beta – Gregory D. Allen

Fama, Eugene F., and James D. MacBeth, “Risk, Return, and Equilibrium Empirical Tests,” The Journal of Political Economy, Vol 81, No 3 (1973), 607-636.

Purpose: To test market efficiency, the two-period portfolio model, and the tradeoff between risk and expected return.

Motivation:

In the two-parameter portfolio model, the risk of an individual asset is proportional to its contribution to the total portfolio’s ratio of expected value to dispersion (typically, standard deviation).
- Each asset’s risk depends upon its weight in the portfolio, its covariance with other portfolio assets, and its standard deviation. The same asset, therefore, can have different risk levels in different portfolios.
- Risk-averse investors choose assets and weights to form an “efficient portfolio,” or one that maximizes expected return for any level of dispersion.

Theoretical Background:

The expected return-dispersion model, , makes three testable predictions:
- In an efficient portfolio, an asset’s relationship between expected return and risk should be linear.
- $\beta_i = \frac{cov(R_i, R_m)}{\sigma^2(R_m)}$ should measure the total risk of security i in the portfolio m.
- Higher risk should be associated with higher expected return.
In an efficient capital market, investors should form efficient portfolios that fit the model given above.

Data/Methods:

Data are NYSE monthly stock returns for 1926-1968
Market return is estimated by the equal-weighted NYSE return
Form portfolios of stocks
- estimated portfolio betas exhibit less error than the sum of individual security betas if the individual betas are not perfectly positively correlated
- to avoid bunching positive and negative errors in the portfolios, stocks are sorted into portfolios by their betas in one period, then data from a different period are used to calculate portfolio betas.
  - Use 1926-1929 data to sort NYSE stocks into 20 portfolios
  - Use 1930-1934 data to calculate portfolio betas in 1935; use 1930-1935 data to calculate portfolio betas in 1936, etc.
  - portfolios are reformed every four years
  - e.g, regressions in 1950 are on portfolios formed using 1935-1941 data but with portfolio returns calculated using 1942-1949 data.

Results/Conclusions:

The beta-return relationship is linear for all periods except the five-year post-war period 1951-1955
Beta appears to be a very complete measure of risk in all periods.
The expected return-beta relationship is positive for all periods but the five years 1956-1960, where it is slightly negative.
Findings are consistent with an efficient capital market, where risk-averse investors assemble efficient portfolios.

Fama, Eugene F. and Kenneth R. French, 1992, “The Cross-Section of Expected Stock Returns,” The Journal of Finance 47 (2), 427-465.

Purpose: This paper evaluates the joint effect of market beta, firm size, E/P ratio, leverage, and book-to-market equity in explaining the cross-section of average stock returns on NYSE, AMEX, and NASDAQ.

Findings: Beta does not explain the cross-section of average returns. Size and book-to-market equity each have explanative power both when used alone and in the presence of other variables.

Motivation: The Sharpe, Lintner, and Black asset pricing model (beta) has been very influential, but there are notable exceptions to its premises. Banz (1981) finds a significant size effect. Bhandari (1988) finds a leverage effect. Others have argued for effects of the book-to-market equity ratio and the earnings-to-price ratio. Furthermore, Reinganum (1981) and Lakonishok and Shapiro (1986) find that the beta-return relationship disappears after 1963.

Data/Methods:

Data: Nonfinancial NYSE, AMEX, and NASDAQ firms from 1962-1989
- Monthly return data from CRSP
- Annual accounting data from COMPUSTAT
Create portfolios based on size and pre-ranked beta (using trailing data)
Calculate the beta for each portfolio-year and assign it to each stock in that portfolio-year
Fama-MacBeth Regressions
- Beta-size portfolios
  - For each month, for the entire cross-section, regress average return on beta, ln(ME), ln(BE/ME), ln(A/ME), ln(A/BE), and E/P
  - Sort stocks into 10 size deciles and then into 100 sub-deciles on “pre-ranking” beta
    - pre-ranking beta is each security’s beta for the 60 months prior to portfolio creation (requiring at least 24 months of data for inclusion in any portfolio)
    - Pre-ranking beta cutoffs are established using only NYSE stocks
- Book-to-market portfolios and E/P portfolios
  - formed in a similar manner, with stocks sorted on either BE/ME or E/P
- Size & book-to-market portfolios
  - Match accounting data for fiscal year-ends in calendar year t-1 to returns for the period starting in July of year t and ending in June of year t+1.
  - Use market equity in December of year t-1 to calculate leverage, book-to-market, and E/P ratios.
  - Use market equity in June of year t to measure size.
  - sort stocks into 10 market equity deciles, then into 100 book-to-market sub-deciles.

Conclusions:

Controlling for size, there is no relationship between beta and average return
Size is significant in predicting average returns
Book-to-market equity is also significant in predicting average returns, and has an even bigger effect than size
The effects of leverage and E/P are captured by size and book-to-market equity