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Factor Models: CAPM, APT, and Fama-French
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22 min read
Statistical / ML for Quants
Factor Models
CAPM
Fama-French
Risk Premia
1
Article
2
Notebook
3
Quiz
Quick Quiz
1.
In deriving the CAPM Security Market Line, a portfolio of asset
i
i
i
and the market
M
M
M
is required to be tangent at
α
=
0
\alpha=0
α
=
0
. What is the economic rationale for evaluating at
α
=
0
\alpha=0
α
=
0
?
Because
α
=
0
\alpha=0
α
=
0
minimises tracking error to the market portfolio
M
M
M
already holds
i
i
i
, so adding more of
i
i
i
helps nothing at the optimum
Because the drift cancels at
α
=
0
\alpha=0
α
=
0
by Girsanov's theorem
Because investors hold zero weight in individual assets in equilibrium
2.
APT derives
E
[
R
i
]
=
λ
0
+
∑
k
β
i
,
k
λ
k
\mathbb{E}[R_i]=\lambda_0+\sum_k\beta_{i,k}\lambda_k
E
[
R
i
]
=
λ
0
+
∑
k
β
i
,
k
λ
k
without naming the factors. What minimal assumption does it need, and what does it NOT require that CAPM does?
Only no-arbitrage and a factor structure for returns
APT requires positive factor premia; CAPM does not
APT requires a risk-free asset; CAPM does not
APT requires normally distributed returns; CAPM does not
3.
In the Fama-MacBeth second pass,
R
i
,
t
R_{i,t}
R
i
,
t
is regressed on estimated
β
^
i
,
k
\hat\beta_{i,k}
β
^
i
,
k
. What is the premium's standard error, and why does the Shanken (1992) correction inflate it?
SE
=
std
(
γ
^
k
,
t
)
/
T
\text{SE}=\text{std}(\hat\gamma_{k,t})/\sqrt T
SE
=
std
(
γ
^
k
,
t
)
/
T
; Shanken corrects cross-sectional heteroskedasticity
SE
=
std
(
R
i
,
t
)
/
T
\text{SE}=\text{std}(R_{i,t})/\sqrt T
SE
=
std
(
R
i
,
t
)
/
T
; Shanken corrects the factor correlation structure
SE
=
1
/
N
\text{SE}=1/\sqrt N
SE
=
1/
N
(
N
N
N
assets); Shanken corrects time-series autocorrelation
SE
=
std
(
γ
^
k
,
t
)
/
T
\text{SE}=\text{std}(\hat\gamma_{k,t})/\sqrt T
SE
=
std
(
γ
^
k
,
t
)
/
T
; Shanken adjusts for the errors-in-variables bias from noisy
β
^
\hat\beta
β
^
4.
Roll's critique (1977) holds that CAPM is untestable because the true market portfolio (including human capital, real estate, private and foreign assets) is unobservable, so any test with a proxy is a joint test of CAPM and the proxy's adequacy.
True
False
5.
SMB and HML premia are unexplained by CAPM beta. Between the 'risk factor' and 'mispricing' interpretations, which is falsifiable and what evidence discriminates?
Neither is testable because asset-pricing models are always observationally equivalent
The mispricing story is falsifiable; the risk story is definitionally true since any return difference reflects some risk
Both are equally falsifiable; only international evidence can discriminate
The risk view is falsifiable via consumption-CAPM tests; mispricing is not; the global value premium supports risk
6.
A Fama-French regression gives
α
^
=
0.4
%
\hat\alpha=0.4\%
α
^
=
0.4%
/month with
t
=
1.8
t=1.8
t
=
1.8
over 48 months. Is
α
\alpha
α
significant at 5% (two-sided), and holding the information ratio fixed, what track record gives two-sided 5% significance?
Significant at 5%; ~36 months would already have sufficed
Significant at 5%; the track record is irrelevant once
t
>
1.645
t>1.645
t
>
1.645
Not significant; significance requires the information ratio to exceed 2.0
Not significant at 5% (1.8 < 1.96); significant at 10%; need ~57 months
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