Factor Regression Analysis with finm

Factor Regression Analysis with finm#

This notebook demonstrates how to use the finm package to perform factor regression analysis, including CAPM and Fama-French 3-factor models.

Learning Objectives#

  1. Understand Jensen’s alpha and factor betas

  2. Run CAPM regression and interpret results

  3. Run Fama-French 3-factor regression

  4. Interpret statistical significance of alpha and betas

Key Functions#

  • finm.run_factor_regression() - Core OLS regression with full statistics

  • finm.run_capm_regression() - Convenience wrapper for CAPM

  • finm.run_fama_french_regression() - Convenience wrapper for FF3

import os
from pathlib import Path

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from dotenv import load_dotenv

import finm
from finm.data import fama_french

load_dotenv()

DATA_DIR = Path(os.environ.get("DATA_DIR", "./_data"))
DATA_DIR.mkdir(parents=True, exist_ok=True)

print(f"Data directory: {DATA_DIR}")

Data directory: /Users/jbejarano/GitRepositories/finm/_data

1. Load Fama-French Factors#

The Fama-French 3-factor model includes:

  • Mkt-RF: Market excess return (market return minus risk-free rate)

  • SMB: Small Minus Big (size factor)

  • HML: High Minus Low (value factor)

  • RF: Risk-free rate

# Load Fama-French data from bundled data
ff_factors = fama_french.load(data_dir=None).to_pandas()
ff_factors = ff_factors.set_index("Date")
print(f"Fama-French factors shape: {ff_factors.shape}")
print(f"Date range: {ff_factors.index.min()} to {ff_factors.index.max()}")
ff_factors.head()

Fama-French factors shape: (1227, 4)
Date range: 2021-01-12 00:00:00 to 2025-11-28 00:00:00
Mkt-RF SMB HML RF
Date
2021-01-12 0.0037 0.0128 0.0124 0.0
2021-01-13 0.0006 -0.0094 -0.0045 0.0
2021-01-14 -0.0012 0.0202 0.0113 0.0
2021-01-15 -0.0086 -0.0050 -0.0074 0.0
2021-01-19 0.0092 0.0088 -0.0079 0.0

2. Create Synthetic Portfolio Returns#

For demonstration, we’ll create synthetic portfolio returns with known characteristics. In practice, you would use actual portfolio or asset returns.

# Create synthetic portfolio returns with known factor exposures
np.random.seed(42)

# Use a subset of dates
dates = ff_factors.index[-120:]  # Last 10 years of monthly data
factors = ff_factors.loc[dates]

# Simulate a portfolio with:
# - Market beta = 1.2
# - SMB beta = 0.5 (tilted toward small caps)
# - HML beta = 0.3 (tilted toward value)
# - Alpha = 0.002 per month (about 2.4% annually)

market_beta = 1.2
smb_beta = 0.5
hml_beta = 0.3
monthly_alpha = 0.002

# Generate portfolio excess returns
noise = np.random.randn(len(dates)) * 0.01
portfolio_excess = (
    monthly_alpha
    + market_beta * factors["Mkt-RF"]
    + smb_beta * factors["SMB"]
    + hml_beta * factors["HML"]
    + noise
)

# Raw returns = excess returns + risk-free rate
portfolio_returns = portfolio_excess + factors["RF"]

print("Synthetic portfolio created with known exposures:")
print(f"  Target Market Beta: {market_beta}")
print(f"  Target SMB Beta: {smb_beta}")
print(f"  Target HML Beta: {hml_beta}")
print(f"  Target Monthly Alpha: {monthly_alpha}")

Synthetic portfolio created with known exposures:
  Target Market Beta: 1.2
  Target SMB Beta: 0.5
  Target HML Beta: 0.3
  Target Monthly Alpha: 0.002

3. CAPM Regression#

The Capital Asset Pricing Model (CAPM) regresses portfolio excess returns on market excess returns:

\[R_p - R_f = \alpha + \beta (R_m - R_f) + \epsilon\]

Where:

  • \(R_p\) is portfolio return

  • \(R_f\) is risk-free rate

  • \(R_m\) is market return

  • \(\alpha\) is Jensen’s alpha (abnormal return)

  • \(\beta\) is market beta (systematic risk)

# Run CAPM regression
capm_result = finm.run_capm_regression(
    excess_returns=portfolio_excess,
    market_excess_returns=factors["Mkt-RF"],
    annualization_factor=12,  # Monthly data, annualize alpha
)

print("CAPM Regression Results")
print("=" * 50)
print(f"Alpha (monthly):     {capm_result.alpha:>10.4f}")
print(f"Alpha (annualized):  {capm_result.alpha_annualized:>10.2%}")
print(f"Alpha t-stat:        {capm_result.alpha_tstat:>10.2f}")
print(f"Alpha p-value:       {capm_result.alpha_pvalue:>10.4f}")
print(f"")
print(f"Market Beta:         {capm_result.betas['Mkt-RF']:>10.2f}")
print(f"Beta t-stat:         {capm_result.beta_tstats['Mkt-RF']:>10.2f}")
print(f"Beta p-value:        {capm_result.beta_pvalues['Mkt-RF']:>10.4f}")
print(f"")
print(f"R-squared:           {capm_result.r_squared:>10.2%}")
print(f"Observations:        {capm_result.n_observations:>10d}")

CAPM Regression Results
==================================================
Alpha (monthly):         0.0010
Alpha (annualized):       1.25%
Alpha t-stat:              1.19
Alpha p-value:           0.2376

Market Beta:               1.47
Beta t-stat:              12.32
Beta p-value:            0.0000

R-squared:               56.25%
Observations:               120

Interpretation of CAPM Results#

  • Alpha: The intercept represents Jensen’s alpha - the average return not explained by market exposure. A positive alpha suggests outperformance.

  • Beta: Measures sensitivity to market movements. Beta > 1 means the portfolio is more volatile than the market.

  • t-statistic: Tests if the coefficient is significantly different from zero. Generally, |t| > 2 is considered statistically significant.

  • p-value: Probability of observing this result if the true coefficient is zero. p < 0.05 is often used as a significance threshold.

  • R-squared: Proportion of variance explained by the model.

4. Fama-French 3-Factor Regression#

The Fama-French 3-factor model extends CAPM by adding size and value factors:

\[R_p - R_f = \alpha + \beta_{MKT}(R_m - R_f) + \beta_{SMB} \cdot SMB + \beta_{HML} \cdot HML + \epsilon\]

This model captures additional sources of systematic risk:

  • SMB (Small Minus Big): Return of small-cap stocks minus large-cap stocks

  • HML (High Minus Low): Return of value stocks minus growth stocks

# Run Fama-French 3-factor regression
ff_result = finm.run_fama_french_regression(
    returns=portfolio_returns,
    factors=factors,
    annualization_factor=12,
)

print("Fama-French 3-Factor Regression Results")
print("=" * 50)
print(f"Alpha (monthly):     {ff_result.alpha:>10.4f}")
print(f"Alpha (annualized):  {ff_result.alpha_annualized:>10.2%}")
print(f"Alpha t-stat:        {ff_result.alpha_tstat:>10.2f}")
print(f"Alpha p-value:       {ff_result.alpha_pvalue:>10.4f}")
print(f"")
print("Factor Betas:")
for factor in ["Mkt-RF", "SMB", "HML"]:
    print(
        f"  {factor:8s}:        {ff_result.betas[factor]:>10.2f}  "
        f"(t={ff_result.beta_tstats[factor]:>6.2f}, p={ff_result.beta_pvalues[factor]:.4f})"
    )
print(f"")
print(f"R-squared:           {ff_result.r_squared:>10.2%}")
print(f"Adj R-squared:       {ff_result.adj_r_squared:>10.2%}")
print(f"Observations:        {ff_result.n_observations:>10d}")

Fama-French 3-Factor Regression Results
==================================================
Alpha (monthly):         0.0009
Alpha (annualized):       1.13%
Alpha t-stat:              1.11
Alpha p-value:           0.2689

Factor Betas:
  Mkt-RF  :              1.45  (t= 10.32, p=0.0000)
  SMB     :              0.19  (t=  1.29, p=0.1985)
  HML     :              0.38  (t=  2.76, p=0.0067)

R-squared:               60.44%
Adj R-squared:           59.42%
Observations:               120

Comparing CAPM and FF3 Results#

Notice how the results change when we account for size and value exposures:

  • The market beta changes when we control for other factors

  • The R-squared increases (more variance explained)

  • Alpha may change as the model explains more of the returns

# Create comparison table
comparison_data = {
    "Statistic": ["Alpha (monthly)", "Alpha (annual)", "Alpha t-stat", "Market Beta", "R-squared"],
    "CAPM": [
        f"{capm_result.alpha:.4f}",
        f"{capm_result.alpha_annualized:.2%}",
        f"{capm_result.alpha_tstat:.2f}",
        f"{capm_result.betas['Mkt-RF']:.2f}",
        f"{capm_result.r_squared:.2%}",
    ],
    "FF3": [
        f"{ff_result.alpha:.4f}",
        f"{ff_result.alpha_annualized:.2%}",
        f"{ff_result.alpha_tstat:.2f}",
        f"{ff_result.betas['Mkt-RF']:.2f}",
        f"{ff_result.r_squared:.2%}",
    ],
}
comparison_df = pd.DataFrame(comparison_data)
print("Model Comparison")
print("=" * 50)
print(comparison_df.to_string(index=False))

Model Comparison
==================================================
      Statistic   CAPM    FF3
Alpha (monthly) 0.0010 0.0009
 Alpha (annual)  1.25%  1.13%
   Alpha t-stat   1.19   1.11
    Market Beta   1.47   1.45
      R-squared 56.25% 60.44%

5. Using the Generic run_factor_regression()#

The run_factor_regression() function is the core function that handles both single-factor and multi-factor regressions. It provides full flexibility for custom factor models.

# Example: Run a custom 2-factor model (Market + SMB only)
custom_factors = factors[["Mkt-RF", "SMB"]]

custom_result = finm.run_factor_regression(
    returns=portfolio_excess,
    factors=custom_factors,
    annualization_factor=12,
)

print("Custom 2-Factor Model (Market + SMB)")
print("=" * 50)
print(f"Alpha (annualized):  {custom_result.alpha_annualized:>10.2%}")
for factor_name, beta in custom_result.betas.items():
    print(f"{factor_name} Beta:         {beta:>10.2f}")
print(f"R-squared:           {custom_result.r_squared:>10.2%}")

Custom 2-Factor Model (Market + SMB)
==================================================
Alpha (annualized):       1.41%
Mkt-RF Beta:               1.32
SMB Beta:               0.30
R-squared:               57.85%

6. Accessing the RegressionResult Dataclass#

The RegressionResult dataclass provides a structured way to access all regression statistics. Here’s a complete view of available attributes:

# Display all available attributes
print("RegressionResult Attributes")
print("=" * 50)
print(f"alpha:             {ff_result.alpha}")
print(f"alpha_tstat:       {ff_result.alpha_tstat}")
print(f"alpha_pvalue:      {ff_result.alpha_pvalue}")
print(f"alpha_se:          {ff_result.alpha_se}")
print(f"betas:             {ff_result.betas}")
print(f"beta_tstats:       {ff_result.beta_tstats}")
print(f"beta_pvalues:      {ff_result.beta_pvalues}")
print(f"beta_ses:          {ff_result.beta_ses}")
print(f"r_squared:         {ff_result.r_squared}")
print(f"adj_r_squared:     {ff_result.adj_r_squared}")
print(f"n_observations:    {ff_result.n_observations}")
print(f"residual_std:      {ff_result.residual_std}")
print(f"alpha_annualized:  {ff_result.alpha_annualized}")
print(f"annualization_factor: {ff_result.annualization_factor}")

RegressionResult Attributes
==================================================
alpha:             0.0009454922708299472
alpha_tstat:       1.1110360410517994
alpha_pvalue:      0.2688504206104958
alpha_se:          0.0008510005399418594
betas:             {'Mkt-RF': np.float64(1.4549760801019018), 'SMB': np.float64(0.18634381266329703), 'HML': np.float64(0.3836527439059588)}
beta_tstats:       {'Mkt-RF': np.float64(10.323167198177247), 'SMB': np.float64(1.2933658504647636), 'HML': np.float64(2.759068027904079)}
beta_pvalues:      {'Mkt-RF': np.float64(0.0), 'SMB': np.float64(0.1984542228344075), 'HML': np.float64(0.006737881235053367)}
beta_ses:          {'Mkt-RF': np.float64(0.14094279906255958), 'SMB': np.float64(0.1440766451320293), 'HML': np.float64(0.13905157104712634)}
r_squared:         0.6044139450741914
adj_r_squared:     0.5941832712399033
n_observations:    120
residual_std:      0.009164980521723246
alpha_annualized:  0.011345907249959366
annualization_factor: 12

Summary#

This notebook demonstrated:

  1. run_capm_regression(): Single-factor model for market beta and alpha

  2. run_fama_french_regression(): Three-factor model with size and value

  3. run_factor_regression(): Generic function for custom factor models

  4. RegressionResult: Dataclass containing all regression statistics

Key Takeaways#

  • Alpha represents the abnormal return not explained by factor exposures

  • Betas measure sensitivity to each factor

  • T-statistics and p-values indicate statistical significance

  • R-squared shows how much variance is explained by the model

  • Adding more factors typically increases R-squared but may reduce alpha

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