Correlation Analysis

Why Correlation Matters

• Diversification: holding correlated assets provides no diversification benefit — you are effectively holding one concentrated position
• Risk management: portfolio risk depends on the correlation structure, not just individual asset volatility
• Pairs trading: highly correlated assets that temporarily diverge create mean-reversion opportunities
• Portfolio construction: optimal allocation requires accurate correlation estimates
• Crash protection: understanding tail dependence reveals whether assets crash together

Correlation Methods

Pearson Correlation

import pandas as pd
import numpy as np

# Always compute on returns, never on prices
returns_a = prices_a.pct_change().dropna()
returns_b = prices_b.pct_change().dropna()

pearson_corr = returns_a.corr(returns_b)  # default is Pearson

• Range: -1 (perfect inverse) to +1 (perfect co-movement)
• Assumes: linear relationship, normally distributed returns, no outliers
• Limitation: crypto returns are heavy-tailed — Pearson underestimates extreme co-movement

Spearman Rank Correlation

spearman_corr = returns_a.corr(returns_b, method='spearman')

• More robust to outliers and non-linear relationships
• Better for crypto due to heavy-tailed return distributions
• Slightly lower power than Pearson when normality holds

Kendall Tau Correlation

kendall_corr = returns_a.corr(returns_b, method='kendall')

• Most robust to outliers of the three methods
• Computationally slower on large datasets
• Best for small samples or heavily skewed data

Rolling Correlation

Window-Based Rolling Correlation

# Rolling Pearson correlation
rolling_corr = returns_a.rolling(window=60).corr(returns_b)

# Multiple windows for different time horizons
windows = {
    'short': 20,    # ~1 month of trading days
    'medium': 60,   # ~3 months
    'long': 120,    # ~6 months
}
for label, w in windows.items():
    df[f'corr_{label}'] = returns_a.rolling(w).corr(returns_b)

EWMA Correlation

def ewma_correlation(x: pd.Series, y: pd.Series, span: int = 60) -> pd.Series:
    """Compute EWMA correlation between two return series."""
    cov_xy = x.mul(y).ewm(span=span).mean() - x.ewm(span=span).mean() * y.ewm(span=span).mean()
    std_x = x.ewm(span=span).std()
    std_y = y.ewm(span=span).std()
    return cov_xy / (std_x * std_y)

Typical Windows

Correlation Matrix Analysis

Computing the Full Matrix

# Build return matrix for multiple assets
returns = pd.DataFrame({
    'BTC': btc_returns,
    'ETH': eth_returns,
    'SOL': sol_returns,
    'AVAX': avax_returns,
})

# Correlation matrix (Pearson)
corr_matrix = returns.corr()

# Spearman (better for crypto)
spearman_matrix = returns.corr(method='spearman')

Eigenvalue Decomposition

eigenvalues, eigenvectors = np.linalg.eigh(corr_matrix.values)

# Sort descending
idx = eigenvalues.argsort()[::-1]
eigenvalues = eigenvalues[idx]
eigenvectors = eigenvectors[:, idx]

# First eigenvalue = market factor (explains most variance)
# Subsequent eigenvalues = sector/style factors
market_factor_pct = eigenvalues[0] / eigenvalues.sum() * 100

• First eigenvector: the market factor — when this dominates (>60% variance), everything moves together
• Subsequent eigenvectors: sector or style factors
• Small eigenvalues: noise / idiosyncratic risk

Minimum Variance Portfolio

from numpy.linalg import inv

cov_matrix = returns.cov()
ones = np.ones(len(cov_matrix))
inv_cov = inv(cov_matrix.values)

# Minimum variance weights
weights = inv_cov @ ones / (ones @ inv_cov @ ones)

Hierarchical Clustering

from scipy.cluster.hierarchy import linkage, fcluster
from scipy.spatial.distance import squareform

# Convert correlation to distance
dist_matrix = np.sqrt(2 * (1 - corr_matrix.values))
np.fill_diagonal(dist_matrix, 0)

# Hierarchical clustering
condensed = squareform(dist_matrix)
linkage_matrix = linkage(condensed, method='ward')

# Cut at threshold to get clusters
clusters = fcluster(linkage_matrix, t=1.0, criterion='distance')

• Sector detection: assets in the same cluster behave similarly
• Diversification: select one asset per cluster for maximum diversification
• Risk allocation: allocate risk budget across clusters, not individual assets

Tail Dependence

Lower Tail Dependence

def tail_dependence(x: pd.Series, y: pd.Series, quantile: float = 0.05) -> float:
    """Estimate lower tail dependence coefficient.

    Measures P(Y < q | X < q) for quantile q.
    Higher values mean assets crash together more often.
    """
    threshold_x = x.quantile(quantile)
    threshold_y = y.quantile(quantile)
    joint_extreme = ((x < threshold_x) & (y < threshold_y)).sum()
    marginal_extreme = (x < threshold_x).sum()
    return joint_extreme / marginal_extreme if marginal_extreme > 0 else 0.0

Crypto-Specific Tail Behavior

• Normal correlation of 0.6 between two altcoins might have tail dependence of 0.8
• During market panics, correlations spike toward 1.0 across all risk assets
• This means diversification benefits disappear exactly when needed most

Regime-Dependent Correlation

Detecting Correlation Regime Shifts

def correlation_zscore(rolling_corr: pd.Series, lookback: int = 252) -> pd.Series:
    """Z-score of rolling correlation vs its own history."""
    mean = rolling_corr.rolling(lookback).mean()
    std = rolling_corr.rolling(lookback).std()
    return (rolling_corr - mean) / std

# Flag regime shift when z-score exceeds threshold
zscore = correlation_zscore(rolling_corr_60d)
regime_shift = zscore.abs() > 2.0

Crypto-Specific Correlation Patterns

Typical Correlation Ranges

Key Observations

• Most altcoins are highly correlated with BTC (0.6–0.9) — the market factor dominates
• Meme and PumpFun tokens show lower normal correlation but higher tail dependence
• SOL ecosystem tokens correlate strongly with SOL price
• Stablecoins should be uncorrelated with risk assets — if correlation appears, investigate (depeg risk)
• Correlation tends to increase during high-volatility regimes
• New token launches may show temporarily low correlation until price discovery stabilizes

Integration with Other Skills

• risk-management: use correlation to compute portfolio-level VaR and stress scenarios
• portfolio-analytics: correlation matrix feeds optimal allocation algorithms
• regime-detection: correlation regime shifts are an input to regime classification
• cointegration-analysis: pairs with high correlation are candidates for cointegration testing
• position-sizing: correlation-adjusted sizing prevents correlated concentration

Files

References

• references/methodology.md

  — Correlation formulas, statistical tests, estimation methods

• references/portfolio_applications.md

  — Diversification metrics, pairs trading, risk decomposition

Scripts

• scripts/correlation_matrix.py

  — Multi-asset correlation matrix, clustering, diversification metrics

• scripts/rolling_correlation.py

  — Rolling correlation, regime detection, tail dependence analysis