Thomas J. Sargent

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Research Papers in Economics

FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 301
Commentary
Thomas J. Sargent
and his coauthors begin from the observation that
it is difficult to distinguish that specification
from an alternative one in which the drift in log
consumption growth is itself a highly persistent
covariance stationary process with low condi-
tional volatility but high unconditional volatility.
Thus, the drift itself is almost but not quite a ran-
dom walk. The high unconditional volatility of
the drift confronts the representative consumer
with what Bansal and his coauthors call long-run
risk because the conditional mean of consumption
growth is not constant but wanders.
Bansal also posits that cash flows on particu-
lar portfolios differ in the extent to which they are
subject to long-run risks that are more or less cor-
related with the long-run risk in aggregate con-
sumption. For example, Bansal and coauthors as
well as Hansen, Heaton, and Li (2006) have offered
evidence that the cash flows from Fama and
French’s high book-to-market portfolios have long-
run components that are more highly correlated
with long-run components of consumption than
are the cash flows from low book-to-market port-
folios. Can the need to compensate the representa-
tive consumer for that higher long-run correlation
with consumption explain why those high book-
to-market portfolios have higher returns?
PREFERENCES
The answer is no if the representative con-
sumer’s preferences are usual ones assumed by
I
n several recent papers—including the
paper from this conference, Bansal (2007)—
Ravi Bansal and his coauthors have con-
structed an interpretation of some asset
pricing puzzles that I think macroeconomists
should pay attention to. Why? Because a repre-
sentative agent’s consumption Euler equation
that links a one-period real interest rate to the
consumption growth rate is the “IS curve” that
is central to the policy transmission mechanism
in today’s New Keynesian models. A long list
of empirical failures called puzzles come from
applying the stochastic discount factor implied
by that Euler equation. Until we succeed in get-
ting a consumption-based asset pricing model
that works well, the New Keynesian IS curve is
built on sand.
In several exciting papers, Bansal and his
coauthors propose a way to explain some of those
asset pricing puzzles by (i) specifying the inter-
temporal structure of risks to put long-run risks
into consumption and assets’ cash flows and (ii)
altering preferences to make the representative
consumer care more about those long-run risks.
LONG-RUN RISK
Let ct be the logarithm of aggregate consump-
tion. A workhorse model that does a good job of
fitting per capita U.S. consumption of nondurables
and services makes ct a random walk with con-
stant drift and i.i.d. Gaussian innovations. Bansal
Thomas J. Sargent is a professor of economics and business at New York University, a senior fellow at the Hoover Institution, and the president
of the American Economic Association.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 301-03.
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only with prior written permission of the Federal Reserve Bank of St. Louis.macroeconomists—for example, time-separable
logarithmic preferences with discount factor
β ￿ (0, 1). Why? Those preferences lead to the
usual stochastic discount factor whose logarithm is
which has the property that the representative
consumer just doesn’t care enough about those
long-run risks to pump up the returns on those
high book-to-market portfolios. Therefore, Bansal
and his coauthors adopt a preference specifica-
tion of Epstein and Zin (1989) that separates the
intertemporal elasticity of substitution (IES) from
the reciprocal of the coefficient of relative risk
aversion, both of which are unity for the log
preference specification mentioned above. With
the IES held fixed at 1, those preferences imply a
stochastic discount factor whose logarithm is
where γ is the coefficient of relative risk aversion
and Et denotes mathematical expectation condi-
tioned on time-t information. Setting γ > 1 adds
forward-looking terms to the stochastic discount
factor that make the representative consumer care
today about rates of log consumption growth far
in the future. For a γ high enough, the representa-
tive consumer does have to be compensated for
long-run cash flow risk correlated with long-run
consumption growth risk in amounts that are big
enough to explain how the market prices those
Fama and French portfolios. Furthermore,
Tallarini (2000), Bansal and Yaron (2004) and
others have shown that with a high enough γ
this kind of preference specification can provide
a neat explanation for both the risk-free rate and
the equity premium puzzles.
ASSESSMENT
It is possible to reinterpret the above stochas-
tic discount factor with IES = 1 and atemporal
sc
EE c
tt t
tt
j
tj
++
+
=
+
=
( ) ( ) ∑
11
1
0
, log￿
–– 1
β
γ
–
–
∆
∆
`
+ +
++ +
=
( ) − ( )




∑
1
2
11
0
1
2
1 – – γβ vart
j
tt t j
j
EE c ∆
`
 
,
sc tt t ++ = 11 , log￿ , β – ∆
risk aversion γ > 1 in terms of a representative
consumer who has IES and risk aversion both
equal to 1, but where now γ > 1 measures his
doubts about the stochastic specification of his
model for consumption growth and cash flows.
This reinterpretation is achieved by noting that
the continuation value in Epstein and Zin’s for-
mulas equals the indirect utility function for a
robust valuation problem in which a malevolent
nature helps the decisionmaker construct valua-
tions that are robust to misspecification by choos-
ing a worst-case model from a set of models
surrounding the decisionmaker’s approximation
model. Now γ acquires the interpretation of a
penalty parameter on the relative entropy between
the approximating and the distorted model. The
additional terms in the log stochastic discount
factor that appear when γ > 1 encode the likeli-
hood ratio of the worst case to the approximating
model.1 This reinterpretation is of special interest
for the work of Bansal and his coauthors, who
reason as follows:
1. Statistically, it is difficult to distinguish a
stochastic specification with long run-risk
from one without it.
2. Therefore, without attributing wacky ideas
to their representative consumer, Bansal
and coauthors can assume that the repre-
sentative consumer assigns probability 1
to the long-run risk model and takes our
original i.i.d. log consumption growth
model off the table.
3. Besides, by using the rational expectations
cross-equation restrictions associated with
the consumption Euler equation, we can
infer that the representative consumer has
to believe the long-run risk specification
to explain the asset pricing data.
There is more to say here. Although long-run
risks are difficult to detect (assertion 1), Epstein-
Zin preferences or concerns about robustness
make it vital for the representative consumer to
302 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
1 Barillas, Hansen, and Sargent (2007) show that by interpreting γ
as measuring fear of model misspecification rather than risk-
aversion, a moderate fear of model misspecification can do most
of the job of the large risk-aversion parameters of Tallarini (2000)
and Bansal and Yaron (2004) in explaining the equity premium.
Sargentcare about them. The tenuous part of this argu-
ment is how the representative consumer can
come to be sure about the presence of these long-
run risks when they are so difficult to detect sta-
tistically. Assertion 3 differs from arguments in
the least-squares learning literature that typically
have an agent learn about a forcing process by
way of a least-squares learning algorithm on that
process itself, not by using prices and the rational
expectations cross-equation restrictions to reverse
engineer what that process must be.
The robustness interpretation can help with
these learning issues. Hansen and Sargent (2007)
address these in the context of a model with a
representative consumer who responds to asser-
tion 1 by leaving both the i.i.d. and long-run risk
models for log consumption growth on the table,
attaching a prior initialized at the equal ignorance
value of 0.5 to the long-run risk model, then
updating by Bayes’ law. We show that a consumer
who distrusts both submodels and the posterior
over submodels that emerges from Bayes’ law
will behave in a way that supports much of what
Bansal and his coauthors do. In particular, because
the long-run risk model is worse for the repre-
sentative consumer, his worst-case probabilities
become slanted toward that model and possibly
put almost all of the mass on that model. This is
nice because it provides an alternative defense
of Bansal’s assumption that the representative
consumer acts as if he puts probability 1 on the
long-run risk model.
But the structure of Hansen and Sargent (2007)
yields other interesting outcomes, too. Even when
the robust investor slants his worst-case probabil-
ity to put probability 1 on the long-run risk model,
the gap between the ordinary Bayesian probabil-
ity and this worst-case probability contributes a
potential source of time-varying countercyclical
risk premia.
CONCLUSION
Bansal and Yaron’s idea of stressing long-run
risks that are difficult to detect but, with Epstein-
Zin preferences or fear of model misspecification,
easy to care about is worth taking seriously. When
many macroeconomists are now busy attaching
loosely interpreted shocks or wedges to agents’
first-order condition to make our dynamic sto-
chastic general equilibrium (DSGE) models fit
better, I welcome Bansal’s new approach.
REFERENCES
Bansal, Ravi. “Long-Run Risks and Financial
Markets.” Federal Reserve Bank of St. Louis
Review, July/August 2007, 89(4), pp. 283-99.
Bansal, Ravi and Yaron, Amir. “Risks for the Long
Run: A Potential Resolution of Asset Pricing
Puzzles.” Journal of Finance, August 2004, 59(4),
pp. 1481-509.
Barillas, Francisco; Hansen, Lars Peter and Sargent,
Thomas J. “Doubts or Variability?” Working paper,
University of Chicago and New York University,
2007.
Epstein, Larry G. and Zin, Stanley E. “Substitution,
Risk Aversion, and the Temporal Behavior of
Consumption and Asset Returns: A Theoretical
Framework.” Econometrica, July 1989, 57(4),
pp. 937-69.
Hansen, Lars Peter; Heaton, John and Li, Nan.
“Consumption Strikes Back: Measuring Long Run
Risk.” Working paper, University of Chicago, 2006.
Hansen, Lars Peter and Sargent, Thomas J. “Fragile
Beliefs and the Price of Model Uncertainty.”
Unpublished manuscript, 2007.
Tallarini, Thomas D. Jr. “Risk-Sensitive Real Business
Cycles.” Journal of Monetary Economics, June
2000, 45(3), pp. 507-32.
Sargent
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 303304 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

Consumption Strikes Back: Measuring Long Run Risk.” Working paper,

Doubts or Variability?” Working paper,

Fragile Beliefs and the Price of Model Uncertainty.”

Long-Run Risks and Financial Markets.” Federal Reserve Bank of St. Louis Review,

Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework.” Econometrica,

Risk-Sensitive Real Business Cycles.”

Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles.”

Commentary on "Long-run risks and financial markets"

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Commentary on "Long-run risks and financial markets"

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