Asset Pricing
Zheng Zhenlong
Chapter 8
Conditioning Information
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Structure
•
•
•
•
•
•
•
Introduction
8.0 An Important Theorem
8.1 Scaled Payoffs
8.2 Sufficiency of Adding Scaled Returns
8.3 Conditional and Unconditional Models
8.4 Scaled Factors:A Partial Solution
8.5 Summary
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Introduction
pt  Et (mt 1 xt 1 )
• If payoffs and discount factors were independent and
identically distributed (i.i.d.) over time, then conditional
expectations would be the same as unconditional
expectations.
• But stock price/dividend ratios, bond and option prices all
change over time, which must reflect changing
conditional moments.
• Then how can we do?
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Approach
• One approach is to specify and estimate explicit statistical
models of conditional distributions of asset payoffs and
discount factor variables (e.g. consumption growth). This
approach is sometimes used, and is useful in some
applications, but it is usually cumbersome. Because the
number of required parameters is so large that may quickly
exceed the number of observations.
• More importantly, we(economists or would-be social
planners) obviously don’t even observe all the conditioning
information used by economic agents, and we can’t include
even a fraction of observed conditioning information in our
models.
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Method to settle
• If we don’t want to model conditional distributions explicitly,
and if we want to avoid assuming that investors only see the
variables that we include in an empirical investigation, we
eventually have to think about unconditional moments, or at
least moments conditioned on less information than agents
see.
Advantage of the unconditional
moment
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• We may be interested in finding out why the
unconditional mean returns on some stock portfolios
are higher than others, even if every agent
fundamentally seeks high conditional mean returns.
• Most statistical estimation essentially amounts to
characterizing unconditional means, as we will see in
the chapter on GMM.
8.0 An Important Theorem
the Law of Iterated Expectation
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• Specification:If you take an expected value using less
information of an expected value that is formed on more
information, you get back the expected value using less
information.
• Your best forecast today of your best forecast tomorrow is the
same as your best forecast today.
• Some useful
Et 1guises:
( Et ( xt 1 ))  Et 1 ( xt 1 )
（1）
E[ E ( x | ) | I  ]  E[ x | I ]
E ( Et ( x ))  E ( x )
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8.1 Scaled Payoffs
• From
pt  Et (mt 1 xt 1 )
or
pt  E[mt 1 xt 1 | I t ]
We can take unconditional expectations:
E ( pt )  E (mt 1 xt 1 )
（2）
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Conditioning Down
•
pt  It
pt  E[mt 1 xt 1 | ]
 E[ pt | I  ]  E[mt 1 xt 1 | I  ]
 pt  E[mt 1 xt 1 | I t  t ]
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Instruments and Managed
Portfolios
•
pt zt  Et (mt 1 xt 1 zt )
(3)
E ( pt zt )  E (mt 1 xt 1 zt )
(4)
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E ( pt zt )  E (mt 1 xt 1 zt )
•
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Remark 1
• Actually, every test practically uses managed portfolios. For
example, the size, beta, industry, and so forth portfolios of
stocks are all managed portfolios. Investors changes their
composition every year in response to conditioning
information-the size, beta, etc. of the individual stocks.
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Remark 2 :Nonlinear
•
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Conclusion
• If we want to incorporate the extra information in
conditioning information, we just add managed portfolio
payoffs, and proceed with unconditional moments as if
conditioning information did not exist.
• We can thus incorporate conditioning information while still
looking at unconditional moments instead of conditional
moments, without any of the statistical machinery of explicit
models with time-varying moments.
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Subtleties
• The set of asset payoffs expands dramatically, since we
can consider all managed portfolios as well as basic assets,
potentially multiplying every asset return by every
information variable.
8.2 Sufficiency of Adding Scaled
Returns
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•
E[(mt 1 xt 1  pt ) zt ]  0
 E[(mt 1 xt 1  pt ) | I t ]  0
(5)
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Remark 1
•
E ( pt )  E (mt 1 xt 1 )xt 1  X t 1
 pt  E (mt 1 xt 1 | I t )
(6)
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Remark 2
• [Theoretical]“All linear and nonlinear transformations of all
variables observed at time t” sounds like a lot of instruments,
and it is.But there is a practical limit to the number of
instruments
one needs to scale by,since only variables that
forecast returns or m add any information.
zt
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Remark 3
• [Practical] Implicitly, one feels that the chosen payoffs do a
pretty good job of spanning the set of available risk loadings
or mean returns, and hence that adding additional assets will
not affect the results.
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Remark 4
• Of course, the other way to incorporate conditioning
information is by constructing explicit parametric models of
conditional distrubutions.But the parametric model may be
incorrect,or may not reflect some variable used by investors.
8.3 Conditional and unconditional
models
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•
mt 1  1/ RtW1
pt  Et (mt 1 xt 1 )  E ( pt )  E (mt 1 xt 1 )
Conditional vs. Unconditional
Factor Models in Discount Factor
Language
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•
(7)
1

W
a


bE
(
R
t
t 1 )
f
W
W
W

Rt
1  Et (mt 1 Rt 1 ) 1  Et [(a  bRt 1 ) Rt 1 ] 



W
f
f
W
f
E
(
R
)

R
1

E
(
m
)
R
1

E
(
a

bR
)
R


t
b  t t 1
t
t 1
t
t
t 1
t

Rt f  t2 ( RtW1 )
(8)
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Rt f
• Here a>0 and b>0,and asEt ( RtW1 ) ,t2 ( RtW1 ) ,and
vary over
time, a and b must vary over time.
• If it is to price assets conditionally, the CAPM must be a
linear factor model with time-varying weights:
mt 1  at  bt RtW1
(9)
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• This fact means that we can no longer transparently
condition down. That is:
(10)
1  Et [(at  bt RtW1 ) Rt 1 ]
• does not imply that we can find constants a and b so
that
1  E[( a  bRtW1 ) Rt 1 ]
(11)
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Proof
• taking unconditional expectations,
1  E[(at  bt RtW1 ) Rt 1 ]  E[at Rt 1  bt RtW1Rt 1 ]
 E (at ) E ( Rt 1 )  E (bt ) E ( RtW1Rt 1 )  cov(at , Rt 1 )  cov(bt , RtW1Rt 1 )
• While the unconditional model is:
1  E[( E (at )  E (bt ) RtW1 ) Rt 1 ]
 E[ E (at ) Rt 1  E (bt ) RtW1Rt 1 ]
 E (at ) E ( Rt 1 )  E (bt ) E ( RtW1Rt 1 )
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Et ( RtW1 )  Rt f
1
W
at  f  bEt ( Rt 1 ), bt 
Rt
Rt f  t2 ( RtW1 )
•
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Remark
•
1  Et [(a  bRtW1 ) Rt 1 ]  1  E[(a  bRtW1 ) Rt 1 ]
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Conditional vs. Unconditional in an
Expected Return-Beta Model
• We begin with beta pricing language,
Et ( Rti1 )  Rt f   ti t
• But it does not imply that
E ( Rti1 )     i 
• Taking unconditional expectations of (14):
(14)
(15)
E ( Rti1 )  E ( Rt f   ti t )
 E ( Rt f )  E (  ti ) E (t )  cov(  ti , t )
(16)
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A Precise Statement
•
pt  Et (mt 1 xt 1 )
E ( pt )  E (mt 1 xt 1 )
xt 1  X
xt 1  X
mt 1  at  bt' ft 1
mt 1  a  b' ft 1
Remark 1
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• Actually, an unconditional factor pricing model
might be more appropriately called a fixed-weight
factor pricing model.
• So, the unconditional model is just a special case of
the conditional model, one that happens to have fixed
weights.
• Thus, a conditional factor model does not imply an
unconditional factor model (because the weights
may vary) but an unconditional factor model does
imply a conditional factor model.
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Remark 2
• Consider the relation between a conditional factor
pricing model using a fine information set (like
investors’ information sets) and conditional factor
pricing models using coarser information sets (like
ours). If a set of factors prices assets with respect to
investors’ information, that does not mean the same set
of factors prices assets with respect to our, coarser,
information sets.
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Mean-Variance Frontiers
•
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Relation between two Frontiers
• These two frontiers are related by:
 If a return is on the unconditional mean-variance frontier,
it is on the conditional mean-variance frontier.
 If a return is on the conditional mean-variance frontier, it
need not be on the unconditional mean-variance frontier.
• Several ways to see this relation:
Using the Connection to Factor
Models
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mt 1  at＋bt RtW1
•
RtW1
mt 1  a＋bRtW1
RtW1
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Using the Orthogonal
Decomposition
•
R
mv
 R R
*
e*
*
e*
Rtmv

R


R
1
t 1
t t 1
*
e*
Rtmv

R


R
1
t 1
t 1
(17)
(18)
(19)
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Proof 2(续)
• Constants are in the time-t information set; but time-t random
variables are not necessarily constant.
• Thus again, unconditional efficiency (including managed
portfolios) implies conditional efficiency but not vice versa.
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Brute Force and Examples
• If a return is on the unconditional MVF it must be on the
conditional MVF at each date.
• The unconditional MVF solves:
min E ( R 2 ) s.t.E ( R)  
(20)
• Writing the unconditional moment in terms of conditional
moments:
min E[ Et ( R 2 )]s.t.E[ Et ( R )]  
(21)
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Remark
•
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Example 1
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Example Presentation
• Return B is conditionally mean-variance efficient. It
also has zero unconditional variance, so it is the
unconditionally mean-variance efficient return at the
expected return shown. Return A is on the conditional
mean-variance frontiers, and has the same
unconditional expected return as B. But return A has
some unconditional variance, and so is inside the
unconditional mean-variance frontier.
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Example 2(Answer to Pro. 1)
•
R f  R*  R f Re*
(22)
R*   R e*
Implications: Hansen-Richard
Critique
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• Models such as the CAPM imply a conditional linear factor
model with respect to investors’ information sets. However,
the best we can hope to do is to test implications conditioned
down on variables that we can observe and include in a test.
Thus, a conditional linear factor model is not testable!
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• Here call this observation the “Hansen-Richard critique” by
analogy to the “Roll Critique.” Roll pointed out, among other
things, that the wealth portfolio might not be observable, making
tests of the CAPM impossible. Hansen and Richard point out
that the conditioning information of agents might not be
observable, and that one cannot omit it in testing a conditional
model. Thus, even if the wealth portfolio were observable, the
fact that we cannot observe agents’ information sets dooms tests
of the CAPM.
8.4 Scaled Factors: a Partial
Solution
•
z t2
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Scaled Factors
•
mt 1  a( zt )  b( zt ) f t 1
 a0  a1 zt  (b0  b1 zt ) f t 1
 a0  a1 zt  b0 f t 1  b1 ( zt f t 1 )
(24)
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•
pt  Et [(a0  a1 zt  b0 f t 1  b1 ( zt f t 1 )) xt 1 ]
 E ( pt )  E[(a0  a1 zt  b0 f t 1  b1 ( zt f t 1 )) xt 1 ]
(25)
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• For example, consider the conditional CAPM:
mt 1  at  bt RtW1
• We can add some scaled factors, say, the
dividend/price ratio and term premium, thus the
conditional CAPM implies an unconditional, fivefactor (plus constant) model. The factors are a
constant, the market return, the dividend/price ratio,
the term premium, and the market return times the
dividend-price ratio and the term premium.
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• The unconditional pricing implications of such a five-factor
model could, of course, be summarized by a single-beta
representation. The reference portfolio would not be the market
portfolio, of course, but a mimicking portfolio of the five factors.
However, the single mimicking portfolio would not be easily
interpretable in terms of a single factor conditional model and
two instruments. In this case, it might be more interesting to look
at a multiple-beta or multiple-factor representation.
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Kronecker Product
•
m  b1 f1  b2 f1 z1  b3 f1 z2 
 bN 1 f 2  bN 2 f 2 z1 
(26)
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8.5 Summary
• It is rather troublesome when facing conditioning
information. But now we get a simple method. To express
the conditional implications of a given model, all we have to
do is include some scaled or managed portfolio returns, and
then pretend we never heard about conditioning information.
• Some factor models are conditional models, and have
coefficients that are functions of investors’ information sets. In
general, there is no way to test such models, but if we are
willing to assume that the relevant conditioning information is
well summarized by a few variables, then we can just add new
factors, equal to the old factors scaled by the conditioning
variables, and again forget that we ever heard about
conditioning information.
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