>>>>
Decomposition of Stochastic Discount Factor and their Volatility Bounds
20122012年年 1111 月月 2121日日
>>>> FrameworkFramework
• Motivation• Decomposition of SDF• Permanent and Transitory Bounds• Comparisons with Alvarez & Jermann (2005) • Eigenfunction and Eigenvalue Method• Asset Pricing Models Representation• Empirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 2
>>>> MotivationMotivation
• Economic Intuitions:• Explanation Inability of Equilibrium Asset Pricing Model - Various Puzzles (Return, Volatility)- Frequency Mismatch (Daniel & Marshall,1997)- Features of Investor Preference: Local Durability, Habit
Persistence or Long Run Risk• Unit Root Contributions of Macroeconomic Variables• Econometric Similarity:- Beveridge-Nelson Decomposition
2012/12/17 Asset Pricing 3
>>>> Decomposition of SDFDecomposition of SDF
• No Arbitrage Opportunities in Frictionless Market if and only if a strictly positive Pricing Kernel exists:
• So SDF for any gross return on a generic portfolio held from to
• Define as the gross return from holding from time to a claim to one unit of the numeraire to be delivered at time
2012/12/17 Asset Pricing 4
( )( ) t t k t k
t t kt
E M DV D
M+ +
+ =
{ }tM
1t
t
MM
+
t 1t+1
11 tt t
t
ME R
M+
+æ ö÷ç ÷= ×ç ÷ç ÷çè ø
t1t+
1,t kR +
t k+
>>>> Decomposition of SDFDecomposition of SDF
• So risk-free return:
• Long term bond return:
2012/12/17 Asset Pricing 5
11,
( )( )
t t kt k
t t k
V IR
V I+ +
++
=
1 11,1
( ) 1( ) ( )
t tt
t t k t t k
V IR
V I V I+ +
++ +
= =
1 11, 1,
( )lim lim
( )t t
t t kk kt t k
V IR R
V I+ +
+ ¥ +®¥ ®¥+= =
>>>> Decomposition of SDFDecomposition of SDF
• Assumptions:- SDF and Return Jointly Stationary and Ergodic- There is a number such that
- For each there is a random variable such that
with finite for all
2012/12/17 Asset Pricing 6
b
( )0 lim , t t k
kk
V Ifor all t
b+
®¥< <¥
1t+ 1tx +
1 1 111
( ) . .t t t k
tt k
M V Ix a s
b b+ + + +
++ £
1t tE x + k
>>>> Decomposition of SDFDecomposition of SDF
• Unique Decomposition (Alvarez & Jermann,2005)
and:
with:
2012/12/17 Asset Pricing 7
P Tt t tM M M=
1P P
t t tE M M+ =
limP t t kt t kk
E MM
b+
+®¥= lim
( )
t kTt k
t t k
MV Ib +
®¥ +=
1,1
Tt
t Tt
MRM+ ¥
+=
>>>> Decomposition of SDFDecomposition of SDF
• How to link transitory component to Long term bond?• No cash flow uncertainty
2012/12/17 Asset Pricing 8
1 1
1 1 11,
1 1
1 1
1
/( )
lim lim lim/( )
/lim
/
lim
t kt t k t t k
t t k t tt t kk k kt t k t t kt t k
t t
t k Pt t k t
Tkt t t
t k P Tt t k t t
ktt
E M E MV I M M
R E M E MV IM M
E M MM M M
E M M MMM
b
b
b
b
++ + + +
+ + + ++ ¥ +®¥ ®¥ ®¥+ ++
++ + +
®¥ + ++
+ +®¥
= = =
= = =
:Proof
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 9
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 10
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
• Inequality (6) bounds the variance of the permanent component of the SDF, useful for understanding what time-series assumptions are necessary to achieve consistent risk pricing across multiple asset markets
• is receptive to an interpretation as in Hansen & Jagannathan (1991) bound:
• So can be interpreted as the maximum Sharpe ratio, but relative to the long-term bond
2012/12/17 Asset Pricing 11
2pc
11 1,
1,
R log(R ) log( )tt t
t
RR
2pc
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 12
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 13
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
• The transitory component equals the inverse of the gross return of an infinite-maturity discount bond and governs the behavior of interest rates
• The quantity on the right-hand side of equation (9) is tractable and computable from the return data. And the bound in (9) is a parabola in space.
• is positively associated with the square of the Sharpe ratio of the long-term bound.
• (9) to assess the bound market implications of asset pricing models.
2012/12/17 Asset Pricing 14
21 ,Tt
tcTt
ME
M
2
tc
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 15
>>>> Permanent and Transitory BoundsPermanent and Transitory Bounds
2012/12/17 Asset Pricing 16
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
• In Alvarez & Jermann, L-measure (entropy) a random variable u as a measure of volatility:
• One-to-one correspondence exists between variance and L-measure when is log-normally distributed
• Such discrepancies between the two measures can get magnified under departures from log-normality.
2012/12/17 Asset Pricing 17
[ ] log[ ( )] (log[ ])L u E u E u
u1[ ] [log( )]2
L u Var u
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 18
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 19
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 20
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 21
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 22
>>>> Comparisons with Alvarez & Jermann (2005) Comparisons with Alvarez & Jermann (2005)
2012/12/17 Asset Pricing 23
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• Continuous Time Version (Luttmer,2003):• Consider State-Price Process:
• Suppose:
• For Any , and is bounded for all , the dominated convergence theorem implies that
2012/12/17 Asset Pricing 24
,t
t t tt
drdt dWsL
L =- +L
lim [ ]Pt t TT
E®¥
L = L
0t > [ ]t TE t+ L T
[ ] lim [ ] lim [ ]P Pt t t t T t T tT T
E E E Et t+ +®¥ ®¥é ùL = L = L =Lê úë û
>>>>
• The process is referred to as the permanent component of SDF
• Define to be the residual, So:• And suppose:
• As we all know, it also can be decomposed:
2012/12/17 Asset Pricing 25
TtL
PtL
P Tt t tL =L L
,P Pt t P t td dWsL =L
,, , , ,[( ( ) ( ) ]T Tt t t t P t t P t tP td r dt dWs s ss sL LL =L - - - + -
Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
µ( ) ( )rkt ktt t k t t k t k
t
rk rtt kt
t
MV D E D e E D
M
M dQdP e dQ M eM dP
-++ + +
- -+
é ùê ú= =ê úë ûÞ = Þ =
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• So How to Decompose? What’s ?• Hansen & Scheinkman (2009, Econometrica) • Let be a Banach space, and let be a
family of operators on . If: 1, for all 2, Positive if for any whenever 3, For each , Then is a semi-group.
2012/12/17 Asset Pricing 26
b
{ }: 0tM ³
0( ( ) | )t tE M X X x Ly = Î
LL
0 , t s t s+M =I M =M M , 0s t ³0, 0tt y³ M ³ 0y ³
Ly Î{ }: 0tM ³
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• Consider General Multiplicative Semi-group:
• Extended Generator: a Boral function belongs to the domain of the extended generator of the multiplica- tive function if there exists a Boral function such that is a local martingale wrt. filtration . In this case, the extended generator assigns function to and write
• Associates to function a function such that is the expected time derivative of
2012/12/17 Asset Pricing 27
0 0( ) ( ) ( )
t
t t t s sN M X X M X dsy y c= - - ò
yA
tM c
tIy c y=A
0: ( ) [ ( ) | ]t tt x E M X X x
y
c
c ( )t tM Xc( )t tM X
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• A Borel function is an eigenfunction of the extended generator with eigenvalue if .
• Intuitively, So if is an eigenfunction, the expected time derivative of is . Hence, the expected time derivative of is zero.
• How to get ?
• Expected time derivative is zero Local Martingale
2012/12/17 Asset Pricing 28
A
( )t tM X ( )t tM X exp( ) ( )t tt M X
( )( )1
( )t t
t t
M XM
dEdt X
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
2012/12/17 Asset Pricing 29
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• 6.1Proof: let , so:
• And:
• Interpretation:- : Growth rate of multiplicative functional- : Transient or Stationary Component- : Martingale Component, Distort the drift
2012/12/17 Asset Pricing 30
0 0( ) ( ) ( )
t
t t t s s
t t t t t t
N M X X M X ds
dN dY Y dt dY dN Y dt
y y c
r r
= - -Þ = - Þ = +
ò( )t t tY M X
0 0 0 0exp( ) exp( ) exp( ) exp( )
t t t
t s s st Y Y s Y ds s dY s dNr r r r r- - =- - + - = -ò ò ò
tM
¶tM
0( ( )) / ( ( ))tX X
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• Further more:
• If we treat as a numeraire, similar to the risk-neutral pricing in finance.
• Decomposition Existence and Uniqueness is given in Proposition 7.2 (Hansen & Scheinkman,2009)
• Congruity of Bakshi & Chabi-Yo Decomposition
2012/12/17 Asset Pricing 31
0 0
( )[ ( ) | ] exp( ) ( ) |
( )t
t tt
XE M X X x t x E X x
X
exp( ) ( )tt X
1: exp( ),( )
et
t
let v MX
1
1
1tt e e
t t t
M vEM M M
, /T t e P Tt t t t tM v M M M M
0( )exp( ) ,( )
ttt
XM t MX
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• Example: consider a multiplicative process :
• And :
• Guess an eigenfunction of the form
2012/12/17 Asset Pricing 32
exp( )M A
( )f o f f ot f t o t t f t o tdA X X dt X dB dB
,f oX X( )f f f f
ft f t t f tdX x X dt X dB
( )o o oot o t o tdX x X dt dB
( ) exp( )t f f o oX c X c X
( )1(ln ) (ln ( )) (ln( ( ))) ( )( )t t
t t
t tt t
dd M d X d M X Ed
Mt
XM X
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
2012/12/17 Asset Pricing 33
>>>> Eigenfunction and Eigenvalue MethodEigenfunction and Eigenvalue Method
• define a new probability measure, resulting distorted drift of :
2012/12/17 Asset Pricing 34
¶tM
,f oX X:fX:oX
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
• Consider the modification of the long-run risk model proposed in Kelly (2009).
• The distinguishing attribute: the model incorporates heavy-tailed shocks to the evolution of nondurable consumption growth (log), governed by a tail risk state variable .
2012/12/17 Asset Pricing 35
t
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
• While the transitory component of SDF is distributed log-normally, the permanent component of SDF and SDF itself are not log-normally distributed.
• The non-gaussian shock are meant to amplify the tails of the permanent component of SDF and SDF.
2012/12/17 Asset Pricing 36
gW
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
2012/12/17 Asset Pricing 37
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
2012/12/17 Asset Pricing 38
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
2012/12/17 Asset Pricing 39
>>>> Asset Pricing Models RepresentationAsset Pricing Models Representation
2012/12/17 Asset Pricing 40
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 41
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 42
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 43
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 44
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 45
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 46
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 47
>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models
2012/12/17 Asset Pricing 48
>>>>
Thank you for listening andThank you for listening and
Comments are welcome.Comments are welcome.
20122012年年 1111 月月 2121日日