Factor Models¶

Understanding what drives your portfolio returns is crucial for informed investing. Factor models decompose returns into systematic components, helping you distinguish between alpha (skill) and beta (market exposure).

What Are Factor Models?¶

Factor models explain asset returns as a linear combination of common risk factors plus an idiosyncratic component:

\[ R_i - R_f = \alpha_i + \beta_1 F_1 + \beta_2 F_2 + ... + \epsilon_i \]

Where:

\(R_i - R_f\) = Excess return of asset \(i\)
\(\alpha_i\) = Alpha (unexplained return, often attributed to skill)
\(\beta_k\) = Sensitivity to factor \(k\)
\(F_k\) = Factor return
\(\epsilon_i\) = Idiosyncratic (asset-specific) return

Available Models¶

CAPM (Capital Asset Pricing Model)¶

The simplest model with one factor: the market.

from portfolio_analysis.factors import FactorRegression

results = regression.run_regression('capm')
print(f"Market Beta: {results.betas['Mkt-RF']:.3f}")

Factors: Market excess return (Mkt-RF)

Fama-French 3-Factor Model¶

Adds size and value factors to CAPM.

results = regression.run_regression('ff3')

Factors:

Factor	Description	Interpretation
Mkt-RF	Market excess return	Equity market exposure
SMB	Small Minus Big	Small-cap tilt (+ = small, - = large)
HML	High Minus Low	Value tilt (+ = value, - = growth)

Fama-French 5-Factor Model¶

Adds profitability and investment factors.

# Requires FF5 data
ff5 = factor_loader.get_ff5_factors(start, end)
regression = FactorRegression(returns, ff5)
results = regression.run_regression('ff5')

Additional Factors:

Factor	Description	Interpretation
RMW	Robust Minus Weak	Quality/profitability tilt
CMA	Conservative Minus Aggressive	Investment tilt

Carhart 4-Factor Model¶

FF3 plus momentum.

carhart = factor_loader.get_carhart_factors(start, end)
regression = FactorRegression(returns, carhart)
results = regression.run_regression('carhart')

Additional Factor:

Factor	Description	Interpretation
MOM	Momentum	Winners minus losers (12-1 month)

Loading Factor Data¶

Factor data is automatically fetched from Kenneth French's Data Library:

from portfolio_analysis.factors import FactorDataLoader

loader = FactorDataLoader()

# Daily or monthly frequency
ff3_daily = loader.get_ff3_factors('2020-01-01', '2024-01-01', frequency='daily')
ff3_monthly = loader.get_ff3_factors('2020-01-01', '2024-01-01', frequency='monthly')

# Data is cached locally for 7 days

Running Regressions¶

Basic Regression¶

from portfolio_analysis.factors import FactorRegression

regression = FactorRegression(portfolio_returns, factor_data)
results = regression.run_regression('ff3')

# Access results
print(f"Alpha (annual): {results.alpha:.2%}")
print(f"Alpha p-value: {results.alpha_pvalue:.4f}")
print(f"R-squared: {results.r_squared:.4f}")

# Factor betas
for factor, beta in results.betas.items():
    pval = results.beta_pvalues[factor]
    sig = "***" if pval < 0.01 else ("**" if pval < 0.05 else "")
    print(f"{factor}: {beta:.3f}{sig}")

Interpreting Results¶

print(results.summary())

============================================================
Factor Regression Results: FF3
============================================================
Observations: 1257
R-squared: 0.9234
Adj R-squared: 0.9231
Residual Std: 5.23% (annualized)

Coefficient      Value     T-stat    P-value
--------------------------------------------
Alpha            0.02%       0.45     0.6521
Mkt-RF           0.682      45.23     0.0000
SMB              0.123       4.56     0.0000
HML              0.087       3.21     0.0014
============================================================

How to read this:

R-squared = 0.92: Factors explain 92% of return variance
Alpha = 0.02% (p=0.65): No statistically significant alpha
Mkt-RF = 0.68: Defensive positioning (less than 1.0)
SMB = 0.12: Slight small-cap tilt
HML = 0.09: Slight value tilt

Rolling Regressions¶

Track how factor exposures change over time:

# 60-day rolling window
rolling = regression.run_rolling_regression('ff3', window=60)

# Plot results
import matplotlib.pyplot as plt

fig, axes = plt.subplots(3, 1, figsize=(12, 8), sharex=True)
for ax, factor in zip(axes, ['Mkt-RF', 'SMB', 'HML']):
    ax.plot(rolling.index, rolling[factor])
    ax.axhline(y=rolling[factor].mean(), linestyle='--', color='red')
    ax.set_ylabel(factor)
plt.show()

Model Comparison¶

Compare how well different models explain your returns:

comparison = regression.compare_models()
print(comparison)

  Model  Alpha (%)  Alpha p-value  R-squared  Mkt Beta   SMB   HML   MOM
  CAPM      0.15           0.42       0.89      0.72   NaN   NaN   NaN
   FF3      0.02           0.65       0.92      0.68  0.12  0.09   NaN
CARHART     0.01           0.78       0.93      0.67  0.11  0.08  0.05

Return Attribution¶

Decompose your total return into factor contributions:

from portfolio_analysis.factors import FactorAttribution

attribution = FactorAttribution(returns, factor_data)
decomp = attribution.decompose_returns('ff3')

print(f"Total Return: {decomp['total']:.2%}")
print(f"  Risk-Free:  {decomp['risk_free']:.2%}")
print(f"  Market:     {decomp['Mkt-RF']:.2%}")
print(f"  Size (SMB): {decomp['SMB']:.2%}")
print(f"  Value (HML):{decomp['HML']:.2%}")
print(f"  Alpha:      {decomp['alpha']:.2%}")

Risk Attribution¶

Understand where your portfolio risk comes from:

risk_decomp = attribution.decompose_risk('ff3')

print(f"Total Variance: {risk_decomp['total']:.6f}")
print(f"  Market:       {risk_decomp['Mkt-RF']/risk_decomp['total']:.1%}")
print(f"  Size:         {risk_decomp['SMB']/risk_decomp['total']:.1%}")
print(f"  Value:        {risk_decomp['HML']/risk_decomp['total']:.1%}")
print(f"  Idiosyncratic:{risk_decomp['idiosyncratic']/risk_decomp['total']:.1%}")

Characteristic-Based Tilts¶

Estimate factor exposures from portfolio characteristics (no regression needed):

from portfolio_analysis.factors import FactorExposures

exposures = FactorExposures(
    tickers=['VTI', 'VBR', 'VTV', 'VUG', 'BND'],
    weights=[0.3, 0.15, 0.2, 0.2, 0.15]
)

tilts = exposures.get_all_tilts()
print(f"Size tilt:       {tilts['size']:.2f}")  # -1=large, +1=small
print(f"Value tilt:      {tilts['value']:.2f}") # -1=growth, +1=value
print(f"Momentum tilt:   {tilts['momentum']:.2f}")
print(f"Quality tilt:    {tilts['quality']:.2f}")
print(f"Investment tilt: {tilts['investment']:.2f}")

Visualization¶

from portfolio_analysis.factors import FactorVisualization

# Factor exposure bar chart
FactorVisualization.plot_factor_exposures(results)

# Rolling betas over time
FactorVisualization.plot_rolling_betas(rolling)

# Return attribution waterfall
FactorVisualization.plot_return_attribution(decomp)

# Factor tilts radar chart
FactorVisualization.plot_factor_tilts(tilts)

Best Practices¶

1. Use Sufficient Data¶

Daily data: At least 1 year (252+ observations)
Monthly data: At least 3 years (36+ observations)

2. Check Statistical Significance¶

Don't over-interpret insignificant betas:

for factor in results.factors:
    if results.beta_pvalues[factor] < 0.05:
        print(f"{factor}: Significant at 5% level")

3. Consider Multiple Models¶

No single model is "correct". Compare models and use domain knowledge.

4. Watch for Regime Changes¶

Use rolling regressions to detect when factor exposures shift.

5. Distinguish Alpha from Factor Timing¶

"Alpha" in a static regression may actually be factor timing (changing betas over time).