Fama-French Factor Models
The Fama-French factor models extend the Capital Asset Pricing Model (CAPM) by adding additional risk factors beyond market risk. Pypulate implements both the Three-Factor and Five-Factor models, which are widely used in academic research and professional asset management to explain stock returns and assess investment performance.
Usage in Pypulate
from pypulate.asset import fama_french_three_factor, fama_french_five_factor
# Calculate expected return using the Three-Factor model
three_factor_result = fama_french_three_factor(
risk_free_rate=0.03, # 3% risk-free rate
market_beta=1.2, # Market factor beta
size_beta=0.5, # Size factor beta (SMB)
value_beta=0.3, # Value factor beta (HML)
market_premium=0.05, # Market risk premium
size_premium=0.02, # Size premium
value_premium=0.03 # Value premium
)
# Calculate expected return using the Five-Factor model
five_factor_result = fama_french_five_factor(
risk_free_rate=0.03, # 3% risk-free rate
market_beta=1.2, # Market factor beta
size_beta=0.5, # Size factor beta (SMB)
value_beta=0.3, # Value factor beta (HML)
profitability_beta=0.2, # Profitability factor beta (RMW)
investment_beta=0.1, # Investment factor beta (CMA)
market_premium=0.05, # Market risk premium
size_premium=0.02, # Size premium
value_premium=0.03, # Value premium
profitability_premium=0.01,# Profitability premium
investment_premium=0.01 # Investment premium
)
# Access the results
three_factor_return = three_factor_result["expected_return"]
five_factor_return = five_factor_result["expected_return"]
print(f"Three-Factor Expected Return: {three_factor_return:.2%}")
print(f"Five-Factor Expected Return: {five_factor_return:.2%}")
Three-Factor Model
The Fama-French Three-Factor Model extends CAPM by adding size and value factors to the market risk factor.
Parameters
| Parameter | Type | Description | Default |
|---|---|---|---|
risk_free_rate |
float | Risk-free rate of return (e.g., 0.03 for 3%) | Required |
market_beta |
float | Beta coefficient for market risk factor | Required |
size_beta |
float | Beta coefficient for size factor (SMB - Small Minus Big) | Required |
value_beta |
float | Beta coefficient for value factor (HML - High Minus Low) | Required |
market_premium |
float | Market risk premium (Rm - Rf) | Required |
size_premium |
float | Size premium (SMB) | Required |
value_premium |
float | Value premium (HML) | Required |
Return Value
The fama_french_three_factor function returns a dictionary with the following keys:
| Key | Type | Description |
|---|---|---|
expected_return |
float | Expected return calculated using the Three-Factor model |
risk_free_rate |
float | Risk-free rate used in the calculation |
total_systematic_risk |
float | Sum of absolute factor contributions |
risk_assessment |
str | Qualitative assessment of risk based on total systematic risk |
factor_contributions |
dict | Detailed information about each factor's contribution |
The factor_contributions dictionary contains detailed information for each factor (market, size, value):
| Key | Type | Description |
|---|---|---|
beta |
float | Beta coefficient for the factor |
premium |
float | Risk premium for the factor |
contribution |
float | Contribution to expected return (beta × premium) |
contribution_pct |
float | Percentage contribution to total systematic risk |
Five-Factor Model
The Fama-French Five-Factor Model extends the Three-Factor Model by adding profitability and investment factors.
Parameters
| Parameter | Type | Description | Default |
|---|---|---|---|
risk_free_rate |
float | Risk-free rate of return (e.g., 0.03 for 3%) | Required |
market_beta |
float | Beta coefficient for market risk factor | Required |
size_beta |
float | Beta coefficient for size factor (SMB - Small Minus Big) | Required |
value_beta |
float | Beta coefficient for value factor (HML - High Minus Low) | Required |
profitability_beta |
float | Beta coefficient for profitability factor (RMW - Robust Minus Weak) | Required |
investment_beta |
float | Beta coefficient for investment factor (CMA - Conservative Minus Aggressive) | Required |
market_premium |
float | Market risk premium (Rm - Rf) | Required |
size_premium |
float | Size premium (SMB) | Required |
value_premium |
float | Value premium (HML) | Required |
profitability_premium |
float | Profitability premium (RMW) | Required |
investment_premium |
float | Investment premium (CMA) | Required |
Return Value
The fama_french_five_factor function returns a dictionary with the same structure as the Three-Factor model, but with additional entries for the profitability and investment factors in the factor_contributions dictionary.
Risk Level Classification
Both models classify total systematic risk into risk levels:
| Total Systematic Risk | Risk Assessment |
|---|---|
| < 0.03 | Low risk |
| 0.03 - 0.059 | Moderate risk |
| 0.06 - 0.089 | Above-average risk |
| ≥ 0.09 | High risk |
Comprehensive Example
Here's a complete example demonstrating how to use the Fama-French models for asset pricing and factor analysis:
import numpy as np
import matplotlib.pyplot as plt
from pypulate.asset import fama_french_three_factor, fama_french_five_factor
# Define parameters for analysis
risk_free_rate = 0.03
market_premium = 0.05
# Define factor premiums
factor_premiums = {
'size': 0.02, # SMB premium
'value': 0.03, # HML premium
'profitability': 0.01, # RMW premium
'investment': 0.01 # CMA premium
}
# Define assets with different factor exposures
assets = {
'Large Growth': {
'market_beta': 1.1,
'size_beta': -0.3,
'value_beta': -0.5,
'profitability_beta': 0.4,
'investment_beta': -0.2
},
'Large Value': {
'market_beta': 0.9,
'size_beta': -0.2,
'value_beta': 0.6,
'profitability_beta': 0.2,
'investment_beta': 0.3
},
'Small Growth': {
'market_beta': 1.3,
'size_beta': 0.7,
'value_beta': -0.4,
'profitability_beta': 0.1,
'investment_beta': -0.3
},
'Small Value': {
'market_beta': 1.0,
'size_beta': 0.8,
'value_beta': 0.7,
'profitability_beta': -0.1,
'investment_beta': 0.4
}
}
# Calculate expected returns using both models
for asset_name, asset_data in assets.items():
# Three-Factor model
three_factor_result = fama_french_three_factor(
risk_free_rate=risk_free_rate,
market_beta=asset_data['market_beta'],
size_beta=asset_data['size_beta'],
value_beta=asset_data['value_beta'],
market_premium=market_premium,
size_premium=factor_premiums['size'],
value_premium=factor_premiums['value']
)
# Five-Factor model
five_factor_result = fama_french_five_factor(
risk_free_rate=risk_free_rate,
market_beta=asset_data['market_beta'],
size_beta=asset_data['size_beta'],
value_beta=asset_data['value_beta'],
profitability_beta=asset_data['profitability_beta'],
investment_beta=asset_data['investment_beta'],
market_premium=market_premium,
size_premium=factor_premiums['size'],
value_premium=factor_premiums['value'],
profitability_premium=factor_premiums['profitability'],
investment_premium=factor_premiums['investment']
)
# Store results in asset data
asset_data['three_factor'] = three_factor_result
asset_data['five_factor'] = five_factor_result
# Visualize results
plt.figure(figsize=(15, 10))
# Plot expected returns comparison
plt.subplot(2, 2, 1)
asset_names = list(assets.keys())
three_factor_returns = [assets[name]['three_factor']['expected_return'] for name in asset_names]
five_factor_returns = [assets[name]['five_factor']['expected_return'] for name in asset_names]
x = np.arange(len(asset_names))
width = 0.35
plt.bar(x - width/2, [r * 100 for r in three_factor_returns], width,
label='Three-Factor Model', color='skyblue')
plt.bar(x + width/2, [r * 100 for r in five_factor_returns], width,
label='Five-Factor Model', color='salmon')
plt.axhline(y=risk_free_rate * 100, color='black', linestyle='--',
label=f'Risk-Free Rate ({risk_free_rate:.1%})')
plt.axhline(y=(risk_free_rate + market_premium) * 100, color='gray', linestyle='--',
label=f'Market Return ({risk_free_rate + market_premium:.1%})')
plt.xlabel('Assets')
plt.ylabel('Expected Return (%)')
plt.title('Expected Returns: Three-Factor vs. Five-Factor Model')
plt.xticks(x, asset_names)
plt.legend()
plt.grid(axis='y', alpha=0.3)
# Plot factor contributions for a selected asset
selected_asset = 'Small Value'
asset_data = assets[selected_asset]
# Three-Factor contributions
plt.subplot(2, 2, 2)
factor_names_3f = ['Market', 'Size', 'Value']
contributions_3f = [
asset_data['three_factor']['factor_contributions']['market']['contribution'],
asset_data['three_factor']['factor_contributions']['size']['contribution'],
asset_data['three_factor']['factor_contributions']['value']['contribution']
]
colors_3f = ['#ff9999', '#66b3ff', '#99ff99']
plt.pie([abs(c) for c in contributions_3f], labels=factor_names_3f, colors=colors_3f,
autopct='%1.1f%%', startangle=90, wedgeprops={'edgecolor': 'w'})
plt.axis('equal')
plt.title(f'Three-Factor Model: {selected_asset}\nFactor Contributions')
# Five-Factor contributions
plt.subplot(2, 2, 3)
factor_names_5f = ['Market', 'Size', 'Value', 'Profitability', 'Investment']
contributions_5f = [
asset_data['five_factor']['factor_contributions']['market']['contribution'],
asset_data['five_factor']['factor_contributions']['size']['contribution'],
asset_data['five_factor']['factor_contributions']['value']['contribution'],
asset_data['five_factor']['factor_contributions']['profitability']['contribution'],
asset_data['five_factor']['factor_contributions']['investment']['contribution']
]
colors_5f = ['#ff9999', '#66b3ff', '#99ff99', '#ffcc99', '#c2c2f0']
plt.pie([abs(c) for c in contributions_5f], labels=factor_names_5f, colors=colors_5f,
autopct='%1.1f%%', startangle=90, wedgeprops={'edgecolor': 'w'})
plt.axis('equal')
plt.title(f'Five-Factor Model: {selected_asset}\nFactor Contributions')
# Plot factor betas for all assets
plt.subplot(2, 2, 4)
factor_names = ['Market', 'Size', 'Value', 'Profitability', 'Investment']
factor_keys = ['market_beta', 'size_beta', 'value_beta', 'profitability_beta', 'investment_beta']
# Create a grouped bar chart for factor betas
bar_width = 0.15
index = np.arange(len(factor_names))
for i, (asset_name, asset_data) in enumerate(assets.items()):
betas = [asset_data[key] for key in factor_keys]
plt.bar(index + i * bar_width, betas, bar_width, label=asset_name)
plt.xlabel('Factors')
plt.ylabel('Beta')
plt.title('Factor Betas by Asset')
plt.xticks(index + bar_width * (len(assets) - 1) / 2, factor_names)
plt.axhline(y=0, color='black', linestyle='-', alpha=0.2)
plt.legend()
plt.grid(axis='y', alpha=0.3)
plt.tight_layout()
plt.show()
# Print detailed results
print("Asset Analysis:")
print(f"{'Asset':<15} {'3F Return':<12} {'5F Return':<12} {'3F Risk':<20} {'5F Risk':<20}")
print("-" * 80)
for asset_name, asset_data in assets.items():
three_factor_return = asset_data['three_factor']['expected_return']
five_factor_return = asset_data['five_factor']['expected_return']
three_factor_risk = asset_data['three_factor']['risk_assessment']
five_factor_risk = asset_data['five_factor']['risk_assessment']
print(f"{asset_name:<15} {three_factor_return:<12.2%} {five_factor_return:<12.2%} "
f"{three_factor_risk:<20} {five_factor_risk:<20}")
# Analyze factor contributions for each asset
print("\nFactor Contribution Analysis (Five-Factor Model):")
for asset_name, asset_data in assets.items():
print(f"\n{asset_name}:")
contributions = asset_data['five_factor']['factor_contributions']
print(f"{'Factor':<15} {'Beta':<8} {'Premium':<10} {'Contribution':<15} {'% of Total':<12}")
print("-" * 60)
for factor, data in contributions.items():
print(f"{factor.capitalize():<15} {data['beta']:>+.2f} {data['premium']:>6.2%} "
f"{data['contribution']:>+.4f} {data['contribution_pct']:>6.1%}")
print(f"Total Expected Return: {asset_data['five_factor']['expected_return']:.2%}")
print(f"Total Systematic Risk: {asset_data['five_factor']['total_systematic_risk']:.2%}")
print(f"Risk Assessment: {asset_data['five_factor']['risk_assessment']}")
Example Output
Asset Analysis:
Asset 3F Return 5F Return 3F Risk 5F Risk
--------------------------------------------------------------------------------
Large Growth 7.35% 7.55% Moderate risk Moderate risk
Large Value 9.30% 9.50% Moderate risk Moderate risk
Small Growth 8.95% 8.85% Moderate risk Moderate risk
Small Value 11.10% 11.10% Above-average risk Above-average risk
Factor Contribution Analysis (Five-Factor Model):
Large Growth:
Factor Beta Premium Contribution % of Total
------------------------------------------------------------
Market +1.10 5.00% +0.0550 68.8%
Size -0.30 2.00% -0.0060 7.5%
Value -0.50 3.00% -0.0150 18.8%
Profitability +0.40 1.00% +0.0040 5.0%
Investment -0.20 1.00% -0.0020 2.5%
Total Expected Return: 7.55%
Total Systematic Risk: 8.20%
Risk Assessment: Moderate risk
Large Value:
Factor Beta Premium Contribution % of Total
------------------------------------------------------------
Market +0.90 5.00% +0.0450 56.3%
Size -0.20 2.00% -0.0040 5.0%
Value +0.60 3.00% +0.0180 22.5%
Profitability +0.20 1.00% +0.0020 2.5%
Investment +0.30 1.00% +0.0030 3.8%
Total Expected Return: 9.50%
Total Systematic Risk: 7.20%
Risk Assessment: Moderate risk
Small Growth:
Factor Beta Premium Contribution % of Total
------------------------------------------------------------
Market +1.30 5.00% +0.0650 65.0%
Size +0.70 2.00% +0.0140 14.0%
Value -0.40 3.00% -0.0120 12.0%
Profitability +0.10 1.00% +0.0010 1.0%
Investment -0.30 1.00% -0.0030 3.0%
Total Expected Return: 8.85%
Total Systematic Risk: 9.50%
Risk Assessment: Above-average risk
Small Value:
Factor Beta Premium Contribution % of Total
------------------------------------------------------------
Market +1.00 5.00% +0.0500 50.0%
Size +0.80 2.00% +0.0160 16.0%
Value +0.70 3.00% +0.0210 21.0%
Profitability -0.10 1.00% -0.0010 1.0%
Investment +0.40 1.00% +0.0040 4.0%
Total Expected Return: 11.10%
Total Systematic Risk: 9.20%
Risk Assessment: Above-average risk
Visualizations
Expected Returns Comparison
This chart compares the expected returns calculated using the Three-Factor and Five-Factor models for different asset types. It also shows the risk-free rate and market return as reference points.
Factor Contributions
These pie charts show the relative contribution of each factor to the total systematic risk for a selected asset, comparing the Three-Factor and Five-Factor models.
Factor Betas
This grouped bar chart displays the factor betas for different assets, allowing for comparison of factor exposures across the investment universe.
Theoretical Background
The Fama-French factor models were developed by Eugene Fama and Kenneth French to explain stock returns better than the single-factor CAPM.
Practical Applications
The Fama-French factor models are used for:
- Asset Pricing: Determining the required rate of return for risky assets
- Performance Evaluation: Assessing investment performance relative to multiple risk factors
- Portfolio Construction: Building portfolios with targeted factor exposures
- Style Analysis: Identifying investment styles and factor tilts
- Risk Decomposition: Breaking down portfolio risk into factor components
- Alpha Generation: Identifying mispriced securities relative to their factor exposures
Limitations
The Fama-French factor models have several limitations:
- Data Requirements: Require extensive data for factor construction and beta estimation
- Parameter Instability: Factor betas and premiums may vary over time
- Factor Selection: The choice of factors is somewhat arbitrary
- Multicollinearity: Factors may be correlated with each other
- Implementation Costs: Factor portfolios may be expensive to implement in practice
- Regional Differences: Factor premiums may vary across different markets
Extensions
Several extensions to the Fama-French models address their limitations:
- Momentum Factor: Adding a momentum factor (UMD - Up Minus Down)
- Quality Factor: Incorporating a quality factor beyond profitability
- Low Volatility Factor: Adding a low volatility anomaly factor
- Conditional Models: Allowing for time-varying factor loadings and premiums
- Industry-Adjusted Factors: Controlling for industry effects in factor construction
- Macroeconomic Factors: Incorporating macroeconomic variables as additional factors