Correlation

Quick Reference

Misura quanto due asset si muovono insieme. Range da -1 (perfettamente opposti) a +1 (perfettamente insieme).

Definizione

Correlation coefficient (ρ): Misura relazione lineare tra due serie di rendimenti.

ρ = Cov(X,Y) / (σ_X × σ_Y)

Range e Interpretazione

ρ = +1.0 (Perfect Positive)

Assets muovono perfettamente insieme: - X sale 5% → Y sale 5% - X scende 3% → Y scende 3% - No diversification benefit

ρ = 0.0 (Zero Correlation)

Assets indipendenti: - X sale → Y può salire/scendere/flat - Maximum diversification benefit

ρ = -1.0 (Perfect Negative)

Assets perfettamente opposti: - X sale 5% → Y scende 5% - X scende 3% → Y sale 3% - Perfect hedge

Typical Values

-0.3 to +0.3: Low correlation (good diversification) +0.3 to +0.7: Moderate correlation +0.7 to +1.0: High correlation (poor diversification)

Esempi Reali

Within Asset Classes

Equity indices: - S&P 500 vs FTSE 100: ρ ≈ 0.80 - S&P 500 vs Nikkei: ρ ≈ 0.60 - US vs Emerging: ρ ≈ 0.70

FX pairs: - EURUSD vs GBPUSD: ρ ≈ 0.70 - EURUSD vs USDJPY: ρ ≈ -0.50 - AUDUSD vs NZDUSD: ρ ≈ 0.90

Commodities: - Gold vs Silver: ρ ≈ 0.70 - Crude vs Gas: ρ ≈ 0.50 - Wheat vs Corn: ρ ≈ 0.60

Across Asset Classes

Equities vs Bonds: ρ ≈ -0.20 to +0.30 (varies!) Equities vs Gold: ρ ≈ -0.10 to +0.10 Equities vs Commodities: ρ ≈ 0.30

Key insight: Cross-asset correlations lower → better diversification!

Correlation e Diversification

Portfolio Variance

Due assets, equal weight:

σ_portfolio² = 0.5²σ_A² + 0.5²σ_B² + 2×0.5×0.5×ρ×σ_A×σ_B

Esempio (σ_A = σ_B = 20%, equal weight):

ρ = 0: σ_portfolio = 14.1% (29% reduction!) ρ = 0.5: σ_portfolio = 17.3% (14% reduction) ρ = 1.0: σ_portfolio = 20% (no reduction)

IDM Relationship

IDM ≈ √[N / (1 + (N-1)ρ)]

Higher correlation → Lower IDM → Less diversification benefit.

Time-Varying Correlation

Normal Markets

Typical correlations: - Equity markets: ρ ≈ 0.50-0.70 - Cross-asset: ρ ≈ 0.00-0.30

Crisis Markets

Correlations spike: - 2008: Equity correlations → 0.90+ - 2020 COVID: Similar spike - "Diversification fails when you need it most"

Recovery

Post-crisis, correlations mean-revert back to normal.

Implication: Use long-term average ρ, not recent spike.

Correlation vs Causation

Critical distinction:

Correlation: X and Y move together Causation: X causes Y to move

Esempio: - Ice cream sales e drowning correlati - But: Ice cream non causa drowning - Real cause: Estate (temperature)

Trading: Correlation sufficiente, causation non necessaria.

Measuring Correlation

Sample Correlation

ρ = Σ[(X_i - μ_X)(Y_i - μ_Y)] / √[Σ(X_i - μ_X)² × Σ(Y_i - μ_Y)²]

Requires: Almeno 30-50 data points per stima stabile.

Rolling Correlation

Window mobile (es. 60 giorni): - Shows how ρ cambia over time - Useful per identificare regime changes

EWMA Correlation

Exponentially weighted: - More weight a dati recenti - Smoother di rolling window

Correlation Matrix

N assets → N×(N-1)/2 correlations:

Esempio 5 assets:

       A     B     C     D     E
A   1.00  0.60  0.40  0.30  0.20
B   0.60  1.00  0.50  0.45  0.35
C   0.40  0.50  1.00  0.55  0.40
D   0.30  0.45  0.55  1.00  0.50
E   0.20  0.35  0.40  0.50  1.00

10 unique correlations (5×4/2).

Using Correlations

Portfolio Construction

Select assets con low ρ: - Better diversification - Higher IDM - Lower portfolio risk

Risk Management

Monitor correlation changes: - Rising ρ → diversification eroding - Consider rebalancing

Forecast Diversification

Multiple trading rules con low forecast correlation: - Trend vs Carry: ρ ≈ 0.00 - Trend vs Mean reversion: ρ ≈ -0.30 - Combine for diversification

Spurious Correlation

Random correlations:

Con 100 asset pairs, ~5 avranno |ρ| > 0.30 per caso!

Solution: Don't data mine correlations.

Non-Linear Relationships

Correlation cattura solo linear:

Esempio: Options - Stock vs Call option: Strong relationship - But: Non-linear (convex) - Low measured ρ despite strong link

Solution: Use other measures (e.g., beta, copulas) se needed.

Correlation Breakdown

Tail Correlation

During extremes, correlation spesso diversa:

Esempio: - Normal times: Stock A vs B, ρ = 0.50 - Crashes: Both drop together, ρ → 0.90 - Tail correlation > normal correlation

Implication: Diversification works less in tail events.

Optimal Number Assets

Diversification benefit vs complexity:

Formula (assuming equal ρ):

σ_portfolio = σ_asset × √[(1/N) + ρ(1 - 1/N)]

Diminishing returns: - 1 → 4 assets: Large reduction - 4 → 10 assets: Moderate reduction - 10 → 30 assets: Small reduction - 30+ assets: Minimal reduction

Optimal: ~10-20 uncorrelated assets.

Forecast Correlation

Trading rules correlation:

Trend following rules (different speeds): - EWMAC 16/64 vs 64/256: ρ ≈ 0.80 - Moderate diversification

Different styles: - Trend vs Carry: ρ ≈ 0.00 - Excellent diversification

Practical Guidelines

Target Correlations

Within portfolio: - Average ρ < 0.50: Good - Average ρ < 0.30: Excellent - Average ρ > 0.70: Poor diversification

Red Flags

• All assets ρ > 0.80: Pseudo-diversification
• Crisis ρ → 1.0: Expect this, plan accordingly
• Negative ρ claims: Usually spurious, verify carefully

Correlation vs Covariance

Related but different:

Covariance: Cov(X,Y) = E[(X-μ_X)(Y-μ_Y)] - Units: Depends on X and Y units - Hard to interpret

Correlation: ρ = Cov / (σ_X × σ_Y) - Unitless, -1 to +1 - Easy to interpret

Use correlation for interpretation, covariance for calculations.

Errori Comuni

• Assuming correlations static: They change over time
• Ignoring crisis spikes: Plan for ρ → 1.0 durante stress
• Too few data points: Need 30+ for reliable estimate
• Confusing with causation: Correlation ≠ causation
• Data mining correlations: Finding spurious relationships
• Ignoring non-linearity: Some relationships non-linear
• Using recent ρ only: Long-term average more reliable

Concetti Correlati

• [[Diversification Multiplier]] - funzione di correlation
• [[IDM Calculation]] - usa correlation matrix
• [[Portfolio Construction]] - correlation drives allocation
• [[Skew]] - correlations can be asymmetric by tail