CORREL (Pearson's Correlation Coefficient)

Overview

The Pearson's Correlation Coefficient (CORREL) is a statistical indicator that measures the linear relationship between two price series over a specified period. It quantifies how closely two assets move together, producing values between -1 and +1. This indicator is essential for pair trading, portfolio diversification, and identifying correlation breakdowns that can signal trading opportunities.

Parameters

Parameter	Type	Default	Description
prices1	Array	Required	Array of price values for first asset (typically close prices)
prices2	Array	Required	Array of price values for second asset (typically close prices)
period	Integer	30	Number of periods over which to calculate correlation

Parameter Details

Note: Array elements should be ordered from oldest to newest (chronological order)

prices1 - First price series for correlation calculation - Should be synchronized with prices2 (same timestamps) - Typically closing prices, but can use any price type - Must have same length as prices2

prices2 - Second price series for correlation calculation - Should be synchronized with prices1 (same timestamps) - Typically closing prices, but can use any price type - Must have same length as prices1

period - Lookback window for correlation calculation - Shorter periods (10-20): More responsive, detects rapid correlation changes - Medium periods (20-40): Balanced view of correlation stability - Longer periods (50-100): Stable long-term correlation trends - Default 30 provides good balance for most pair trading applications

Usage

Basic Usage

require 'sqa/tai'

# Calculate correlation between two assets
spy_prices = [440.12, 441.35, 439.88, 442.17, 443.50, 442.90, 444.25,
              445.10, 444.75, 446.30, 445.85, 447.20, 446.95, 448.10]

qqq_prices = [368.45, 369.80, 368.20, 370.15, 371.25, 370.85, 372.10,
              372.90, 372.45, 373.80, 373.40, 374.60, 374.35, 375.40]

correlation = SQA::TAI.correl(spy_prices, qqq_prices, period: 10)

# Current correlation value
current_corr = correlation.last
puts "SPY/QQQ Correlation: #{current_corr.round(3)}"

if current_corr > 0.7
  puts "Strong positive correlation - assets moving together"
elsif current_corr < -0.7
  puts "Strong negative correlation - assets moving opposite"
else
  puts "Weak correlation - assets moving independently"
end

Practical Application

# Pair trading setup - monitor correlation for trading opportunities
stock_a = [100.0, 101.5, 102.0, 101.0, 102.5, 103.0, 102.5, 104.0,
           105.0, 104.5, 106.0, 105.5, 107.0, 106.5, 108.0]

stock_b = [50.0, 50.8, 51.0, 50.5, 51.2, 51.5, 51.2, 52.0,
           52.5, 52.2, 53.0, 52.7, 53.5, 53.2, 54.0]

# Calculate rolling correlation
correlation = SQA::TAI.correl(stock_a, stock_b, period: 10)

# Track correlation trend
recent_corr = correlation.last(5)
avg_corr = recent_corr.sum / recent_corr.size
current_corr = correlation.last

puts "Average correlation (5-period): #{avg_corr.round(3)}"
puts "Current correlation: #{current_corr.round(3)}"

# Detect correlation breakdown
if avg_corr > 0.8 && current_corr < 0.5
  puts "CORRELATION BREAKDOWN DETECTED"
  puts "Potential pair trading opportunity - investigate spread"
end

Understanding the Indicator

What It Measures

CORREL measures the strength and direction of the linear relationship between two price series. It answers the question: "How closely do these two assets move together?"

The correlation coefficient ranges from -1 to +1: - +1: Perfect positive correlation (assets move in lockstep) - 0: No correlation (assets move independently) - -1: Perfect negative correlation (assets move in opposite directions)

Why It's Important

Understanding correlation between assets is crucial for:

Portfolio Management: - Diversification: Lower correlation reduces portfolio risk - Risk assessment: High correlation increases concentration risk - Hedging: Negative correlation provides natural hedges

Pair Trading: - Identify cointegrated pairs with high correlation - Detect temporary correlation breakdowns for mean reversion trades - Monitor correlation stability for pair selection

Market Analysis: - Sector rotation: Correlation shifts indicate changing market dynamics - Risk-on/risk-off: Correlation patterns reveal market sentiment - Intermarket relationships: Bonds, currencies, commodities

Calculation Method

The Pearson correlation coefficient is calculated using the formula:

r = Σ[(Xi - X̄)(Yi - Ȳ)] / sqrt[Σ(Xi - X̄)² × Σ(Yi - Ȳ)²]

Where: - Xi, Yi = Individual price values for assets 1 and 2 - X̄, Ȳ = Mean prices over the period - Σ = Sum over the period

Step-by-step process:

1. Calculate means of both price series over the period
2. Compute deviations from mean for each price point
3. Calculate covariance (product of deviations)
4. Calculate standard deviations for each series
5. Divide covariance by product of standard deviations
6. Result is correlation coefficient between -1 and +1

Key Concept: The correlation coefficient is normalized, meaning it measures relationship strength independent of the actual price levels or volatility of the assets.

Indicator Characteristics

• Range: -1.0 to +1.0
• Type: Statistical, relationship indicator
• Lag: Moderate - requires lookback period to calculate
• Best Used: All market conditions, especially for relative value trading

Interpretation

Value Ranges

Understanding correlation strength:

• Perfect Positive Correlation (+0.90 to +1.00)
• Assets move almost identically
• Common in same-sector stocks or index vs constituents
• Limited diversification benefit

• Strong pair trading candidates

• Strong Positive Correlation (+0.70 to +0.89)

• Assets generally move together
• Some independent price action
• Moderate diversification benefit

• Good for momentum pair trades

• Moderate Positive Correlation (+0.40 to +0.69)

• Noticeable relationship but significant independence
• Better diversification benefit
• Less reliable for pair trading

• Consider spread volatility

• Weak Correlation (-0.39 to +0.39)

• Assets move mostly independently
• Good diversification benefit
• Poor pair trading candidates

• Focus on individual asset analysis

• Moderate Negative Correlation (-0.69 to -0.40)

• Assets tend to move in opposite directions
• Natural hedging characteristics
• Interesting for spread trading

• Risk-on/risk-off dynamics

• Strong Negative Correlation (-0.89 to -0.70)

• Assets consistently move opposite
• Excellent hedging properties
• Consider inverse pair trades

• Common in gold vs stocks, bonds vs stocks

• Perfect Negative Correlation (-1.00 to -0.90)

• Assets move in exact opposite directions
• Rare in practice
• Perfect hedge if maintained
• Usually synthetic (leveraged ETFs)

Correlation Changes

Increasing Correlation: When correlation coefficient rises over time: - Assets becoming more synchronized - Sector or market-wide forces strengthening - Diversification benefit decreasing - Increased systemic risk

Decreasing Correlation: When correlation coefficient falls over time: - Assets diverging in behavior - Company-specific factors dominating - Diversification benefit increasing - Potential pair trading opportunities

Correlation Breakdown: Rapid drop from high to low correlation: - Unexpected divergence in historically correlated assets - Often signals fundamental change in one asset - Prime opportunity for mean reversion trades (if temporary) - Or signal to exit pair trade (if permanent)

Stability Analysis

• Stable correlation (changes gradually): Reliable relationship for trading
• Volatile correlation (rapid swings): Unstable relationship, use with caution
• Trending correlation (steady increase/decrease): Changing market regime

Trading Signals

Pair Trading Setup

Classic Pair Trading Strategy:

# Identify highly correlated pair
stock_a_prices = [...]  # Your data
stock_b_prices = [...]  # Your data

# Calculate correlation
correlation = SQA::TAI.correl(stock_a_prices, stock_b_prices, period: 60)
current_corr = correlation.last

# Qualify pair for trading
if current_corr > 0.80
  puts "PAIR QUALIFIED: Correlation = #{current_corr.round(3)}"

  # Calculate price spread
  spread = stock_a_prices.zip(stock_b_prices).map { |a, b| a - b }

  # Calculate spread statistics
  spread_mean = spread.sum / spread.size
  spread_stddev = Math.sqrt(spread.map { |x| (x - spread_mean)**2 }.sum / spread.size)

  current_spread = spread.last
  z_score = (current_spread - spread_mean) / spread_stddev

  puts "Spread Z-Score: #{z_score.round(2)}"

  # Generate signals based on spread
  if z_score > 2.0
    puts "SELL SIGNAL: Spread overextended"
    puts "Action: Short Stock A, Long Stock B"
  elsif z_score < -2.0
    puts "BUY SIGNAL: Spread overextended"
    puts "Action: Long Stock A, Short Stock B"
  elsif z_score.abs < 0.5
    puts "EXIT SIGNAL: Spread returned to mean"
  end
else
  puts "Pair not suitable - correlation too weak: #{current_corr.round(3)}"
end

Correlation Breakdown Trading

Entry Signals:

1. Mean Reversion Entry
2. Historical correlation > 0.75
3. Current correlation drops below 0.50
4. Verify correlation drop is temporary, not fundamental

5. Enter spread trade expecting correlation to recover

6. Divergence Trade

7. Long-term correlation stable and high
8. Short-term divergence in price action
9. One asset lags the other's move
10. Trade the laggard to catch up

Example Strategy:

# Monitor correlation stability
long_correlation = SQA::TAI.correl(prices1, prices2, period: 60)
short_correlation = SQA::TAI.correl(prices1, prices2, period: 10)

long_corr = long_correlation.last
short_corr = short_correlation.last

puts "Long-term correlation (60): #{long_corr.round(3)}"
puts "Short-term correlation (10): #{short_corr.round(3)}"

# Detect breakdown
if long_corr > 0.75 && short_corr < 0.50
  puts "CORRELATION BREAKDOWN DETECTED"

  # Calculate which asset is outlier
  returns1 = prices1.each_cons(2).map { |a, b| (b - a) / a }
  returns2 = prices2.each_cons(2).map { |a, b| (b - a) / a }

  recent_return1 = returns1.last(5).sum
  recent_return2 = returns2.last(5).sum

  if recent_return1 > recent_return2 + 0.02
    puts "Asset 1 outperforming - consider shorting Asset 1 / buying Asset 2"
  elsif recent_return2 > recent_return1 + 0.02
    puts "Asset 2 outperforming - consider buying Asset 1 / shorting Asset 2"
  end

  puts "Expected correlation reversion within 20-30 periods"
end

Sector Rotation Trading

Use correlation to identify sector shifts:

# Compare individual stock to sector ETF
stock_prices = [...]
sector_etf_prices = [...]

correlation = SQA::TAI.correl(stock_prices, sector_etf_prices, period: 30)
corr_trend = correlation.last(10)

# Calculate correlation trend
recent_corr = corr_trend.last
past_corr = corr_trend.first

if recent_corr > past_corr + 0.20
  puts "INCREASING CORRELATION: Stock becoming sector-driven"
  puts "Strategy: Use sector momentum for stock direction"
elsif recent_corr < past_corr - 0.20
  puts "DECREASING CORRELATION: Stock developing independent story"
  puts "Strategy: Focus on stock-specific fundamentals"
end

Portfolio Hedging

Identify hedging opportunities:

# Find negative correlation for hedging
portfolio_prices = [...]  # Your portfolio value
hedge_candidate = [...]   # Potential hedge (e.g., VIX, gold, bonds)

correlation = SQA::TAI.correl(portfolio_prices, hedge_candidate, period: 40)
current_corr = correlation.last

if current_corr < -0.60
  puts "HEDGE CANDIDATE IDENTIFIED"
  puts "Correlation: #{current_corr.round(3)}"
  puts "Strong negative correlation provides downside protection"

  # Calculate hedge ratio
  portfolio_stddev = calculate_stddev(portfolio_prices)
  hedge_stddev = calculate_stddev(hedge_candidate)

  hedge_ratio = current_corr * (portfolio_stddev / hedge_stddev)
  puts "Suggested hedge ratio: #{hedge_ratio.abs.round(3)}"
  puts "For $100k portfolio, hedge with $#{(100000 * hedge_ratio.abs).round(0)} of hedge asset"
end

Best Practices

Optimal Use Cases

When CORREL works best: - Liquid pairs: High-volume stocks or ETFs - Same market hours: Both assets trade simultaneously - Similar sectors: Related industries or asset classes - Daily to weekly timeframes: More stable correlations than intraday - Established relationships: Historical precedent for correlation

Less effective in: - Very short timeframes (< 5 minutes) - too much noise - Low-liquidity assets - correlation may be spurious - Fundamentally unrelated assets - correlation likely random - Crisis periods - correlations converge to 1 (all assets fall together)

Combining with Other Indicators

With Spread Analysis:

# Correlation + spread for complete pair trading view
correlation = SQA::TAI.correl(prices1, prices2, period: 30)

# Calculate spread
spread = prices1.zip(prices2).map { |a, b| a - b }
spread_sma = SQA::TAI.sma(spread, period: 20)
spread_stddev = SQA::TAI.stddev(spread, period: 20)

# Bollinger Bands on spread
upper_band = spread_sma.zip(spread_stddev).map { |m, s| m + (2 * s) }
lower_band = spread_sma.zip(spread_stddev).map { |m, s| m - (2 * s) }

current_spread = spread.last
current_upper = upper_band.last
current_lower = lower_band.last
current_corr = correlation.last

if current_corr > 0.75
  if current_spread > current_upper
    puts "SELL SPREAD: High correlation + spread at upper band"
  elsif current_spread < current_lower
    puts "BUY SPREAD: High correlation + spread at lower band"
  end
end

With Beta Analysis:

# Correlation tells you relationship strength, beta tells you magnitude
correlation = SQA::TAI.correl(stock_prices, market_prices, period: 60)
beta = SQA::TAI.beta(stock_prices, market_prices, period: 60)

current_corr = correlation.last
current_beta = beta.last

puts "Correlation: #{current_corr.round(3)}"
puts "Beta: #{current_beta.round(3)}"

if current_corr > 0.70
  if current_beta > 1.2
    puts "High correlation + high beta = Amplified market moves"
    puts "Strategy: Use for leveraged market exposure"
  elsif current_beta < 0.8
    puts "High correlation + low beta = Defensive with market"
    puts "Strategy: Use for lower-risk market participation"
  end
end

With Cointegration Tests:

# Correlation measures relationship, cointegration measures mean reversion
# High correlation is necessary but not sufficient for pair trading
# Need to verify cointegration separately (ADF test, etc.)

correlation = SQA::TAI.correl(prices1, prices2, period: 60)

if correlation.last > 0.80
  puts "High correlation detected: #{correlation.last.round(3)}"
  puts "Next step: Perform cointegration test (ADF/Johansen)"
  puts "Only trade pairs that are both correlated AND cointegrated"
end

Common Pitfalls

1. Confusing Correlation with Causation
2. Correlation does not imply one asset causes the other to move
3. Both may respond to common external factors

4. Use correlation for relationship measurement, not explanation

5. Ignoring Changing Correlations

6. Correlation is not static - it changes over time
7. Market regime changes alter correlations
8. Crisis periods see correlations spike (diversification fails when needed most)

9. Always monitor correlation stability

10. Using Only Correlation for Pair Trading

11. High correlation doesn't guarantee mean reversion
12. Must also check cointegration
13. Need to verify spreads are stationary

14. Correlation alone is insufficient

15. Period Selection Issues

16. Too short: Noisy, unreliable correlations
17. Too long: Misses recent regime changes
18. Use multiple periods for comprehensive view

19. Typical range: 20-60 periods

20. Spurious Correlations

21. Random correlations can appear with enough data
22. Verify with economic logic
23. Test across different time periods
24. Don't trade based on correlation alone

Parameter Selection Guidelines

For Day Trading: - Use very short periods (5-20) - Monitor intraday correlation changes - Focus on highly liquid pairs - Expect more volatile correlations

For Swing Trading: - Use medium periods (20-40) - Look for multi-day correlation stability - Focus on established sector relationships - Balance responsiveness with stability

For Position Trading: - Use longer periods (40-100) - Focus on long-term correlation trends - Suitable for portfolio hedging decisions - More stable but slower to adapt

For Pair Trading: - Use multiple periods: 20, 40, 60 - Require correlation > 0.75 on 60-period - Monitor 20-period for entry timing - Longer periods for pair selection, shorter for trade timing

Practical Example

Complete pair trading system example:

require 'sqa/tai'

# Historical price data for two correlated stocks
# Example: XLE (Energy ETF) and CVX (Chevron)
xle_prices = [
  78.50, 79.20, 78.90, 79.80, 80.50, 80.20, 81.00, 81.50,
  81.20, 82.00, 82.50, 82.20, 83.00, 82.70, 83.50, 83.20,
  84.00, 84.50, 84.20, 85.00, 85.50, 85.20, 86.00, 85.70,
  86.50, 86.20, 87.00, 87.50, 87.20, 88.00, 88.50, 88.20,
  89.00, 88.70, 89.50, 89.20, 90.00, 89.70, 90.50, 90.20,
  91.00, 90.70, 91.50, 91.20, 92.00, 91.70, 92.50, 92.20,
  93.00, 92.70, 93.50, 93.20, 94.00, 93.70, 94.50, 94.20,
  95.00, 94.70, 95.50, 95.20, 96.00
]

cvx_prices = [
  145.00, 146.50, 145.80, 147.20, 148.50, 148.00, 149.50, 150.20,
  149.80, 151.00, 151.80, 151.20, 152.50, 152.00, 153.20, 152.80,
  154.00, 154.80, 154.20, 155.50, 156.20, 155.80, 157.00, 156.50,
  157.80, 157.20, 158.50, 159.20, 158.80, 160.00, 160.80, 160.20,
  161.50, 161.00, 162.20, 161.80, 163.00, 162.50, 163.80, 163.20,
  164.50, 164.00, 165.20, 164.80, 166.00, 165.50, 166.80, 166.20,
  167.50, 167.00, 168.20, 167.80, 169.00, 168.50, 169.80, 169.20,
  170.50, 170.00, 171.20, 170.80, 172.00
]

# Step 1: Calculate correlation across multiple periods
long_correlation = SQA::TAI.correl(xle_prices, cvx_prices, period: 60)
medium_correlation = SQA::TAI.correl(xle_prices, cvx_prices, period: 30)
short_correlation = SQA::TAI.correl(xle_prices, cvx_prices, period: 10)

long_corr = long_correlation.last
medium_corr = medium_correlation.last
short_corr = short_correlation.last

puts "=== CORRELATION ANALYSIS ==="
puts "Long-term (60): #{long_corr.round(3)}"
puts "Medium-term (30): #{medium_corr.round(3)}"
puts "Short-term (10): #{short_corr.round(3)}"
puts ""

# Step 2: Qualify pair for trading
if long_corr > 0.75
  puts "PAIR QUALIFIED: Strong long-term correlation"
  puts ""

  # Step 3: Calculate spread
  spread = xle_prices.zip(cvx_prices).map { |xle, cvx| xle - (cvx * 0.55) } # Ratio adjusted

  # Step 4: Spread statistics
  spread_mean = spread.last(30).sum / 30.0
  spread_values = spread.last(30)
  spread_stddev = Math.sqrt(
    spread_values.map { |x| (x - spread_mean)**2 }.sum / 30.0
  )

  current_spread = spread.last
  z_score = (current_spread - spread_mean) / spread_stddev

  puts "=== SPREAD ANALYSIS ==="
  puts "Current spread: #{current_spread.round(3)}"
  puts "30-period mean: #{spread_mean.round(3)}"
  puts "Standard deviation: #{spread_stddev.round(3)}"
  puts "Z-Score: #{z_score.round(2)}"
  puts ""

  # Step 5: Check for correlation breakdown opportunity
  if long_corr > 0.75 && short_corr < 0.50
    puts "=== CORRELATION BREAKDOWN DETECTED ==="
    puts "This may be a mean reversion opportunity"
    puts ""
  end

  # Step 6: Generate trading signals
  puts "=== TRADING SIGNALS ==="

  if z_score > 2.0
    stop_loss_z = z_score + 0.5
    take_profit_z = 0.0

    puts "SELL SPREAD SIGNAL"
    puts "Action: Short XLE / Long CVX"
    puts "Rationale: Spread #{z_score.round(2)} std devs above mean"
    puts "Entry Z-Score: #{z_score.round(2)}"
    puts "Stop Loss Z-Score: #{stop_loss_z.round(2)}"
    puts "Take Profit Z-Score: #{take_profit_z.round(2)}"
    puts "Expected duration: 10-20 periods for mean reversion"

  elsif z_score < -2.0
    stop_loss_z = z_score - 0.5
    take_profit_z = 0.0

    puts "BUY SPREAD SIGNAL"
    puts "Action: Long XLE / Short CVX"
    puts "Rationale: Spread #{z_score.abs.round(2)} std devs below mean"
    puts "Entry Z-Score: #{z_score.round(2)}"
    puts "Stop Loss Z-Score: #{stop_loss_z.round(2)}"
    puts "Take Profit Z-Score: #{take_profit_z.round(2)}"
    puts "Expected duration: 10-20 periods for mean reversion"

  elsif z_score.abs < 0.5
    puts "EXIT SIGNAL: Spread near mean"
    puts "Close any open positions - spread has reverted"

  else
    puts "NO SIGNAL: Spread at #{z_score.round(2)} std devs"
    puts "Wait for |Z-Score| > 2.0 for entry"
  end

  # Step 7: Risk assessment
  puts ""
  puts "=== RISK ASSESSMENT ==="

  if short_corr < 0.50 && long_corr > 0.75
    puts "WARNING: Short-term correlation breakdown"
    puts "Risk: Pair relationship may be unstable"
    puts "Recommendation: Reduce position size by 50%"
  elsif short_corr > 0.70
    puts "STABLE: Short and long correlations aligned"
    puts "Risk: Normal pair trading risk"
    puts "Recommendation: Use standard position size"
  end

  # Calculate position sizing
  portfolio_value = 100000
  risk_per_trade = 0.02 # 2% risk

  # Risk based on stop loss
  risk_amount = portfolio_value * risk_per_trade
  stop_distance = spread_stddev * 0.5 # 0.5 std dev stop

  xle_position_size = risk_amount / stop_distance
  cvx_position_size = xle_position_size * 0.55 # Ratio adjusted

  puts ""
  puts "=== POSITION SIZING ==="
  puts "Portfolio value: $#{portfolio_value}"
  puts "Risk per trade: #{(risk_per_trade * 100).round(1)}%"
  puts "XLE position: #{xle_position_size.round(0)} shares"
  puts "CVX position: #{cvx_position_size.round(0)} shares"

else
  puts "PAIR NOT SUITABLE"
  puts "Reason: Long-term correlation too weak (#{long_corr.round(3)})"
  puts "Requirement: Correlation > 0.75 for pair trading"
end

# Step 8: Monitor correlation stability
puts ""
puts "=== CORRELATION STABILITY ==="

recent_corrs = long_correlation.last(10)
corr_stddev = Math.sqrt(
  recent_corrs.map { |c| (c - recent_corrs.sum/10.0)**2 }.sum / 10.0
)
corr_cv = corr_stddev / (recent_corrs.sum / 10.0)

if corr_cv < 0.10
  puts "STABLE: Correlation coefficient of variation = #{corr_cv.round(3)}"
  puts "Pair relationship is reliable"
else
  puts "UNSTABLE: Correlation coefficient of variation = #{corr_cv.round(3)}"
  puts "WARNING: Pair relationship is volatile - trade with caution"
end

Related Indicators

Statistical Family

• BETA: Measures systematic risk and return sensitivity
• VAR: Sample variance for volatility measurement
• STDDEV: Standard deviation for dispersion analysis
• TSF: Time series forecast for trend projection
• LINEARREG: Linear regression for trend analysis

Complementary Indicators

• ATR: Use for volatility-adjusted position sizing
• BBANDS: Apply to spread for entry/exit signals
• RSI: Additional confirmation for pair divergence
• ADX: Measure trend strength in spread

Alternative Correlation Approaches

While CORREL uses Pearson's coefficient, consider: - Spearman rank correlation (for non-linear relationships) - Rolling correlations with varying windows - Conditional correlations (regime-dependent) - Dynamic Conditional Correlation (DCC-GARCH)

Advanced Topics

Multi-Asset Correlation Analysis

# Analyze correlation matrix for portfolio
assets = {
  'SPY' => spy_prices,
  'TLT' => tlt_prices,
  'GLD' => gld_prices,
  'USO' => uso_prices
}

puts "=== CORRELATION MATRIX ==="
puts "        " + assets.keys.join("    ")

assets.each do |name1, prices1|
  row = "#{name1}   "
  assets.each do |name2, prices2|
    corr = SQA::TAI.correl(prices1, prices2, period: 60).last
    row += "#{corr.round(2)}  "
  end
  puts row
end

# Identify best diversification pairs
# Look for correlations near 0 or negative

Time-Varying Correlation

# Track how correlation changes over time
correlation_series = SQA::TAI.correl(prices1, prices2, period: 30)

# Calculate correlation trend
recent_trend = correlation_series.last(10)
older_trend = correlation_series[-20..-11]

recent_avg = recent_trend.sum / recent_trend.size
older_avg = older_trend.sum / older_trend.size

correlation_change = recent_avg - older_avg

if correlation_change > 0.15
  puts "RISING CORRELATION: Assets converging (+#{correlation_change.round(3)})"
  puts "Implication: Reduced diversification benefit"
elsif correlation_change < -0.15
  puts "FALLING CORRELATION: Assets diverging (#{correlation_change.round(3)})"
  puts "Implication: Increased diversification benefit or breakdown"
end

Correlation-Based Portfolio Optimization

# Find optimal portfolio weights using correlation
# Lower correlation = better diversification = higher allocation

asset_returns = {
  'Asset A' => [0.01, 0.02, -0.01, 0.03],
  'Asset B' => [-0.01, 0.01, 0.02, -0.02]
}

correlation = SQA::TAI.correl(
  asset_returns['Asset A'],
  asset_returns['Asset B'],
  period: 30
).last

# Simple allocation based on correlation
if correlation < 0.30
  puts "Low correlation - equal weight portfolio optimal"
  puts "Asset A: 50%, Asset B: 50%"
elsif correlation > 0.70
  puts "High correlation - concentrated portfolio acceptable"
  puts "Consider single asset or reduce allocation to weaker performer"
end

Statistical Significance Testing

# Test if correlation is statistically significant
# Rule of thumb: For 95% confidence, need |r| > 2/sqrt(n)

correlation = SQA::TAI.correl(prices1, prices2, period: 30)
current_corr = correlation.last
n = 30 # sample size

critical_value = 2.0 / Math.sqrt(n)

puts "Correlation: #{current_corr.round(3)}"
puts "Critical value (95% confidence): #{critical_value.round(3)}"

if current_corr.abs > critical_value
  puts "STATISTICALLY SIGNIFICANT: Correlation is likely real"
else
  puts "NOT SIGNIFICANT: Correlation may be random noise"
  puts "Need larger sample or stronger correlation"
end

Crisis Alpha - Correlation in Extreme Markets

# Detect correlation regime shifts during volatility spikes
normal_correlation = SQA::TAI.correl(prices1, prices2, period: 60)
recent_correlation = SQA::TAI.correl(prices1, prices2, period: 10)

# Calculate volatility
returns = prices1.each_cons(2).map { |a, b| (b - a) / a }
recent_volatility = Math.sqrt(returns.last(10).map { |r| r**2 }.sum / 10)
normal_volatility = Math.sqrt(returns.map { |r| r**2 }.sum / returns.size)

if recent_volatility > normal_volatility * 2.0
  puts "=== HIGH VOLATILITY REGIME ==="
  puts "Recent vol: #{(recent_volatility * 100).round(2)}%"
  puts "Normal vol: #{(normal_volatility * 100).round(2)}%"

  if recent_correlation.last > normal_correlation.last + 0.20
    puts "WARNING: Correlation spike during stress"
    puts "Diversification benefits reduced in crisis"
    puts "Consider reducing overall exposure"
  end
end

References

• "Quantitative Trading" by Ernest Chan - Chapter on pair trading and correlation
• "Algorithmic Trading" by Ernest Chan - Statistical arbitrage using correlation
• "Pairs Trading: Quantitative Methods and Analysis" by Ganapathy Vidyamurthy
• "Active Portfolio Management" by Grinold & Kahn - Correlation in portfolio construction
• Original Research: Karl Pearson (1896) - Mathematical theory of correlation
• TA-Lib Documentation: CORREL Function

See Also

• Statistical Indicators Overview

• Pair Trading Guide

• Portfolio Optimization
• API Reference