Chapter 17: Conditioning
information
Fan Longzhen
Part 3: deep discussion: Estimating and evaluating
asset pricing models
Discount model is in terms of
conditional moments
? The first order condition is
? The expectation is conditional expectation on investor’s time t
information;
? The basic pricing equation is
? If the payoff were independent and identically distributed over time,
then the conditional expectation is the same as unconditional
expectation, we would not worried about the distinction;
? Because the information may be not observable to economist,
describing the conditional distribution is not possible, we eventually
have to think about unconditional information;
? Unconditional moments are interesting themselves, for example, we
may concern why average stock return are high than risk free rate.
])('[)('
11 ++
=
ttttt
xcuEcup β
)(
11 ++
=
tttt
xmEp
Scaled payoffs
? From the pricing model
? Take unconditional expectations to obtain
? Multiply the payoff and price by any variable or instruments
observed at time t, and take expectations
? Sufficiency of adding scaled returns:
? If
)(
11 ++
=
tttt
xmEp
)()(
11 ++
=
ttt
xmEpE
t
z
)()(
11 ttttt
zxmEzpE
++
=
)()()(
1111 ttttttttttt
IxmEpIzzxmEpzE
++++
=?∈?=
Conditional and unconditional models in discount
factor language
? As an example, consider the CAPM,
? Where is the return on the market or wealth portfolio
? From equation
? Above equation shows explicitly that a and b must vary over time, as
? vary over time;
? If it is to price asset conditionally, the CAPM must be a linear factor
model with time-varying weights of the form
? Form the conditional pricing model
? And take unconditional expectation, we have
W
bRam ?=
W
R
?
?
?
?
?
?
?
?
=
+=
?
?
?
?
?
?
=
=
+
+
+
+
++
)(
)(
)(
1
)(1
)(1
1
2
1
1
1
11
W
tt
f
t
f
t
W
tt
W
tt
f
t
f
ttt
W
ttt
RR
RRE
b
RbE
R
a
RmE
RmE
σ
f
t
W
tt
W
tt
RRRE ),(),(
1
2
1 ++
σ
W
tttt
Rbam
11 ++
?=
])[(1
11 ++
?=
t
W
tttt
RRbaE
continued
? The conditional model
? only holds if the covariance terms happen to be zero, this is generally
not right.
? On the other hand, suppose that it is true that are constant over
time, even if are not i.i.d., can be constant over time.
Then
? Does implies
),cov(),cov()()()()(
])[(1
111111
11
++++++
++
?+?=
+=
t
W
ttttt
W
tttt
t
W
ttt
RRbRaRREbEREaE
RRbaE
]))()([(1
11 ++
?=
t
W
ttt
RRbEaEE
tt
ba ,
W
t
f
t
RR , tt
ba ,
])[(1
11 ++
?=
t
W
tt
RbRaE
])[(1
11 ++
?=
t
W
t
RbRaE
Conditional and unconditional
in an expected return-beta model
? To put the observation into beta pricing language
? Does not imply
? Even if is a constant, you cannot derive the pricing
model also, because
t
i
t
f
t
i
tt
RRE λβ+=
+
)(
1
λβγ
ii
tt
RE +=
+
)(
1
λλ =
t
)](var/),([cov)var(/),cov(
111111 ++++++
≠=
ttt
i
tttt
i
t
i
ffREffRβ
A partial solution: scaled factors
? How to handle the parameters change with time ;
? A partial solution is to model the dependence of parameters
on variables in the time t information set;
? Let be a vector of variables observed at time t;
? In particular, a linear function of ;
? The discount factor is now as follows:
?
? In place of the one-factor model with time-varying coefficients, we
have a three-factor with fixed coefficients.
tt
ba ,
tt
ba ,
t
z
t
z
)(
)(
)()(
111010
11010
11
++
+
++
+++=
+++=
+=
tttt
ttt
tttt
fzbfbzaa
fzbbzaa
fzbzam
),,(
11 ++ tttt
fzfz
continued
? Since the coefficients are now fixed, we can use the scale-factor model
with unconditional moments:
]))([()(
]))([(
1111010
1111010
+++
+++
+++=
?+++=
tttttt
ttttttt
xfzbfbzaaEpE
xfzbfbzaaEp