gr5215 fall 2021 lec 12 Consumption and asset pricing
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Consumption and asset pricing
Miller GR5215
Consumption and asset pricing
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Hall looked at uncertainty about income –
but can also look at risky assets. Setup is
similar but let’s now look at how consumers
would respond to different risky assets
In other words, what if there’s not just
income uncertainty, but also interest rate
uncertainty?
Question was addressed in GE by Breeden
(1979) and Lucas (1978), and in PE by
Stiglitz (1970)
A risky asset
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Dt+1
i ,D
t+2
i ,…
Suppose we have a bunch of assets,
indexed by i
One unit of asset i produces a stream of
(real) random payoffs:
And has price: Pt
i
D is for dividend. Assets like this are
sometimes called “fruit trees” – every year
they produce a crop of consumable goods.
The Euler equation
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Apply the usual perturbation argument; here
the consumer must be neutral between
buying another unit of any asset and not.
U = E
1
(1+ ρ)t
u(C
t
)
t=1
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥ ρ>0
′u (C
t
)P
t
i = E
t
1
(1+ ρ)k
′u (C
t+k
)D
t+k
i
k=1
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥
And this has to hold for every asset i
The Euler equation (II)
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′u (C
t
)P
t
i = E
t
1
(1+ ρ)k
′u (C
t+k
)D
t+k
i
k=1
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥
Suppose we hold for one period and sell.
Define:
r is the return to the strategy of buying a
unit of asset i, collecting next period’s
dividend and then reselling the asset
r
t+1
i ≡
D
t+1
i +P
t+1
i
P
t
i
−1
The general Euler equation
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Then:
′u (C
t
)= 1
1+ ρ
E
t
[(1+ r
t+1
i ) ′u (C
t+1)]
So:
′u (C
t
)= 1
1+ ρ
E
t
[1+ r
t+1
i ]E
t
[ ′u (C
t+1)] + cov(1+ rt+1
i , ′u (C
t+1)){ }
Where the covariance is conditional on
time t information
r
t+1
i ≡
D
t+1
i +P
t+1
i
P
t
i
−1
Regard the model as GE
7
′u (C
t
)= 1
1+ ρ
E
t
[1+ r
t+1
i ]E
t
[ ′u (C
t+1)] + cov(1+ rt+1
i , ′u (C
t+1)){ }
Note that this expression needs to hold for every
risky asset that is held in positive amounts: it is
more general than the fruit tree model (the model
where the trees are the only source of output)
But, in the fruit tree model, if all individuals are
the same, we can now solve for the price of assets
in economy
Asset prices
8
Recall:
Thus:
′u (C
t
)P
t
i = E
t
1
(1+ ρ)k
′u (C
t+k
)D
t+k
i
k=1
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥
P
t
i = E
t
1
(1+ ρ)k
′u (C
t+k
)
′u (C
t
)
D
t+k
i
k=1
∞
∑
⎡
⎣
⎢
⎤
⎦
⎥
The factors in front of the D under the
summation are together called the
stochastic discount factor or the asset
pricing kernel
Called the consumption-based capital asset
pricing model
Quadratic utility- an informative
special case
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Recall:
With quadratic utility:
′u (C
t
)= 1
1+ ρ
E
t
[1+ r
t+1
i ]E
t
[ ′u (C
t+1)] + cov(1+ rt+1
i , ′u (C
t+1)){ }
E
t
[1+ r
t+1
i ]= 1
E
t
[ ′u (C
t+1)]
[(1+ρ) ′u (C
t
)+acov(1+ r
t+1
i ,(C
t+1)]
Where a is the utility function parameter as
in Hall
Quadratic utility (II)
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If there is a risk-free asset, then:
So:
i.e.: expected return premium relative to the risk-
free rate is proportional to the covariance of its
return with consumption
1+ r
t+1 =
(1+ ρ) ′u (C
t
)
E
t
[ ′u (C
t+1)]
E
t
[r
t+1
i ]− r
t+1 =
acov(1+ r
t+1
i ,C
t+1)
E
t
[ ′u (C
t+1)]
E
t
[1+ r
t+1
i ]= 1
E
t
[ ′u (C
t+1)]
[(1+ρ) ′u (C
t
)+acov(1+ r
t+1
i ,(C
t+1)]
Empirical issues with CB-CAPM
11
The equity premium puzzle: Mehra and
Prescott (1985)
Recall in the quadratic
case
M+P worked in a more general case, but the key
issue is that the equity premium is historically about
6% and there isn’t enough variance in aggregate
consumption to explain that using our Euler equation
and a reasonable coefficient of RRA (see Romer)
E
t
[r
t+1
i ]− r
t+1 =
acov(1+ r
t+1
i ,C
t+1)
E
t
[ ′u (C
t+1)]
Empirical issues with CB-CAPM
(II)
12
But if you assume the coefficient of relative
risk aversion is high enough (or try to fix
the model in similar ways):
You find that implies the risk-free rate is
very high (“the risk-free rate puzzle”)
1+ r
t+1 =
(1+ ρ) ′u (C
t
)
E
t
[ ′u (C
t+1)]
Empirical issues with CB-CAPM
(III)
13
Hansen and Singleton (1983) went after the
Euler equation directly:
If you have a specific utility function, you can
estimate a system of these equations for
different assets, and aggregate consumption
′u (C
t
)= 1
1+ ρ
E
t
[1+ r
t+1
i ]E
t
[ ′u (C
t+1)] + cov(1+ rt+1
i , ′u (C
t+1)){ }
Empirical issues with CB-CAPM
(IV)
14
Strategy of estimation is to set up something
that has expected value of zero conditional
on time t info and then use time t variables
as instrumental variables (GMM)
The fact that there are many possible
instruments allows for testing overidentifying
restrictions – these are rejected strongly
They also find a very low estimated
intertemporal elasticity of substitution