> ## Documentation Index
> Fetch the complete documentation index at: https://docs.chicago.global/llms.txt
> Use this file to discover all available pages before exploring further.

# Correlation

> Understanding correlation - how assets move together and the foundation of diversification

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.

## Beginner

### 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

### Portfolio Examples

| Stock Pair  | Correlation | What It Means                                 |
| ----------- | ----------- | --------------------------------------------- |
| Stock A & B | +0.9        | When A goes up, B strongly tends to go up too |
| Stock A & C | -0.7        | When A goes up, C strongly tends to go down   |
| Stock A & D | 0.0         | A's movements tell you nothing about D        |

<Note>
  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.
</Note>

### 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.

***

## Advanced

### 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
```

### Interpretation Scale

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

### 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.

### 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

### 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)
```

### Data Requirements

| Requirement | Duration                   | Notes                              |
| ----------- | -------------------------- | ---------------------------------- |
| Minimum     | 36 months (3 years)        | Basic correlation estimate         |
| Preferred   | 60+ months (5 years)       | Portfolio construction decisions   |
| Updates     | Quarterly or semi-annually | Correlations change across regimes |

<Warning>
  "Correlation goes to 1 in a crisis" - diversification fails when you need it most. Update correlations frequently.
</Warning>

### 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

### Alternatives

| Alternative                     | Description                                                    |
| ------------------------------- | -------------------------------------------------------------- |
| **Rank Correlation (Spearman)** | More robust to outliers, captures monotonic relationships      |
| **Tail Dependence**             | Measures correlation specifically during extreme events        |
| **Copulas**                     | Capture full dependency structure beyond linear correlation    |
| **Rolling Correlation**         | Time-varying correlation that adapts to changing relationships |

### Correlation Breakdown in Crises

**Empirical Reality:**

| Market Condition   | Average 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.

<Tip>
  **Practical Guidance**: Stress test portfolios using correlation = 0.8, not historical average. Diversification works in calm markets but fails in crises.
</Tip>

### Related Terms

<CardGroup cols={3}>
  <Card title="Beta" href="/glossary/beta">
    Correlation with market specifically
  </Card>

  <Card title="Standard Deviation" href="/glossary/standard-deviation">
    Individual asset volatility
  </Card>

  <Card title="Drawdown" href="/glossary/drawdown">
    What happens when correlations spike
  </Card>
</CardGroup>