Statistical Arbitrage

Statistical Arbitrage

name: stat-arb description: Statistical arbitrage — PCA-based, factor-neutral, sector-neutral portfolio construction. Use for stat-arb strategy development.

When to Activate

• User wants to build a statistical arbitrage portfolio
• Decomposing returns using PCA to extract residual alpha
• Constructing sector-neutral or factor-neutral portfolios
• Trading mean reversion of residual returns
• Estimating capacity and managing risk of stat-arb strategies

First Questions

1. What is the investment universe (US equities, global, sector-specific)?
2. Are you targeting residual mean reversion, or do you have a specific alpha signal?
3. What neutrality constraints are required (market, sector, factor)?
4. What is the target holding period and rebalancing frequency?
5. What AUM and capacity requirements exist?

Core Concepts

What is Statistical Arbitrage?

Statistical arbitrage (stat arb) is a market-neutral strategy that profits from temporary mispricings in cross-sectional return relationships. Unlike pairs trading (2 assets), stat arb operates across an entire universe simultaneously.

Core principle:
  1. Decompose stock returns into systematic (factor) and idiosyncratic (residual) components
  2. The residual should be mean-reverting (if it is not, there is no stat arb)
  3. Trade the residual: buy stocks with abnormally negative residuals, sell stocks with abnormally positive residuals
  4. Neutralize factor exposures so the portfolio profits ONLY from residual mean reversion

PCA for Return Decomposition

Principal Component Analysis extracts the dominant systematic risk factors from the return covariance matrix:

R (T x N return matrix) = U * S * V'   (SVD decomposition)

First K principal components explain most variance:
  PC1: ~30-40% (market factor)
  PC2: ~5-8% (sector/size)
  PC3: ~3-5% (value/growth)
  PCs 4-10: ~1-3% each
  Residual: ~40-50% of variance (idiosyncratic)

Eigenportfolios:
  Each column of V corresponds to a portfolio (eigenportfolio)
  First eigenportfolio ≈ market portfolio
  Subsequent eigenportfolios capture sector tilts, style factors

Factor return estimation:
  f_t = R_t * V_K   (project returns onto top K eigenvectors)
  Factor loadings: B = V_K
  Residual returns: e_t = R_t - f_t * B'

Residual Returns and Mean Reversion

The residual return e_{i,t} is the stock's return after removing systematic factor exposure. In stat arb, the key assumption is that cumulative residuals are mean-reverting.

Cumulative residual: E_{i,t} = sum(e_{i,s}) for s = t-L to t

If E_{i,t} is abnormally negative: Stock has underperformed relative to its factor exposure → BUY
If E_{i,t} is abnormally positive: Stock has outperformed relative to its factor exposure → SELL

This is the "residual reversal" effect — one of the most established statistical regularities in equity markets.

Detailed Methodology

Building a PCA-Based Stat Arb Portfolio

Step 1: Universe definition
  - Liquid stocks (top 500-1000 by market cap)
  - Minimum ADV filter ($5M+)
  - Remove recent IPOs (< 1 year of data)
  - Exclude stocks pending corporate actions

Step 2: Return matrix construction
  - Use daily log returns, 60-120 trading days
  - Demean returns (subtract cross-sectional mean each day)

Step 3: PCA decomposition
  - Compute covariance matrix of demeaned returns
  - Extract top K eigenvectors (K = 5-15, determined by scree plot or cross-validation)
  - Alternative: use pre-specified factors (Fama-French, Barra) instead of PCA

Step 4: Compute residual returns
  - For each stock: e_{i,t} = r_{i,t} - sum_k(beta_{i,k} * f_{k,t})
  - Cumulative residual over lookback window L (typically 5-20 days)

Step 5: Signal construction
  - s_{i,t} = -E_{i,t} / sigma(E_i)   (negative because we mean-revert)
  - Z-score normalize cross-sectionally

Step 6: Portfolio construction
  - Raw weights: w_i proportional to s_i
  - Apply neutrality constraints (see below)
  - Scale to target volatility or gross exposure

Neutrality Constraints

Neutrality ensures the portfolio profits from alpha, not from unintended factor bets:

Market neutrality:
  sum(w_i) = 0  (dollar neutral)
  sum(w_i * beta_i) = 0  (beta neutral)

Sector neutrality:
  For each sector k: sum(w_i * I(sector_i = k)) = 0
  Ensures no net sector bet

Factor neutrality:
  For each factor k: sum(w_i * beta_{i,k}) = 0
  Neutralize market, size, value, momentum, quality exposure

Implementation:
  Solve constrained optimization:
    max sum(w_i * s_i)
    subject to:
      sum(w_i) = 0
      sum(|w_i|) <= Gross_max
      For each factor k: |sum(w_i * beta_{i,k})| < tolerance
      For each sector: |sum(w_i * I_sector)| < sector_tolerance
      |w_i| <= max_position_weight

Position Construction and Sizing

Position weight determination:
  1. Start with signal strength: w_i^raw = s_i
  2. Cap outliers: winsorize at +/- 3 standard deviations
  3. Make dollar-neutral: w_i = w_i^raw - mean(w_i^raw)
  4. Apply sector/factor neutralization via optimization
  5. Scale to target: w_i = w_i * target_gross / sum(|w_i|)

Gross exposure targets:
  Conservative: 100% long + 100% short = 200% gross
  Moderate: 150% + 150% = 300% gross
  Aggressive: 200% + 200% = 400% gross
  Typical institutional stat arb: 200-400% gross, 3-6x leverage

Number of positions:
  Typical: 200-500 long + 200-500 short
  More positions = more diversification, lower idiosyncratic risk
  Fewer positions = higher conviction, higher volatility

Portfolio-Level Risk Management

Risk decomposition:
  Total risk = factor risk + specific risk
  For a well-constructed stat arb: specific risk >> factor risk
  If factor risk is dominant, neutralization constraints are too loose

Daily risk monitoring:
  - Factor exposures (should be near zero)
  - Sector exposures (should be near zero)
  - Gross and net exposure
  - Portfolio volatility (ex-ante vs realized)
  - Concentration (top 10 positions as % of gross)
  - Beta to market (should be near zero)

Drawdown management:
  - Reduce gross exposure after X% drawdown (e.g., halve at -5%)
  - De-risk faster than re-risk (asymmetric response)
  - Circuit breaker: flatten at -10% drawdown, review model

Turnover management:
  - Daily turnover: 5-15% of gross (typical for 5-10 day holding period)
  - Annual turnover: 15-30x gross exposure
  - Transaction costs: 3-10 bps per trade (depends on universe)
  - Net alpha must exceed turnover * cost

Capacity Estimation

Capacity is the maximum AUM before market impact erodes alpha:

Method 1: ADV-based
  For each stock: max_position = participation_rate * ADV * days_to_trade
  Portfolio capacity = sum(max_positions)
  Typical participation rate: 1-5% of ADV
  Days to trade: 1-3 days

Method 2: Impact cost estimation
  Market impact = kappa * sigma * sqrt(shares / ADV)
  kappa: impact coefficient (0.1-0.5 for liquid stocks)
  Breakeven capacity: where impact cost = expected alpha

Method 3: Empirical
  Plot backtest Sharpe vs AUM assumption (increasing impact)
  Capacity = AUM where Sharpe drops below hurdle (e.g., 0.5)

Typical capacity:
  US large-cap stat arb: $500M - $5B
  US mid/small-cap: $100M - $1B
  Global: $1B - $10B
  Capacity erodes as more capital enters stat arb (crowding)

Templates / Examples

Stat Arb Strategy Report

Strategy: PCA Residual Reversal
Universe: S&P 500 | Rebalance: Daily | Period: 2010-2024

Model:
  PCA factors: 10 (explaining 55% of return variance)
  Lookback for residual: 10 days
  Signal: Negative cumulative residual, z-scored
  Constraints: Dollar-neutral, beta-neutral, sector-neutral
  Gross exposure: 300% ($150M long + $150M short on $100M capital)

--- Performance ---
Annualized Return:     8.5% (on capital, not gross)
Annualized Volatility: 5.2%
Sharpe Ratio:          1.63
Max Drawdown:          -7.8%
Calmar Ratio:          1.09
Market Correlation:    0.03 (effectively zero)

--- Risk ---
Ex-ante vol:           5.0%
Factor risk fraction:  12% (well-controlled)
Specific risk fraction:88% (dominant, as desired)
Average positions:     280 long, 260 short
Top 10 concentration:  18% of gross

--- Costs ---
Daily turnover:        8% of gross
Annual turnover:       20x gross
Avg trade cost:        5 bps
Cost drag:             -5.0% annualized
Gross alpha:           13.5% → Net alpha: 8.5%

--- Capacity ---
Estimated capacity:    $800M (at 2% ADV participation)
Current AUM:           $100M
Capacity headroom:     8x

Factor Exposure Report

Factor Exposure Report — [Date]

Factor      | Exposure | t-stat | Limit  | Status
Market      | +0.02    | 0.8    | 0.05   | OK
Size        | -0.03    | -1.1   | 0.10   | OK
Value       | +0.01    | 0.4    | 0.10   | OK
Momentum    | -0.04    | -1.5   | 0.10   | OK
Quality     | +0.02    | 0.7    | 0.10   | OK
Volatility  | -0.01    | -0.3   | 0.10   | OK

Sector      | Net ($M) | % of Gross | Limit  | Status
Technology  | +$1.2M   | 0.8%       | 3%     | OK
Healthcare  | -$0.8M   | 0.5%       | 3%     | OK
Financials  | +$0.3M   | 0.2%       | 3%     | OK
Energy      | -$0.5M   | 0.3%       | 3%     | OK
...

All exposures within limits.

Quality Gate

Before deploying a stat arb strategy:

• PCA or factor model explains significant variance (>50%) with reasonable number of factors
• Residual returns are demonstrably mean-reverting (ADF test, Hurst exponent)
• Neutrality constraints enforced: market, sector, and key factor exposures near zero
• Out-of-sample Sharpe exceeds 1.0 (stat arb should be high Sharpe, market neutral)
• Transaction costs properly estimated and net performance is attractive
• Turnover is manageable (< 20x annual gross)
• Capacity analysis confirms strategy can absorb target AUM
• Risk system monitors factor exposures, drawdown, and concentration in real-time
• Drawdown management rules defined (de-risk triggers)
• Crowding risk assessed (are other stat arb funds trading similar signals?)
• Execution infrastructure supports daily rebalancing of 500+ positions