Normality of Daily Stock Returns
Evidence from Apple, Microsoft, and NVIDIA using daily logarithmic returns from January 2020 through October 22, 2025.
The central finding
The study tests whether a single normal distribution provides an adequate description of daily stock returns. Across all three companies, descriptive statistics, graphical diagnostics, and formal tests point in the same direction.
Do daily stock returns follow a normal distribution?
Four layers of evidence
Rather than relying on a single statistic, the analysis combines descriptive measures, graphical diagnostics, and two formal tests of normality.
Construct returns
Calculate daily logarithmic returns from consecutive available Close prices.
Describe the shape
Measure mean, standard deviation, skewness, kurtosis, minimum, and maximum.
Inspect graphically
Use histograms, normal Q-Q plots, and box plots to inspect the distributions.
Test formally
Apply Shapiro–Wilk and Jarque–Bera tests at a nominal 5% significance level.
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Three distinct return profiles
NVIDIA has the highest volatility and the widest observed return range. Microsoft has the highest kurtosis, showing that volatility and tail behavior describe different statistical characteristics.
Apple
Microsoft
NVIDIA
Kurtosis vs. Normal Benchmark
Pearson kurtosis · Normal benchmark = 3
Mean Return vs. Volatility
Daily mean return compared with standard deviation
Observed Daily Return Range
Minimum to maximum daily logarithmic return
The statistical evidence agrees
Both Shapiro–Wilk and Jarque–Bera report p-values below 0.001 for every company. At the nominal 5% significance level, the normality null hypothesis is rejected for all three stocks.
Tests reject normality
Jarque–Bera and Shapiro–Wilk reach the same decision for Apple, Microsoft, and NVIDIA.
| Stock | Jarque–Bera | Shapiro–Wilk | P-value | 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 strongest departure appears in the tails.
Skewness remains fairly close to zero for all three stocks, yet normality is strongly rejected.
A distribution can therefore be approximately symmetric while still producing extreme observations more frequently than a Gaussian model.
What this study does not claim
Rejecting normality answers a specific statistical question. It does not establish an optimal alternative distribution or imply that future stock returns can be predicted.
Three stocks only
The sample cannot represent every industry, market, firm size, or asset class.
One pooled period
Nearly six years are combined, including the unusually volatile COVID-19 period.
No fitted alternative
Student’s t and other heavy-tailed distributions are discussed but not formally fitted.
Dependence remains possible
Test p-values do not adjust for possible volatility clustering or serial dependence.
A single normal distribution is an inadequate description of these pooled daily returns.
AAPL, MSFT, and NVDA all display Pearson kurtosis well above the normal benchmark of 3, substantial tail departures in the Q-Q diagnostics, and p-values below 0.001 in both formal tests. For this sample, a single fitted normal model may give too little weight to extreme daily movements.
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.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity.
Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues.
Fama, E. F. (1965). The behavior of stock-market prices.
Mandelbrot, B. (1963). The variation of certain speculative prices.
Peiró, A. (1999). Skewness in financial returns.
Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality.
Zhu, D., & Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution.