TSLA daily-return tails vs Gaussian over the last ~3 years
TSLA’s daily returns over the last ~3 years show a clear breakdown of the bell curve: we observed 7 two‑sided |z|≥3 sessions versus 2.03 expected (about 3.45× inflation), and a binomial test gives p = 0.0048 — far less likely than chance. The return series also displays visible heavy tails and positive skew, so extreme moves are both large and asymmetric.
This study resampled minute bars to daily closes, standardized close‑to‑close returns, counted two‑tailed exceedances and measured excess kurtosis. The short thesis is simple: normal‑based daily‑return models materially understate how often TSLA detonates into true 3σ events. The charts, full stats and significance tests that support this finding follow below.
For TSLA over the past ~3 years, do its daily returns break the bell curve — how many 3-sigma-plus sessions actually occur versus the ~2 a normal distribution predicts over this many trading days, and how heavy are the tails and kurtosis? Thesis: TSLA logs several times more extreme days than Gaussian math allows, so any risk model built on normal returns badly understates how often the stock detonates in either direction.
How this was measured
Resampled TSLA minute bars to daily closes, computed close-to-close returns, and standardized them to z-scores using the in-sample mean and sample standard deviation over the last ~3 years (ending at the latest available date). Counted two-tailed exceedances beyond 2σ and 3σ and compared against normal-theory tail probabilities p=2·(1−Φ(k)). Binomial tests (greater-tail) assess whether observed exceedances are significantly above Gaussian expectations. Excess kurtosis (Fisher) summarizes tail heaviness.
The key numbers
Reading the numbers
Over 751 trading days TSLA produced 7 two‑sided 3σ sessions versus about 2 expected under a Gaussian (Observed/Expected = 3.45), and the binomial p=0.0048 says that excess is unlikely to be random — extreme days are materially more common than a normal model predicts.
The charts
This histogram plots TSLA daily returns after converting them to z‑scores (n=751, mean centered at 0). What to look at are the far ends: the leftmost return reaches −4.8846 and the rightmost reaches 6.191, so there are very large moves on both sides rather than a tight bell. Those long tails, combined with the positive skew and excess kurtosis reported in the headline stats, show that returns are more extreme and a bit right‑leaning compared with a normal bell curve.
The two bars compare observed counts to Gaussian expectations for |z|≥2 and |z|≥3. For |z|≥2 TSLA had 32 observed events versus ~34.17 expected (so roughly in line), but for |z|≥3 the stock produced 7 observed events versus only ~2.03 expected — about 3.45× more 3σ days. Note the asymmetry in direction (3σ upper=4, lower=3; 2σ upper=20, lower=12) and the binomial p=0.0048 from the headlines, which reinforces that the excess 3σ tail frequency is not just sampling noise.
Tail exceedance summary (two-tailed)
| threshold | n_days | p_gaussian | expected_count | observed_count | obs_over_exp | upper_tail | lower_tail | binom_p_greater |
|---|---|---|---|---|---|---|---|---|
| |z| ≥ 2 | 751 | 0.0455 | 34.17 | 32 | 0.94 | 20 | 12 | 0.6724 |
| |z| ≥ 3 | 751 | 0.0027 | 2.03 | 7 | 3.45 | 4 | 3 | 0.0048 |
Top 10 absolute z-score days (most extreme)
| date | return | z_score | tail_sign |
|---|---|---|---|
| 2025-04-09 | 0.231 | 6.191 | 1 |
| 2025-03-10 | -0.1799 | -4.8846 | -1 |
| 2024-04-23 | 0.1591 | 4.2542 | 1 |
| 2024-04-29 | 0.1445 | 3.8585 | 1 |
| 2025-03-24 | 0.132 | 3.523 | 1 |
| 2025-06-05 | -0.1273 | -3.4659 | -1 |
| 2025-04-04 | -0.1225 | -3.3375 | -1 |
| 2024-11-11 | 0.109 | 2.9025 | 1 |
| 2024-11-06 | 0.1083 | 2.8844 | 1 |
| 2024-10-23 | 0.101 | 2.6864 | 1 |
The takeaway
Short answer: yes — TSLA produces far more extreme one-day moves than a Gaussian would predict. Over 751 trading days we saw 7 two‑sided |z|≥3 sessions versus 2.03 expected (about a 3.45× inflation), and a binomial test gives p=0.0048 — only about a 5‑in‑1,000 chance this gap is random. By contrast the |z|≥2 band is not elevated: 32 observed vs 34.17 expected (obs/exp ≈0.94). The distribution is visibly heavy‑tailed (excess kurtosis = 3.33) and skewed to the upside, and the largest single‑day moves reached +23.10% and −17.99%, so extremes are both big and more frequent at the far tail. Practical takeaway: normal‑based daily‑return models materially understate how often TSLA detonates into 3σ‑class moves; the signal for extreme‑tail fatness is statistically strong, even though the absolute count of extremes is modest.
The fine print
- Z‑scores use the in‑sample mean and sample std for the same 3‑year window; volatility clustering widens thresholds and biases against finding exceedances.
- Close‑to‑close returns ignore intraday spikes; a high‑low or realized‑volatility view would typically show even fatter tails.
- Counts are two‑tailed and lump all days; earnings/regime days can concentrate extremes and deserve separate treatment.
- Only seven 3σ events occurred — statistically significant but a small absolute count to characterize rare‑event structure.