> ## 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.

# Style Analysis R-Squared

> Understanding R-squared in style analysis - measuring how much factor exposures explain returns

Style Analysis R-squared measures how much of a portfolio's performance can be explained by systematic factors like market, value, size, and momentum. It reveals whether returns come from factor exposures or genuine stock selection.

## Beginner

### What It Means

R-squared tells you what percentage of your portfolio's returns are explained by known factors. Higher R-squared means your returns mostly come from factor exposures, not unique stock picking.

### Portfolio Examples

| Portfolio       | R-Squared | Interpretation                                           |
| --------------- | --------- | -------------------------------------------------------- |
| Index Fund      | 99%       | Almost entirely explained by market factor               |
| "Active" Fund A | 95%       | Closet indexer - paying active fees for passive exposure |
| Active Fund B   | 70%       | Mix of factor exposure and stock selection               |
| Hedge Fund      | 40%       | Mostly unique strategies, less factor-driven             |

### Why It Matters

R-squared helps identify "closet indexers" - funds charging active fees but delivering passive-like returns. If 95% of returns are explained by factors you could get cheaply through ETFs, why pay active management fees?

***

## Advanced

### Mathematical Definition

```
R-squared = 1 - (SS_residual / SS_total)
          = Variance explained by factors / Total return variance

From regression:
Rp = α + Σ(βi × Fi) + ε

R-squared = 1 - Var(ε) / Var(Rp)
```

### Interpreting R-Squared with Alpha

| R-Squared | Alpha | Interpretation                                 |
| --------- | ----- | ---------------------------------------------- |
| High      | Low   | Closet indexer - consider cheaper alternatives |
| Low       | High  | Genuine skill creating value                   |
| Low       | Low   | Active bets not adding value                   |
| High      | High  | Skilled factor timing (rare)                   |

### Factor Models Used

Common models for style analysis:

| Model             | Factors Included            |
| ----------------- | --------------------------- |
| **CAPM**          | Market only                 |
| **Fama-French 3** | Market, Size, Value         |
| **Carhart 4**     | + Momentum                  |
| **Fama-French 5** | + Profitability, Investment |

<Note>
  R-squared depends on which factors you include. Adding more factors typically increases R-squared. Use a consistent model for fair comparisons.
</Note>

### Typical R-Squared Values

| Fund Type         | Typical R-Squared |
| ----------------- | ----------------- |
| Index Funds       | 98-99%            |
| Enhanced Index    | 95-98%            |
| Large-Cap Active  | 85-95%            |
| Small-Cap Active  | 75-90%            |
| Long/Short Equity | 50-80%            |
| Market Neutral    | 10-40%            |

### Detecting Closet Indexers

Warning signs of closet indexing:

| Metric                     | Concern Level |
| -------------------------- | ------------- |
| R-squared above 95%        | High concern  |
| Tracking error under 2%    | High concern  |
| Holdings overlap above 80% | High concern  |
| Active share under 20%     | High concern  |

<Warning>
  If a fund has R-squared above 90%, you're likely paying active fees for returns you could replicate with cheap factor ETFs.
</Warning>

### R-Squared vs. Active Share

| Metric           | Based On | Measures                                  |
| ---------------- | -------- | ----------------------------------------- |
| **R-Squared**    | Returns  | How much return variance factors explain  |
| **Active Share** | Holdings | How different holdings are from benchmark |

Both together give a complete picture:

| R-Squared | Active Share | Manager Type              |
| --------- | ------------ | ------------------------- |
| High      | Low          | Closet indexer            |
| High      | High         | Factor bettor             |
| Low       | Low          | Diversified stock picker  |
| Low       | High         | Concentrated stock picker |

### Practical Applications

**Due Diligence Process:**

1. Run style regression on manager returns
2. If R-squared above 90%: Question if active fees justified
3. Examine factor loadings for style consistency
4. Compare R-squared over rolling windows to detect style drift
5. Combine with active share analysis

### Style Drift Detection

Changes in R-squared over time may indicate:

* Strategy changes
* Manager turnover
* Capacity constraints forcing index-like positions
* Deliberate shift in investment approach

### Data Requirements

| Requirement  | Details                            |
| ------------ | ---------------------------------- |
| Minimum      | 36 months for stable estimate      |
| Preferred    | 60+ months                         |
| Factor Data  | Need factor return series          |
| Model Choice | Results depend on factors included |

### Related Terms

<CardGroup cols={3}>
  <Card title="Alpha" href="/glossary/alpha">
    What remains after factor adjustment
  </Card>

  <Card title="Beta" href="/glossary/beta">
    Key factor exposure
  </Card>

  <Card title="Factor Investing" href="/glossary/factor-investing">
    The factors being measured
  </Card>
</CardGroup>