Lars Peter Hansen
Scott F. Richard
The Role of Conditioning Information in
Deducing Testable Restrictions Implied by
Dynamic Asset Pricing Models
Econometrica, 55, 1987
Also based on Cochrane Chapter 5
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©Eric Ghysels
BIG PICTURE RESULTS
EXISTENCE OF SDF
Law of one price
If two portfolios have same payoffs in all states of
nature then they must have the same price
∃m:p=Em x
iff law holds
Absence of arbitrage
No arbitrage opportunities
Iff : ∃ m > 0 p = E m x
©Eric Ghysels
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Back to the basic questions
Instead of starting from general equilibrium
consumption - based asset pricing model.
Let us view discount factor as stochastic process
satisfying
p=Emx
• Does there always exist such an m ?
• What markets structure supports this
fundamental equation?
©Eric Ghysels
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• Many aspects of payoff space can be
conveniently captured by restrictions on the
discount factor.
• Check and impose restrictions on discount
factor is easier than on all possible portfolios
priced by discount factor.
©Eric Ghysels
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CONTINGENT CLAIMS
• Suppose S states of nature tomorrow
• Contingent claim pays $1 ( 1 unit of
consumption)
• pc(s) : price today of contingent claim state “s”
©Eric Ghysels
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COMPLETE MARKET
• Any contingent claim can be bought
• Note that each contract does not have to be traded
explicitly
• It can be spanned by other securities (so called
synthesized securities e.g. European options with
every possible strike)
If a complete set of contingent claims exists, then a
discount factor exists and it is equal to the contingent
claim prices divided by state probabilities
©Eric Ghysels
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Let x (s) denote Payoff ⇒ p(x) =Σ pc(s) x(s)
s
 pc(s ) 
p( x) = ∑ π ( s ) 
x( s )

s
 π (s) 
Where π (s) is probability of states
Then
pc( s )
m( s ) =
π (s)
p = Σπ (s)m(s) x (s) = Em x
s
©Eric Ghysels
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Therefore in complete markets the stochastic
discount factor m exists with p=Emx
m(s) is state price pc(s) rescaled by probability of
state, therefore m(s) : State Price Density
Risk neutral probabilities
π * (s) ≡ R f m(s)π (s) = R f pc(s) =
E* ( x)
and p( x) =
Rf
pc(s)
E(m)
©Eric Ghysels
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• Asset pricing becomes as if there is risk
neutrality with re-scaled probabilities π *
• Deep issue : risk aversion is equivalent to
paying more attention to certain
(unpleasant) states
π * (s) > π (s)
In “bad” states high subjective probabilities
m( s )
π * ( s) =
π (s)
E ( m)
©Eric Ghysels
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• In incomplete markets there are many m,
but one is positive
• m>0 but is not unique
• When m is not unique there might be some
negative discount factors.
©Eric Ghysels
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Hilbert Spaces
A non-empty set H is called a Hilbert space if H is a real
linear vector space, together with a real-valued function
<.,.> from H x H into C having the following properties:
• <x,x> is nonnegative and <x,x> = 0 iff x = 0
• <x+y,z> = <x,z> + <y,z> for all x,y,z
• <ax,y> = a<x,y> for all x,y in H and a real-valued
• <x,y> = <y,x> for all x,y in H
• If {xn} is a sequence in H with lim < xn- xm ,xn- xm> → 0
as n,m → ∞ then there is an element x in H such that
lim < xn- x ,xn- x> = 0.
©Eric Ghysels
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Inner product and norm
• The function <.,.> is called the inner product
1/ 2
x
=<
x
,
x
>
• The real-valued function
is called
the norm.
• Schwartz inequality < x , y > ≤ x y
• Two points x and y in a Hilbert space are
orthogonal if <x,y> = 0
• Riesz representation theorem: for every bounded
linear functional x* on H there is a unique z in H
*
*
x
= z
such that x (x) = <x,z> for all x in H and
©Eric Ghysels
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Notations:
Gt - all the info at time t
It - all the random variables with outcomes in Gt (i.e.
measurable functions)
Pt+1 - all payoffs pt+1 on assets at time t + 1
Bt(pt+1): Pt+1 → It price of p at time t
wt in It: portfolio weights
p1, p2 in P → w1p1 + w2p2 in P
©Eric Ghysels
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Objectives
• Construct a general asset pricing function
• This construction will allow for a very nice
decomposition of any return into a sum of three
orthogonal components
• The decomposition yields an easy solution to the
mean-variance problem
• Discuss conditional vs.unconditional approach
• Bridge to empirical work
©Eric Ghysels
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One-period setting
• One period setting suffices because of stationarity
• For any p in I1 consider P + = { p ∈ I1 : E[ p 2 | G] < ∞}
• Conditional counterpart of inner product on P+ for
any pa and pb in P+ < p a , pb > G = E[ p a pb | G ]
• We introduce a subset P of P+ that is restricted to
satisfy conditional counterparts of linearity and
completeness (i.e. linear combinations of elements
of P belong to P and conditional Cauchy
sequences converge in P)
©Eric Ghysels
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Technical (and Economic)
Assumptions
Assumption A.1 (Law of one price)
For any p1, p2 in P and w1, w2 in I
B(w1p1 + w2p2) = w1B(p1) + w2B(p2)
Assumption A.2 For any p$0, B(p)$0
Assumption A.3 › po: Pr{B(po) = 0} = 0
©Eric Ghysels
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Technical Assumption
Implications
The conditional version of the Riesz representation
theorem allows us to write B(p) = E(pp*|G) for
some p* > 0 with Pr = 1.
Moreover, we have that
A.1:
B(w1p1+ w2p2) = E((w1p1+w2p2)p*|G)
= E(w1p1p* | G) + E(w2p2p*|G)
= w1B(p1) + w2B(p2)
A.2:
p $ 0 B(p) = E(pp*|G) $ 0
A.3:
po=1 B(po) = E(p*|G) > 0
©Eric Ghysels
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Theorem
A.1-3 + conditional Riesz theorem are satisfied ⇒
exists unique p* in P:
B(p) = E(pp*|G) = <p,p*>G for all p in P
Pr {B(p*) = 0} = 0
i.e.
ALL pricing functions have such a representation
©Eric Ghysels
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Lemma (Conditional Projection Theorem)
Find ho that minimizes
||p - h||G
ho is unique iff
p - ho is orthogonal to H
i.e. E((p-ho) h|G) = 0
call ho is the projection of p onto H
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©Eric Ghysels
Sketch of the proof:
Take po that satisfies A.3
zo is the projection of po onto Z the conditionally
complete linear subspace of P satisfying {p in P: B(p) = 0}
and hence B(zo) = 0.
p0 − z 0
p0 − z 0
r :=
=
⇒ r b orthogonal to Z
π ( p0 − z 0 ) π ( p0 )
b
Note that B(rb) =1 and rb is called the benchmark return and rb is
conditionally orthogonal to Z. Note also that:
p - B(p) rb 0 Z: B(p - B(p)rb) = B(p) - B(p) B(rb) = 0
©Eric Ghysels
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Since p - B(p) rb 0 Z
⇒ E(([p - B(p)rb] rb |G) = <p - B(p)rb,rb>G = 0
and therefore
E(prb|G) = <p,rb>G = π (p)||r b||2G
b
moreover, the technical assumptions imply that Pr{ r







π (p) = E p
b
G
= 0} = 0







r
|G
b 2
||r ||
G
p* ↑
©Eric Ghysels
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Definition:
B has no arbitrage opportunities if for any p ∈ P
Pr {p $ 0} = 1
Pr({B(p) # 0} 1 {p > 0}) = 0
Lemma: B has no arbitrage iff
Pr {p* > 0} = 1
r*:
p*
= π(p*)
- return on p*
©Eric Ghysels
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There are three sets that are of special interest:
R = {p in P: B(p) = 1} space of returns with price 1
Z = {p in P: B(p) = 0} space of returns with price 0
N = {z in Z: E[z|G] = 0} space of idiosyncratic risk
We will show that
R = {r : r = r* + wz* + n}
Where r* = p*/ B(p*) is return on benchmark asset, projection
of stochastic discount factor onto space of assets.
(wz* + n) is in Z and n is in N
©Eric Ghysels
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Lemma: (P,B) satisfies A.1-3 + technical assumptions
(i) E(r*z|G) = 0
(ii) ||r*||G ≤ ||r||G, r ∈ R
R = {r : r = r* + z}
©Eric Ghysels
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Some implications of results obtained so far:
1. ||r*||G ≤ ||r||G, r ∈ R implies that r* conditional
minimum mean-variance portfolio (more on this
later)
2. When Z is non-empty asset pricing does not coincide
with risk neutral pricing
©Eric Ghysels
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Problem C min r 2G
s.t. E(r|G) = w ∈ I
*|G)
−
w
E
(
r
*
w =
E(z*|G)
E(r|G) = w ⇔ r = r* + w*z* + n
⇒ n = 0 solves the problem
What is the effect of omitting the info?
Consider R*, Z*, N*
R* = {r : r = r* + cz* + n*}
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©Eric Ghysels
Problem U min ||r||2
s.t. E(r) = c
c* = [c − E (r * )] / E ( z * )
E(r) = c ⇔ r = r* + c*z* + n*
n* = 0 solves the problem
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©Eric Ghysels
Back to Big Picture
Want to test H0 : ri is on the frontier
Method: Test CAPM via regression
However, when we do not take into account
conditioning information Hansen and Richard
show that rejection of CAPM does not necessarily
mean that a portfolio is not mean-variance
efficient.
©Eric Ghysels
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Conditional vs unconditional factor models in
discount factor language
Consider CAPM
r*=m*= a - bRw
Where Rw is the return on the market or wealth
portfolio 1 = E m * R W
t
t +1
t +1
1 = E t m t*+ 1 R t f+ 1
We can solve for a and b
1
a = f + bEt ( RtW+1 )
Rt
b = ( Et ( RtW+1 ) − Rt f ) / Rt f σ t2 ( RtW+1 )
But this means a and b vary over time
©Eric Ghysels
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If it is to price assets conditionally, the CAPM
model must be linear factor model with
time-varying weights:
m
*
t +1
= a t + bt R
W
t +1
Therefore
1 = E t [( a t + bt R tW+ 1 ) R t + 1 ]
While we can say:
1 = E [( a t + bt R
W
t +1
©Eric Ghysels
) R t +1 ]
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We cannot say
W
1 = E [( a + bR t +1 ) Rt +1 ]
Instead:
W
1 = E ( a t ) E ( Rt +1 ) + E (bt ) ER t +1 Rt
+ Cov ( a t , Rt +1 )
+ Cov (bt , R Rt +1 )
W
t +1
Only when the covariances are zero then,
1 = E [ E ( a t ) + E (bt ) R ) Rt +1 ]
W
t +1
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©Eric Ghysels
In general
1
W
Cov ( f + bt E t ( Rt +1 ), Rt +1 ) ≠ 0
Rt
E t ( R ) − Rt
W
Cov ( f
, R t +1 R t + 1 ) ≠ 0
W
R t σ ( R t +1 )
W
t +1
2
t
f
Of course if:
a t = a , bt = b
Then,
1 = E ( a + bR ) Rt +1
W
t +1
©Eric Ghysels
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CONDITIONAL VS CONDITIONAL IN AN
EXPECTED RUTURN/BETA MODEL
Et ( Ri ) = Rt f + β t λt
Does not imply
E ( Ri ) = α + βλ
If returns & factors are i.i.d. then Cov(.)=
Covt(.), Var(.) = Vart(.)
So that,
βt = β
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©Eric Ghysels