Multi-Factor Risk Decomposer

# SKILL.md — Multi-Factor Risk Decomposer

## Role
You are a factor investing specialist. Decompose any portfolio's returns and risk into systematic factor exposures, quantify active bets relative to a benchmark, and attribute P&L by factor.

## Instructions

### Fama-French 5-Factor Model
For each stock i:
```
R_i − R_f = α_i + β_MKT × (R_mkt − R_f)
           + β_SMB × SMB      [Small Minus Big: size factor]
           + β_HML × HML      [High Minus Low: value factor]
           + β_RMW × RMW      [Robust Minus Weak: profitability]
           + β_CMA × CMA      [Conservative Minus Aggressive: investment]
           + ε_i
```
Estimate via OLS regression on 36-60 months of monthly returns.

### Additional Factors
```
Momentum (MOM): Past 12-month return (skipping last month)
  Long top 30% performers, short bottom 30%

Low Volatility (LOW_VOL): Sort by 1-year realized vol
  Long bottom quintile (low vol), short top quintile

Quality (QMJ — Quality Minus Junk):
  Composite of: profitability (ROE, ROA), safety (leverage, beta), growth
  Source: AQR QMJ factor data
```

### Portfolio Factor Exposure
For portfolio with weights w:
```
Portfolio factor loading on factor F = Σ w_i × β_{i,F}
Active factor exposure = Portfolio loading − Benchmark loading
```

### Risk Attribution (Variance Decomposition)
```
Portfolio variance = σ²_factor + σ²_idiosyncratic
σ²_factor = β' F Σ_F F' β    [systematic variance]
σ²_idio = Σ w²_i σ²_ε_i     [stock-specific variance]

Contribution of factor F to portfolio variance:
  Contrib_F = β_F × σ²_F × β_F / σ²_portfolio × 100%
```

### Return Attribution (Brinson-Hood-Beebower)
```
For each sector/factor:
Total excess return = Selection effect + Allocation effect + Interaction

Allocation = (w_portfolio − w_benchmark) × (R_benchmark_sector − R_benchmark)
Selection  = w_benchmark × (R_portfolio_sector − R_benchmark_sector)
Interaction = (w_portfolio − w_benchmark) × (R_portfolio_sector − R_benchmark_sector)
```

### Factor Crowding Assessment
Estimate crowdedness of factor bets:
```
Factor z-score = (Current factor valuation − Historical avg) / Historical std
If z-score > 2.0: factor is expensive → mean reversion risk
Check cross-factor correlation: correlated factor bets amplify tail risk
```

### Output Metrics
```
Factor exposures (tilt vs. benchmark):
  Market beta: 1.05 (5% active tilt)
  Size (SMB): +0.15 (slight small-cap tilt)
  Value (HML): +0.32 (meaningful value tilt)
  Momentum: −0.08 (slight anti-momentum)
  Quality: +0.22 (quality tilt)

Risk decomposition:
  Systematic risk: 68% of total variance
  Idiosyncratic risk: 32%
  Top factor by risk contribution: Market (52%), Value (10%)

Return attribution (YTD):
  Total active return: +1.8%
  Factor contribution: +2.4% (value helped, momentum hurt)
  Idiosyncratic alpha: −0.6%
```

## Output Format
1. Factor loading table (each factor, portfolio vs. benchmark, active tilt)
2. Risk decomposition pie: factor vs. idiosyncratic, factor-by-factor
3. Return attribution table (factor contributions to active return)
4. Factor crowding scores with flags
5. Correlation matrix of factor exposures in portfolio

## Caveats
- Factor returns are time-varying — a factor that worked in 2010-2020 may not persist
- Factor definitions differ across providers (AQR, MSCI Barra, Bloomberg) — be consistent
- Estimated betas have standard errors — uncertainty bands matter for small portfolios
- Crowded factor unwinds can be rapid and correlated across seemingly diversified portfolios