Fama-French Three-Factor Model

Core components of the model

The Fama-French Three-Factor Model expresses expected returns using three key factors:

1. Market factor (excess return) - Similar to CAPM
2. Size factor (SMB - Small Minus Big)
3. Value factor (HML - High Minus Low)

The mathematical expression is:

$R_i - R_f = \alpha_i + \beta_i(R_m - R_f) + s_i\text{SMB} + h_i\text{HML} + \epsilon_i$

Where:

• $R_i$ = Return of investment i
• $R_f$ = Risk-free rate
• $R_m$ = Market return
• SMB = Size premium (Small Minus Big)
• HML = Value premium (High Minus Low)
• $\beta_i, s_i, h_i$ = Factor sensitivities
• $\alpha_i$ = Excess return
• $\epsilon_i$ = Error term

Factor definitions and construction

Market factor

The market factor represents the excess return of the market portfolio over the risk-free rate. This is identical to the market risk premium in CAPM.

Size factor (SMB)

SMB captures the historical observation that smaller companies tend to outperform larger ones:

1. Sort stocks by market capitalization
2. Calculate the difference between:
   • Average return of small-cap portfolios
   • Average return of large-cap portfolios

Value factor (HML)

HML reflects the tendency of value stocks to outperform growth stocks:

1. Sort stocks by book-to-market ratio
2. Calculate the difference between:
   • Average return of high book-to-market portfolios
   • Average return of low book-to-market portfolios

Applications in portfolio management

Risk factor exposure analysis

The model helps portfolio managers:

• Decompose returns into factor contributions
• Identify systematic biases in investment strategies
• Adjust portfolio exposures to target specific factors

Performance attribution

Managers can attribute portfolio performance to:

• Market movements
• Size effects
• Value effects
• Stock-specific returns (alpha)

Model limitations and extensions

Key limitations

1. Assumes linear relationships between factors
2. May not capture all relevant risk factors
3. Factor premiums can vary over time
4. Model parameters are estimated using historical data

Modern extensions

1. Carhart Four-Factor Model (adds momentum)
2. Fama-French Five-Factor Model (adds profitability and investment)
3. Various industry and region-specific adaptations

Implementation considerations

Data requirements

• Historical price data
• Market capitalization figures
• Book-to-market ratios
• Risk-free rate history
• Market index returns

Statistical methodology

1. Calculate factor returns
2. Estimate factor loadings using regression
3. Test for statistical significance
4. Monitor factor stability over time

Portfolio construction

The model informs:

• Factor targeting strategies
• Risk management approaches
• Portfolio optimization techniques
• Performance benchmarking

Practical applications in trading

Strategy development

1. Factor rotation strategies
2. Smart beta product design
3. Risk-adjusted portfolio optimization
4. Systematic trading approaches

Risk management

• Factor exposure monitoring
• Portfolio rebalancing triggers
• Risk decomposition analysis
• Stress testing scenarios

Model validation and testing

Backtesting considerations

1. Out-of-sample testing
2. Transaction costs
3. Market impact
4. Regime changes

Performance metrics

• Sharpe Ratio analysis
• Jensen's Alpha calculation
• Information ratio
• Factor exposure stability

Modern adaptations and extensions

Machine learning enhancements

1. Non-linear factor relationships
2. Dynamic factor weightings
3. Alternative data integration
4. Adaptive parameter estimation

ESG integration

• Environmental factor overlays
• Social responsibility metrics
• Governance quality indicators
• Sustainability scoring