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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."
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tags: [stocks, mean-reversion, cluster, statistical-arbitrage]
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---
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# Mean-Reversion — Single Cluster (and Multiple Clusters)
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**Section**: 3.9 / 3.9.1 | **Asset Class**: Stocks | **Type**: Mean-Reversion / Statistical Arbitrage
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## Overview
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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.
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## Construction / Signal
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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.
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Cluster mean return and demeaned returns:
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```
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R_bar = (1/N) * sum_{i=1}^{N} R_i (293)
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R_tilde_i = R_i - R_bar (294)
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```
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Short stocks with positive `R_tilde_i` (outperformers), buy stocks with negative `R_tilde_i` (underperformers).
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**Dollar-neutrality constraints**:
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```
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sum_{i=1}^{N} P_i |Q_i| = I (295)
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sum_{i=1}^{N} P_i Q_i = 0 (296)
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```
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A simple prescription for dollar positions `D_i = P_i Q_i` proportional to demeaned returns:
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```
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D_i = -gamma * R_tilde_i (297)
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```
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where `gamma > 0` (short outperformers, buy underperformers). Eq. (296) is automatically satisfied; Eq. (295) fixes gamma:
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```
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gamma = I / sum_{i=1}^{N} |R_tilde_i| (298)
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```
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## Entry / Exit Rules
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- **Entry**: Compute demeaned returns over a short measurement window; enter positions D_i = -gamma * R_tilde_i.
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- **Exit**: Close when demeaned returns converge back toward zero, or at a predefined time horizon.
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## Key Parameters
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- **Cluster definition**: Industry group, sector, or any set of historically correlated stocks
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- **Measurement window**: Short-term (days to weeks) for mean-reversion
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- **Position sizing**: Dollar-neutral via gamma normalization (Eq. 298)
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- **Weights**: Uniform modulus, or non-uniform (e.g., suppressed by volatility)
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## Variations
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### 3.9.1 — Mean-Reversion: Multiple Clusters
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Generalize to K > 1 clusters, where stocks within each cluster are historically highly correlated. Treat clusters independently and combine via linear regression (unified approach).
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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`.
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Run a linear regression of stock returns R_i on Lambda_{iA} (no intercept, unit weights):
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```
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R_i = sum_{A=1}^{K} Lambda_{iA} f_A + epsilon_i (303)
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```
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Regression coefficients (cluster mean returns):
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```
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f = Q^{-1} Lambda^T R, Q = Lambda^T Lambda (304, 305)
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Q_{AB} = N_A delta_{AB} (307)
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R_bar_A = (1/N_A) sum_{j in J_A} R_j (308)
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```
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Demeaned return (residual) for stock i:
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```
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epsilon_i = R_i - R_bar_{G(i)} = R_tilde_i (309)
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```
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where G(i) is the cluster to which stock i belongs. These residuals are cluster-neutral:
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```
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sum_{i=1}^{N} R_tilde_i Lambda_{iA} = 0, A = 1,...,K (310)
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```
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Also automatically: `sum_{i=1}^{N} R_tilde_i = 0` (dollar-neutral).
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Investments can be allocated uniformly across the K independent cluster strategies.
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## Notes
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- The single-cluster strategy (3.9) is the natural generalization of pairs trading to N stocks.
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- The multiple-cluster (3.9.1) formulation uses linear regression to compute all cluster means simultaneously in a unified framework.
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- The binary loadings matrix Lambda_{iA} ensures each stock belongs to exactly one cluster (no overlap assumed).
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- The intercept is automatically included via the constraint that each stock belongs to one cluster (sum_A Lambda_{iA} = 1).
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- Mean-reversion strategies work best when stocks are truly co-integrated or highly correlated; sector/industry groupings provide natural clusters.
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