decomposition of stochastic discount factor and their volatility bounds

>>>>

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


>>>> 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


( )( ) 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+


• So risk-free return:

• Long term bond return:


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+ +

+ ¥ +®¥ ®¥+= =


• 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


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


• Unique Decomposition (Alvarez & Jermann,2005)

and:

with:


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+ ¥

+=


• How to link transitory component to Long term bond?• No cash flow uncertainty


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



• 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


2pc

11 1,

1,

R log(R ) log( )tt t

t

RR

2pc


• 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.


21 ,Tt

tcTt

ME

M

2

tc

>>>> 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.


[ ] log[ ( )] (log[ ])L u E u E u

u1[ ] [log( )]2

L u Var u

>>>> 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


,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:


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 =


• 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.


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


• Consider General Multiplicative Semi-group:

• Extended Generator: a Boral function belongs to the domain of the extended generator of the multiplicative 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


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


• 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


A

( )t tM X ( )t tM X exp( ) ( )t tt M X

( )( )1

( )t t

t t

M XM

dEdt X


• 6.1Proof: let , so:

• And:

• Interpretation:- : Growth rate of multiplicative functional- : Transient or Stationary Component- : Martingale Component, Distort the drift


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


• 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


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


• Example: consider a multiplicative process :

• And :

• Guess an eigenfunction of the form


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


• define a new probability measure, resulting distorted drift of :


¶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 .


t


• 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.


gW

>>>> Empirical Application to Asset Pricing ModelsEmpirical Application to Asset Pricing Models


>>>>

Thank you for listening andThank you for listening and

Comments are welcome.Comments are welcome.

20122012年年 1111 月月 2121日日

decomposition of stochastic discount factor and their volatility bounds

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