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Decomposition of Stochastic Discount Factor
and their Volatility Bounds
2012年11月21日
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Framework
• 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
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Motivation
• 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
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Decomposition of SDF
• No Arbitrage Opportunities in Frictionless Market if and
only if a strictly positive Pricing Kernel{M }exists:
t
Vt ( Dt + k ) =
M t+ 1
Mt
Et (M t + k Dt + k )
Mt
• So SDF
for any gross return on a generic portfolio
held from t to t + 1
æ
ö
M
÷
1 = Et ççç t + 1 ×Rt + 1 ÷
÷
÷
M
è t
ø
• DefineRt + 1,k as the gross return from holding from time t
to t + 1 a claim to one unit of the numeraire to be
delivered at time t + k
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Decomposition of SDF
Rt + 1,k
Vt + 1 ( I t + k )
=
Vt ( I t + k )
• So risk-free return:
Vt + 1 ( It + 1 )
1
Rt + 1,1 =
=
Vt ( It + k ) Vt ( It + k )
• Long term bond return:
Rt + 1,¥
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Vt + 1 ( It + 1 )
= lim
= lim Rt + 1,k
k® ¥ V (I
k® ¥
t
t+ k )
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Decomposition of SDF
• Assumptions:
-
SDF and Return Jointly Stationary and Ergodic
- There is a number such
b that
0 < lim
k® ¥
- For each
Vt ( I t + k )
< ¥ , for all t
k
b
t+ 1
there
is a random variable
M t + 1 Vt + 1 ( I t + 1+ k )
£ xt + 1
t+ 1
k
b
b
with
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Et finite
xt + 1 for all
xt + 1 that
such
a.s.
k
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Decomposition of SDF
• Unique Decomposition (Alvarez & Jermann,2005)
Mt = MtP MtT
and:
Et M t + k
M = lim
k® ¥
b t+ k
P
t
b t+ k
M = lim
k® ¥ V (I
t
t+ k )
T
t
with:
Et M
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P
t+ 1
=M
P
t
Asset Pricing
Rt + 1,¥
M tT
= T
M t+ 1
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Decomposition of SDF
Proof :
Rt + 1,¥
Et + 1M t + k
Et + 1M t + k / b t + k
V (I )
M t+ 1
M t+ 1
= lim t + 1 t + k = lim
= lim
t+ k
k® ¥ V (I
k® ¥
k® ¥
E
M
E
M
/
b
)
t
t
+
k
t
t+ k
t
t+ k
Mt
Mt
Et + 1M t + k / b t + k
M tP+ 1
lim
k® ¥
M t+ 1
M t+ 1
M tT
=
=
=
t+ k
P
Et M t + k / b
Mt
M tT+ 1
lim
k® ¥
Mt
Mt
• How to link transitory component to Long term bond?
• No cash flow uncertainty
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Permanent and Transitory Bounds
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Permanent and Transitory Bounds
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Permanent 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
2
•  pc is receptive to an interpretation as in Hansen &
Jagannathan (1991) bound:
R t 1
   log(R t 1 )  log( Rt 1, )
Rt 1,
• So  pc2 can be interpreted as the maximum Sharpe ratio,
but relative to the long-term bond
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Permanent and Transitory Bounds
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Permanent and Transitory Bounds
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Permanent 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
T
bound in (9) is a parabola in  E  M t T1  ,  tc2  space.


 Mt 


2
•  tc 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.
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Permanent and Transitory Bounds
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Permanent and Transitory Bounds
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Comparisons with Alvarez & Jermann (2005)
• In Alvarez & Jermann, L-measure (entropy) a random
variable u as a measure of volatility:
L[u]  log[ E (u)]  E (log[u])
• One-to-one correspondence exists between variance
and L-measure when u is log-normally distributed
1
L[u ]  Var[log(u )]
2
• Such discrepancies between the two measures can
get magnified under departures from log-normality.
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Comparisons with Alvarez & Jermann (2005)
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Comparisons with Alvarez & Jermann (2005)
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Comparisons with Alvarez & Jermann (2005)
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Comparisons with Alvarez & Jermann (2005)
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Comparisons with Alvarez & Jermann (2005)
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Comparisons with Alvarez & Jermann (2005)
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Eigenfunction and Eigenvalue Method
• Continuous Time Version (Luttmer,2003):
• Consider State-Price Process:
• Suppose:
dLt
= - rt dt + s L ,t dWt
Lt
L tP = lim Et [L T ]
T® ¥
• For Any t > 0 , and Et + t [L T ] is bounded for all T , the
dominated convergence theorem implies that
P
Et [L tP+ t ] = Et éêlim Et + t [L T ]ù
=
lim
E
[
L
]
=
L
t
ú
ëT ® ¥
û T® ¥ t T
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Eigenfunction and Eigenvalue Method
P
L
• The process t is referred to as the permanent
component of SDF
• Define L Tt to be the residual, So: L t = L t L t
P
• And suppose:
T
d L tP = L tPs P,t dWt
dLTt = LTt [(- rt - s P,t (s L ,t - s P,t )dt + (s L ,t - s P,t )dWt ]
• As we all know, it also can be decomposed:
éM t + k
ù - rk
µt ( D )
ê
Vt ( Dt + k ) = Et
Dt + k ú= e E
t+ k
ê Mt
ú
ë
û
M t+ k
dQ
Þ
dP = e- rk dQ Þ M t = e- rt
Mt
dP
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Eigenfunction and Eigenvalue Method
• So How to Decompose? What’s b ?
• Hansen & Scheinkman (2009, Econometrica)
• Let L be a Banach space, and let {M : t ³ 0} be a family
of operators on L . If:
1, M0 = I, Mt + s = Mt Ms for all s, t ³ 0
2, Positive if for any t ³ 0, Mt y ³ 0 whenever y ³ 0
3, For eachy Î L , E(Mt y ( X t ) | X 0 = x) Î L
Then {M : t ³ 0} is a semi-group.
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Eigenfunction and Eigenvalue Method
• Consider General Multiplicative Semi-group:
t : ( x)  E[Mt ( X t ) | X 0  x]
• Extended Generator: a Boral function y belongs to the
domain of the extended generator A of the multiplicative function M t if there exists ta Boral function c such
that Nt = M t y ( X t ) - y ( X 0 ) - ò0 M s c ( X s )ds is a local
martingale wrt. filtration I t . In this case, the extended
generator assigns function c to y and write c = Ay
• Associates to function y a function c such that Mt c ( X t )
is the expected time derivative of Mt ( X t )
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Eigenfunction and Eigenvalue Method
• A Borel function  is an eigenfunction of the extended
generator A with eigenvalue  if    .

• Intuitively,
So if is an eigenfunction, the expected time
derivative of M t ( X t ) is  M t ( X t ). Hence, the expected
time derivative of exp(t )Mt ( X t ) is zero.
• How to get
 ?
1  d ( M t ( X t )) 
  E

dt  M t ( X t ) 
• Expected time derivative is zero
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Local Martingale
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Eigenfunction and Eigenvalue Method
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Eigenfunction and Eigenvalue Method
• 6.1Proof: let Yt  M t ( X t ) , so:
Nt = M t y ( X t ) - y ( X 0 ) -
ò
t
0
M s c ( X s )ds
Þ dNt = dYt - r Yt dt Þ dYt = dNt + r Yt dt
• And:
t
exp(- r t )Yt - Y0 = -
t
t
ò r exp(- r s)Y ds + ò exp(- r s)dY = ò exp(- r s)dN
0
s
0
s
0
s
• Interpretation:
-
: Growth rate of multiplicative functional
-
( ( X 0 )) / ( ( X t )): Transient or Stationary Component
¶: tMartingale Component, Distort the drift
M
-
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Mt
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Eigenfunction and Eigenvalue Method
• Further more:
 ( X t )

E[ M t ( X t ) | X 0  x]  exp( t ) ( x) E 
| X 0  x
 (Xt )

• If we treat exp(t ) ( X t ) 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
 M t 1 1  v
(X0 )
 e
M t  exp(  t ) M t
, Et 
e 
(Xt )
 M t M t 1  M t
1
e
let : v  exp(  ), M t 
 MtT  vt Mte , MtP  Mt / MtT
( Xt )
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Eigenfunction and Eigenvalue Method
• Example: consider a multiplicative process
M  exp( A) :
dAt  (    f X t f  o X to )dt  X t f  f dBt f   o dBto
• And X f , X o :
dX t f   f ( x f  X t f )dt  X t f  f dBt f
dX to  o ( xo  X to )dt   odBto
• Guess an eigenfunction of the form  ( X t )  exp(c f X f
d (ln M t )  d (ln  ( X t ))  d (ln( M t ( X t )))   
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 co X o )
1  d ( M t ( X t )) 
E

dt  M t ( X t ) 
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Eigenfunction and Eigenvalue Method
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Eigenfunction and Eigenvalue Method
¶ t define a new probability measure, resulting distorted
•M
drift of X f , X o :
Xf :
Xo :
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Asset 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 .
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Asset 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 Wg are meant to amplify the
tails of the permanent component of SDF and SDF.
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Asset Pricing Models Representation
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Asset Pricing Models Representation
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Asset Pricing Models Representation
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Asset Pricing Models Representation
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Empirical Application to Asset Pricing Models
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Thank you for listening and
Comments are welcome.
2012年11月21日