# Risk, Return, and Equilibrium Empirical Tests

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, $E(R_i) = E(R_0) + \beta[E(R_m) - E(R_0)]$, 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.