CAPM anomalies and multi-factor models
Sebastien Plante
Sebastien Plante
CAPM anomalies and multi-factor models
1 / 27
Outline
Previously, we saw how to estimate βi and test CAPM:
- estimate the linear regression model
ri,t − rf ,t = αi + βi rm,t − rf ,t + εi,t
- the CAPM predicts that α̂i = 0
We identied two anomalies
1
2
beta anomaly
size anomaly
In this lecture, we:
- perform additional tests of the CAPM
- identify more empirical deviations from CAPM predictions
- discuss possible explanations
Sebastien Plante
CAPM anomalies and multi-factor models
2 / 27
Empirical test of CAPM
Recall the simple portfolio sorting strategy typically used to test
CAPM:
each year, sort stocks into groups based on some characteristics; e.g.,
rm size
2
for each group, build an equal (or value) weighted portfolio of the
stocks in that group
3
measure the realized returns of the portfolios over a long horizon
4
estimate the α̂ s of the portfolios, and test if α̂ s are statistically
dierent from 0
1
We use this approach to study additional famous CAPM anomalies
- value vs. growth
- momentum
Researchers have identied dozens of additional anomalies
Sebastien Plante
CAPM anomalies and multi-factor models
3 / 27
Value vs. growthdenition
Value rms:
- companies that are valuable because of their assets in place
- think of well established companies like Coca-Cola, ExxonMobil, and
General Motors
- much of their value comes from established business lines
- not much from their growth potential
Growth rms:
- companies that are valuable because of their growth potential
- think of companies like Tesla and Twitter
- much of their value come from expected future business lines
- not much from their existing assets
Sebastien Plante
CAPM anomalies and multi-factor models
4 / 27
Identifying value vs. growth companies
In the data, we identify growth vs. value rms by looking at their
book-to-market ratio (B/M)
B
Book value of physical assets
=
M
Market value of nancial assets
- growth rms have low B/M
- value rms have high B/M
Are there variations in expected returns between growth and value
rms?
If so, can CAPM explain the variations?
Sebastien Plante
CAPM anomalies and multi-factor models
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The value vs. growth anomaly
Results for equal weighted B/M portfolios (1964-2021):
Growth stocks appear riskier and have lower expected returns
Sebastien Plante
CAPM anomalies and multi-factor models
6 / 27
Size and B/M anomalies explanations
Why is the CAPM failing here?
- the truth is, we don't have a denitive answer
Rational explanation:
- small and value stocks are exposed to additional risk factors beyond
market risk
Behavioral explanation:
- salience: investors overlook small rms, which drives prices below
fundamental value and imply higher returns in the future
- overoptimism: investors are overoptimistic about the prospect of
growth stocks, bid the price of those stocks up, which leads to
underperformance in the long run
As with all the anomalies studied in this lecture, the jury is still out as
to the right explanation
Sebastien Plante
CAPM anomalies and multi-factor models
7 / 27
Price based anomalies
Next, we study an anomaly related to prior returns
- momentum
We'll see that, in some way, this price based anomalies is even more
puzzling than the size and the B/M anomalies
Sebastien Plante
CAPM anomalies and multi-factor models
8 / 27
Momentum denition
Rank stocks based on their past market return over the last year
(exclude last month realized return)
- stocks with high realized return in the previous year are said to have
high momentum
- stocks with low realized return in the previous year are said to have low
momentum
In other words,
- high momentum stocks are last year's winners
- low momentum stocks are last year's losers
Are there variations in expected returns between high and low
momentum stocks?
If so, can CAPM explain the variations?
Sebastien Plante
CAPM anomalies and multi-factor models
9 / 27
The momentum anomaly
Results for equal weighted momentum portfolios (1964-2021):
High momentum stocks have positive alpha
Sebastien Plante
CAPM anomalies and multi-factor models
10 / 27
Price based anomalies explanations
Why is the CAPM failing?
- again, we do have a denitive answer
Rational explanations:
- exposure to missing risk factors
- but it is not clear why a stock's risk exposure be greater or lower based
on previous return
- the case is easier to make for size and B/M anomalies which are
related to fundamental characteristics of the rm
Behavioral explanations:
- focus on investors' overreaction or underreaction to new information
Sebastien Plante
CAPM anomalies and multi-factor models
11 / 27
Key takeaways
The data reveal several empirical deviations from CAPM
Stocks with the following characteristics are underpriced:
- low beta
- small market capitalization
- value rms
In the remainder of this lecture we explore possible risk-based
explanation of these anomalies
Sebastien Plante
CAPM anomalies and multi-factor models
12 / 27
CAPM anomaly interpretations
There are at least three possible explanations for CAPM anomalies:
1
We didn't really test CAPM
- our empirical tests are awed
2
CAPM is not the right model of risk
- if we knew the right model, there would be no mispricing
- i.e., if we had the right model, the α s would be 0
3
The market is inecient
- some assets are genuinely undervalued/overvalued
- clever investors can identify mispriced securities and prot
We elaborate on these ideas below
Sebastien Plante
CAPM anomalies and multi-factor models
13 / 27
Anomaly explanationswe didn't really test CAPM
There are several issues with our empirical tests:
1
Beta coecients are not observed
- if betas change over time, simple historical estimates of betas are not
likely to be accurate
- there is indeed evidence that betas change over time
2
Expected returns are not observed
- average returns do not need to match expected returns
- it can take many years of data to obtain accurate expected return
estimates
3
The US stock market is not the accurate proxy for the market
portfolio
- US stock market captures less than 50% of securities traded in the US
- using the wrong proxy for market return might be the reason behind
the empirical failures of the CAPM
Sebastien Plante
CAPM anomalies and multi-factor models
14 / 27
Anomaly explanationsCAPM is not the right model of risk
CAPM states that only market risk matters for investors
But it is possible that investors also care about other sources of risk
Maybe the right model for expected returns looks like
r¯i − rf = βm,i (¯
rm − rf ) +βF ,i F̄
where
- F̄ is the premium on an additional risk factor
- βF ,i measures the exposure of asset i to that risk factor
If this is true, the α̂i measured in a CAPM regression may reect the
asset's exposure βF ,i to the additional risk factor F
- see Python workbook "Multifactor_estimation.ipynb" for example
Sebastien Plante
CAPM anomalies and multi-factor models
15 / 27
Anomaly explanationsCAPM is not the right model of risk
small firms more exposed to
other risk factors, higher return
What is this other risk factor F ?
- remains elusive
- still an area of active research
- challenging to identify additional risk factors from rst principle
- instead, we rely on ad hoc proxies
Ad hoc proxies:
- typically originate from CAPM asset pricing anomalies
- are not grounded in economic theory
- evaluated on their ability to eliminate alpha in asset pricing regressions
Sebastien Plante
CAPM anomalies and multi-factor models
16 / 27
Anomaly explanationsCAPM is not the right model of risk
An example of proxy: the small minus big (SMB) factor
The idea behind the proxy is as follows
- portfolios of small rms have larger CAPM α than portfolios of big
rms because they are more exposed to the additional risk factor
- in other words, the returns of small rms are more strongly correlated
with the additional risk factor than the returns of big rms
- as a result, the dierence between the return on portfolios of small
rms and big rms can be used as a proxy for the actual (but
unknown) risk factor
Two-factor model:
r¯i − rf =βm,i (¯
rm − rf ) +βSMB,i r¯small − r¯big
=βm,i (¯
rm − rf ) +βSMB,i SMB
Sebastien Plante
CAPM anomalies and multi-factor models
17 / 27
Anomaly explanationsCAPM is not the right model of risk
What if we suspect there is more than one additional risk factor?
We can extend the previous insight and obtain additional proxies
- value vs. growth portfolios
- high vs. low momentum portfolios
higher return:
more exposure to other unknown risk factors
low return than sml:
less or negative correlation with the unknown risk factor
=> capture the unkown risk factors with the proxy
the proxy is the difference of return between high and low return
Sebastien Plante
CAPM anomalies and multi-factor models
18 / 27
Anomaly explanationsCAPM is not the right model of risk
The FamaFrench three-factor model (FF3F)
r¯i − rf =βm,i (¯
rm − rf ) + βSMB,i r¯small − r¯big + βHML,i r¯value − r¯growth
=βm,i (¯
rm − rf ) + βSMB,i SMB + βHML,i HML
The Carhart four-factor model
r¯i − rf = βm,i (¯
rm − rf ) + βSMB,i r¯small − r¯big + βHML,i r¯value − r¯growth
+ βMOM,i (¯
rHmom − r¯Lmom )
= βm,i (¯
rm − rf ) + βSMB,i SMB + βHML,i HML + βMOM,i MOM
Those are the two most popular multi-factor models in nance
Sebastien Plante
CAPM anomalies and multi-factor models
19 / 27
Anomaly explanationsCAPM is not the right model of risk
How can we estimate the beta coecients and the alpha coecient of
multifactor models?
- answer: multivariate regression
For the case of the FF3F model, estimate the regression model
ri,t − rf = αi + βm,i (rm,t − rf ) + βSMB,i SMBt + βHML,i HMLt + εi,t
If FF3F is a good risk model, then αi = 0
Obtain estimates α̂i , βˆm,i , βˆSML,i , βˆHML,i using linear regression
- test if α̂i = 0
Sebastien Plante
CAPM anomalies and multi-factor models
20 / 27
Anomaly explanationsCAPM is not the right model of risk
Let's test if the FF3F model can resolve the CAPM anomalies we have
studied
- beta
- size
- B/M
- momentum
- reversal
Strategy:
1
2
estimate α̂i , βˆm,i , βˆSMB,i , βˆHML,i from the FF3F model
calculate FF3F predicted average return
ˆr̄i = rf + βˆm,i (¯
rm − rf ) + βˆSMB,i SMB + βˆHML,i HML
3
compare ˆr̄i with the realized average return of the asset (portfolio)
Sebastien Plante
CAPM anomalies and multi-factor models
21 / 27
Beta anomalyCAPM (left) vs. FF3F (right)
FF3F provides some improvement
- more portfolios have non signicant α̂
But low beta portfolios still appear underpriced
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CAPM anomalies and multi-factor models
22 / 27
Beta and size anomalyCAPM (left) vs. FF3F (right)
Overall, FF3F brings some improvement over CAPM since realized
average returns are more aligned with predicted average returns
Sebastien Plante
CAPM anomalies and multi-factor models
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B/M anomalyCAPM (left) vs. FF3F (right)
FF3F improves upon CAPM in the sense that higher predicted returns
are associated with higher realized average returns
Yet, most FF3F α̂ s remain statistically signicant
This is particularly obvious for the extreme portfolios
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CAPM anomalies and multi-factor models
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Momentum anomalyCAPM (left) vs. FF3F (right)
Both CAPM and FF3F perform quite poorly with respect to
momentum portfolios
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CAPM anomalies and multi-factor models
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Key takeaways
In general, the FF3F improves upon the CAPM
- notable improvement for size and value
- small improvement for beta
- no clear improvement for momentum anomaly
The lack of success of FF3F for momentum portfolios led to the
creation of the Carhart 4-factor model
The big picture: multi-factor models perform better than
CAPM at explaining the average return of assets based on
exposure to risk-factors
However, multi-factor models are not completely satisfying:
- assume there are several relevant risk factors beyond market risk
- but they do not identify the economic nature of those additional factors
Sebastien Plante
CAPM anomalies and multi-factor models
26 / 27
Anomaly explanationsmarket is inecient
Another possibility to explain deviations from CAPM is that nancial
markets are inecient
- CAPM may be true, but investors make mistakes when determining the
price of assets
- perhaps because of cognitive biases
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CAPM anomalies and multi-factor models
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