6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Multiperiod Market Equilibrium
George Pennacchi
University of Illinois
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
1/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Introduction
The …rst order conditions from an individual’s multiperiod
consumption and portfolio choice problem can be interpreted
as equilibrium conditions for asset pricing.
A particular equilibrium asset pricing model where asset
supplies are exogenous is the Lucas (1978) endowment
economy.
We also consider bubbles: nonfundamental asset price
dynamics.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
2/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Asset Pricing in the Multiperiod Model
In the multiperiod model, the individual’s objective is
"T 1
#
X
max Et
U (Cs ; s) + B (WT ; T )
C s ;f! is g;8s ;i
(1)
s =t
which is solved as a series of single period problems using the
Bellman equation:
J (Wt ; t) =
max U (Ct ; t) + Et [J (Wt+1 ; t + 1)]
C t ;f! i ;t g
(2)
This led to the …rst-order conditions
UC (Ct ; t) = Rf ;t Et [JW (Wt+1 ; t + 1)]
= JW (Wt ; t)
(3)
Et [Rit JW (Wt+1 ; t + 1)] = Rf ;t Et [JW (Wt+1 ; t + 1)] ; i = 1; :::; n
(4)
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
3/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Multiperiod Pricing Kernel
This model has equilibrium implications even if assumptions
about utility, income, and asset return distributions do not
lead to explicit formulas for Ct and ! it .
Substituting the envelope condition UC (Ct ; t) = JW (Wt ; t)
at t + 1 into the right-hand side of the …rst line of (3),
UC (Ct ; t) = Rf ;t Et [JW (Wt+1 ; t + 1)]
= Rf ;t Et UC Ct+1 ; t + 1
(5)
Furthermore, substituting (4) into (3) and, again, using the
envelope condition at date t + 1 allows us to write
UC (Ct ; t) = Et [Rit JW (Wt+1 ; t + 1)]
= Et Rit UC Ct+1 ; t + 1
(6)
or
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
4/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Multiperiod Pricing Kernel cont’d
1 = Et [mt;t+1 Rit ]
= Rf ;t Et [mt;t+1 ]
(7)
where mt;t+1 UC Ct+1 ; t + 1 =UC (Ct ; t) is the SDF
(pricing kernel) between dates t and t + 1.
The relationship derived in the single-period context holds
more generally: Updating equation (6) for risky asset j one
period, UC Ct+1 ; t + 1 = Et+1 Rj ;t+1 UC Ct+2 ; t + 2 ,
and substituting in the right-hand side of the original (6), one
obtains
UC (Ct ; t) = Et Rit Et+1 Rj ;t+1 UC Ct+2 ; t + 2
= Et Rit Rj ;t+1 UC Ct+2 ; t + 2
George Pennacchi
Multiperiod Market Equilibrium
(8)
University of Illinois
5/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Multiperiod Pricing Kernel cont’d
or
1 = Et [Rit Rj ;t+1 mt;t+2 ]
(9)
where mt;t+2 UC Ct+2 ; t + 2 =UC (Ct ; t) is the marginal
rate of substitution, or the SDF, between dates t and t + 2.
By repeated substitution, (9) can be generalized to
1 = Et [Rt;t+k mt;t+k ]
(10)
where mt;t+k UC Ct+k ; t + k =UC (Ct ; t) and Rt;t+k is
the return from any trading strategy involving multiple assets
over the period from dates t to t + k.
Equation (10) says that equilibrium expected marginal utilities
are equal across all time periods and assets.
These moment conditions are often tested using a Generalized
Method of Moments technique.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
6/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Lucas Model of Asset Pricing
Lucas (1978) derives equilibrium asset prices for an
endowment economy where the random process generating
the economy’s real output is exogenous.
Output obtained at a particular date cannot be reinvested and
must be consumed immediately.
Assets representing ownership claims on this exogenous
output are in …xed supply.
Assuming all individuals are identical (representative), the
endowment economy assumptions …x the process for
individual consumption, making the SDF exogenous.
Exogenous output implies the market portfolio’s payout
(dividend) is exogenous, which makes it easy to solve for the
equilibrium price.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
7/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Including Dividends in Asset Returns
Let the return on the i th risky asset, Rit , include a dividend
payment made at date t + 1, di ;t+1 , along with a capital gain,
Pi ;t+1 Pit :
di ;t+1 + Pi ;t+1
(11)
Rit =
Pit
Substituting (11) into (7) and rearranging gives
#
"
UC Ct+1 ; t + 1
Pit = Et
(di ;t+1 + Pi ;t+1 )
UC (Ct ; t)
(12)
Similar to what was done in equation (8), substitute for Pi ;t+1
using equation (12) updated one period to solve forward this
equation.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
8/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Including Dividends in Asset Returns cont’d
P it
=
=
Et
Et
"
"
U C C t+1 ; t + 1
U C (C t ; t)
U C C t+1 ; t + 1
U C (C t ; t)
d i ;t+1 +
d i ;t+1 +
U C C t+2 ; t + 2
U C C t+1 ; t + 1
U C C t+2 ; t + 2
U C (C t ; t)
d i ;t+2 + P i ;t+2
d i ;t+2 + P i ;t+2
#
!#
(13)
Repeating this type of substitution, that is, solving forward
the di¤erence equation (13), gives us
3
2
T U
X
C Ct+j ; t + j
UC Ct+T ; t + T
Pit = Et 4
di ;t+j +
Pi ;t+T 5
UC (Ct ; t)
UC (Ct ; t)
j =1
(14)
where the integer T re‡ects a large number of future periods.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
9/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Including Time Preference
If utility is of the form U (Ct ; t) = t u (Ct ), where
= 1+1 < 1, then (14) becomes
2
3
T
uC Ct+j
X
u
C
C
t+T
j
Pit = Et 4
di ;t+j + T
Pi ;t+T 5
uC (Ct )
uC (Ct )
j =1
(15)
Assuming individuals that have in…nite lives or a bequest
u (C
)
motive and limT !1 Et T CuC (Ct+T) Pi ;t+T = 0 (no
t
speculative price “bubbles”), then
3
2
1
uC Ct+j
X
j
di ;t+j 5
Pit = Et 4
uC (Ct )
(16)
j =1
George Pennacchi
Multiperiod Market Equilibrium
10/ 27
University of Illinois
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Dividends and Consumption
j
In terms of the SDF mt; t+j
uC Ct+j =uC (Ct ):
3
2
1
X
Pit = Et 4
mt; t+j di ;t+j 5
(17)
j =1
Lucas (1978) makes (17) into a general equilibrium model by
assuming an in…nitely-lived representative individual and
where risky asset i pays a real dividend of dit at date t.
The dividend is nonstorable and non-reinvestable. With no
wage income, aggregate consumption equals the total
dividends paid by all of the n assets at that date:
Ct =
n
X
dit
(18)
i =1
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
11/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Examples
If the representative individual is risk-neutral, so that
u(C ) = C and uC is a constant (1), then (17) becomes
3
2
1
X
j
di ;t+j 5
Pit = Et 4
(19)
j =1
If utility
Pnis logarithmic (u (Ct ) = ln Ct ) and aggregate dividend
dt = i =1 dit , the price of risky asset i is given by
3
2
1
X
j Ct
di ;t+j 5
Pit = Et 4
Ct+j
j =1
2
3
1
X
d
t
j
= Et 4
di ;t+j 5
(20)
dt+j
j =1
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
12/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Examples cont’d
Under logarithmic utility, we can price the market portfoio
without making assumptions regarding the distribution of dit
in (20).
If Pt is the value of aggregate dividends, then from (20):
2
3
1
X
d
t
j
Pt = Et 4
dt+j 5
dt+j
j =1
= dt
1
(21)
Note that higher expected future dividends, dt+j , are exactly
o¤set by a lower expected marginal utility of consumption,
mt; t+j = j dt =dt+j , leaving the value of a claim on this
output process unchanged.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
13/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Examples cont’d
For more general power utility, u (Ct ) = Ct = , we have
3
2
1
1
X
d
t+j
j
dt+j 5
Pt = Et 4
dt
j =1
2
3
1
X
j
= dt1 Et 4
d 5
t+j
(22)
j =1
which does depend on the distribution of future dividends.
The value of a hypothetical riskless asset that pays a
one-period dividend of $1 is
"
#
1
1
dt+1
Pft =
= Et
(23)
Rft
dt
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
14/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Examples cont’d
We can view the Mehra and Prescott (1985) …nding in its true
multiperiod context: they used equations such as (22) and
(23) with dt = Ct to see if a reasonable value of produces a
risk premium (excess average return over a risk-free return)
that matches that of market portfolio of U.S. stocks’historical
average excess returns.
Reasonable values of could not match the historical risk
premium of 6 %, a result they described as the equity
premium puzzle.
As mentioned earlier, for reasonable levels of risk aversion,
aggregate consumption appears to vary too little to justify the
high Sharpe ratio for the market portfolio of stocks.
The moment conditions in (22) and (23) require a highly
negative value of to …t the data.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
15/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Labor Income
We can add labor income to the market endowment
(Cecchetti, Lam and Mark, 1993). Human capital pays a
wage payment of yt at date t, also non-storable. Hence,
equilibrium aggregate consumption equals
Ct = dt + yt
(24)
so that equilibrium consumption no longer equals dividends.
The value of the market portfolio is:
2
3
P1 j uC Ct+j
Pt = Et 4 j =1
dt+j 5
uC (Ct )
#
"
1
P1 j Ct+j
dt+j
(25)
= Et
j =1
Ct
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
16/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Labor Income cont’d
Now specify separate lognormal processes for dividends and
consumption:
ln Ct+1 =Ct
=
c
+
c t+1
ln (dt+1 =dt ) =
d
+
d "t+1
(26)
where the error terms are serially uncorrelated and distributed
as
0
1
t
~N
;
(27)
"t
0
1
Now what is the equilibrium price of the market portfolio?
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
17/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Labor Income cont’d
When e < 1, the expectation in (25) equals
e
Pt = dt
1
e
where
i
1h
(1
) c+
(1
)2 2c + 2d
(1
d
2
(28)
)
c
d
(29)
(See Exercise 6.3.)
Equation (28) equals (21) when = 0, d = c , c = d ,
and = 1, which is the special case of log utility and no labor
income.
With no labor income ( d = c , c = d , = 1) but 6= 0,
we have = c + 12 2 2c , which is increasing in the growth
rate of dividends (and consumption) when 1 > > 0.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
18/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Labor Income cont’d
When > 0, greater dividend growth leads individuals to
desire increased savings due to high intertemporal elasticity
(" = 1= (1
) > 1). Market clearing requires the value of
the market portfolio to rise, raising income or wealth to make
desired consumption rise to equal the …xed supply.
The reverse occurs when < 0, as the income or wealth e¤ect
will exceed the substitution e¤ect.
For the general case of labor income where is given by
equation (29), a lower correlation between consumption and
dividends (decline in ) increases .
Since @Pt =@ > 0, lower correlation raises the value of the
market portfolio because it is a better hedge against uncertain
labor income.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
19/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Rational Asset Price Bubbles
De…ne pt Pit uC (Ct ), the product of the asset price and the
marginal utility of consumption.
Then equation (12) is
Et [pt+1 ] =
1
pt
Et uC Ct+1 di ;t+1
(30)
1
where
= 1 + > 1 where is the subjective rate of time
preference. The solution (17) to this equation is referred to
as the fundamental solution, which we denote as ft :
2
3
1
X
j
pt = ft
Et 4
uC Ct+j di ;t+j 5
(31)
j =1
The sum in (31) converges if the marginal utility-weighted
dividends are expected to grow more slowly than the time
preference discount factor.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
20/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Rational Asset Price Bubbles cont’d
There are other solutions to (30) of the form pt = ft + bt
where the bubble component bt is any process that satis…es
1
Et [bt+1 ] =
bt = (1 + ) bt
(32)
This is easily veri…ed by substitution into (30):
Et [ft+1 + bt+1 ] =
1
Et [ft+1 ] + Et [bt+1 ] =
1
ft +
Et [bt+1 ] =
1
bt = (1 + ) bt
(ft + bt )
1
bt
Et uC Ct+1 di ;t+1
Et uC Ct+1 di ;t+1
(33)
where in the last line of (33) uses the fact that ft satis…es the
di¤erence equation. Since 1 > 1, bt explodes in expected
value:
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
21/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Bubble Examples
lim Et [bt+i ] = lim (1 + )i bt =
i !1
i !1
+1 if bt > 0
1 if bt < 0
(34)
The exploding nature of bt provides a rationale for interpreting
the general solution pt = ft + bt , bt 6= 0, as a bubble solution.
Suppose that bt follows a deterministic time trend:
bt = b0 (1 + )t
(35)
pt = ft + b0 (1 + )t
(36)
Then the solution
implies that the marginal utility-weighted asset price grows
exponentially forever.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
22/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Bubble Examples cont’d
Next, consider a possibly more realistic modeling of a
"bursting" bubble proposed by Blanchard (1979):
bt+1 =
1+
q
bt + et+1
zt+1
with probability q
with probability 1
q
(37)
with Et [et+1 ] = Et [zt+1 ] = 0.
The bubble continues with probability q each period but
“bursts” with probability 1 q.
This process satis…es the condition in (32), so that
pt = ft + bt is again a valid bubble solution, and the expected
return conditional on no crash is higher than in the in…nite
bubble.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
23/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Likelihood of Rational Bubbles
Additional economic considerations may rule out many
rational bubbles: consider negative bubbles where bt < 0.
From (34) individuals must expect that at some future date
> t that p = f + b can be negative.
Since marginal utility is always positive, Pit = pt =uC (Ct ),
must be expected to become negative, which is inconsistent
with limited-liability securities.
Similarly, bubbles that burst and start again can be ruled out.
Note that a general process for a bubble can be written as
bt = (1 + )t b0 +
t
X
(1 + )t
s
"s
(38)
s =1
where "s , s = 1; :::; t are mean-zero innovations.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
24/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Likelihood of Rational Bubbles cont’d
To avoid negative values of bt (and negative expected future
prices), realizations of "t must satisfy
"t
(1 + ) bt
1,
8t
0
(39)
This is due to
"t
= bt
(1 + ) bt
1
bt
= "t + (1 + ) bt
1
"t
(1 + ) bt
>0
1
For example, if bt = 0 so that a bubble does not exist at date
t, then from (39) and the requirement that "t+1 have mean
zero, it must be the case that "t+1 = 0 with probability 1.
Hence, if a bubble currently does not exist, it cannot get
started.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
25/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Likelihood of Rational Bubbles cont’d
Moreover, the bursting and then restarting bubble in (37)
could only avoid a negative value of bt+1 if zt+1 = 0 with
probability 1 and et+1 = 0 whenever bt = 0. Hence, this
type of bubble would need to be positive on the …rst trading
day, and once it bursts it could never restart.
Tirole (1982) considers an economy model with a …nite
number of agents and shows that rational individuals will not
trade assets at prices above their fundamental values.
Santos and Woodford (1997) consider rational bubbles in a
wide variety of economies and …nd only a few examples of the
overlapping generations type where they can exist.
If conditions for rational bubbles are limited yet bubbles seem
to occur, some irrationality may be required.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
26/ 27
6.1: Pricing
6.2: Lucas
6.3: Bubbles
6.4: Summary
Summary
If an asset’s dividends are modeled explicitly, the asset’s price
satis…es a discounted dividend formula.
The Lucas endowment economy takes this a step further by
equating aggregate dividends to consumption, simplifying
valuation of claims on aggregate dividends.
In an in…nite horizon model, rational asset price bubbles are
possible but additional aspects of the economic environment
can often rule them out.
George Pennacchi
Multiperiod Market Equilibrium
University of Illinois
27/ 27