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Documentation Index

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Correlation measures whether two investments tend to move together, move in opposite directions, or move independently. It’s the foundation of diversification and portfolio construction.

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Beginner

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What It Means

Correlation tells you the relationship between how two investments move:
  • Positive correlation (+1): They move together
  • Negative correlation (-1): They move in opposite directions
  • Zero correlation (0): Their movements are unrelated

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Portfolio Examples

Stock PairCorrelationWhat It Means
Stock A & B+0.9When A goes up, B strongly tends to go up too
Stock A & C-0.7When A goes up, C strongly tends to go down
Stock A & D0.0A’s movements tell you nothing about D
Correlation doesn’t tell you the magnitude of movements, only the direction and strength of the relationship. The actual percentage moves depend on each investment’s volatility.

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Why It Matters

Lower correlation between your holdings means better diversification. When one investment falls, others may hold steady or rise, reducing overall portfolio risk. This is the only “free lunch” in investing.

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Advanced

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Mathematical Definition

Correlation (ρ) = Cov(Ra, Rb) / (σa × σb)

Where:
- Cov(Ra, Rb) = Covariance of returns between assets A and B
- σa, σb = Standard deviations of assets A and B
- Range: -1.0 to +1.0

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Interpretation Scale

CorrelationInterpretation
+1.0Perfect positive (move in lockstep)
+0.7 to +0.9Strong positive
+0.4 to +0.6Moderate positive
-0.1 to +0.3Weak/no correlation
-0.4 to -0.6Moderate negative
-0.7 to -0.9Strong negative
-1.0Perfect negative (perfect hedges)

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Historical Context

Correlation’s importance in portfolio theory stems from Markowitz’s Modern Portfolio Theory (1952). He showed mathematically that portfolio risk depends not just on individual asset risks, but critically on how assets correlate with each other. This insight revolutionized portfolio construction.

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What Makes It Useful

  • Diversification Quantification: Lower correlation = greater diversification benefits
  • Portfolio Risk Reduction: Portfolio with N uncorrelated assets has risk reduced by factor of √N
  • Risk Decomposition: Identify which holdings contribute most to portfolio risk
  • Hedging Strategy: Find negative correlation assets for portfolio protection
  • Multi-Asset Allocation: Construct portfolios spanning stocks, bonds, commodities based on correlation matrix

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Diversification Math

Two-asset portfolio variance:
σp² = wa²σa² + wb²σb² + 2wawbσaσbρab

Where ρab is correlation between assets A and B

If ρab = 1.0:  No diversification benefit (risk is weighted average)
If ρab = 0.0:  Significant diversification (portfolio risk < weighted average)
If ρab = -1.0: Maximum diversification (can create zero-risk portfolio)

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Data Requirements

RequirementDurationNotes
Minimum36 months (3 years)Basic correlation estimate
Preferred60+ months (5 years)Portfolio construction decisions
UpdatesQuarterly or semi-annuallyCorrelations change across regimes
“Correlation goes to 1 in a crisis” - diversification fails when you need it most. Update correlations frequently.

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Limitations

  • Instability Over Time: Correlations increase during market stress
  • Linear Relationship Only: Doesn’t capture non-linear dependencies
  • Outlier Sensitivity: Extreme events disproportionately influence correlation
  • Assumes Stationarity: Historical correlation may not persist

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Alternatives

AlternativeDescription
Rank Correlation (Spearman)More robust to outliers, captures monotonic relationships
Tail DependenceMeasures correlation specifically during extreme events
CopulasCapture full dependency structure beyond linear correlation
Rolling CorrelationTime-varying correlation that adapts to changing relationships

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Correlation Breakdown in Crises

Empirical Reality:
Market ConditionAverage Stock Correlation
Calm (VIX < 15)~0.25
Normal (VIX 15-25)~0.35
Stress (VIX > 30)0.65-0.80
2008 Crisis~0.85
Key Insight: Longin and Solnik (2001) documented extreme correlation asymmetry - correlations spike during crashes but not rallies.
Practical Guidance: Stress test portfolios using correlation = 0.8, not historical average. Diversification works in calm markets but fails in crises.

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Related Terms

Beta

Correlation with market specifically

Standard Deviation

Individual asset volatility

Drawdown

What happens when correlations spike