有关随机贴现因子的一些思考

Cochrane(1996,2000)
β
Kan and
Zhou(1999)
β
β β
Kan and Zhou(1999)
Kan and
Zhou(1999)
u'( Ct
1)
M = +
t+ 1
β
u'( C )
β t
β β

m
β
β
β
:
Er ( ) = r + β [ Er ( ) −r
],
β
j
j f j m f
cov( rj, rm)
=
var( r )
m
r = Er ( ) + β f + ε , f f = r −Er
( )
j j j j
k q
pj = Ekx (
q j)
Ekx (
q j)
kr
q j
= kEr
q
(
j)
+ β
jfkq + kqε
jEkr
(
q j) = = 1
p
1 = Ek ( ) Er ( ) + β E( k f ) + Ek ( ε )
q j j q q j
1 Ekf (
q
) Ek (
qε
j)
Er (
j) = + β
j( − ) −
Ek ( ) Ek ( ) Ek ( )
q q q
Ek ( ) = 1/ r ,
Er ( ) = r − β r p( f ), p( f)
j f j f
q
f
β = cov( rj, f ) cov( rj, rm)
j
var( f) = var( r )
β
Er ( )
m
q( f) = Ek (
q( rm − Er (
m)) = 1−
r
Er ( ) = r + β ( Er ( ) − r )
j f m f
m
r = Er ( ) + β f + ε = r + β( Er ( ) − r ) + β f + ε ,
j j j f m f j
f
j
f
r − r = β( Er ( ) − r ) + β f + ε λ = Er ( ) − r , β
j f m f j
β r = r + β
Er ( ) − r ) + η
j f j m f j
rj = rf + β
j( Er (
m) − rf)
+ β
jf
+ ε
j
rj
r j
− r f
m
f
m
m

α β rj = βα
j
+ ηj cov( r , ) j
f
β
j
=
var( f )
r = βα + β f + ε
j j j j
f
f ' = β
σ ( f )
'
β = cov( r, f ') = Erf ( ') E( f ') = 0
1
j j j
β r = βα '
' + η β ' = Erf ( ')
j j j
' '
r = βα'
+ β f ' + ε
j j j j
j
j
GMM
( rt − rf
t tλ)
t=
β
GMM
g
1
1T
=
T
T
∑
Tg : N(0 , S )
1 T N 1
argmin
λ
gT( λ)' W1
TgT( λ)
λ ) 1
λ
) −
= ( D ' W D ) ( D ' W r )
2
T 1T T T 1T T
D
T
=
T
∑
rf
t
t=
1
T
t
r
=
T
T
∑
r
t
t=
1
T
W S −1
1T
1
GMM E( εt) = Er (
t
−βλ − β ft) = 0;
E( ε f ) = E[( r − βλ − β f ) f ] = 0
t t t t t
T
1
g ( λβ , ) = ∑[ z ⊗( r −βλ −β
f )] Tg 2 T( λβ , ):
(0 2 N, S 2
), zt = [1, ft]
T
2T t t t
t=
1
λβ , argmin
λβ ,
g2 T( λβ , )' W2Tg2T( λβ , )
β GMM
1 ' j
β = α ' = ασ ( f )
2 rt
1
j
β
σ ( f )
N ×
t N

Erf ( ) = 1/ T∑ rf
GMM
t t t t
t=
1
T
β
β
Cochrane(2000)
M
t
( ErM
t t) = 0Mt
= δ0 −δ1f
3 t
ErM (
t t) = Er [
t( δ0 − δ1ft)] = 0, Er [
t(1 − λft)] = 0, λ = δ1/
δ0
Er ( ) = Erf ( λ) = Erf ( ) λ
β t t t t t
β
Kan and Zhou(1999)
β
Kan and Zhou1999
f t
[ Ert(1 − λ ft)] = 0
ft
+ nt
1g = , λ = λ 1 + σ
2
1 + σ
2
t g n
n
−1
β ' ∑ εt
2h
=
−1
β'
∑ β
−1
λh
= λ β'
∑ β
n t
, g h
t
t
β
δ
0
= 1/ rf
EMt
δ0
3
( ) = = 1/ r
f

ft
+ nt
1 g , 1 2
t
= λ
2
g
= λ + σ
n
1 + σ
n
ˆ f cov( , )
cov( ,
t
+ nt r )
t
ft + nt
β
βg
= rt
= =
1+ σ 1+ σ 1+
σ
2 2 2
n n n
cov( rn , ) = 0
t
t
βλ
ĝ
g
= βλ
−1
β ' ∑ εt
2
h =
−1
β'
∑ β
−1
λh
= λ β'
∑ β
−1
β ' εt
β
βˆ = cov( , ) =
)
βλ
h
h
r t
∑
∑ ∑
−1 −1
β ' β β ' β
= βλ
f
t
β
β
ϕ
4
Ek (
qε
j)
= Er ( ) −r −β( Er ( ) − r ) =− = − rqε
f
(
j)
Ek ( )
j j f m f
F F
kq
F, k = k + η, k ∈ F,
η⊥ F
j
q
q
q q q
Ek ( ε ) = E( ηε ),
q j j
,
| E( ηε )| ≤|| η |||| ε ||
q
j
j
F
F
| ϕ
j
| ≤ rfσ ε
|| kq − kq
||, || k − k ||
k q
0
4 Leroy and Werner(2000)

CAPM β
0
α
= βλ
5 CAPM
CAPM
u'( Ct+
)
CAPM δ 1
u'( C )
t
α = βλ + ϕ
CAPM
Cochrane, J. H., 1996, “A Cross Sectional Test of an Investment-based Asset Pricing Models”,
Journal of Political Economy 104, 574-621.
Cochrane, J. H., 2000, Asset Pricing, Princeton University Press.
Kan, R. and G. Zhou, 1999, “A Critique of the Stochastic Discount Factor Methodology”,
Journal of Finance, vol. LIV, 1221-1248.
Press.
Leroy, S.F., and J. Werner, 2000, Principles of Financial Economics, Cambridge University
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