1
The equity premium puzzle
Consider a one period model of consumption choice in Arrow Debreu Markets
u(c0 ) + β
max
c ,c
s
0
subject to :
X
c0 +
q(s)cs ≤ W0
"
X
#
p(s)u(cs )
s
s
where p(s) is the probability of state s and q(s) is the price of an “Arrow
Debreu” security that pays 1 unit of consumption if state s is realized and 0
otherwise. Attach a Lagrange Multiplier λ:
max
u(c0 ) + β
c ,c
0
s
"
X
s
#
"
p(s)u(cs ) − λ c0 +
X
s
q(s)cs − W0
#
First order condition:
u0 (c0 ) = λ
βp(s)u0 (cs ) = λq(s) for all s
Combining:
β
u0 (cs )
q(s)
=
for all s
u0 (c0 )
p(s)
Adding up over all s gives:
β
X
s
p(s)
u0 (cs ) X
q(s) X
1
p(s)
q(s)
=
=
=
u0 (c0 )
p(s)
1 + rf
s
s
(1)
The last equality follows since a riskless bond pays 1 in all states of the world
(irrespective of s). Equation (1) can be re-written as:
³
β 1+r
f
´
"
#
u0 (cs )
E 0
=1
u (c0 )
(2)
Moreover, letting Dt+1 (s) denote the dividends of a stock and Pt its price, we
have that:
1
Pt =
X
q(s) [Dt+1 (s) + Pt+1 (s)] =
s
X
q(s)
[Dt+1 (s) + Pt+1 (s)] =
p(s)
s
X
u0 (cs )
=
[Dt+1 (s) + Pt+1 (s)]
p(s)β 0
u (c0 )
s
=
p(s)
(3)
The gross rate of return is defined as:
Rt+1 =
Dt+1 (s) + Pt+1 (s)
Pt
(4)
Combining (3) with (4) implies the stochastic Euler Equation:
"
#
u0 (cs )
Rt+1 (s) = 1
E β 0
u (c0 )
(5)
Combine (2) and (5) to obtain:
"
#
³
´i
u0 (cs ) h
Rt+1 (s) − 1 + rf
=0
E 0
u (c0 )
(6)
Define:
rs = Rt+1 − 1
to rewrite (6) as:
"
#
i
u0 (cs ) h s
E β 0
r − rf = 0
u (c0 )
Now:
"
Ã
#
!
Ã
i
³
´
u0 (cs ) h s
u0 (cs )
u0 (cs )
r − rf = E rs − rf E β 0
−cov rs − rf , β 0
0=E β 0
u (c0 )
u (c0 )
u (c0 )
!
Divide appropriately to obtain:
³
s
E r −r
f
´
Ã
u0 (cs )
= cov r − r ,
Eu0 (cs )
s
f
!
<–– Consumption CAPM
To show the equity premium puzzle, assume that agents have constant relative
risk aversion, so that:
c1−γ
u(c) =
1−γ
and assume that returns and consumption are joint lognormal:
2
σ2S
+ σ S ε1 , ε1 ∼ N (0, 1)
2
σ2
log (ct+1 ) = log(ct ) + µc − c + σ c ε2 , ε2 ∼ N (0, 1)
2
log (Rt+1 ) = µ −
(7)
(8)
and
cov (ε1 , ε2 ) = κ
The goal will be to use (2) and (5) to obtain the riskfree rate rf and the equity
premium µ − rf in such an economy.
To obtain the riskfree rate one can substitute into (2) and use (8) to obtain:
³
1+r
f
´
"
³
log(β)−γ µc −
E e
σ2
c +σ ε
c 2
2
´#
=1
By properties of the lognormal:
"
³
log(β)−γ µc −
E e
so that:
³
log 1 + r
So
f
´
|
{z
}
´#
Ã
³
log(β)−γ µc −
=e
Ã
σ2
+ log(β) − γ µc − c
2
rf = − log(β) +
time pref. effect
σ2
c +σ ε
c 2
2
Ã
σ2
γ µc − c
2
|
{z
!
´
2
+ γ2 σ2c
!
γ2
+ σ 2c = 0
2
!
}
σ2
c
2
Borrowing against future growth
γ2 2
σ
2 c}
| {z
−
Precautionary savings
Following similar steps, one can substitute (7) and (8) into (5) to obtain:
µ − rf = γσ c κσ S
Some back of the envelope calculations based on US data (data for most free
market economies are similar):
σ c = 0.03
σS = 0.22
κ = 0.1 − 0.3
µ − r = 0.05
r = 0.02
Using these values implies a γ close at about 37. < − Equity premium puzzle.
Riskless rate implied by that γ is at least 9.4%. With a rate of time preference
equal to 1% the implied interest rate is 10.4% <–— Low risk free rate puzzle.
3
2
Incomplete markets - Limited Participation
Let there be two categories: stockholders (S) and non-stockholders (B). Both
have logarithmic preferences:
Et
Z T
t
e−r(s−t) log(cs )ds
and the model is set up in continuous time for simplicity. Consider now a finite
T Lucas Tree economy (i.e. no labor income, no investment) with a single tree
paying dividends equal to the aggregate endowment Dt :
dDt
= µD dt + σ D dZt
Dt
where µD is the rate of growth of the endowment, σ D the standard deviation
of the endowment, and dZt represents the increment to a brownian motion,
i.e.:
√
Zt − Zs ∼ N(0, t − s) for all t > s
Bonds are in zero net supply. The only positive supply asset is the tree with
price Pt .Take all the shareholders and define their wealth as
WtS = PtS + BtS
where PtS represents the value of the shares of the tree that is held by stockholders and BtS denotes the holdings of bonds by shareholders. Similarly, for
bondholders:
WtB = PtB + BtB = BtB
where the last equality follows from the fact that bondholders hold no stock
and hence all of their wealth is in the form of bonds. Aggregate wealth is given
as:
Wt ≡ WtS + WtB = PtS + BtS + BtB = Pt
(9)
where the last equality follows from the fact that bonds are in 0 net supply so
that:
(10)
BtS + BtB = 0
and the stockholders hold the entire stock market:
PtS = Pt
Equilibrium in the goods market implies that:
CtS + CtB = Dt
(11)
We will use the following fact about logarithmic preferences: Irrespective of
whether an agent is a shareholder or not their optimal consumption is:
4
CtS =
CtB =
ρ
1−
e−ρ(T −t)
ρ
1 − e−ρ(T −t)
WtS
WtB
Hence, combining (11) with the previous two equations and invoking (9) gives:
Dt = CtS + CtB =
³
ρ
1−
e−ρ(T −t)
Accordingly
Pt =
ÃZ
T
´
WtS + WtB = R T
−ρ(s−t)
e
t
t
1
e−ρ(s−t) ds
Pt
!
ds Dt
Time differentiate
Ã
!
Z T
d Z T −ρ(s−t)
e
ds = −1 + ρ
e−ρ(s−t) ds
dt t
t
Hence:
Ã
dPt = −1 + ρ
Z T
−ρ(s−t)
e
t
!
ds Dt dt +
ÃZ
T
−ρ(s−t)
e
t
!
ds Dt (µD dt + σ D dZt ) =
= −Dt dt + ρPt dt + Pt (µD dt + σ D dZt )
Hence, the rate of return in the stock marker is
dRt ≡
dPt + Dt dt
= ρdt + µD dt + σ D dZt
Pt
Take expectations to find the expected return per unit of time dt :
µ≡
E(dRt )
= ρ + µD
dt
(12)
Similarly, the volatility in the stock market is
σ ≡ stdev(dRt ) = σ D
(13)
Having found the expected return and the volatility in the stock market, it remains to determine the riskfree rate in order to determine the equity premium
µ − r. To do that, we shall use the following property for the log shareholder
that we give without proof: The fraction π of wealth that a log investor invests
in the stock market is given by:
π=
µ−r
σ2
5
Therefore the amount of bonds that are held by a stockholder equals:
BtS
= (1 −
π)WtS
µ
Ã
¶
!
µ−r
ρ + µD − r
S
= 1−
W
=
1
−
WtS
t
2
2
σ
σD
where the last equality follows from (12) and (13). Clearing in the bond market
(10) implies that:
BtB
+
BtS
=
WtB
Ã
!
ρ + µD − rt
+ 1−
WtS = 0
σ 2D
(14)
It will be convenient to define:
xt =
WtB
Wt
so that equation (14) becomes:
Ã
!
ρ + µD − rt
(1 − xt ) = 0
xt + 1 −
σ 2D
or:
or:
(1 − xt ) (ρ + µD − rt ) = σ 2D
ρ + µD − rt =
σ 2D
1 − xt
rt = ρ + µD −
σ 2D
1 − xt
or:
Compare with the case where everyone holds stock:
rt = ρ + µD − σ 2D
Note that non-participation does the right thing. It pushes down the real
interest rate. Similarly, using (12) the equity premium is
µ − rt = (ρ − µD ) − (ρ − µD ) +
σ 2D
σ 2D
=
1 − xt
1 − xt
Compare with the case where everyone participates:
µ − rt = σ 2D
As xt → 1 the equity premium becomes unboundedly large, which helps resolve
the equity premium puzzle
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Remark 1 Some problems with limited stock market participation:
Wealth distribution imlpied by the model is non-stationary (asymptotically
non-participants die out)
Analytically intractable if risk aversion of both agents 6= 1.
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