Rate of return on S&P 500 (the market portfolio proxy) during period t


Variance and Covariance

International Business Machines Corp., calculation of variance and covariance of returns

Microsoft Excel LibreOffice Calc
t Date RIBM,t RS&P 500,t (RIBM,tRIBM)2 (RS&P 500,tRS&P 500)2 (RIBM,tRIBM)×(RS&P 500,tRS&P 500)
1. Feb 28, 2017 3.84% 3.72% 13.20 5.58 8.58
2. Mar 31, 2017 -3.16% -0.04% 11.32 1.95 4.70
3. Apr 30, 2017 -7.95% 0.91% 66.56 0.20 3.66
. . . . . . .
. . . . . . .
. . . . . . .
58. Nov 30, 2021 -5.08% -0.83% 27.98 4.80 11.59
59. Dec 31, 2021 14.14% 4.36% 194.23 9.02 41.86
Total (Σ): 3,065.40 1,164.17 1,320.75
t Date RIBM,t RS&P 500,t (RIBM,tRIBM)2 (RS&P 500,tRS&P 500)2 (RIBM,tRIBM)×(RS&P 500,tRS&P 500)
1. Feb 28, 2017 3.84% 3.72% 13.20 5.58 8.58
2. Mar 31, 2017 -3.16% -0.04% 11.32 1.95 4.70
3. Apr 30, 2017 -7.95% 0.91% 66.56 0.20 3.66
4. May 31, 2017 -3.84% 1.16% 16.39 0.04 0.81
5. Jun 30, 2017 0.79% 0.48% 0.34 0.77 -0.51
6. Jul 31, 2017 -5.95% 1.93% 37.94 0.33 -3.55
7. Aug 31, 2017 -0.10% 0.05% 0.09 1.70 0.39
8. Sep 30, 2017 1.43% 1.93% 1.51 0.33 0.70
9. Oct 31, 2017 6.19% 2.22% 35.81 0.74 5.15
10. Nov 30, 2017 0.92% 2.81% 0.50 2.10 1.03
11. Dec 31, 2017 -0.36% 0.98% 0.32 0.14 0.21
12. Jan 31, 2018 6.70% 5.62% 42.19 18.15 27.67
13. Feb 28, 2018 -3.89% -3.89% 16.78 27.59 21.52
14. Mar 31, 2018 -1.54% -2.69% 3.05 16.37 7.06
15. Apr 30, 2018 -5.52% 0.27% 32.78 1.18 6.22
16. May 31, 2018 -1.43% 2.16% 2.69 0.64 -1.32
17. Jun 30, 2018 -1.14% 0.48% 1.81 0.76 1.17
18. Jul 31, 2018 3.74% 3.60% 12.52 5.04 7.94
19. Aug 31, 2018 2.15% 3.03% 3.79 2.78 3.25
20. Sep 30, 2018 3.23% 0.43% 9.14 0.86 -2.81
21. Oct 31, 2018 -23.66% -6.94% 569.67 68.86 198.06
22. Nov 30, 2018 9.02% 1.79% 77.67 0.18 3.77
23. Dec 31, 2018 -8.53% -9.18% 76.30 111.00 92.03
24. Jan 31, 2019 18.25% 7.87% 325.78 42.39 117.51
25. Feb 28, 2019 3.93% 2.97% 13.86 2.61 6.01
26. Mar 31, 2019 2.15% 1.79% 3.78 0.19 0.85
27. Apr 30, 2019 -0.59% 3.93% 0.63 6.62 -2.04
28. May 31, 2019 -8.31% -6.58% 72.55 62.97 67.59
29. Jun 30, 2019 8.59% 6.89% 70.32 30.64 46.42
30. Jul 31, 2019 7.50% 1.31% 53.19 0.00 -0.33
31. Aug 31, 2019 -7.48% -1.81% 59.08 10.03 24.34
32. Sep 30, 2019 7.30% 1.72% 50.30 0.13 2.55
33. Oct 31, 2019 -8.04% 2.04% 67.96 0.47 -5.65
34. Nov 30, 2019 1.75% 3.40% 2.39 4.19 3.16
35. Dec 31, 2019 -0.30% 2.86% 0.26 2.25 -0.77
36. Jan 31, 2020 7.23% -0.16% 49.34 2.31 -10.68
37. Feb 29, 2020 -8.32% -8.41% 72.70 95.43 83.29
38. Mar 31, 2020 -14.77% -12.51% 224.19 192.37 207.67
39. Apr 30, 2020 13.19% 12.68% 168.56 128.29 147.05
40. May 31, 2020 0.77% 4.53% 0.32 10.05 1.80
41. Jun 30, 2020 -3.31% 1.84% 12.33 0.23 -1.69
42. Jul 31, 2020 1.80% 5.51% 2.53 17.24 6.61
43. Aug 31, 2020 1.63% 7.01% 2.02 31.91 8.03
44. Sep 30, 2020 -1.33% -3.92% 2.36 27.89 8.11
45. Oct 31, 2020 -8.23% -2.77% 71.11 17.01 34.78
46. Nov 30, 2020 12.08% 10.75% 141.04 88.30 111.60
47. Dec 31, 2020 1.91% 3.71% 2.91 5.54 4.01
48. Jan 31, 2021 -5.38% -1.11% 31.17 6.11 13.80
49. Feb 28, 2021 1.22% 2.61% 1.02 1.57 1.27
50. Mar 31, 2021 12.05% 4.24% 140.28 8.33 34.18
51. Apr 30, 2021 6.47% 5.24% 39.23 15.09 24.33
52. May 31, 2021 2.47% 0.55% 5.11 0.65 -1.83
53. Jun 30, 2021 1.98% 2.22% 3.16 0.75 1.53
54. Jul 31, 2021 -3.84% 2.27% 16.37 0.84 -3.71
55. Aug 31, 2021 0.72% 2.90% 0.27 2.38 0.80
56. Sep 30, 2021 -1.00% -4.76% 1.46 37.39 7.40
57. Oct 31, 2021 -9.95% 6.91% 103.22 30.87 -56.45
58. Nov 30, 2021 -5.08% -0.83% 27.98 4.80 11.59
59. Dec 31, 2021 14.14% 4.36% 194.23 9.02 41.86
Total (Σ): 3,065.40 1,164.17 1,320.75

Show all

VarianceIBM = Σ(RIBM,tRIBM)2 ÷ (59 – 1)
= 3,065.40 ÷ (59 – 1)
= 52.85

VarianceS&P 500 = Σ(RS&P 500,tRS&P 500)2 ÷ (59 – 1)
= 1,164.17 ÷ (59 – 1)
= 20.07

CovarianceIBM, S&P 500 = Σ(RIBM,tRIBM)×(RS&P 500,tRS&P 500) ÷ (59 – 1)
= 1,320.75 ÷ (59 – 1)
= 22.77


Systematic Risk (β) Estimation

Microsoft Excel LibreOffice Calc
VarianceIBM 52.85
VarianceS&P 500 20.07
CovarianceIBM, S&P 500 22.77
Correlation coefficientIBM, S&P 5001 0.70
βIBM2 1.13
αIBM3 -1.34%

Calculations

1 Correlation coefficientIBM, S&P 500
= CovarianceIBM, S&P 500 ÷ (Standard deviationIBM × Standard deviationS&P 500)
= 22.77 ÷ (7.27% × 4.48%)
= 0.70

2 βIBM
= CovarianceIBM, S&P 500 ÷ VarianceS&P 500
= 22.77 ÷ 20.07
= 1.13

3 αIBM
= AverageIBM – βIBM × AverageS&P 500
= 0.21%1.13 × 1.36%
= -1.34%


Expected Rate of Return

Microsoft Excel LibreOffice Calc
Assumptions
Rate of return on LT Treasury Composite1 RF 3.19%
Expected rate of return on market portfolio2 E(RM) 12.22%
Systematic risk (β) of International Business Machines Corp. common stock βIBM 1.13
 
Expected rate of return on International Business Machines Corp. common stock3 E(RIBM) 13.43%

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).

2 See details »

3 E(RIBM) = RF + βIBM [E(RM) – RF]
= 3.19% + 1.13 [12.22%3.19%]
= 13.43%