International Business Machines Corp. (NYSE:IBM)

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 International Business Machines Corp. common stock.

Rates of Return
Variance and Covariance
Systematic Risk (β) Estimation
Expected Rate of Return

Rates of Return

International Business Machines Corp., monthly rates of return

		International Business Machines Corp. (IBM)			Standard & Poor’s 500 (S&P 500)
t	Date	Price_IBM,t¹	Dividend_IBM,t¹	R_IBM,t²	Price_{S&P 500,t}	R_{S&P 500,t}³
	Jan 31, 2020	$143.73			3,225.52
1.	Feb 29, 2020	$130.15	$1.62	-8.32%	2,954.22	-8.41%
2.	Mar 31, 2020	$110.93		-14.77%	2,584.59	-12.51%
3.	Apr 30, 2020	$125.56		13.19%	2,912.43	12.68%
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58.	Nov 30, 2024	$227.41	$1.67	10.82%	6,032.38	5.73%
59.	Dec 31, 2024	$219.83		-3.33%	5,881.63	-2.50%
Average (R):				1.46%		1.16%
Standard deviation:				6.87%		5.28%

		International Business Machines Corp. (IBM)			Standard & Poor’s 500 (S&P 500)
t	Date	Price_IBM,t¹	Dividend_IBM,t¹	R_IBM,t²	Price_{S&P 500,t}	R_{S&P 500,t}³
	Jan 31, 2020	$143.73			3,225.52
1.	Feb 29, 2020	$130.15	$1.62	-8.32%	2,954.22	-8.41%
2.	Mar 31, 2020	$110.93		-14.77%	2,584.59	-12.51%
3.	Apr 30, 2020	$125.56		13.19%	2,912.43	12.68%
4.	May 31, 2020	$124.90	$1.63	0.77%	3,044.31	4.53%
5.	Jun 30, 2020	$120.77		-3.31%	3,100.29	1.84%
6.	Jul 31, 2020	$122.94		1.80%	3,271.12	5.51%
7.	Aug 31, 2020	$123.31	$1.63	1.63%	3,500.31	7.01%
8.	Sep 30, 2020	$121.67		-1.33%	3,363.00	-3.92%
9.	Oct 31, 2020	$111.66		-8.23%	3,269.96	-2.77%
10.	Nov 30, 2020	$123.52	$1.63	12.08%	3,621.63	10.75%
11.	Dec 31, 2020	$125.88		1.91%	3,756.07	3.71%
12.	Jan 31, 2021	$119.11		-5.38%	3,714.24	-1.11%
13.	Feb 28, 2021	$118.93	$1.63	1.22%	3,811.15	2.61%
14.	Mar 31, 2021	$133.26		12.05%	3,972.89	4.24%
15.	Apr 30, 2021	$141.88		6.47%	4,181.17	5.24%
16.	May 31, 2021	$143.74	$1.64	2.47%	4,204.11	0.55%
17.	Jun 30, 2021	$146.59		1.98%	4,297.50	2.22%
18.	Jul 31, 2021	$140.96		-3.84%	4,395.26	2.27%
19.	Aug 31, 2021	$140.34	$1.64	0.72%	4,522.68	2.90%
20.	Sep 30, 2021	$138.93		-1.00%	4,307.54	-4.76%
21.	Oct 31, 2021	$125.10	$8.00	-4.20%	4,605.38	6.91%
22.	Nov 30, 2021	$117.10	$1.64	-5.08%	4,567.00	-0.83%
23.	Dec 31, 2021	$133.66		14.14%	4,766.18	4.36%
24.	Jan 31, 2022	$133.57		-0.07%	4,515.55	-5.26%
25.	Feb 28, 2022	$122.51	$1.64	-7.05%	4,373.94	-3.14%
26.	Mar 31, 2022	$130.02		6.13%	4,530.41	3.58%
27.	Apr 30, 2022	$132.21		1.68%	4,131.93	-8.80%
28.	May 31, 2022	$138.84	$1.65	6.26%	4,132.15	0.01%
29.	Jun 30, 2022	$141.19		1.69%	3,785.38	-8.39%
30.	Jul 31, 2022	$130.79		-7.37%	4,130.29	9.11%
31.	Aug 31, 2022	$128.45	$1.65	-0.53%	3,955.00	-4.24%
32.	Sep 30, 2022	$118.81		-7.50%	3,585.62	-9.34%
33.	Oct 31, 2022	$138.29		16.40%	3,871.98	7.99%
34.	Nov 30, 2022	$148.90	$1.65	8.87%	4,080.11	5.38%
35.	Dec 31, 2022	$140.89		-5.38%	3,839.50	-5.90%
36.	Jan 31, 2023	$134.73		-4.37%	4,076.60	6.18%
37.	Feb 28, 2023	$129.30	$1.65	-2.81%	3,970.15	-2.61%
38.	Mar 31, 2023	$131.09		1.38%	4,109.31	3.51%
39.	Apr 30, 2023	$126.41		-3.57%	4,169.48	1.46%
40.	May 31, 2023	$128.59	$1.66	3.04%	4,179.83	0.25%
41.	Jun 30, 2023	$133.81		4.06%	4,376.86	4.71%
42.	Jul 31, 2023	$144.18		7.75%	4,588.96	4.85%
43.	Aug 31, 2023	$146.83	$1.66	2.99%	4,507.66	-1.77%
44.	Sep 30, 2023	$140.30		-4.45%	4,288.05	-4.87%
45.	Oct 31, 2023	$144.64		3.09%	4,193.80	-2.20%
46.	Nov 30, 2023	$158.56	$1.66	10.77%	4,567.80	8.92%
47.	Dec 31, 2023	$163.55		3.15%	4,769.83	4.42%
48.	Jan 31, 2024	$183.66		12.30%	4,845.65	1.59%
49.	Feb 29, 2024	$185.03	$1.66	1.65%	5,096.27	5.17%
50.	Mar 31, 2024	$190.96		3.20%	5,254.35	3.10%
51.	Apr 30, 2024	$166.20		-12.97%	5,035.69	-4.16%
52.	May 31, 2024	$166.85	$1.67	1.40%	5,277.51	4.80%
53.	Jun 30, 2024	$172.95		3.66%	5,460.48	3.47%
54.	Jul 31, 2024	$192.14		11.10%	5,522.30	1.13%
55.	Aug 31, 2024	$202.13	$1.67	6.07%	5,648.40	2.28%
56.	Sep 30, 2024	$221.08		9.38%	5,762.48	2.02%
57.	Oct 31, 2024	$206.72		-6.50%	5,705.45	-0.99%
58.	Nov 30, 2024	$227.41	$1.67	10.82%	6,032.38	5.73%
59.	Dec 31, 2024	$219.83		-3.33%	5,881.63	-2.50%
Average (R):				1.46%		1.16%
Standard deviation:				6.87%		5.28%

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

t	Date	R_IBM,t	R_{S&P 500,t}	(R_IBM,t–R_IBM)²	(R_{S&P 500,t}–R_{S&P 500})²	(R_IBM,t–R_IBM)×(R_{S&P 500,t}–R_{S&P 500})
1.	Feb 29, 2020	-8.32%	-8.41%	95.59	91.63	93.59
2.	Mar 31, 2020	-14.77%	-12.51%	263.20	186.96	221.83
3.	Apr 30, 2020	13.19%	12.68%	137.65	132.78	135.19
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58.	Nov 30, 2024	10.82%	5.73%	87.62	20.87	42.77
59.	Dec 31, 2024	-3.33%	-2.50%	22.94	13.40	17.53
Total (Σ):				2,736.67	1,618.62	1,262.36

t	Date	R_IBM,t	R_{S&P 500,t}	(R_IBM,t–R_IBM)²	(R_{S&P 500,t}–R_{S&P 500})²	(R_IBM,t–R_IBM)×(R_{S&P 500,t}–R_{S&P 500})
1.	Feb 29, 2020	-8.32%	-8.41%	95.59	91.63	93.59
2.	Mar 31, 2020	-14.77%	-12.51%	263.20	186.96	221.83
3.	Apr 30, 2020	13.19%	12.68%	137.65	132.78	135.19
4.	May 31, 2020	0.77%	4.53%	0.47	11.34	-2.30
5.	Jun 30, 2020	-3.31%	1.84%	22.68	0.46	-3.23
6.	Jul 31, 2020	1.80%	5.51%	0.12	18.91	1.48
7.	Aug 31, 2020	1.63%	7.01%	0.03	34.17	1.00
8.	Sep 30, 2020	-1.33%	-3.92%	7.76	25.85	14.16
9.	Oct 31, 2020	-8.23%	-2.77%	93.76	15.43	38.03
10.	Nov 30, 2020	12.08%	10.75%	112.90	92.03	101.93
11.	Dec 31, 2020	1.91%	3.71%	0.21	6.51	1.16
12.	Jan 31, 2021	-5.38%	-1.11%	46.71	5.18	15.55
13.	Feb 28, 2021	1.22%	2.61%	0.06	2.10	-0.35
14.	Mar 31, 2021	12.05%	4.24%	112.21	9.50	32.65
15.	Apr 30, 2021	6.47%	5.24%	25.13	16.66	20.46
16.	May 31, 2021	2.47%	0.55%	1.02	0.38	-0.62
17.	Jun 30, 2021	1.98%	2.22%	0.28	1.12	0.56
18.	Jul 31, 2021	-3.84%	2.27%	28.05	1.24	-5.90
19.	Aug 31, 2021	0.72%	2.90%	0.54	3.02	-1.27
20.	Sep 30, 2021	-1.00%	-4.76%	6.06	35.02	14.56
21.	Oct 31, 2021	-4.20%	6.91%	31.95	33.10	-32.52
22.	Nov 30, 2021	-5.08%	-0.83%	42.77	3.98	13.04
23.	Dec 31, 2021	14.14%	4.36%	160.93	10.24	40.59
24.	Jan 31, 2022	-0.07%	-5.26%	2.32	41.21	9.78
25.	Feb 28, 2022	-7.05%	-3.14%	72.39	18.47	36.56
26.	Mar 31, 2022	6.13%	3.58%	21.85	5.84	11.29
27.	Apr 30, 2022	1.68%	-8.80%	0.05	99.14	-2.27
28.	May 31, 2022	6.26%	0.01%	23.10	1.34	-5.56
29.	Jun 30, 2022	1.69%	-8.39%	0.06	91.26	-2.26
30.	Jul 31, 2022	-7.37%	9.11%	77.83	63.21	-70.14
31.	Aug 31, 2022	-0.53%	-4.24%	3.93	29.22	10.72
32.	Sep 30, 2022	-7.50%	-9.34%	80.30	110.27	94.10
33.	Oct 31, 2022	16.40%	7.99%	223.20	46.58	101.97
34.	Nov 30, 2022	8.87%	5.38%	54.90	17.76	31.22
35.	Dec 31, 2022	-5.38%	-5.90%	46.72	49.82	48.25
36.	Jan 31, 2023	-4.37%	6.18%	33.97	25.14	-29.22
37.	Feb 28, 2023	-2.81%	-2.61%	18.16	14.23	16.08
38.	Mar 31, 2023	1.38%	3.51%	0.01	5.49	-0.17
39.	Apr 30, 2023	-3.57%	1.46%	25.26	0.09	-1.52
40.	May 31, 2023	3.04%	0.25%	2.50	0.83	-1.44
41.	Jun 30, 2023	4.06%	4.71%	6.78	12.62	9.25
42.	Jul 31, 2023	7.75%	4.85%	39.61	13.58	23.19
43.	Aug 31, 2023	2.99%	-1.77%	2.35	8.60	-4.50
44.	Sep 30, 2023	-4.45%	-4.87%	34.85	36.40	35.62
45.	Oct 31, 2023	3.09%	-2.20%	2.68	11.28	-5.50
46.	Nov 30, 2023	10.77%	8.92%	86.78	60.17	72.26
47.	Dec 31, 2023	3.15%	4.42%	2.86	10.64	5.52
48.	Jan 31, 2024	12.30%	1.59%	117.50	0.18	4.64
49.	Feb 29, 2024	1.65%	5.17%	0.04	16.09	0.78
50.	Mar 31, 2024	3.20%	3.10%	3.06	3.77	3.39
51.	Apr 30, 2024	-12.97%	-4.16%	208.00	28.33	76.77
52.	May 31, 2024	1.40%	4.80%	0.00	13.26	-0.22
53.	Jun 30, 2024	3.66%	3.47%	4.84	5.32	5.07
54.	Jul 31, 2024	11.10%	1.13%	92.92	0.00	-0.28
55.	Aug 31, 2024	6.07%	2.28%	21.27	1.26	5.18
56.	Sep 30, 2024	9.38%	2.02%	62.71	0.74	6.80
57.	Oct 31, 2024	-6.50%	-0.99%	63.22	4.63	17.10
58.	Nov 30, 2024	10.82%	5.73%	87.62	20.87	42.77
59.	Dec 31, 2024	-3.33%	-2.50%	22.94	13.40	17.53
Total (Σ):				2,736.67	1,618.62	1,262.36

Show all

Variance_IBM = Σ(R_IBM,t–R_IBM)² ÷ (59 – 1)
= 2,736.67 ÷ (59 – 1)
= 47.18

Variance_{S&P 500} = Σ(R_{S&P 500,t}–R_{S&P 500})² ÷ (59 – 1)
= 1,618.62 ÷ (59 – 1)
= 27.91

Covariance_{IBM, S&P 500} = Σ(R_IBM,t–R_IBM)×(R_{S&P 500,t}–R_{S&P 500}) ÷ (59 – 1)
= 1,262.36 ÷ (59 – 1)
= 21.76

Systematic Risk (β) Estimation

Variance_IBM	47.18
Variance_{S&P 500}	27.91
Covariance_{IBM, S&P 500}	21.76
Correlation coefficient_{IBM, S&P 500}¹	0.60
β_IBM²	0.78
α_IBM³	0.55%

Calculations

¹ Correlation coefficient_{IBM, S&P 500}
= Covariance_{IBM, S&P 500} ÷ (Standard deviation_IBM × Standard deviation_{S&P 500})
= 21.76 ÷ (6.87% × 5.28%)
= 0.60

² β_IBM
= Covariance_{IBM, S&P 500} ÷ Variance_{S&P 500}
= 21.76 ÷ 27.91
= 0.78

³ α_IBM
= Average_IBM – β_IBM × Average_{S&P 500}
= 1.46% – 0.78 × 1.16%
= 0.55%

Expected Rate of Return

Assumptions
Rate of return on LT Treasury Composite¹	R_F	4.74%
Expected rate of return on market portfolio²	E(R_M)	14.86%
Systematic risk (β) of International Business Machines Corp. common stock	β_IBM	0.78

Expected rate of return on International Business Machines Corp. common stock³	E(R_IBM)	12.63%

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(R_IBM) = R_F + β_IBM [E(R_M) – R_F]
= 4.74% + 0.78 [14.86% – 4.74%]
= 12.63%