---
description: "Generalizes pairs trading to N>2 historically correlated stocks within a cluster (e.g., an industry), buying underperformers and shorting overperformers relative to the cluster mean."
tags: [stocks, mean-reversion, cluster, statistical-arbitrage]
---

# Mean-Reversion — Single Cluster (and Multiple Clusters)

**Section**: 3.9 / 3.9.1 | **Asset Class**: Stocks | **Type**: Mean-Reversion / Statistical Arbitrage

## Overview
This strategy generalizes pairs trading to N > 2 stocks that are historically highly correlated — for example, stocks belonging to the same industry or sector. Each stock's return is demeaned relative to the cluster mean, and positions are taken proportional to negative demeaned returns: buy stocks that underperformed the cluster and short stocks that outperformed it.

## Construction / Signal
Let `R_i = ln(P_i(t_2) / P_i(t_1))` be the log return for stock i in the cluster of N stocks.

Cluster mean return and demeaned returns:

```
R_bar = (1/N) * sum_{i=1}^{N} R_i                         (293)
R_tilde_i = R_i - R_bar                                   (294)
```

Short stocks with positive `R_tilde_i` (outperformers), buy stocks with negative `R_tilde_i` (underperformers).

**Dollar-neutrality constraints**:

```
sum_{i=1}^{N} P_i |Q_i| = I                               (295)
sum_{i=1}^{N} P_i Q_i = 0                                 (296)
```

A simple prescription for dollar positions `D_i = P_i Q_i` proportional to demeaned returns:

```
D_i = -gamma * R_tilde_i                                  (297)
```

where `gamma > 0` (short outperformers, buy underperformers). Eq. (296) is automatically satisfied; Eq. (295) fixes gamma:

```
gamma = I / sum_{i=1}^{N} |R_tilde_i|                     (298)
```

## Entry / Exit Rules
- **Entry**: Compute demeaned returns over a short measurement window; enter positions D_i = -gamma * R_tilde_i.
- **Exit**: Close when demeaned returns converge back toward zero, or at a predefined time horizon.

## Key Parameters
- **Cluster definition**: Industry group, sector, or any set of historically correlated stocks
- **Measurement window**: Short-term (days to weeks) for mean-reversion
- **Position sizing**: Dollar-neutral via gamma normalization (Eq. 298)
- **Weights**: Uniform modulus, or non-uniform (e.g., suppressed by volatility)

## Variations

### 3.9.1 — Mean-Reversion: Multiple Clusters

Generalize to K > 1 clusters, where stocks within each cluster are historically highly correlated. Treat clusters independently and combine via linear regression (unified approach).

Let `Lambda_{iA}` be the N×K binary loadings matrix: `Lambda_{iA} = 1` if stock i belongs to cluster A, else 0. Cluster sizes: `N_A = sum_{i=1}^{N} Lambda_{iA} > 0`, `N = sum_{A=1}^{K} N_A`.

Run a linear regression of stock returns R_i on Lambda_{iA} (no intercept, unit weights):

```
R_i = sum_{A=1}^{K} Lambda_{iA} f_A + epsilon_i           (303)
```

Regression coefficients (cluster mean returns):

```
f = Q^{-1} Lambda^T R,   Q = Lambda^T Lambda               (304, 305)
Q_{AB} = N_A delta_{AB}                                   (307)
R_bar_A = (1/N_A) sum_{j in J_A} R_j                      (308)
```

Demeaned return (residual) for stock i:

```
epsilon_i = R_i - R_bar_{G(i)} = R_tilde_i                (309)
```

where G(i) is the cluster to which stock i belongs. These residuals are cluster-neutral:

```
sum_{i=1}^{N} R_tilde_i Lambda_{iA} = 0,  A = 1,...,K    (310)
```

Also automatically: `sum_{i=1}^{N} R_tilde_i = 0` (dollar-neutral).

Investments can be allocated uniformly across the K independent cluster strategies.

## Notes
- The single-cluster strategy (3.9) is the natural generalization of pairs trading to N stocks.
- The multiple-cluster (3.9.1) formulation uses linear regression to compute all cluster means simultaneously in a unified framework.
- The binary loadings matrix Lambda_{iA} ensures each stock belongs to exactly one cluster (no overlap assumed).
- The intercept is automatically included via the constraint that each stock belongs to one cluster (sum_A Lambda_{iA} = 1).
- Mean-reversion strategies work best when stocks are truly co-integrated or highly correlated; sector/industry groupings provide natural clusters.