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The Cross Section of Expected Returns

Fama and French, 1992

I’ve had to go and read Fama (1972) first. Will return and complete these notes afterwards.


Size and P/B account for the variation in stock returns.

Bottom line result

Beta does not explain returns. Size and P/B explain returns, and absorb the effects of leverage and P/E for 1963-1990.


Central prediction of CAPM is that market portfolio is mean-variance efficient. This implies:

  • Expected returns are linearly related to betas
  • Market betas are enough to explain variation in returns

However, fails to explain:

  • The size effect (smaller caps return more than their betas would suggest).
  • Also fails to capture leverage effect - higher leverage get higher returns than their beta + size predictions.
  • Also fails to account for higher returns of low P/B stocks
  • P/E also explains some additional variation in beta + size models

So CAPM is not sufficient to explain variation in returns. One theory is that each of these (beta, leverage, P/B, P/E, size) may be different ways to scale the price; that is, they may be redundant information.

SO: Which ones should we keep, which should we drop?

What is P/B? Two possibilities. (1) Distress factor (2) Higher return of low P/B is just irrational pricing correcting itself.


Their theory is that, if assets are priced rationally, risks are multidimensional, and that one dimension is proxies by size, another by by P/E.

Stock Universe

  • Used NYSE, AMEX and NASDAQ data from CRSP and COMPUSTAT for fundamentals.
  • Excluded financial sticks because leverage for them is higher than others.
  • COMPUSTAT pre-1962 has serious survivorship bias problems
  • Accounting Data Lags

  • To avoid look-ahead bias they lag annual financials until June next year. So a September 1965 financial report will be used from June 1966 until May 1967. SEC requires companies to file within 90 days, but 20% duo not comply, which is why they are conservative.

  • Market equity at December t-1 is used to compute ratios for year t-1; market equity for June of year t is used for size calc.

  • Data required for inclusion: return data for 24 of last 60 months, Total Assets, BV Equity, Earnings for previous fiscal year.

  • Using December market cap is objectionable; rather use market cap at time of financial statement. BUT then ratios reflect general cross section of ratios across the year: P/B may be generally lower in June because of, say, an oil price scare or whatever. So rather use December market cap. BUT they tested both and it has little impact. ** Note: if you can’t decide, just do both.

  • Also, different fiscal year ends mean that some info is more ‘stale’ than others - this may have an effect. BUT they tested with just December year-end stocks will little effect. ** Same thing: run the other possible test and see if there is a difference.

  • Breakpoints are the ways that the stocks are bucketed into different populations. For this paper, NYSE stocks are ranked by size at the end of June and then bucketed into 10 sizes. Then they put all the AMEX and NASDAQ stocks in these buckets as well.
  • Why only NYSE used? Because otherwise most buckets would have only small stocks (size is not uniformly distributed on NASDAQ and AMEX).
  • Why bucket by size? Because Chan and Chen (1988) find size produces a wide spread of returns and betas.
  • Problem: size and beta are tightly correlated, so cannot separate beta from size effect.
  • To deal with this, they divide each decile into 10 more groups by beta rankings. Betas are estimated on 24 of the last 60 monthly returns. Beta breakpoints are also only computed for NYSE stocks. SO there are 10x10 = 100 portfolios.


  • They equal-weight returns in each size+beta portfolio.
  • They compute annual returns beginning July -> end June.

Portfolio Betas

  • Asset pricing tests are cross-sectional regression (see Fama 1972 - double sort). Each month cross section of returns is regressed on variavbles hypothesized to explain expected returns. ** GO READ THE PAPER **
  • Once they have the 100 portfolios, they calculate each portfolio’s monthly return and then compute each portfolio’s beta relative to the market, where market return is defined as the value weighted return of all NYSE, AMEX and NASDAQ stocks in CRSP.
  • Betas are calculated as sum of slopes of regression on return on a portfolio on the current and prior months return. Sum betas adjust for asynchronous trading.
  • THe range of beta-divided size portfolios shows significantly larger range than simple size-bucketed betas. That is, the finer grain shows increased dispersion as you drill down.
  • In each size decile the post-ranking beta are close to the pre-ranking betas.
  • In for each size decile, the beta sub portfolios do not sort by size; that is, betas don’t sort by size naturally in the deciles. It produces variation in returns that is not related to size.

Beta and size

  • when sorted by size alone, there appears to be a relation between beta and return (ie risk and return). BUT since size and betas are correlated, it’s unclear whether this is because of size effects or beta.
  • When subdividing each decile into betas, there is no relation to return and beta.

Historical Factoids

Relationship between beta and return holds pre-1969. 1969-1990 is does not hold. In 1941-1990 the relation between beta and average return does not hold.

BUT relation between average return and above ratios (size, P/B, P/E, leverage) do hold.

Flagged for further reading

CAPM foundational papers: Sharpe 1964, Lintner 1965, Black 1972 - referred to the SLB model in the paper On using size breakpoints: Chan and Chen 1988 Cross-Sectional Regression: Fama and Macbeth, 1973