Q-Q Plots for Financial Data: Detecting Non-Normality in Asset Returns¶

Overview¶

Q-Q plots are particularly valuable in finance for diagnosing departures from normality in asset returns. Many financial models assume returns follow a normal distribution, but empirical data often exhibit heavy tails and skewness, leading to significant underestimation of tail risk. This section focuses on using Q-Q plots to visualize these departures in financial data.

Why Normality Matters in Finance¶

Financial models rely heavily on distributional assumptions:

Option pricing (Black-Scholes) assumes log-normal asset prices, equivalent to normally distributed log-returns
Value at Risk (VaR) and Expected Shortfall (ES) rely on tail behavior assumptions
Portfolio optimization uses variance-covariance approaches that assume multivariate normality

When actual returns deviate from normality, these models systematically underestimate the probability of large losses, especially during market stress periods.

Real-World Example: Netflix (NFLX) Log-Returns¶

Netflix stock provides an excellent case study of non-normal financial returns. The following example uses daily log-returns computed from closing prices:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

# Simulate NFLX-like log-returns (or load real data)
np.random.seed(42)
# Use a distribution with slightly heavier tails than normal
# (Actual NFLX returns are even heavier-tailed)
returns = np.random.standard_t(df=8, size=1000) * 0.02

# Create Q-Q plot against normal distribution
fig, ax = plt.subplots(figsize=(8, 6))
stats.probplot(returns, dist="norm", plot=ax)

ax.set_title("Q-Q Plot: Daily Log-Returns vs Normal Distribution", fontsize=12)
ax.set_xlabel("Theoretical Normal Quantiles", fontsize=11)
ax.set_ylabel("Sample Quantiles (Observed Returns)", fontsize=11)

# Enhance aesthetics
ax.spines[["top", "right"]].set_visible(False)
ax.grid(True, alpha=0.3, linestyle='--')

plt.tight_layout()
plt.show()

Interpreting the Q-Q Plot for Financial Data¶

Perfect Normality¶

When data follow a normal distribution exactly, the Q-Q plot shows all points tightly clustered along the 45-degree reference line.

Heavy Tails (The Financial Reality)¶

Real stock returns exhibit heavy tails: both the left tail (large negative returns/losses) and right tail (large positive returns/gains) contain more observations than the normal distribution predicts.

Visual signature: The Q-Q plot "bends upward" in the right tail and "bends downward" in the left tail. This creates an S-shaped pattern.

# Simulate heavy-tailed returns (e.g., using Student's t distribution)
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)
df = 5  # degrees of freedom; lower = heavier tails
heavy_tailed_returns = stats.t.rvs(df=df, scale=0.02, size=2000)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Histogram with normal overlay
ax1.hist(heavy_tailed_returns, bins=50, density=True, alpha=0.6, label='Observed Returns')
x = np.linspace(heavy_tailed_returns.min(), heavy_tailed_returns.max(), 100)
ax1.plot(x, stats.norm.pdf(x, loc=heavy_tailed_returns.mean(),
                            scale=heavy_tailed_returns.std()),
         'r-', lw=2, label='Normal PDF')
ax1.set_xlabel('Daily Log-Return', fontsize=11)
ax1.set_ylabel('Density', fontsize=11)
ax1.set_title('Distribution Shape: Heavy Tails vs Normal', fontsize=12)
ax1.legend()
ax1.spines[["top", "right"]].set_visible(False)

# Right: Q-Q plot
stats.probplot(heavy_tailed_returns, dist="norm", plot=ax2)
ax2.set_title("Q-Q Plot: Revealing Heavy Tails", fontsize=12)
ax2.set_xlabel('Theoretical Normal Quantiles', fontsize=11)
ax2.set_ylabel('Sample Quantiles', fontsize=11)
ax2.spines[["top", "right"]].set_visible(False)
ax2.grid(True, alpha=0.3, linestyle='--')

plt.tight_layout()
plt.show()

Consequences of Ignoring Non-Normality¶

Underestimating Tail Risk¶

The normal distribution has a kurtosis of 3. Real financial returns typically have excess kurtosis > 1, meaning the tails are heavier than normal.

Example: If the normal distribution predicts a 5% loss with 0.1% probability, actual heavy-tailed returns might exhibit this loss with 0.5% probability — a 5x underestimation!

Implication for Risk Management¶

VaR models based on normality underestimate losses at the 99th or 99.9th percentile
Hedging strategies based on normal assumptions leave portfolios vulnerable to tail events
Capital requirements (e.g., Basel III) may be insufficient if built on normal assumptions

Practical Workflow: Diagnosing Return Distributions¶

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

# Step 1: Load or simulate returns
np.random.seed(42)
returns = np.random.standard_t(df=6, size=1500) * 0.025

# Step 2: Compute summary statistics
mean_ret = returns.mean()
std_ret = returns.std()
skewness = stats.skew(returns)
kurtosis = stats.kurtosis(returns)  # Excess kurtosis

print(f"Mean:     {mean_ret:.4f}")
print(f"Std Dev:  {std_ret:.4f}")
print(f"Skewness: {skewness:.4f}")
print(f"Ex. Kurtosis: {kurtosis:.4f}")

# Step 3: Normality tests
_, p_ks = stats.kstest(returns, 'norm', args=(mean_ret, std_ret))
_, p_ad = stats.anderson(returns, dist='norm')
_, p_jb = stats.jarque_bera(returns)

print(f"\nKolmogorov-Smirnov test p-value: {p_ks:.4f}")
print(f"Jarque-Bera test p-value: {p_jb:.4f}")

# Step 4: Q-Q plot
fig, ax = plt.subplots(figsize=(8, 6))
stats.probplot(returns, dist="norm", plot=ax)
ax.set_title("Diagnostics: Are Returns Normal?", fontsize=12)
ax.spines[["top", "right"]].set_visible(False)
plt.show()

Adjusting for Non-Normality¶

1. Alternative Distributions¶

Instead of normal, fit Student's \(t\) or Generalized Hyperbolic distributions:

from scipy.stats import t as student_t

# Fit Student's t distribution
df, loc, scale = student_t.fit(returns)
print(f"Fitted df: {df:.2f} (lower df → heavier tails)")

2. Non-Parametric Methods¶

Bootstrap and quantile-based approaches make no distributional assumptions.

3. Modified Risk Measures¶

Use Expected Shortfall (CVaR) instead of VaR; it better captures tail behavior for non-normal distributions.

Summary¶

Q-Q plots visually compare sample quantiles to theoretical quantiles, revealing distributional shape
Financial returns exhibit heavy tails, showing an S-shaped pattern in Q-Q plots
Ignoring non-normality leads to systematic underestimation of tail risk
Practical solutions: use alternative distributions, non-parametric methods, or robust risk measures tailored to observed tail behavior