Q-Q Plot & Jarque-Bera — Are Your Returns Actually Normal?

4-section structure: Concept / How We Compute / How to Read / Caveats.

1. Concept

Many finance textbooks, risk models and portfolio theories assume normal returns:

Markowitz efficient frontier (mean + variance sufficient)
Black-Scholes option pricing (normal)
Parametric VaR (mean − 1.645σ)
Classical Sharpe Ratio (symmetry assumption)

Empirically, stock returns are not normal. Two characteristic deviations:

Fat tails — extreme moves (±3σ+) happen far more often than normal predicts
Skewness — typically left-skewed (down moves more violent)

Two complementary tools to test this:

Tool	Nature	Question
Q-Q Plot	Visual	"How far do points stray from the 45° line?"
Jarque-Bera	Statistical test	"p-value for rejecting normality?"

2. How We Compute

2.1 Q-Q Plot

1. Standardize: (r − mean) / σ (ddof=1)
2. Sort
3. For i-th sample (i = 0..n-1), plotting position p = (i + 0.5) / n
4. Theoretical quantile = Φ⁻¹(p)  (via statistics.NormalDist)
5. Plot (theoretical, sample) and overlay y=x reference line

Reading:

All points on green 45° line → perfectly normal
Tails straying upward → fat tails
Left-tail pulled down → negative skew
Widespread deviation → severely non-normal

Red dots mark points where |sample − theoretical| > 0.8. We downsample to 60 points for clean visuals.

2.2 Jarque-Bera Test

JB = n/6 × (skewness² + (excess_kurtosis)² / 4)
H₀: sample is normal
H₁: not normal
JB asymptotically ~ χ²(df=2)

For df=2, there's a closed-form p-value = exp(−JB/2) — no scipy dependency.

Verdicts (α = 0.05, critical = 5.991):

p ≥ 0.10 → fail_to_reject
0.05 ≤ p < 0.10 → marginal
0.001 ≤ p < 0.05 → reject
p < 0.001 → strong_reject

2.3 Data & Window

Past 252 trading days' simple daily returns
Sample < 60 → available: false

2.4 Fisher Excess Kurtosis (Normal = 0)

Consistent with v2.42 platform-wide fix. excess_kurt > 0 directly means "fatter than normal", more intuitive than raw kurtosis.

3. How to Read

3.1 Q-Q Plot Reading

Pattern	Meaning
Points on 45° line	Perfectly normal (rare)
Center fits, tails stray up	Fat tails (most common in stocks)
Center fits, left drops	Left skew (heavy down tail)
Widespread deviation	Seriously non-normal (e.g., leveraged ETF)

Red dots (|theoretical − sample| > 0.8) usually cluster in the tails. More red + farther = more severe deviation.

3.2 Skewness & Excess Kurtosis Scales

Metric	Range	Reading
Skewness	−0.5 to +0.5	Near-symmetric
	< −1	Clear left skew (heavy down tail)
	> +1	Clear right skew (rare)
Excess Kurtosis	< 1	Near-normal
	1 to 5	Typical stock
	> 5	Severe fat tails

3.3 Decision Impact

If reject normality (most cases):

Parametric VaR underestimates risk → use historical simulation (our choice)
Sharpe incomplete risk picture → pair with Sortino + CVaR
Portfolio correlations spike in extremes → P0.2 matrix misses this in normal times

If near normal (rare):

Trust Sharpe + parametric VaR
Typically liquid large-caps or steady dividend financials

4. Caveats

⚠️ JB Too Sensitive in Large Samples

With n > 1000, even near-normal samples almost always reject. Big-sample test issue.

Mitigation: look at absolute values of skew & excess kurt, not just p-value. If skew within ±0.5, excess kurt < 1, treat as "near-normal enough" even when p < 0.05.

⚠️ JB Is Asymptotic

Chi-square approximation breaks at small n. MIN_SAMPLE_DAYS = 60 ensures validity.

⚠️ Q-Q Plot Subjectivity

"How much deviation is significant?" — no hard answer. |Δ| > 0.8 threshold is visual aid, not formal test. Q-Q excels at comparing two samples (Stock A vs B) more than absolute judgments.

⚠️ Not for Extreme Samples

One-sided movement (only rising) → Q-Q tails distort
Structural breaks (M&A, business pivot) → mixing two distributions

JB rejection here means "history is heterogeneous", not "future is non-normal".

⚠️ Non-Normality Doesn't Invalidate Theory

Normal is the basis of Markowitz / Sharpe / Black-Scholes. Deviation means know the limits, not "abandon everything":

Fails during extreme events
Understates "true" risk
Portfolios may collapse in tail events

This awareness beats blanket rejection.

Try It

Stock Analysis → Risk: scroll to "Return Normality Diagnostic"
Watch tail deviations and red dot count
Compare to skew / excess kurtosis values
Compare different stock types: blue-chip vs speculative — huge JB differences
Click 📐 for formulas, thresholds, data sources