MSFT - Detail Report

Quick Overview

Timeframe	C-C DoR σ (Historical)	H-L DoR σ (Historical)	O-C DoR σ (Historical)	5Y ATRP (Avg)	Binomial IV (Period)	Yahoo IV (Period)	1σ Lower (68.27%)	1σ Upper	2σ Lower (95.45%)	2σ Upper
Daily	2.09%	1.65%	1.80%	2.16%	1.82%	1.89%	390.06	404.53	383.01	411.97
Weekly	4.57%	4.31%	4.52%	5.12%	4.01%	4.17%	381.61	413.49	366.61	430.41
Monthly	9.36%	9.28%	8.75%	10.87%	8.35%	8.68%	365.41	431.82	336.14	469.43
Quarterly	15.99%	16.05%	13.02%	18.70%	14.46%	15.03%	343.74	459.04	297.45	530.47
Annual	—	—	—	—	28.93%	30.07%	297.45	530.47	222.74	708.41
Option Time Horizon	—	—	—	—	11.13%	11.56%	355.40	443.98	317.98	496.23

Calculation Notes: Computed at 2026-02-22T14:16 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $397.23 | Strike (K): $400.00 (first OTM call ≥ spot) | Observed Mid-Price: $17.0750 (Bid $16.9500 / Ask $17.2000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.87% | Binomial IV (Annual): 28.93% (CRR binomial tree, American, n=20 steps) | BS IV: 28.91% (European Black-Scholes) | Yahoo IV (Annual): 30.07% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Historical Dividend Payouts

Ex-Dividend Date	Amount
2025-11-20	$0.9100
2025-08-21	$0.8300
2025-05-15	$0.8300
2025-02-20	$0.8300
2024-11-21	$0.8300
2024-08-15	$0.7500
2024-05-15	$0.7500
2024-02-14	$0.7500
2023-11-15	$0.7500
2023-08-16	$0.6800
2023-05-17	$0.6800
2023-02-15	$0.6800
2022-11-16	$0.6800
2022-08-17	$0.6200
2022-05-18	$0.6200
2022-02-16	$0.6200
2021-11-17	$0.6200
2021-08-18	$0.5600
2021-05-19	$0.5600
2021-02-17	$0.5600
2020-11-18	$0.5600
2020-08-19	$0.5100
2020-05-20	$0.5100
2020-02-19	$0.5100
2019-11-20	$0.5100
2019-08-14	$0.4600
2019-05-15	$0.4600
2019-02-20	$0.4600
2018-11-14	$0.4600
2018-08-15	$0.4200
2018-05-16	$0.4200
2018-02-14	$0.4200
2017-11-15	$0.4200
2017-08-15	$0.3900
2017-05-16	$0.3900
2017-02-14	$0.3900
2016-11-15	$0.3900
2016-08-16	$0.3600
2016-05-17	$0.3600
2016-02-16	$0.3600
2015-11-17	$0.3600
2015-08-18	$0.3100
2015-05-19	$0.3100
2015-02-17	$0.3100
2014-11-18	$0.3100
2014-08-19	$0.2800
2014-05-13	$0.2800
2014-02-18	$0.2800
2013-11-19	$0.2800
2013-08-13	$0.2300
2013-05-14	$0.2300
2013-02-19	$0.2300
2012-11-13	$0.2300
2012-08-14	$0.2000
2012-05-15	$0.2000
2012-02-14	$0.2000
2011-11-15	$0.2000
2011-08-16	$0.1600
2011-05-17	$0.1600
2011-02-15	$0.1600
2010-11-16	$0.1600
2010-08-17	$0.1300
2010-05-18	$0.1300
2010-02-16	$0.1300
2009-11-17	$0.1300
2009-08-18	$0.1300
2009-05-19	$0.1300
2009-02-17	$0.1300
2008-11-18	$0.1300
2008-08-19	$0.1100
2008-05-13	$0.1100
2008-02-19	$0.1100
2007-11-13	$0.1100
2007-08-14	$0.1000
2007-05-15	$0.1000
2007-02-13	$0.1000
2006-11-14	$0.1000
2006-08-15	$0.0900
2006-05-15	$0.0900
2006-02-15	$0.0900
2005-11-15	$0.0800
2005-08-15	$0.0800
2005-05-16	$0.0800
2005-02-15	$0.0800
2004-11-15	$3.0800
2004-08-23	$0.0800
2003-10-15	$0.1600
2003-02-19	$0.0800

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.10. Further, an Excess Kurtosis of 10.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.17%	70	0.70%	0.70%
-6.17% to -4.60%	79	0.79%	1.48%
-4.60% to -3.03%	338	3.36%	4.84%
-3.03% to -1.46%	1148	11.41%	16.25%
-1.46% to 0.11%	3582	35.60%	51.85%
0.11% to 1.68%	3143	31.24%	83.08%
1.68% to 3.25%	1161	11.54%	94.62%
3.25% to 4.82%	345	3.43%	98.05%
4.82% to 6.39%	120	1.19%	99.24%
Greater than 6.39%	76	0.76%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.98%	2.20%	77.71%	68.27%
±2σ	-4.07%	4.29%	95.18%	95.45%
±3σ	-6.17%	6.39%	98.55%	99.73%

Descriptive Statistics

Mean	0.110%
Standard Error	0.021%
Median	0.038%
Mode	0.000%
Standard Deviation	2.092%
Sample Variance	4.376%
Kurtosis	10.85
Skewness	-0.10
Range	49.681%
Minimum	-30.116%
Maximum	19.565%
Count	10062

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.528%	5102	50.71%	0.775%
Negative	-1.409%	4748	47.19%	-0.665%
Zero	0.000%	212	2.11%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	1.66%
20	1 Month	3.18%
60	1 Quarter	2.22%
250	1 Year	2.06%
750	3 Years	1.97%
1,250	5 Years	2.16%
2,500	10 Years	2.09%
5,000	20 Years	2.16%
12,500	50 Years	2.64%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.82. Further, an Excess Kurtosis of 45.29 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -2.45%	0	0.00%	0.00%
-2.45% to -1.21%	0	0.00%	0.00%
-1.21% to 0.03%	0	0.00%	0.00%
0.03% to 1.27%	1808	17.97%	17.97%
1.27% to 2.51%	4408	43.80%	61.77%
2.51% to 3.75%	2286	22.72%	84.49%
3.75% to 4.99%	892	8.86%	93.35%
4.99% to 6.23%	360	3.58%	96.93%
6.23% to 7.47%	140	1.39%	98.32%
Greater than 7.47%	169	1.68%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.86%	4.17%	84.46%	68.27%
±2σ	-0.79%	5.82%	96.25%	95.45%
±3σ	-2.45%	7.47%	98.32%	99.73%

Descriptive Statistics

Mean	2.513%
Standard Error	0.016%
Median	2.096%
Mode	1.786%
Standard Deviation	1.654%
Sample Variance	2.735%
Kurtosis	45.29
Skewness	3.82
Range	40.735%
Minimum	0.376%
Maximum	41.111%
Count	10063

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.513%	10063	100.00%	2.513%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.83. Further, an Excess Kurtosis of 19.80 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.43%	69	0.69%	0.69%
-5.43% to -4.08%	119	1.18%	1.87%
-4.08% to -2.73%	357	3.55%	5.42%
-2.73% to -1.38%	1160	11.53%	16.94%
-1.38% to -0.03%	3265	32.45%	49.39%
-0.03% to 1.32%	3381	33.60%	82.99%
1.32% to 2.67%	1148	11.41%	94.40%
2.67% to 4.02%	396	3.94%	98.33%
4.02% to 5.37%	99	0.98%	99.31%
Greater than 5.37%	69	0.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.83%	1.77%	77.09%	68.27%
±2σ	-3.63%	3.57%	94.96%	95.45%
±3σ	-5.43%	5.37%	98.63%	99.73%

Descriptive Statistics

Mean	-0.031%
Standard Error	0.018%
Median	0.000%
Mode	0.000%
Standard Deviation	1.801%
Sample Variance	3.244%
Kurtosis	19.80
Skewness	0.83
Range	49.525%
Minimum	-13.613%
Maximum	35.912%
Count	10063

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.286%	4802	47.72%	0.614%
Negative	-1.295%	5006	49.75%	-0.644%
Zero	0.000%	255	2.53%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.10. Further, an Excess Kurtosis of 4.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.17%	10	0.48%	0.48%
-13.17% to -9.75%	28	1.34%	1.82%
-9.75% to -6.32%	67	3.21%	5.04%
-6.32% to -2.90%	281	13.48%	18.52%
-2.90% to 0.53%	679	32.58%	51.10%
0.53% to 3.95%	658	31.57%	82.68%
3.95% to 7.37%	237	11.37%	94.05%
7.37% to 10.80%	85	4.08%	98.13%
10.80% to 14.22%	23	1.10%	99.23%
Greater than 14.22%	16	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.04%	5.09%	75.24%	68.27%
±2σ	-8.61%	9.66%	94.77%	95.45%
±3σ	-13.17%	14.22%	98.75%	99.73%

Descriptive Statistics

Mean	0.525%
Standard Error	0.100%
Median	0.460%
Mode	0.000%
Standard Deviation	4.565%
Sample Variance	20.836%
Kurtosis	4.61
Skewness	0.10
Range	64.739%
Minimum	-36.713%
Maximum	28.026%
Count	2084

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.444%	1153	55.33%	1.905%
Negative	-3.140%	916	43.95%	-1.380%
Zero	0.000%	15	0.72%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	7.30%
12	1 Quarter	5.22%
52	1 Year	4.95%
156	3 Years	4.64%
260	5 Years	5.12%
520	10 Years	4.87%
1,040	20 Years	5.01%
2,600	50 Years	6.19%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.76. Further, an Excess Kurtosis of 34.83 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.54%	0	0.00%	0.00%
-6.54% to -3.31%	0	0.00%	0.00%
-3.31% to -0.08%	0	0.00%	0.00%
-0.08% to 3.15%	375	17.99%	17.99%
3.15% to 6.38%	913	43.79%	61.77%
6.38% to 9.62%	477	22.88%	84.65%
9.62% to 12.85%	182	8.73%	93.38%
12.85% to 16.08%	76	3.65%	97.03%
16.08% to 19.31%	31	1.49%	98.51%
Greater than 19.31%	31	1.49%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.08%	10.69%	84.60%	68.27%
±2σ	-2.23%	15.00%	95.88%	95.45%
±3σ	-6.54%	19.31%	98.51%	99.73%

Descriptive Statistics

Mean	6.385%
Standard Error	0.094%
Median	5.348%
Mode	4.762%
Standard Deviation	4.308%
Sample Variance	18.561%
Kurtosis	34.83
Skewness	3.76
Range	65.149%
Minimum	0.962%
Maximum	66.111%
Count	2085

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.385%	2085	100.00%	6.385%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.33. Further, an Excess Kurtosis of 16.30 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.81%	10	0.48%	0.48%
-13.81% to -10.42%	19	0.91%	1.39%
-10.42% to -7.02%	75	3.60%	4.99%
-7.02% to -3.63%	257	12.33%	17.31%
-3.63% to -0.24%	730	35.01%	52.33%
-0.24% to 3.16%	630	30.22%	82.54%
3.16% to 6.55%	245	11.75%	94.29%
6.55% to 9.94%	75	3.60%	97.89%
9.94% to 13.34%	27	1.29%	99.18%
Greater than 13.34%	17	0.82%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.76%	4.29%	76.45%	68.27%
±2σ	-9.29%	8.81%	95.01%	95.45%
±3σ	-13.81%	13.33%	98.71%	99.73%

Descriptive Statistics

Mean	-0.237%
Standard Error	0.099%
Median	-0.435%
Mode	0.000%
Standard Deviation	4.524%
Sample Variance	20.463%
Kurtosis	16.30
Skewness	1.33
Range	78.574%
Minimum	-18.905%
Maximum	59.669%
Count	2085

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.282%	935	44.84%	1.472%
Negative	-3.142%	1134	54.39%	-1.709%
Zero	0.000%	16	0.77%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.70. Further, an Excess Kurtosis of 3.00 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.80%	1	0.21%	0.21%
-25.80% to -18.78%	2	0.42%	0.63%
-18.78% to -11.76%	22	4.59%	5.22%
-11.76% to -4.74%	72	15.03%	20.25%
-4.74% to 2.28%	150	31.32%	51.57%
2.28% to 9.30%	153	31.94%	83.51%
9.30% to 16.32%	52	10.86%	94.36%
16.32% to 23.34%	11	2.30%	96.66%
23.34% to 30.36%	11	2.30%	98.96%
Greater than 30.36%	5	1.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.08%	11.64%	74.53%	68.27%
±2σ	-16.44%	21.00%	95.41%	95.45%
±3σ	-25.80%	30.35%	98.75%	99.73%

Descriptive Statistics

Mean	2.278%
Standard Error	0.428%
Median	2.111%
Mode	0.000%
Standard Deviation	9.359%
Sample Variance	87.585%
Kurtosis	3.00
Skewness	0.70
Range	85.907%
Minimum	-34.353%
Maximum	51.554%
Count	479

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.747%	288	60.12%	4.658%
Negative	-6.065%	188	39.25%	-2.380%
Zero	0.000%	3	0.63%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	9.25%
12	1 Year	10.22%
36	3 Years	9.98%
60	5 Years	10.87%
120	10 Years	10.43%
240	20 Years	10.71%
600	50 Years	13.45%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.67. Further, an Excess Kurtosis of 28.37 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.34%	0	0.00%	0.00%
-13.34% to -6.38%	0	0.00%	0.00%
-6.38% to 0.57%	0	0.00%	0.00%
0.57% to 7.53%	76	15.83%	15.83%
7.53% to 14.49%	221	46.04%	61.88%
14.49% to 21.44%	109	22.71%	84.58%
21.44% to 28.40%	45	9.38%	93.96%
28.40% to 35.36%	16	3.33%	97.29%
35.36% to 42.31%	6	1.25%	98.54%
Greater than 42.31%	7	1.46%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.21%	23.76%	83.96%	68.27%
±2σ	-4.06%	33.04%	96.88%	95.45%
±3σ	-13.34%	42.31%	98.54%	99.73%

Descriptive Statistics

Mean	14.488%
Standard Error	0.423%
Median	12.176%
Mode	6.392%
Standard Deviation	9.275%
Sample Variance	86.020%
Kurtosis	28.37
Skewness	3.67
Range	109.902%
Minimum	2.849%
Maximum	112.751%
Count	480

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.488%	480	100.00%	14.488%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.23. Further, an Excess Kurtosis of 1.78 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.54%	2	0.42%	0.42%
-27.54% to -20.97%	6	1.25%	1.67%
-20.97% to -14.41%	18	3.75%	5.42%
-14.41% to -7.84%	58	12.08%	17.50%
-7.84% to -1.28%	173	36.04%	53.54%
-1.28% to 5.28%	126	26.25%	79.79%
5.28% to 11.85%	65	13.54%	93.33%
11.85% to 18.41%	23	4.79%	98.12%
18.41% to 24.98%	6	1.25%	99.38%
Greater than 24.98%	3	0.62%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-10.03%	7.47%	73.75%	68.27%
±2σ	-18.78%	16.22%	94.17%	95.45%
±3σ	-27.54%	24.98%	98.96%	99.73%

Descriptive Statistics

Mean	-1.280%
Standard Error	0.399%
Median	-1.900%
Mode	-6.750%
Standard Deviation	8.752%
Sample Variance	76.599%
Kurtosis	1.78
Skewness	0.23
Range	69.411%
Minimum	-34.017%
Maximum	35.394%
Count	480

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.686%	194	40.42%	2.702%
Negative	-6.731%	284	59.17%	-3.983%
Zero	0.000%	2	0.42%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.49. Further, an Excess Kurtosis of 4.97 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.31%	0	0.00%	0.00%
-41.31% to -29.32%	0	0.00%	0.00%
-29.32% to -17.33%	4	2.50%	2.50%
-17.33% to -5.34%	29	18.12%	20.62%
-5.34% to 6.65%	53	33.12%	53.75%
6.65% to 18.64%	47	29.38%	83.12%
18.64% to 30.63%	18	11.25%	94.38%
30.63% to 42.62%	4	2.50%	96.88%
42.62% to 54.61%	3	1.88%	98.75%
Greater than 54.61%	2	1.25%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.34%	22.64%	78.12%	68.27%
±2σ	-25.32%	38.62%	95.62%	95.45%
±3σ	-41.31%	54.61%	98.75%	99.73%

Descriptive Statistics

Mean	6.650%
Standard Error	1.264%
Median	5.758%
Mode	-28.736%
Standard Deviation	15.986%
Sample Variance	255.541%
Kurtosis	4.97
Skewness	1.49
Range	117.445%
Minimum	-28.736%
Maximum	88.709%
Count	160

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.923%	101	63.12%	9.420%
Negative	-7.513%	59	36.88%	-2.770%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	19.39%
12	3 Years	17.21%
20	5 Years	18.70%
40	10 Years	18.37%
80	20 Years	18.60%
200	50 Years	23.98%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.26. Further, an Excess Kurtosis of 7.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.38%	0	0.00%	0.00%
-21.38% to -9.35%	0	0.00%	0.00%
-9.35% to 2.69%	0	0.00%	0.00%
2.69% to 14.73%	26	16.15%	16.15%
14.73% to 26.77%	80	49.69%	65.84%
26.77% to 38.81%	30	18.63%	84.47%
38.81% to 50.85%	13	8.07%	92.55%
50.85% to 62.88%	6	3.73%	96.27%
62.88% to 74.92%	2	1.24%	97.52%
Greater than 74.92%	4	2.48%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.72%	42.82%	85.09%	68.27%
±2σ	-5.33%	58.87%	95.65%	95.45%
±3σ	-21.38%	74.92%	97.52%	99.73%

Descriptive Statistics

Mean	26.770%
Standard Error	1.265%
Median	21.847%
Mode	8.040%
Standard Deviation	16.051%
Sample Variance	257.626%
Kurtosis	7.42
Skewness	2.26
Range	104.711%
Minimum	8.040%
Maximum	112.751%
Count	161

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	26.770%	161	100.00%	26.770%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.02. Further, an Excess Kurtosis of 1.14 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -43.06%	1	0.62%	0.62%
-43.06% to -33.29%	2	1.24%	1.86%
-33.29% to -23.52%	5	3.11%	4.97%
-23.52% to -13.75%	25	15.53%	20.50%
-13.75% to -3.99%	55	34.16%	54.66%
-3.99% to 5.78%	41	25.47%	80.12%
5.78% to 15.55%	22	13.66%	93.79%
15.55% to 25.32%	8	4.97%	98.76%
25.32% to 35.08%	1	0.62%	99.38%
Greater than 35.08%	1	0.62%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.01%	9.04%	74.53%	68.27%
±2σ	-30.03%	22.06%	95.03%	95.45%
±3σ	-43.06%	35.08%	98.76%	99.73%

Descriptive Statistics

Mean	-3.987%
Standard Error	1.026%
Median	-5.098%
Mode	-46.666%
Standard Deviation	13.023%
Sample Variance	169.591%
Kurtosis	1.14
Skewness	-0.02
Range	87.885%
Minimum	-46.666%
Maximum	41.219%
Count	161

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.455%	63	39.13%	3.308%
Negative	-12.110%	97	60.25%	-7.296%
Zero	0.000%	1	0.62%	0.000%

MSFT Deep Analytical Report