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Fama-French Three-Factor Model

Fama-French三因子模型

Overview

概述

Fama and French (1993) extended CAPM by adding two factors — size (SMB) and value (HML) — to explain cross-sectional variation in stock returns that CAPM alone cannot capture. The model shows that small-cap and high book-to-market stocks earn systematic premiums.
Fama和French（1993）在CAPM基础上进行扩展，新增了规模因子（SMB）和价值因子（HML）两个因子，用以解释CAPM无法覆盖的股票收益截面差异。该模型表明，小盘股和高市净率股票能够获得系统性溢价。

When to Use

适用场景

  • Explaining why CAPM alpha is nonzero for certain portfolios
  • Evaluating fund manager skill after controlling for factor exposures
  • Constructing factor-tilted portfolios
  • Academic research on asset pricing anomalies
  • 解释为何某些投资组合的CAPM阿尔法不为零
  • 在控制因子暴露后评估基金经理的能力
  • 构建因子倾斜型投资组合
  • 资产定价异象的学术研究

When NOT to Use

不适用场景

  • For fixed income or derivatives pricing (equity-focused factors)
  • When factor data is unavailable for the market in question
  • As a complete model — profitability and investment factors may also matter (five-factor)
  • 固定收益或衍生品定价（该模型聚焦股票类因子）
  • 目标市场的因子数据不可用时
  • 作为完整模型使用——盈利因子和投资因子也可能有影响（可参考五因子模型）

Assumptions

假设条件

IRON LAW: Single-factor models (CAPM) underestimate expected returns
for small-cap and value stocks. Size and value represent systematic
risk factors that command their own premia.
Key assumptions:
  1. SMB and HML capture systematic risk, not mispricing
  2. Factor premia are persistent across time periods and markets
  3. Factors are constructed from observable, rebalanced portfolios
IRON LAW: 单因子模型（如CAPM）会低估小盘股和价值股的预期收益。规模和价值代表了可获得自身溢价的系统性风险因子。
核心假设：
  1. SMB和HML代表系统性风险，而非定价偏差
  2. 因子溢价在不同时间段和市场中具有持续性
  3. 因子由可观测、定期再平衡的投资组合构建而成

Methodology

方法步骤

Step 1 — Obtain Factor Data

步骤1——获取因子数据

  • Rm-Rf: market excess return
  • SMB (Small Minus Big): return of small-cap portfolio minus large-cap portfolio
  • HML (High Minus Low): return of high B/M portfolio minus low B/M portfolio
  • Rm-Rf：市场超额收益
  • SMB（Small Minus Big）：小盘股投资组合收益减去大盘股投资组合收益
  • HML（High Minus Low）：高市净率投资组合收益减去低市净率投资组合收益

Step 2 — Run Time-Series Regression

步骤2——运行时间序列回归

Ri - Rf = ai + bi(Rm-Rf) + si(SMB) + hi(HML) + ei. See 
references/
for construction details.
Ri - Rf = ai + bi(Rm-Rf) + si(SMB) + hi(HML) + ei。构建细节请参考
references/
目录。

Step 3 — Interpret Factor Loadings

步骤3——解读因子载荷

  • bi: market sensitivity (same as CAPM beta)
  • si: size exposure (positive = small-cap tilt)
  • hi: value exposure (positive = value tilt, negative = growth tilt)
  • bi：市场敏感度（与CAPM贝塔一致）
  • si：规模暴露（为正表示投资组合倾向小盘股）
  • hi：价值暴露（为正表示倾向价值股，为负表示倾向成长股）

Step 4 — Evaluate Alpha

步骤4——评估阿尔法（Alpha）

If alpha (ai) is statistically insignificant, returns are explained by factor exposures — no manager skill.
若阿尔法（ai）在统计上不显著，则收益可由因子暴露解释，说明基金经理无超额能力。

Output Format

输出格式

markdown
undefined
markdown
undefined

Fama-French Analysis: [Fund / Portfolio]

Fama-French分析：[基金/投资组合]

Regression Results

回归结果

FactorLoadingt-statInterpretation
Market (Rm-Rf)x.xxx.xx[market exposure]
SMBx.xxx.xx[size tilt]
HMLx.xxx.xx[value tilt]
Alphax.xx%x.xx[skill or luck]
因子载荷t统计量解读
市场（Rm-Rf）x.xxx.xx[市场暴露情况]
SMBx.xxx.xx[规模倾斜情况]
HMLx.xxx.xx[价值倾斜情况]
阿尔法x.xx%x.xx[能力或运气]

R-squared

拟合优度（R-squared）

  • Three-factor R2: x% vs CAPM R2: x%
  • 三因子模型R²：x% vs CAPM模型R²：x%

Conclusions

结论

  • [Factor attribution summary]
  • [Manager skill assessment]
undefined
  • [因子归因总结]
  • [基金经理能力评估]
undefined

Gotchas

注意事项

  • Factor premia vary across countries and time periods — not guaranteed to persist
  • HML has weakened post-publication; some attribute this to arbitrage
  • Five-factor model (2015) adds profitability (RMW) and investment (CMA) — three-factor may be insufficient
  • Factor construction methodology matters; different breakpoints yield different results
  • High R-squared does not mean the model is "correct" — it means factors explain variance
  • Debate persists whether factors represent risk or mispricing
  • 因子溢价会因国家和时间段不同而变化，无法保证持续存在
  • HML在模型发表后效应有所减弱；部分观点认为这是套利行为导致的
  • 2015年提出的五因子模型新增了盈利因子（RMW）和投资因子（CMA）——三因子模型可能不够全面
  • 因子构建方法会影响结果；不同的分界点会产生不同的结果
  • 高R²并不代表模型“正确”——仅表示因子能够解释收益的方差
  • 关于因子代表风险还是定价偏差的争论仍在持续

References

参考文献

  • Fama, E. & French, K. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
  • Fama, E. & French, K. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1), 1-22.
  • Fama, E. & French, K. (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427-465.
  • Fama, E. & French, K. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
  • Fama, E. & French, K. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1), 1-22.
  • Fama, E. & French, K. (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427-465.