CrossSectional Properties of Returns

1
Cross-Sectional Properties of Returns

• Part 1
• Properties of the Cross-Section

2
Outline

• Relationships among returns
• Size, Book-to-Market
• The CAPM
• Estimating betas
• CAPM Performance
• Multi-factor models and the APT
• Estimating factor loadings
• Fama-French 3-factor model performance
• Conditional models
• Using predictive variables to estimate
  conditional betas.

3
The Cross-Section of Returns

• Expected returns vary firm-by-firm.
• There are at least two potentially competing
  explanations for these differences
• Differences in expected returns reflect
  differences in the riskiness of the returns.
• Differences in expected returns are due to
  behavioral biases.
• Well focus on the first explanation.

4
Simple Returns and Portfolios

• When working with cross-sectional models, it is
  more straightforward to work with simple returns
• Portfolio weighted averages of simple returns are
  simple returns (not true for compound returns).

5
Size and B/M Portfolios

• Fama and French (1992) applied the following
  procedure
• In June of each year, sort NYSE firms according
  to their market capitalization. Select nine
  breakpoints that divide the stocks into deciles.
  Assign all stocks (NYSE, AMEX, NASDAQ) to a
  portfolio according to market cap.
• Sort NYSE stocks according to their December
  book-to-market ratios. Select nine breakpoints.
  Assign all stocks according to their B/M ratios.
• Define 100 portfolios that differ according to
  size and book-to-market ratios.

6
Cross-Sectional Properties of Returns

• Part 2
• The CAPM

7
Explanations for Cross-sectional Differences

• Fama and French found that these characteristics
  are correlated with expected returns.
• Size (market capitalization)
• Smaller firms have higher expected returns.
• Book/Market ratio (book value/mkt value of
  equity)
• Value firms (high B/M) have higher returns than
  growth firms (low B/M).

8
Explanation 1 The CAPM

• According to the CAPM, the returns should differ
  only if their market risks differ
• Market Model
• CAPM

9
Estimating the Market Model

• To estimate the market model, we run an OLS
  regression of excess stock returns (or portfolio
  returns) on the excess market return using
  time-series data.

10
Does the CAPM Explain Returns?

• If the CAPM explains the cross-sectional
  differences in returns, the Betas should explain
  the expected returns

11
What FF Found

• Fama and French found that beta does a poor job
  at explaining expected returns.
• Small growth stocks lie below the security market
  line.
• Small value stocks lie above the security market
  line.

12
Why does the CAPM Fail?

• There are many reasons why the CAPM might not do
  a good job at explaining expected returns
• Returns are not related to risk.
• Returns are related to non-market risk.
• This is a bad way to test the CAPM when firms
  betas change over time.

13
Cross-Sectional Properties of Returns

• Part 3
• The APT

14
Multi-Factor Models and the APT

• The FF 3-factor model
• Abnormal returns on size/BM portfolios are
  related to two additional missing non-market risk
  factors
• A size factor (SMB)
• A growth factor (HML)

15
SMB and HML

• The SMB (Small Minus Big) factor is formed by
  creating a long position in small-cap stocks and
  an offsetting short position in large-cap stocks.
• The HML (High Minus Low) factor is formed by
  creating a long position in high B/M stocks
  (Value stocks) and an offsetting short position
  in low B/M stocks (Growth Stocks).

16
The FF 3-factor APT

• According to this explanation, the expected
  returns on all assets should be related to
  market, SMB and HML factor loadings and to the
  risk premia on those factors

17
Cross-Sectional Properties of Returns

• Part 4
• Conditional Models

18
Conditional Models

• An alternative explanation for the failure of the
  CAPM is that
• Betas change over time,
• Risk premia change over time.
• Solution estimate a time-varying beta

19
A Simple Conditional Model

• Weve already seen that dividend yields can be
  used to predict risk premia.
• A simple conditional beta model, based on this
  evidence, has betas linear in dividend yields

20
Estimating a Conditional Model

• This conditional model can be estimated using OLS

21
Conclusion

• The CAPM does not do a good job at explaining
  cross-sectional dispersion in expected returns.
• The failure of the CAPM may be due to
• Behavioral biases among investors,
• Missing priced factors,
• Time-varying risk premia and factor loadings.
• The FF 3-factor model does reasonably at
  explaining dispersion in risk premia and is
  straightforward to apply.