Module 10: Empirical Tests
IF440 - Capital Markets and Uncertainty
Ranik Raaen Wahlstrøm
Overview
• CAPM assumes only one thing matters when it comes
to determining why different securities, offer different
expected rates of return – beta!
– Nearly impossible to conduct definitive test of CAPM given
reliance on unobservable market portfolio
• Recent research suggests the following:
– Risk-return relationship might entail multiple sources of
systematic risk
– Non-risk-related considerations could lead to substantial
variation in expected returns
– Behavior-finance issues can lead to security mispricing
2
Testing CAPM
• Early simple test followed three steps:
1. Setting up sample data
2. Estimating the security characteristic line (SCL)
3. Estimating the security market line (SML)
3
Testing CAPM: (1) Setting up sample data
• Decide on:
–
–
–
–
–
a sample period (e.g., 10 years, 2011 - 2021)
a holding period (e.g., monthly)
a sampling size (e.g., 1000 stocks)
a proxy for 𝑀 (e.g., DJIA)
a proxy for 𝑟𝑓 (e.g., T-bills)
• The example choices above gives us:
–
–
–
–
–
𝑡 = {1,2, … , 120}
𝑖 = {1,2, … , 1000}
𝑟𝑖𝑡 = {1,2, … , 120,000}
𝑟𝑀𝑡 = {1,2, … , 120}
𝑟𝑓𝑡 = {1,2, … , 120}
4
Testing CAPM: (2) Estimating the security
characteristic line (SCL)
First-pass regression:
• Use regression to solve for each stock 𝑖:
𝑟𝑖,𝑡 − 𝑟𝑓,𝑡 = 𝑎𝑖 + 𝑏𝑖 𝑟𝑀𝑡 − 𝑟𝑓𝑡 + 𝑒𝑖,𝑡
Will be fitted for 1000 stocks, each over 120 months, to obtain:
• 𝑏𝑖 – estimate for the 𝛽𝑖
• 𝜎 2 𝑒𝑖
– estimate for nonsystematic risk for each stock
• 𝑟𝑖 − 𝑟𝑓
– sample averages for each stock, over 120 months
• 𝑟𝑀 − 𝑟𝑓
– sample average over 120 months
5
Testing CAPM: (3) Estimating the security
market line (SML)
Second-pass regression:
• Use regression to solve:
𝑟𝑖 − 𝑟𝑓 = 𝛾0 + 𝛾1 𝑏𝑖 + 𝛾2 𝜎 2 𝑒𝑖
Will be fitted over 1000 observations, one for each stock.
If CAPM is true 𝐸 𝑟𝑖 − 𝑟𝑓 = 𝛼𝑖 + 𝛽𝑖 𝐸 𝑟𝑀 − 𝑟𝑓 , we would expect:
• 𝛾0 = 0
• 𝛾1 = 𝑟𝑀 − 𝑟𝑓
• 𝛾2 = 0
– Non-systematic risk should not be priced
6
Testing CAPM
• Early tests (John Lintner, 1965, Merton Miller and Myron Scholes,
1972) was not promising.
𝛾0 = 12.7%
𝛾1 = 4.2%
𝛾2 = 31.0%
𝜎𝛾0 = 0.6%
𝜎𝛾1 = 0.6%
𝜎𝛾2 = 2.6%
𝑟𝑀 − 𝑟𝑓 = 16.5%
• Highlights:
– SML is “too flat” and intercept is “too large”
– 𝛾0 − 0 = 𝟐𝟏𝜎𝛾0
–
𝛾1 − 𝑟𝑀 − 𝑟𝑓 = 𝟐𝟏𝜎𝛾1
–
𝛾2 − 0 = 𝟏𝟐𝜎𝛾2
7
Exercises solved with code
1. Perform the first-pass regressions
for a single-index model and
tabulate the summary statistics.
2. Specify the hypotheses for the
second-pass regression used to
test the SML and the CAPM.
3. Perform the second-pass SML
regression by regressing the
average excess return of each
portfolio on its beta.
4. Summarize your test results and
compare them to the results
reported in the text.
Code and data available at:
https://github.com/ranikrw/IF440-Empirical-tests
Note: coding is not part of the syllabus in this
course. You will not be asked to either write or
interpret code on the exam.
8
Difficulties with approach employed
• Stock returns are extremely volatile, lessening the
precision of any tests of average return
• Fundamental concerns about the validity of the tests
1. The proxy for 𝑀 (for example DJIA) is not the real 𝑀
2. Betas from first-stage are estimated with sampling error
3. Investors cannot borrow at 𝑟𝑓
9
Concerns when testing CAPM:
(1) The proxy for 𝑴 is not the real 𝑴
• Roll's Critique (Richard Roll, 1977):
– The only testable hypothesis for testing CAPM:
The market portfolio is mean-variance efficient
– CAPM cannot be tested, because one cannot observe 𝑀
• Benchmark error: incorrect proxy for 𝑀 → incorrect testing of CAPM
• 𝑀 might be mean-variance efficient, even if a proxy is not
– Leads us to believe the CAPM is not true, even if it is
• A proxy might be mean-variance efficient, even if 𝑀 is not
– Leads us to believe the CAPM is true, even if it is not
10
Concerns when testing CAPM:
(2) Large sampling errors in the first regression
• Recall the first regression:
𝑟𝑖,𝑡 − 𝑟𝑓,𝑡 = 𝑎𝑖 + 𝑏𝑖 𝑟𝑀𝑡 − 𝑟𝑓𝑡 + 𝑒𝑖,𝑡
• Such regression (linear with error term on the right) gives
a downward bias slope 𝑏𝑖 and upward bias intercept 𝑎𝑖
– Showed (early 1970’s): estimated the SCL using random
numbers with averages exactly agreeing with CAPM
– helps us explain the poor empirical results for CAPM, for early
tests
11
Concerns when testing CAPM:
(2) Large sampling errors in the first regression
• Solution - Using portfolios instead of single stocks
→ diversifying away most firm-specific risk 𝑒𝑖𝑡
– New Problem 1: less observations for the second regression
– New Problem 2: likely very small betas
– Still not much results in favor for the CAPM
12
Using portfolios instead of single stocks:
Fama and MacBeth (1973) Procedure
• Step 1: Portfolio formation – for the period 1926-1929
– In order to obtain portfolios with different betas we first compute
betas of individual stocks 𝑖
Cov(𝑟𝑖 , 𝑟𝑀 )
𝛽𝑖 =
𝜎 2 𝑟𝑀
– 𝑟𝑖 = observed returns for each security 𝑖
– 𝑟𝑀 = the market portfolio
– Next, form 20 portfolios based on the sorted 𝛽𝑖
13
Using portfolios instead of single stocks:
Fama and MacBeth (1973) Procedure
• Step 2: Initial Estimation Period (1930-1934)
– For a later time period, recompute the 𝛽𝑖 for each security
– These are averaged across securities within portfolios to obtain
20 initial portfolio 𝛽𝑝
– For the sample, estimate the variance of residuals 𝜎 2 (𝜀𝑖 ) for
each security 𝑖:
𝑟𝑖𝑡 = 𝑎𝑖 + 𝑏𝑖 𝑟𝑀𝑡 + 𝜀𝑖𝑡
– Get the portfolio's variance 𝜎 2 (𝜀𝑝 ) by averaging over the
corresponding securities’ variance 𝜎 2 (𝜀𝑖 )
14
Using portfolios instead of single stocks:
Fama and MacBeth (1973) Procedure
• Step 3: Testing period (1935-1968)
– For each month 𝑡 per year, perform cross-section regression:
𝑟𝑝𝑡 = 𝛾0 + 𝛾1 𝛽𝑝 + 𝛾2 𝛽𝑝2 + 𝛾3 𝜎 2 𝜀𝑝 + 𝜀𝑝𝑡
– 𝑟𝑝𝑡 is the dependent variable of the 20 portfolio returns
– 𝛽𝑝 represents the 20 portfolios betas as explanatory variable
– For each month in 1935 we use the portfolio betas from Step 2
• They are however updated yearly (starting at Jan 1936)
– Calculate the time series averages of 𝛾0 and 𝛾1
– The CAPM predicts: 𝛾0 = 𝑟ഥ𝑓
𝛾1 = 𝑟𝑀 − 𝑟ഥ𝑓
𝛾2 = 𝛾3 = 0
15
Using portfolios instead of single stocks:
Fama and MacBeth (1973) Procedure
(all rates in basis points per month)
Period
Average γ0
t-statistic (testing γ0 = 0)
Average rM − rf
Average γ1
t-statistic (testing γ1 = rM − rf )
Average γ2
t-statistic (testing γ2 = 0)
Average γ3
t-statistic (testing γ3 = 0)
Average R-square
1935/6 to
1968
8
0.20
130
114
1.85
−26
−0.86
516
1.11
0.31
1935 to
1945
10
0.11
195
118
0.94
−9
−0.14
817
0.94
0.31
1946 to
1955
8
0.20
103
209
2.39
−76
−2.16
−378
−0.67
0.32
1955/6 to
1968
5
0.10
95
34
0.34
0
0
960
1.11
16
0.29
Exercises solved with code
5. Group the nine stocks
into three portfolios,
maximizing the
dispersion of the
betas of the three
resultant portfolios.
Repeat the tests of
Exercises 1-4 and
explain any changes Code and data available at:
https://github.com/ranikrw/IF440-Empirical-tests
in the results.
Note: coding is not part of the syllabus in this course. You will
not be asked to either write or interpret code on the exam. 17
Tests of the Multifactor Models
• Three types of factors likely augment market risk factor
in a multifactor SML:
1. Factors that hedge consumption against uncertainty in prices
• For example, housing or energy
2. Factors that hedge future investment opportunities
• For example, interest rates or market volatility
3. Factors that hedge assets missing from market index
• For example, labor income or private business
18
Labour income:
Jagannathan and Wang (1996)
Second-pass regression:
𝐸(𝑅𝑖𝑡 ) = 𝑐0 + 𝑐size log(ME𝑖 ) + 𝑐vw βvw
+ 𝑐credit βcredit
+ 𝑐labor βlabor
𝑖
𝑖
𝑖
•
•
•
•
ME
vw
labor
𝑐𝑟𝑒𝑑𝑖𝑡
– market value of equity
– value-weighted stock index portfolio
– rate of change in aggregate labor income
– business cycle: spread of yields between
high- and low-grade corporate bonds
19
Labour income:
Jagannathan and Wang (1996)
credit + 𝑐
labor
𝐸(𝑅𝑖𝑡 ) = 𝑐0 + 𝑐size log(ME𝑖 ) + 𝑐vw βvw
+
𝑐
β
β
𝑖
credit 𝑖
labor 𝑖
Coefficient
Estimate
t-statistic
Estimate
t-statistic
Estimate
t-statistic
Estimate
t-statistic
𝒄𝟎
1.24
5.16
2.08
5.77
1.24
4.10
1.70
4.14
𝒄vw
−0.10
−0.28
−0.32
−0.94
−0.40
−0.88
−0.40
−1.06
𝒄credit
0.34
1.73
0.20
2.72
𝒄labor
0.22
2.31
0.10
2.09
𝒄size
R2
1.35
−0.11
−2.30
57.56
55.21
−0.07
−1.30
64.73
Findings:
• Rejection of the CAPM: negative 𝑐vw (expected average returns fall with beta)
• Human Capital is important in any version of the CAPM
20
Private (non-traded) business:
Heaton and Lucas (2000)
• Extend the equation of Jagannathan and Wang (1996) to
also include the rate of change in proprietary-business
wealth
• Find that households with higher investments in private
business reduce the fraction of total wealth invested in
equity
21
Empirical Tests of the APT:
Chen, Roll and Ross (1986)
• Identify possible variables that might proxy for
systematic factors:
IP
Growth rate in industrial production
EI
Changes in expected inflation
UI
Unexpected inflation
CG
Unexpected changes in risk premiums on bonds
GB
Unexpected changes in term premium on bonds
𝑟 = 𝑎 + 𝛽𝑀 𝑟𝑀 + 𝛽𝐼𝑃 𝐼𝑃 + 𝛽𝐸𝐼 𝐸𝐼 + 𝛽𝑈𝐼 𝑈𝐼 + 𝛽𝐶𝐺 𝐶𝐺 + 𝛽𝐺𝐵 𝐺𝐵 + 𝑒
22
Empirical Tests of the APT:
Chen, Roll and Ross (1986)
Used the Fama and MacBeth (1973) Procedure
𝑟 = 𝛾0 + 𝛾𝑀 𝛽𝑀 + 𝛾𝐼𝑃 𝛽𝐼𝑃 + 𝛾𝐸𝐼 𝛽𝐸𝐼 + 𝛾𝑈𝐼 𝛽𝑈𝐼 + 𝛾𝐶𝐺 𝛽𝐶𝐺 + 𝛾𝐺𝐵 𝛽𝐺𝐵 + 𝑒
– 𝛾j is the price of risk associated with factor 𝑗
– The APT is valid if 𝛾j is statistically significant
A
B
EWNY
IP
EI
UI
CG
GB
Constant
5.021
14.009
−0.128
−0.848
0.130
−5.017
6.409
(1.218)
(3.774)
(−1.666)
(−2.541)
(2.855)
(−1.576)
(1.848)
VWNY
IP
EI
UI
CG
GB
Constant
−2.403
11.756
−0.123
−0.795
8.274
−5.905
10.713
(−0.633)
(3.054)
(−1.600)
(−2.376)
(2.972)
(−1.879)
(2.755)
VWNY = Return on the value-weighted NYSE index; EWNY = Return on the equally weighted NYSE index
23
Exercises solved with code
Suppose that a second factor is considered.
The values of this factor for years 1 to 12 were
as follows
8. Perform the first-pass regressions as did
Chen, Roll and Ross and tabulate the
relevant summary statistics.
9. Specify the hypothesis for a test of a
second-pass regression for the two-factor
SML
10. Do the data suggest a two-factor economy?
Code and data available at: https://github.com/ranikrw/IF440-Empirical-tests
Note: coding is not part of the syllabus in this course. You will not be asked to
either write or interpret code on the exam.
Year % change in
factor value
1
-9.84
2
6.46
3
16.12
4
-16.51
5
17.82
6
-13.31
7
-3.52
8
8.43
9
8.23
10
7.06
11
-15.74
12
2.03
24
Empirical Tests of the APT:
Fama and French (1993)
Break down stocks based on book-to-market ratio and size (total assets)
Size (total assets)
Small (50%)
Book to
market
ratio
Big (50%)
High (30%)
S/H
B/H
Medium (40%)
S/M
B/M
Low (30%)
S/L
B/L
25
Empirical Tests of the APT:
Fama and French (1993)
• High book-to-market firms experience higher returns
• Smaller firms experience higher returns
𝐸 𝑟𝑖 − 𝑟𝑓 = 𝑎𝑖 + 𝑏𝑖 𝐸 𝑟𝑀 − 𝑟𝑓 + 𝑠𝑖 𝐸 𝑅𝑆𝑀𝐵 + ℎ𝑖 𝐸 𝑅𝐻𝑀𝐿
• 𝐸 𝑟𝑖 − 𝑟𝑓
– Excess return
• SMB
– Small Minus Big: Difference in returns on portfolios of
small and big stocks
– High Minus Low: Difference in returns on portfolios of
high (value) and low (growth) book-to-market stocks
• HML
26
CAPM versus FF Model
Goyal (2012): CAPM versus the Fama and French model. The figure
plots the average actual returns versus returns predicted by CAPM and
the FF model for 25 size and book-to-market double-sorted portfolios.
US stocks, 1946-2010.
27
• Firm size (SMB) and book-to-market (HML) are firm
characteristics. Why do these predict stock returns?
• Two suggested possibilities:
1. Capture some aspects of the business cycle
2. Behavioral explanations
28
Risk-Based Interpretations
• Liew and Vassalou (2000): Returns on style portfolios
(HML and SMB) seems to predict GDP growth, and thus
may capture some aspects of business cycle risk
Difference in return to
factor portfolios in year
prior to above-average
versus below-average
GDP growth. Both SMB
and HML portfolio
returns tend to be higher
in years preceding better
GDP growth.
29
Behavioral Explanations
• Value premiums is a manifestation of market irrationality
• “Glamour firms”
– Recent good performance
– High prices
– Lower book-to-market ratios
• Overreaction
– High past growth is extrapolated and then impounded in price
• Extrapolation error
– Market ignores evidence that past growth cannot be extrapolated
far into the future
30
Momentum: A Fourth Factor
• Carhart (1997) adds a momentum factor to the
FF 3-factor model:
𝐸 𝑟𝑖 − 𝑟𝑓 = 𝑎𝑖 + 𝑏𝑖 𝐸 𝑟𝑀 − 𝑟𝑓 + 𝑠𝑖 𝐸 𝑅𝑆𝑀𝐵 + ℎ𝑖 𝐸 𝑅𝐻𝑀𝐿 + 𝑤𝑖 𝐸 𝑅𝑊𝑀𝐿
• WML: Winners Minus Loosers
• Motivation:
– Empirical evidence shows that stocks that have been doing well in
the recent past (about 12 months) continue to do so
– Stocks that have been doing poorly in the recent past also continue
31
to do so
Characteristics versus Factor Sensitivities
These are factors that have joined as additions to
the CAPM:
– Size
– Asset growth
– Book-to-market
– Dividend yield
– Momentum
– Volatility
– Stock issues
– Turnover
– Accruals
– Sales
– Profitability
Liquidity and Asset Pricing
• Liquidity involves the following:
–
–
–
–
–
Trading costs
Ease of sale
Necessary price concessions to effect a quick transaction
Market depth
Price predictability
33
Liquidity and Asset Pricing
• Pástor and Stambaugh (2003) look for evidence of
price reversals, especially following large trades
– If price reversed on the following day, then part of the original
price change the previous day was due to the trades
– That is, concession on the part of trade initiators who needed to
offer higher purchase prices (or accept lower selling prices) to
complete trade in a timely manner
34
Liquidity and Asset Pricing
• Pástor and Stambaugh (2003) used models that ignore
liquidity (CAPM and FF)
– Find the same if also including momentum
35
The Equity Premium Puzzle
• Recall from the derivation of the CAPM that
the equilibrium risk premium is given by:
2
ҧ 𝑀
𝐸 𝑅𝑀 = 𝐸 𝑟𝑀 − 𝑟𝑓 = 𝐴𝜎
– 𝐴ҧ is the average investor risk aversion
Historically, market risk premiums on risky assets in the
U.S. are too large to be consistent with economic theory
• unless we assume implausibly high levels of risk aversion
36
The Equity Premium Puzzle
Possibly reason 1: Actual return > expected returns
• The Equity Premium Puzzle is not because economic
theory and suggested levels of risk aversion is wrong
• Rather, it comes from extraordinary good times after 1949
→Actual returns turned out to be higher than what the
market expected
37
The Equity Premium Puzzle
Possibly reason 2: Survivorship Bias
• The Equity Premium Puzzle comes from studying the
U.S. economy, the most successful economy after WW2
– Ignoring the evidence from stock markets that did not survive for
the full sample period
• This could not have been expected. So, when
considering expected returns beforehand, one must
have considered the whole world economy
→ then the Equity Premium Puzzle would probably
not be present.
38
The Equity Premium Puzzle
Possibly reason 3: Illiquidity
The calculations behind the Equity Premium Puzzle do not
account for illiquidity
→ Part of the Equity Premium Puzzle may come
from a compensation for illiquidity
39
The Equity Premium Puzzle
Possibly reason 4: Irrational investor behavior
• Narrow framing: Investors consider every risk in
isolation, rather than their total risk exposure
→ Investors focus on total volatility rather than the low
correlation of a stock portfolio with other
components of wealth
→ Require higher risk premiums than rational
models would predict
40
The Equity Premium Puzzle
Possibly reason 4: Irrational investor behavior
• Loss aversion: Investors
prefer avoiding losses
more than receiving gains
41