Correlation Analysis

Correlation Analysis

Cross-asset correlation analysis for diversification assessment, risk management, pairs trading signal generation, and portfolio construction.

Why Correlation Matters

Correlation measures how assets move together. In crypto markets this is critical for:

• 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

Linear correlation assuming normality. Most common but least robust for crypto.

pythonCopy code
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

Converts values to ranks, then computes Pearson on ranks. Captures monotonic (not just linear) relationships.

pythonCopy code
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

Counts concordant vs discordant pairs. Most robust to outliers.

pythonCopy code
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

Static correlation hides regime changes. Rolling correlation reveals how relationships evolve.

Window-Based Rolling Correlation

pythonCopy code
# 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

Exponentially weighted — more responsive to recent changes.

pythonCopy code
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

pythonCopy code
# 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

Decompose the correlation matrix to identify driving factors.

pythonCopy code
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

pythonCopy code
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

Group assets by correlation similarity to identify natural clusters.

pythonCopy code
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')

Applications:

• 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

Normal correlation understates co-movement during crashes. Tail dependence measures how often assets experience extreme returns simultaneously.

Lower Tail Dependence

pythonCopy code
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

In crypto markets, tail dependence typically exceeds normal correlation:

• 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

Correlation is not constant — it changes with market regime.

Detecting Correlation Regime Shifts

pythonCopy code
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