Risk, Return, and Equilibrium: Empirical Tests. Authors: Eugene F. Fama and James D. MacBeth (FM).
Source: The Journal of Political Economy, Vol. 81, No. 3 (May – Jun., 1973), pp. 607-636
Research question(s):
CAPM predicts that stock’s expected return depends only on its beta, in FM they test if this is true. In
this paper three implications of CAPM are tested:
- The relation between expected returns and risk is linear (C1)
- Beta is a complete measure of risk of security i in portfolio m (C2)
- Higher risk should be associated with higher expected returns (C3)
Underlying intuition
Section I of the article gives a theoretical background. The underlying intuition is that only
systematic risk is undiversifiable under the CAPM assumptions, and thus beta should capture all risk.
In the CAPM model, all investors (are rational and thus) only hold the market portfolio. Additionally,
investors are risk averse, such that all assets have the same marginal utility k. Define as: 𝐸 𝑟# −
. / *0 1*+
𝑎𝜎#' = 𝑘. If 𝜎#*+ = 0, then 𝑟- = 𝑘. Subsequently, 𝐸 𝑟' − 𝑎𝜎' = 𝑟- , such that 𝑎 = 3 . Then
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for asset i we have 𝐸 𝑟# = 𝑟- +( 𝐸 𝑟' − 𝑟- ) 3 . This implies 𝛽 = 3 .
20 20
Research Methodology & Data
To test the implications C1-C3 the following regression is used: 𝑅#; = 𝛾=; + 𝛾>; 𝛽# + 𝛾.; 𝛽#. +
𝛾?; 𝑠# + 𝜂#; . In this equation 𝑠# represents another risk measure. The following hypothesis are
formed:
• C1: 𝐸 𝛾.; = 0.
• C2: 𝐸 𝛾?; = 0.
• C3: 𝐸 𝛾>; > 0.
• 𝛾=; = 𝑟- (Sharpe-Lintner hypothesis)
To test these hypothesis, monthly returns of all NYSE-stocks from 1926 – 1968 are used. The market
portfolio is the equally weighted average of all stocks (efficient, see Black (1972)), and the risk free
rate is the 1-month treasury yield.
As opposed to doing a cross-section of individual stock’s 𝛽′𝑠, FM use portfolio 𝛽′𝑠 as they are less
noisy.
The FM method in steps:
1. Calculate 𝛽 for each security using the first 4 years of data
2. Rank all securities based on 𝛽 and create 20 portfolios
3. Re-estimate 𝛽 using the next 5 years of data and calculate averages per portfolio (𝛽D )
4. For each next month in the next 4 years update the average portfolio 𝛽 from step 3
5. For each of the next 4 years, re-estimate the average portfolio 𝛽 from step 3
6. Create a measure of non-𝛽 risk, here: idiosyncratic volatility
7. For each month in the next 4 years, estimate the following cross-sectional
.
regression: 𝑅D; = 𝛾=; + 𝛾>; 𝛽D,;1> + 𝛾.; 𝛽D,;1> + 𝛾?; 𝑠D,;1> (𝜖H ) + 𝜂D;
Source: The Journal of Political Economy, Vol. 81, No. 3 (May – Jun., 1973), pp. 607-636
Research question(s):
CAPM predicts that stock’s expected return depends only on its beta, in FM they test if this is true. In
this paper three implications of CAPM are tested:
- The relation between expected returns and risk is linear (C1)
- Beta is a complete measure of risk of security i in portfolio m (C2)
- Higher risk should be associated with higher expected returns (C3)
Underlying intuition
Section I of the article gives a theoretical background. The underlying intuition is that only
systematic risk is undiversifiable under the CAPM assumptions, and thus beta should capture all risk.
In the CAPM model, all investors (are rational and thus) only hold the market portfolio. Additionally,
investors are risk averse, such that all assets have the same marginal utility k. Define as: 𝐸 𝑟# −
. / *0 1*+
𝑎𝜎#' = 𝑘. If 𝜎#*+ = 0, then 𝑟- = 𝑘. Subsequently, 𝐸 𝑟' − 𝑎𝜎' = 𝑟- , such that 𝑎 = 3 . Then
20
270 270
for asset i we have 𝐸 𝑟# = 𝑟- +( 𝐸 𝑟' − 𝑟- ) 3 . This implies 𝛽 = 3 .
20 20
Research Methodology & Data
To test the implications C1-C3 the following regression is used: 𝑅#; = 𝛾=; + 𝛾>; 𝛽# + 𝛾.; 𝛽#. +
𝛾?; 𝑠# + 𝜂#; . In this equation 𝑠# represents another risk measure. The following hypothesis are
formed:
• C1: 𝐸 𝛾.; = 0.
• C2: 𝐸 𝛾?; = 0.
• C3: 𝐸 𝛾>; > 0.
• 𝛾=; = 𝑟- (Sharpe-Lintner hypothesis)
To test these hypothesis, monthly returns of all NYSE-stocks from 1926 – 1968 are used. The market
portfolio is the equally weighted average of all stocks (efficient, see Black (1972)), and the risk free
rate is the 1-month treasury yield.
As opposed to doing a cross-section of individual stock’s 𝛽′𝑠, FM use portfolio 𝛽′𝑠 as they are less
noisy.
The FM method in steps:
1. Calculate 𝛽 for each security using the first 4 years of data
2. Rank all securities based on 𝛽 and create 20 portfolios
3. Re-estimate 𝛽 using the next 5 years of data and calculate averages per portfolio (𝛽D )
4. For each next month in the next 4 years update the average portfolio 𝛽 from step 3
5. For each of the next 4 years, re-estimate the average portfolio 𝛽 from step 3
6. Create a measure of non-𝛽 risk, here: idiosyncratic volatility
7. For each month in the next 4 years, estimate the following cross-sectional
.
regression: 𝑅D; = 𝛾=; + 𝛾>; 𝛽D,;1> + 𝛾.; 𝛽D,;1> + 𝛾?; 𝑠D,;1> (𝜖H ) + 𝜂D;
