Normality of Daily Stock Returns
Evidence from Apple, Microsoft, and NVIDIA using daily logarithmic returns from January 2020 through October 22, 2025.
Does the normal benchmark fit?
A normal distribution is symmetric, centered around its mean, and relatively unlikely to produce extreme observations. The study tests whether actual daily returns behave that way in a recent technology-stock sample.
“Do daily stock returns follow a normal distribution?”
Why it matters. If a fitted normal distribution understates the frequency of very large gains and losses, risk estimates based on that benchmark can give too little weight to extreme daily movements.
Scope. The question is distributional—not predictive. Rejecting normality does not imply that future return directions are forecastable.
Four layers of evidence
The paper combines summary statistics, distributional graphics, Q-Q diagnostics, and two formal normality tests. No observations are winsorized or removed.
Construct returns
Daily logarithmic returns from consecutive available trading-day Close prices.
Describe the shape
Mean, standard deviation, skewness, kurtosis, minimum, and maximum.
Inspect graphically
Histograms, fitted normal curves, normal Q-Q plots, and box plots.
Test formally
Shapiro–Wilk and Jarque–Bera tests at a nominal 5% significance level.
rt = ln(Pt / Pt−1)
Returns are reported in percent in the descriptive results.
Three stocks, three distinct profiles
NVIDIA has the highest volatility and widest observed range, while Microsoft has the highest kurtosis. This distinction shows that dispersion and tail shape capture different features of return behavior.
Apple
Microsoft
NVIDIA
Kurtosis vs. normal benchmark
Pearson kurtosis; normal distribution = 3
Mean return vs. volatility
Daily mean plotted against standard deviation
Observed daily return range
Minimum to maximum daily logarithmic return for each stock
Every test reaches the same decision
Both Shapiro–Wilk and Jarque–Bera report p-values below 0.001 for AAPL, MSFT, and NVDA. At the nominal 5% significance level, the null hypothesis of normality is rejected in all six stock-test combinations.
6 of 6 tests reject normality.
The formal evidence agrees with the descriptive statistics and with the paper’s Q-Q plots, where departures are largest in the tails.
| Stock | Jarque–Bera | Shapiro–Wilk | p-values | Decision |
|---|---|---|---|---|
| AAPL | 2373.04 | 0.9336 | <0.001 | Reject H₀ |
| MSFT | 3710.63 | 0.9289 | <0.001 | Reject H₀ |
| NVDA | 989.18 | 0.9635 | <0.001 | Reject H₀ |
The main departure is in the tails.
Skewness is fairly close to zero for all three stocks, yet normality is still rejected. The paper’s Q-Q plots show lower-tail observations below the fitted line and upper-tail observations above it.
This is the core distinction between symmetry and normality: a distribution can be roughly symmetric while still producing extreme observations much more often than a Gaussian benchmark.
What this study does not claim
The paper deliberately keeps the conclusion narrow. The evidence establishes poor agreement with a single normal distribution for this sample, but it does not identify the best replacement model.
Three stocks only
The sample cannot represent all industries, smaller firms, international equities, or other asset classes.
One pooled period
Nearly six years are combined, including the unusually volatile 2020 COVID-19 episode.
No alternative fit
Student’s t, asymmetric Student-t, and stable distributions are discussed but not estimated and compared.
Dependence remains
Formal p-values do not adjust for possible volatility clustering or serial dependence in return magnitudes.
A single normal distribution is an inadequate description of these pooled daily returns.
AAPL, MSFT, and NVDA all display Pearson kurtosis well above 3, visible tail departures in the paper’s Q-Q diagnostics, and p-values below 0.001 in both formal normality tests. The practical implication is narrow but important: a single fitted normal model may assign too little probability to extreme daily movements in this sample.
Selected references from the paper
Baker, S. R., Bloom, N., Davis, S. J., Kost, K., Sammon, M., & Viratyosin, T. (2020). The unprecedented stock market reaction to COVID-19. The Review of Asset Pricing Studies, 10(4), 742–758.
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Fama, E. F. (1965). The behavior of stock-market prices. The Journal of Business, 38(1), 34–105.
Jarque, C. M., & Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6(3), 255–259.
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Peiró, A. (1999). Skewness in financial returns. Journal of Banking & Finance, 23(6), 847–862.
Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3–4), 591–611.
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