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, 2016 $124.79 1,940.24
1. Feb 29, 2016 $131.03 $1.30 6.04% 1,932.23 -0.41%
2. Mar 31, 2016 $151.45 15.58% 2,059.74 6.60%
3. Apr 30, 2016 $145.94 -3.64% 2,065.30 0.27%
. . . . . . .
. . . . . . .
. . . . . . .
58. Nov 30, 2020 $123.52 $1.63 12.08% 3,621.63 10.75%
59. Dec 31, 2020 $125.88 1.91% 3,756.07 3.71%
Average (R): 0.64% 1.22%
Standard deviation: 7.10% 4.36%
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, 2016 $124.79 1,940.24
1. Feb 29, 2016 $131.03 $1.30 6.04% 1,932.23 -0.41%
2. Mar 31, 2016 $151.45 15.58% 2,059.74 6.60%
3. Apr 30, 2016 $145.94 -3.64% 2,065.30 0.27%
4. May 31, 2016 $153.74 $1.40 6.30% 2,096.95 1.53%
5. Jun 30, 2016 $151.78 -1.27% 2,098.86 0.09%
6. Jul 31, 2016 $160.62 5.82% 2,173.60 3.56%
7. Aug 31, 2016 $158.88 $1.40 -0.21% 2,170.95 -0.12%
8. Sep 30, 2016 $158.85 -0.02% 2,168.27 -0.12%
9. Oct 31, 2016 $153.69 -3.25% 2,126.15 -1.94%
10. Nov 30, 2016 $162.22 $1.40 6.46% 2,198.81 3.42%
11. Dec 31, 2016 $165.99 2.32% 2,238.83 1.82%
12. Jan 31, 2017 $174.52 5.14% 2,278.87 1.79%
13. Feb 28, 2017 $179.82 $1.40 3.84% 2,363.64 3.72%
14. Mar 31, 2017 $174.14 -3.16% 2,362.72 -0.04%
15. Apr 30, 2017 $160.29 -7.95% 2,384.20 0.91%
16. May 31, 2017 $152.63 $1.50 -3.84% 2,411.80 1.16%
17. Jun 30, 2017 $153.83 0.79% 2,423.41 0.48%
18. Jul 31, 2017 $144.67 -5.95% 2,470.30 1.93%
19. Aug 31, 2017 $143.03 $1.50 -0.10% 2,471.65 0.05%
20. Sep 30, 2017 $145.08 1.43% 2,519.36 1.93%
21. Oct 31, 2017 $154.06 6.19% 2,575.26 2.22%
22. Nov 30, 2017 $153.97 $1.50 0.92% 2,647.58 2.81%
23. Dec 31, 2017 $153.42 -0.36% 2,673.61 0.98%
24. Jan 31, 2018 $163.70 6.70% 2,823.81 5.62%
25. Feb 28, 2018 $155.83 $1.50 -3.89% 2,713.83 -3.89%
26. Mar 31, 2018 $153.43 -1.54% 2,640.87 -2.69%
27. Apr 30, 2018 $144.96 -5.52% 2,648.05 0.27%
28. May 31, 2018 $141.31 $1.57 -1.43% 2,705.27 2.16%
29. Jun 30, 2018 $139.70 -1.14% 2,718.37 0.48%
30. Jul 31, 2018 $144.93 3.74% 2,816.29 3.60%
31. Aug 31, 2018 $146.48 $1.57 2.15% 2,901.52 3.03%
32. Sep 30, 2018 $151.21 3.23% 2,913.98 0.43%
33. Oct 31, 2018 $115.43 -23.66% 2,711.74 -6.94%
34. Nov 30, 2018 $124.27 $1.57 9.02% 2,760.17 1.79%
35. Dec 31, 2018 $113.67 -8.53% 2,506.85 -9.18%
36. Jan 31, 2019 $134.42 18.25% 2,704.10 7.87%
37. Feb 28, 2019 $138.13 $1.57 3.93% 2,784.49 2.97%
38. Mar 31, 2019 $141.10 2.15% 2,834.40 1.79%
39. Apr 30, 2019 $140.27 -0.59% 2,945.83 3.93%
40. May 31, 2019 $126.99 $1.62 -8.31% 2,752.06 -6.58%
41. Jun 30, 2019 $137.90 8.59% 2,941.76 6.89%
42. Jul 31, 2019 $148.24 7.50% 2,980.38 1.31%
43. Aug 31, 2019 $135.53 $1.62 -7.48% 2,926.46 -1.81%
44. Sep 30, 2019 $145.42 7.30% 2,976.74 1.72%
45. Oct 31, 2019 $133.73 -8.04% 3,037.56 2.04%
46. Nov 30, 2019 $134.45 $1.62 1.75% 3,140.98 3.40%
47. Dec 31, 2019 $134.04 -0.30% 3,230.78 2.86%
48. Jan 31, 2020 $143.73 7.23% 3,225.52 -0.16%
49. Feb 29, 2020 $130.15 $1.62 -8.32% 2,954.22 -8.41%
50. Mar 31, 2020 $110.93 -14.77% 2,584.59 -12.51%
51. Apr 30, 2020 $125.56 13.19% 2,912.43 12.68%
52. May 31, 2020 $124.90 $1.63 0.77% 3,044.31 4.53%
53. Jun 30, 2020 $120.77 -3.31% 3,100.29 1.84%
54. Jul 31, 2020 $122.94 1.80% 3,271.12 5.51%
55. Aug 31, 2020 $123.31 $1.63 1.63% 3,500.31 7.01%
56. Sep 30, 2020 $121.67 -1.33% 3,363.00 -3.92%
57. Oct 31, 2020 $111.66 -8.23% 3,269.96 -2.77%
58. Nov 30, 2020 $123.52 $1.63 12.08% 3,621.63 10.75%
59. Dec 31, 2020 $125.88 1.91% 3,756.07 3.71%
Average (R): 0.64% 1.22%
Standard deviation: 7.10% 4.36%

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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 29, 2016 6.04% -0.41% 29.21 2.67 -8.83
2. Mar 31, 2016 15.58% 6.60% 223.41 28.93 80.40
3. Apr 30, 2016 -3.64% 0.27% 18.28 0.90 4.06
. . . . . . .
. . . . . . .
. . . . . . .
58. Nov 30, 2020 12.08% 10.75% 130.96 90.91 109.11
59. Dec 31, 2020 1.91% 3.71% 1.62 6.21 3.17
Total (Σ): 2,927.30 1,103.45 1,367.60
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 29, 2016 6.04% -0.41% 29.21 2.67 -8.83
2. Mar 31, 2016 15.58% 6.60% 223.41 28.93 80.40
3. Apr 30, 2016 -3.64% 0.27% 18.28 0.90 4.06
4. May 31, 2016 6.30% 1.53% 32.11 0.10 1.77
5. Jun 30, 2016 -1.27% 0.09% 3.66 1.27 2.16
6. Jul 31, 2016 5.82% 3.56% 26.90 5.48 12.14
7. Aug 31, 2016 -0.21% -0.12% 0.72 1.80 1.14
8. Sep 30, 2016 -0.02% -0.12% 0.43 1.81 0.88
9. Oct 31, 2016 -3.25% -1.94% 15.10 10.00 12.29
10. Nov 30, 2016 6.46% 3.42% 33.91 4.83 12.80
11. Dec 31, 2016 2.32% 1.82% 2.84 0.36 1.01
12. Jan 31, 2017 5.14% 1.79% 20.26 0.32 2.56
13. Feb 28, 2017 3.84% 3.72% 10.25 6.25 8.00
14. Mar 31, 2017 -3.16% -0.04% 14.41 1.59 4.78
15. Apr 30, 2017 -7.95% 0.91% 73.80 0.10 2.67
16. May 31, 2017 -3.84% 1.16% 20.07 0.00 0.28
17. Jun 30, 2017 0.79% 0.48% 0.02 0.55 -0.11
18. Jul 31, 2017 -5.95% 1.93% 43.46 0.51 -4.71
19. Aug 31, 2017 -0.10% 0.05% 0.54 1.36 0.86
20. Sep 30, 2017 1.43% 1.93% 0.63 0.50 0.57
21. Oct 31, 2017 6.19% 2.22% 30.83 1.00 5.55
22. Nov 30, 2017 0.92% 2.81% 0.08 2.52 0.44
23. Dec 31, 2017 -0.36% 0.98% 0.99 0.06 0.24
24. Jan 31, 2018 6.70% 5.62% 36.76 19.34 26.66
25. Feb 28, 2018 -3.89% -3.89% 20.51 26.16 23.16
26. Mar 31, 2018 -1.54% -2.69% 4.74 15.28 8.51
27. Apr 30, 2018 -5.52% 0.27% 37.92 0.90 5.84
28. May 31, 2018 -1.43% 2.16% 4.29 0.89 -1.95
29. Jun 30, 2018 -1.14% 0.48% 3.16 0.54 1.31
30. Jul 31, 2018 3.74% 3.60% 9.65 5.67 7.40
31. Aug 31, 2018 2.15% 3.03% 2.30 3.26 2.74
32. Sep 30, 2018 3.23% 0.43% 6.72 0.63 -2.05
33. Oct 31, 2018 -23.66% -6.94% 590.49 66.59 198.30
34. Nov 30, 2018 9.02% 1.79% 70.24 0.32 4.74
35. Dec 31, 2018 -8.53% -9.18% 84.04 108.11 95.32
36. Jan 31, 2019 18.25% 7.87% 310.36 44.20 117.13
37. Feb 28, 2019 3.93% 2.97% 10.83 3.07 5.77
38. Mar 31, 2019 2.15% 1.79% 2.29 0.33 0.87
39. Apr 30, 2019 -0.59% 3.93% 1.50 7.35 -3.32
40. May 31, 2019 -8.31% -6.58% 80.10 60.81 69.79
41. Jun 30, 2019 8.59% 6.89% 63.26 32.18 45.12
42. Jul 31, 2019 7.50% 1.31% 47.07 0.01 0.64
43. Aug 31, 2019 -7.48% -1.81% 65.91 9.18 24.59
44. Sep 30, 2019 7.30% 1.72% 44.35 0.25 3.32
45. Oct 31, 2019 -8.04% 2.04% 75.28 0.68 -7.14
46. Nov 30, 2019 1.75% 3.40% 1.24 4.77 2.43
47. Dec 31, 2019 -0.30% 2.86% 0.89 2.69 -1.54
48. Jan 31, 2020 7.23% -0.16% 43.45 1.91 -9.12
49. Feb 29, 2020 -8.32% -8.41% 80.26 92.76 86.28
50. Mar 31, 2020 -14.77% -12.51% 237.31 188.57 211.54
51. Apr 30, 2020 13.19% 12.68% 157.53 131.43 143.89
52. May 31, 2020 0.77% 4.53% 0.02 10.94 0.45
53. Jun 30, 2020 -3.31% 1.84% 15.56 0.38 -2.44
54. Jul 31, 2020 1.80% 5.51% 1.34 18.40 4.97
55. Aug 31, 2020 1.63% 7.01% 0.98 33.48 5.72
56. Sep 30, 2020 -1.33% -3.92% 3.87 26.45 10.12
57. Oct 31, 2020 -8.23% -2.77% 78.58 15.89 35.34
58. Nov 30, 2020 12.08% 10.75% 130.96 90.91 109.11
59. Dec 31, 2020 1.91% 3.71% 1.62 6.21 3.17
Total (Σ): 2,927.30 1,103.45 1,367.60

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VarianceIBM = Σ(RIBM,tRIBM)2 ÷ (59 – 1)
= 2,927.30 ÷ (59 – 1)
= 50.47

VarianceS&P 500 = Σ(RS&P 500,tRS&P 500)2 ÷ (59 – 1)
= 1,103.45 ÷ (59 – 1)
= 19.02

CovarianceIBM, S&P 500 = Σ(RIBM,tRIBM)×(RS&P 500,tRS&P 500) ÷ (59 – 1)
= 1,367.60 ÷ (59 – 1)
= 23.58


Systematic Risk (β) Estimation

Microsoft Excel LibreOffice Calc
VarianceIBM 50.47
VarianceS&P 500 19.02
CovarianceIBM, S&P 500 23.58
Correlation coefficientIBM, S&P 5001 0.76
βIBM2 1.24
αIBM3 -0.87%

Calculations

1 Correlation coefficientIBM, S&P 500
= CovarianceIBM, S&P 500 ÷ (Standard deviationIBM × Standard deviationS&P 500)
= 23.58 ÷ (7.10% × 4.36%)
= 0.76

2 βIBM
= CovarianceIBM, S&P 500 ÷ VarianceS&P 500
= 23.58 ÷ 19.02
= 1.24

3 αIBM
= AverageIBM – βIBM × AverageS&P 500
= 0.64%1.24 × 1.22%
= -0.87%


Expected Rate of Return

Microsoft Excel LibreOffice Calc
Assumptions
Rate of return on LT Treasury Composite1 RF 2.08%
Expected rate of return on market portfolio2 E(RM) 11.60%
Systematic risk (β) of International Business Machines Corp. common stock βIBM 1.24
 
Expected rate of return on International Business Machines Corp. common stock3 E(RIBM) 13.87%

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]
= 2.08% + 1.24 [11.60%2.08%]
= 13.87%