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(21.350)
»(st ) =
¯ 1 ’ 2 (st ’ s) ’ 1
¯
Sr
γ
¯ (21.351)
S=σ
1’φ
The extra complication of (21.349) rather than (21.347) means consumption is always above
habit, since S = es > 0. Other habit models can give consumption below habit which leads
to in¬nite or imaginary marginal utility.
St becomes the single state variable in this economy. Time-varying expected returns,
price-dividend ratios, etc. are all functions of this state variable.
Marginal utility is
uc (Ct , Xt ) = (Ct ’ Xt )’γ = St Ct .
’γ ’γ


The model assumes an external habit “ each individual™s habit is determined by everyone
else™s consumption, as in Abel™s (1990) “keeping up with the Joneses” speci¬cation. This
is mostly a technical simpli¬cation, since it allows us to ignore terms by which current con-
sumption affect future habits; the opposite speci¬cation gives very similar results (see prob-
lem 2).
With marginal utility, we now have a discount factor.
µ ¶’γ
uc (Ct+1 , Xt+1 ) St+1 Ct+1
Mt+1 ≡ δ =δ .
uc (Ct , Xt ) St Ct
Since we have a stochastic process for S and C, and each is lognormal, we can evaluate the
conditional mean of the discount factor to evaluate the riskfree rate
1
f
(352)
rt = ’ ln Et (Mt+1 ) = ’ ln(δ) + γg ’ γ(1 ’ φ).
2
We gave up on analytic solutions and evaluated the price-dividend ratio as a function of the
state variable by iteration on a grid:
· µ ¶¸
Pt Ct+1 Pt+1
(st ) = Et Mt+1 1+ (st+1 )
Ct Ct Ct+1
With price-dividend ratios, we can calculate returns, expected returns, etc.
How does it work “ equity premium and predictability
We choose parameters, simulate 100,000 arti¬cial data points, and report standard statis-
tics and tests in arti¬cial data. The parameters g = 1.89, σ = 1.50, rf = 0.94 match their
values in postwar data. The parameter φ = 0.87 matches the autocorrelation of the price-
¯
dividend ratio and the choice γ = 2.00 matches the postwar Sharpe ratio. δ = 0.89, S =
0.57 follow from the model.

426
SECTION 21.2 NEW MODELS


Table 2cc presents means and standard deviations predicted by the model. The model
replicates the postwar Sharpe ratio, with a constant 0.94% risk free rate and a reasonable
subjective discount factor δ < 1. Of course, we picked the parameters to do this, but given
the above equity premium discussion it™s already an achievement that we are able to pick any
parameters to hit these moments.
Some models can replicate the Sharpe ratio, but do not replicate the level of expected
returns and return volatility. E = 1% and σ = 2% will give the right Sharpe ratio, but this
model predicts the right levels as well. The model also gets the level of the price-dividend
ratio about right.

Table 2cc. Means and standard deviations of simulated and historical data.
Consumption Dividend Postwar
Statistic claim claim data
0.50 0.50
E(R ’ R )/σ(R ’ Rf )
f

6.64 6.52 6.69
E(r ’ rf )
15.2 20.0 15.7
σ(r ’ rf )
18.3 18.7 24.7
exp[E(p ’ d)]
0.27 0.29 0.26
σ(p ’ d)

The model is simulated at a monthly frequency; statistics are calculated from arti-
¬cial time-averaged data at an annual frequency. Asterisks (*) denote statistics that
model parameters were chosen to replicate. All returns are annual percentages.

Table 5cc shows how the arti¬cial data match the predictability of returns from price-
dividend ratios. The paper goes on, and shows how the model matches the volatility test result
that almost all return variation is due to variation in expected excess returns, the “leverage
effect” of higher volatility after a big price decline, and several related phenomena.

Table 5cc. Long-horizon return regressions
Horizon Cons. claim Postwar data
(Years) 10—coef. R2 10—coef. R2
1 -2.0 0.13 -2.6 0.18
2 -3.7 0.23 -4.3 0.27
3 -5.1 0.32 -5.4 0.37
5 -7.5 0.46 -9.0 0.55
7 -9.4 0.55 -12.1 0.65

How does it work?
How does this model get around all the equity premium - riskfree rate dif¬culties de-
scribed above, and explain predictability as well?
When a consumer has a habit, local curvature depends on how far consumption is above

427
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


the habit, as well as the power γ,
’Ct ucc (Ct ’ Xt ) γ
·t ≡ =.
uc (Ct ’ Xt ) St
As consumption falls toward habit, people become much less willing to tolerate further falls
in consumption; they become very risk averse. Thus a low power coef¬cient γ can still mean
a high, and time-varying curvature. Recall our fundamental equation for the Sharpe ratio,
f
Et (r) ’ rt
= · t σt (∆c)corrt (∆c, r).
σ t (r)
High curvature ·t means that the model can explain the equity premium, and curvature ·t that
varies over time as consumption rises in booms and falls toward habit in recessions means that
the model can explain a time-varying and countercyclical (high in recessions, low in booms)
Sharpe ratio, despite constant consumption volatility σt (∆c) and correlation corrt (∆c, r).
So far so good, but didn™t we just learn that raising curvature implies high and time-
varying interest rates? This model gets around interest rate problems with precautionary sav-
ing. Suppose we are in a bad time, in which consumption is low relative to habit. People want
to borrow against future, higher, consumption, and this force should drive up interest rates.
(In fact, many habit models have very volatile interest rates.) However, people are also much
more risk averse when consumption is low. This consideration induces them to save more, in
order to build up assets against the event that tomorrow might be even worse. This desire to
save drives down interest rates. Our »(s) speci¬cation makes these two forces exactly offset,
leading to constant real rates.
The precautionary saving motive also makes the model more plausibly consistent with
variation in consumption growth across time and countries. Adding (21.351) to (21.352), we
can write
1 ³ γ ´2 2
f
r = ρ + γg ’ ¯σ
2S
The power coef¬cient γ = 2 controls the relation between consumption growth and inter-
est rates, while the curvature coef¬cient γ/St controls the risk premium. Thus this habit
model allows high “risk aversion” with low “aversion to intertemporal substitution,” and it is
consistent with the consumption and interest rate data.
As advertised, this model explains the equity premium and predictability by fundamen-
tally changing the story for why consumers are afraid of holding stocks. The k’ period
stochastic discount factor is
µ ¶’γ
St+k Ct+k
k
Mt’t+k = δ .
St Ct
covariances with S shocks now drive average returns as well as covariances with C shocks.
S = (C ’ X)/C is a recession indicator “ it is low after several quarters of consumption

428
SECTION 21.2 NEW MODELS


declines and high in booms.
While (Ct+k /Ct )’γ and (St+k /St )’γ enter symmetrically in the formula, the volatility
of (Ct+k /Ct )’γ with γ = 2 is so low that it accounts for essentially no risk premia. There-
fore, it must be true, and it is, that variation in (St+k /St )’γ is much larger, and accounts for
nearly all risk premia. In the Merton language of (21.345) and (21.346), variation across as-
sets in expected returns is driven by variation across assets in covariances with recessions far
more than by variation across assets in covariances with consumption growth.
At short horizons, shocks to St+1 and Ct+1 move together, so the distinction between a
recession state variable and consumption risk is minor; one can regard S as an ampli¬cation
mechanism for consumption risks in marginal utility. dS/‚C ≈ 50, so this ampli¬cation
generates the required volatility of the discount factor.
At long horizons, however, St+k becomes less and less conditionally correlated with
Ct+k . St+k depends on Ct+k relative to its recent past, but the overall level of consumption
may be high or low. Therefore, investors fear stocks because they do badly in occasional
serious recessions, times of recent belt-tightening. These risks are at the long run unrelated
to the risks of long-run average consumption growth.
As another way to digest how this model works, we can substitute in the s process from
(21.349) and write the marginal rate of substitution as
µ ¶’γ
St+1 Ct+1
Mt+1 = δ
St Ct
ln Mt+1 = ln δ ’ γ (st+1 ’ st ) ’ γ(ct+1 ’ ct )
= {ln δ ’ γ(1 ’ φ)¯} + {γ (1 ’ φ) st + γg» (st )} ’ γ [» (st ) + 1] (ct+1 ’ ct )
s
ln Mt+1 = a + b(st ) + d(st )(ct+1 ’ ct )

Up to the question of logs vs. levels, this is a “scaled factor model” of the form we studied in
Chapter 8. It still is a consumption-based model, but the sensitivity of the discount factor to
consumption changes over time.
The long-run equity premium is even more of a puzzle. Most recession state variables,
such as recessions, labor, and instruments for time-varying expected returns (“shifts in the
investment opportunity set”) are stationary. Hence, the standard deviation of their growth
rates eventually stops growing with horizon. At a long enough horizon, the standard deviation
of the discount factor is dominated by the standard deviation of the consumption growth term,
and we return to the equity premium puzzle at a long enough run.
Since this model produces predictability of the right sign, it produces a long run equity
premium puzzle. How it manages this feat with a stationary state variable St is subtle (and
’γ
we didn™t notice it until the penultimate draft!) The answer is that while St is stationary, St
’γ
is not. St has a fat tail approaching zero so the conditional variance of St+k grows without
bound.
While the distinction between stationary S and nonstationary S ’γ seems initially minor, it

429
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


is in fact central. Any model that wishes to explain the equity premium at long and short runs
by means of an additional, stationary state variable must ¬nd some similar transformation so
that the volatility of the stochastic discount factor remains high at long horizons.
This model does have high risk aversion. The utility curvature and value function cur-
vature are both high. Many authors require that a “solution” of the equity premium puzzle
display low risk aversion. This is a laudable goal, and no current model has attained it. No
current model generates the equity premium with a low and relatively constant interest rate,
low risk aversion, and the right pattern of predictability “ high prices forecast low returns, not
high returns, and consumption is roughly a random walk. Constantinides (1990) and Boldrin,
Christiano and Fisher (1997) are habit models with a large equity premium and low risk aver-
sion, but they don™t get the pattern of predictability right. Boldrin, Christiano and Fisher have
highly variable interest rates to keep consumption from being predictable. Constantinides
(1990) has a constant interest rate, but consumption growth that is serially correlated, so con-
sumption rises to meet i.i.d. wealth growth. The long-run equity premium is solved with
counterfactually high long-run consumption volatility.
To get a high equity premium with low risk aversion, we need to ¬nd some crucial char-
acteristic that separates stock returns from wealth bets. This is a dif¬cult task. After all, what
are stocks if not a bet? The answer must be some additional state variable. Stocks must pay
off badly in particularly unfortunate states of the world.
Again, the trouble with predictability is that stocks pay off well in particularly bad states
of the world “ states with low future returns. This makes stocks even more desirable, requir-
ing even higher risk aversion to explain the equity premium. The alternative, not yet found,
is to ¬nd some measure of the state of the world that is particularly bad when stocks pay
off badly, enough to explain not only the standard equity premium, but the long run equity
premium resulting from the fact that stocks are less risky at longer horizons.
I write this not to say that such a model is impossible. The point is to show the hurdle that
must be overcome, in the hope that someone will overcome it.

21.2.3 Heterogeneous agents and idiosyncratic risks

A long, increasing, and important literature in the equity premium attacks the problem instead
with relatively standard preferences, but instead adds uninsured idiosyncratic risk. As with
the preference literature, this literature is interesting beyond the equity premium. We are
learning a lot about who holds stocks and why, what risks they face. We are challenged to
think of new assets and creative ways of using existing assets to share risks better.
Constantinides and Duf¬e (1996) provide a very clever and simple model in which id-
iosyncratic risk can be tailored to generate any pattern of aggregate consumption and asset
prices. It can generate the equity premium, predictability, relatively constant interest rates,
smooth and unpredictable aggregate consumption growth and so forth. Furthermore, it re-
quires no transactions costs, borrowing constraints or other frictions, and the individual con-


430
SECTION 21.2 NEW MODELS


sumers can have any nonzero value of risk aversion. Of course, we still have to evaluate
whether the idiosyncratic risk process we construct to explain asset pricing phenomena are
reasonable and consistent with microeconomic data.
A simple version of the model
I start with a very simpli¬ed version of the Constantinides-Duf¬e model. Each consumer
i has power utility,
X 1’γ
e’δt Cit
U =E
t

Individual consumption growth Cit+1 is determined by an independent, idiosyncratic normal
(0,1) shock ·it ,
µ ¶
Cit+1 12
(353)
ln = ·it+1 yt+1 ’ yt+1
Ci,t 2

where yt+1 is, by construction since it multiplies the shock ·it , the cross-sectional standard
deviation of consumption growth. yt+1 is dated t + 1 since it is the cross-sectional standard
deviation given aggregates at t+1. The aggregates are determined ¬rst, and then the shocks
·it are handed out.
Now, yt+1 is speci¬ed so that people suffer a high cross-sectional variance of consump-
tion growth on dates of a low market return Rt+1 ,
¸s
· µ ¶¯
p
Cit+1 ¯ 2
¯ Rt+1 = (354)
yt+1 = σ ln δ ’ ln Rt+1 .
Cit ¯ γ(γ + 1)

Given this structure, the individual is exactly happy to consume {Cit } without further
trading in the stock. (We can call Cit income Iit , and prove the optimal decision rule is to
consume income Cit = Iit .) His ¬rst-order condition for an optimal consumption-portfolio
decision
" #
µ ¶’γ
Cit+1
1 = Et e’δ Rt+1
Cit

holds, exactly.
To prove this assertion, just substitute in for Cit+1 /Cit and take the expectation:
· ¸
12
1 = Et exp ’δ ’ γ· it+1 yt+1 + γyt+1 + ln Rt+1
2

Since · is independent of everything else, we can use E [f (·y)] = E [E(f (·y|y)] . Now,

431
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


with · normal (0,1),
· ¸
¡ £ ¤ ¢ 122
E exp ’γ·it+1 yt+1 | yt+1 = exp γ yt+1 .
2

Therefore, we have
· ¸
122 12
1 = Et exp ’δ + γ yt+1 + γyt+1 + ln Rt+1 .
2 2

Substituting in from (21.354),
· µ ¶ ¸
1 2
1 = Et exp ’δ + γ(γ + 1) (δ ’ ln Rt+1 ) + ln Rt+1
2 γ(γ + 1)


1 = Et 1!

The general model
In the general model, Constantinides and Duf¬e de¬ne
s r
2 Ct+1
(355)
yt+1 = ln mt+1 + δ + γ ln
γ(γ + 1) Ct

where mt is a strictly positive discount factor that prices all assets under consideration,

pt = Et [mt+1 xt+1 ] for all xt+1 ∈ X. (356)

By starting with a discount factor that can price a large collection of assets, where I used the
’1
discount factor Rt+1 to price the single return Rt+1 in (21.354), idiosyncratic risk can be
constructed to price exactly a large collection of assets. We can exactly match the Sharpe
ratio, return forecastability, and other features of the data.
Then, they let
µ ¶
vit+1 12
ln = ·it+1 yt+1 ’ yt+1
vit 2


Cit+1 = vit+1 Ct+1 .

yt+1 is still the conditional standard deviation of consumption growth, given aggregates “
returns and aggregate consumption. This variation allows uncertainty in aggregate consump-
tion. We can tailor the idiosyncratic risk to and consumption-interest rate facts as well.

432
SECTION 21.2 NEW MODELS


Following exactly the same argument as before, we can now show that
" #
µ ¶’γ
Cit+1
1 = Et e’δ Rt+1
Cit

for all the assets priced by m.
A technical assumption
Astute readers will notice the possibility that the square root term in (21.354) and (21.355)
might be negative. Constantinides and Duf¬e rule out this possibility by assuming that the
discount factor m satis¬es
Ct+1
(357)
ln mt+1 ≥ δ + γ ln
Ct
in every state of nature, so that the square root term is positive.
We can sometimes construct such discount factors by picking parameters a, b in mt+1 =
h ³ ´γ i
max a + b0 xt+1 , eδ CCt to satisfy (21.356). However, neither this construction nor a
t+1


discount factor satisfying (21.357) is guaranteed to exist for any set of assets. The restriction
(21.357) is a tighter form of the familiar restriction that mt+1 ≥ 0 that is equivalent to the
absence of arbitrage in the assets under consideration. Ledoit and Bernardo (1997) show that
the restriction m > a is equivalent to restrictions on the maximum gain/loss ratio available
from the set of assets under consideration. Thus, the theorem really does not apply to any set
of arbitrage-free payoffs.
The example m = 1/R is a positive discount factor that prices a single asset return
1 = E(R’1 R), but does not necessarily satisfy restriction (21.357). For high R, we can have
very negative ln 1/R. This example only works if the distribution of R is limited to R ¤ eδ .
How the model works
As the Campbell-Cochrane model is blatantly (and proudly) reverse-engineered to sur-
mount (and here, to illustrate) the known pitfalls of representative consumer models, the
Constantinides-Duf¬e model is reverse engineered to surmount the known pitfalls of idiosyn-
cratic risk models.
Idiosyncratic risk stories face two severe challenges, as explained in section 1.2. First,
the basic pricing equation applies to each individual. If we are to have low risk aversion and
power utility, the required huge volatility of consumption is implausible for any individual.
Second, if you add idiosyncratic risk uncorrelated with asset returns, it has no effect on
pricing implications. Constantinides and Duf¬e™s central contribution is to very cleverly solve
the second problem.
In idiosyncratic risk models, we cannot specify individual consumption directly as we
do in representative agent endowment economies, and go straight to ¬nding prices. The
endowment economy structure says that aggregate consumption is ¬xed, and prices have
to adjust so that consumers are happy consuming the given aggregate consumption stream.

433
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


However, individuals can always trade consumption with each other. The whole point of
assets is that one individual can sell another some consumption, in exchange for the promise
of some consumption in return in the next period. We have to give individuals idiosyncratic
income shocks, and then either check that they do not want to trade away the idiosyncratic
shock, or ¬nd the equilibrium consumption after they do so.
Early idiosyncratic risk papers found quickly how clever the consumers could be in get-
ting rid of the idiosyncratic risks by trading the existing set of assets. Telmer (1993) and
Lucas (1994) found that if you give people transitory but uninsured income shocks, they
respond borrowing and lending or by building up a stock of savings. As in the classic per-
manent income model, consumption only responds by the interest rate times the change in
permanent income, and at low enough interest rates, not at all. “Self-insurance through stor-
age” removes the extra income volatility and we are back to smooth individual consumption
and an equity premium puzzle.
Constantinides and Duf¬e get around this problem by making the idiosyncratic shocks
permanent. The normal ·it shocks determine consumption growth. In an evaluation in
microeconomic data, this makes us look for sources of permanent shocks.
This, at a deeper level, is why idiosyncratic consumption shocks have to be uncorrelated
with the market. We can give individuals idiosyncratic income shocks that are correlated
with the market. Say, agent A gets more income when the market is high, and agent B gets
more income when it is low. But then A will short the market, B will go long, and they will
trade away any component of the shock that is correlated with the returns on available assets.
I argued above that this effect made idiosyncratic shocks hopeless as candidates to explain
the equity premium puzzle. Shocks uncorrelated with asset returns have no effect on asset
pricing, and shocks correlated with asset returns are quickly traded away.
The only way out is to exploit the nonlinearity of marginal utility. We can give people in-
come shocks that are uncorrelated with returns, so they can™t be traded away. Then we have
a nonlinear marginal utility function turn these shocks into marginal utility shocks that are
correlated with asset returns, and hence can affect pricing implications. This is why Con-
stantinides and Duf¬e specify that the variance of idiosyncratic risk rises when the market
declines. If marginal utility were linear, an increase in variance would have no effect on the
average level of marginal utility. Therefore, Constantinides and Duf¬e specify power utility,
and the interaction of nonlinear marginal utility and changing conditional variance produces
an equity premium.
As a simple calculation that shows the basic idea, start with individuals i with power
utility so



"µ #
¶’γ
i
Ct+1
Re
0=E t+1
i
Ct

434
SECTION 21.2 NEW MODELS

PN
Now aggregate across people by summing over i, with EN = 1
i=1
N
" õ ¶’γ ! #
i
Ct+1
Re
0 = E EN t+1 .
i
Ct

If the cross-sectional variation of consumption growth is lognormally distributed,
·µ ¶ ¸
γ2 2
i i
0 = E e’γEN ∆ct+1 + 2 σN ∆ct+1 Rt+1 e



As you see, the economy displays more risk aversion than would a “representative agent”
with aggregate consumption ∆ca = EN ∆cit+1 . That risk aversion can also vary over time
t+1
if σ N varies over time, and this variation can generate risk premia.
Microeconomic evaluation and risk aversion
Like the Campbell-Cochrane model, this could be either a new view of stock market
(and macroeconomic) risk, or just a clever existence proof for a heretofore troubling class
of models. The ¬rst question is whether the microeconomic picture painted by this model is
correct, or even plausible. Is idiosyncratic risk large enough? Does idiosyncratic risk really
rise when the market falls, and enough to account for the equity premium? Are there enough
permanent idiosyncratic shocks? Do people really shy away from stocks because of stock
returns are low at times of high labor market risk?
This model does not change the ¬rst puzzle. To get power utility consumers to shun
stocks, they still must have tremendously volatile consumption growth or high risk aversion.
The point of this model is to show how consumers can get stuck with high consumption
volatility in equilibrium, already a dif¬cult task.
More seriously than volatility itself, consumption growth variance also represents the
amount by which the distribution of individual consumption and income spreads out over
time, since the shocks must be permanent and independent across people. The 50% or larger
consumption growth volatility that we require to reconcile the Sharpe ratio with risk aversion
of one means that the distribution of consumption (and income) must also spread out by 50%
per year. The distribution of consumption does spread out, but not this much.
For example, Deaton and Paxson (1994) report that the cross-sectional variance of log
consumption within an age cohort rises from about 0.2 at age 20 to 0.6 at age 60. This

estimate means that the cross sectional standard deviation of consumption rises from 0.2 =

. 45 or 45% at age 20 to 0.6 = . 77 or 77% at age 60. (77% means that an individual one
standard deviation better off than the mean consumes 77% more than the mean consumer.)
We are back to about 1% per year.
Finally, and most crucially, the cross-sectional uncertainty about individual income must
not only be large, it must be higher when the market is lower. This risk-factor is after all the
central element of Constantinides and Duf¬e™s explanation for the market premium. Figure
51 shows how the cross-sectional standard deviation of consumption growth varies with the

435
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


market return and risk aversion in my simple version of Constantinides and Duf¬e™s model.
If we insist on low (γ = 1 to 2) risk aversion, the cross-sectional standard deviation of
consumption growth must be extremely sensitive to the level of the market return. Looking
at the γ = 2 line for example, is it plausible that a year with 5% market return would show a
10% cross-sectional variation in consumption growth, while a mild 5% decline in the market
is associated with a 25% cross-sectional variation?




Figure 51. Cross-sectional standard deviation of individual consumption growth as a
function of the market return in q simple version of the Constantinides-Duf¬e model.
the q
The plot is the variable yt = ln Rt + δ + γ ln CCt . Parameter values are
2 1
γ(γ+1) t’1

ρ = 0.05, ln Ct /Ct’1 = 0.01.

All of these empirical problems are avoided if we allow high risk aversion rather than
a large risk to drive the equity premium. The γ = 25 line in Figure 51 looks possible; a
γ = 50 line would look even better. With high risk aversion we do not need to specify highly
volatile individual consumption growth, spreading out of the income distribution, or dramatic
sensitivity of the cross-sectional variance to the market return.
As in any model, a high equity premium must come from a large risk, or from large risk
aversion. Labor market risk correlated with the stock market does not seem large enough to
account for the equity premium without high risk aversion.


436
SECTION 21.3 BIBLIOGRAPHY


The larger set of asset pricing facts has not yet been studied in this model. It is clearly
able to generate return predictability, but that requires a pattern of variation in idiosyncratic
risk that remains to be characterized and evaluated. It can generate cross-sectional patterns
such as value premia if value stocks decline at times of higher cross-sectional volatility; that
too remains to be studied.
Summary
In the end, the Constantinides-Duf¬e model and the Campbell-Cochrane model are quite
similar in spirit. First, both models make a similar, fundamental change in the description of
stock market risk. Consumers do not fear much the loss of wealth of a bad market return per
se. They fear that loss of wealth because it tends to come in recessions, in one case de¬ned as
times of heightened labor market risk, and in the other case de¬ned as a fall of consumption
relative to its recent past. This recession state-variable or risk-factor drives most variation in
expected returns.
Second, both models require high risk aversion. While Constantinides and Duf¬e™s proof
shows that one can dream up a labor income process to rationalize the equity premium for
any risk aversion coef¬cient, we see that even vaguely plausible characterizations of actual
labor income uncertainty require high risk aversion to explain the historical equity premium.
Third, both models provide long-sought demonstrations that it is possible to rationalize
the equity premium in their respective class of models. This existence proof is particularly
stunning in Constantinides and Duf¬e™s case. Many authors (myself included) had come
to the conclusion that the effort to generate an equity premium from idiosyncratic risk was
hopeless because any idiosyncratic risk that would affect asset prices would be traded away.


21.3 Bibliography

Shiller (1982) made the ¬rst calculation that showed either a large risk aversion coef¬cient
or counterfactually large consumption variability was required to explain means and vari-
ances of asset returns. Mehra and Prescott (1985) labeled this fact the “equity premium
puzzle.” However, they described these puzzles in the context of a two-state Markov model
for consumption growth, identifying a stock as a claim to consumption and a risk free bond.
Weil (1989) emphasized the interaction between equity premium and risk-free rate puzzles.
Hansen and Jagannathan (1991) sparked the kind of calculations I report here in a simpli¬ed
manner. Cochrane and Hansen (1992) derived many of the extra discount factor moment re-
strictions I surveyed here, calculating bounds in each case. Luttmer (1996), (1999) tackled
the important extension to transactions costs.
Kocherlakota (1996) is a nice summary of equity premium facts and models. Much of the
material in this Chapter is adapted from a survey in Cochrane (1997). Campbell (1999) and
(2000) are two excellent recent surveys. Ferson (1995) is a nice survey of consumption-based
model variations as well as some of the beta pricing models discussed in the last chapter.
The general picture of all solutions based on changing preferences is that they introduce

437
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


non-separabilities. If the marginal utility of consumption depends on z as well as c, uc (c, z),
then expected returns depend on covariances with z as well. In turn, this happens if we cannot
write the utility function as u(c, z) = v(c) + w(z), the separable form.
Habit persistence introduces a non-time-separable utility function, since u(c, x) and x de-
P
pends on past c“you can™t write a utility function with habits as a sum t vt (ct ), so one pe-
riod™s consumption affects another period™s marginal utility. The Campbell-Cochrane model
I presented here is a tip of an iceberg of habit research, including prominent contributions by
Constantinides (1990), Ferson and Constantinides (1991), Heaton (1995), Abel (1990).
Models can be nonseparable across goods as well. Leisure is the most natural extra
variable to add to a utility function. It™s not clear a priori whether more leisure enhances the
marginal utility of consumption (why bother buying a boat if you™re at the of¬ce all day and
can™t use it) or vice versa (if you have to work all day, it™s more important to come home to
a really nice big TV). However, we can let the data speak on this matter. Explicit versions
of this approach have not been very successful to date. (Eichenbaum, Hansen and Singleton
1989). On the other hand, recent research has found that adding labor income as an extra ad-
hoc “factor” can be useful in explaining the cross section of average stock returns, especially
if it is scaled by a conditioning variable (Jagannathan and Wang 1996, Reyfman 1997, Lettau
and Ludvigson 2000).
The non-state separable utility functions following Epstein and Zin (1989) are a major
omission of this presentation. The expectation E in the standard utility function sums over
states of nature, e.g.
U = prob(rain) — u(C if it rains) + prob(shine) — u(C if it shines).
“Separability” means one adds across states, so the marginal utility of consumption in one
state is unaffected by what happens in another state. But perhaps the marginal utility of a little
more consumption in the sunny state of the world is affected by the level of consumption in
the rainy state of the world. Epstein and Zin and Hansen, Sargent and Tallarini (1997) propose
recursive utility functions of the form
£ ¤
1’γ
+ βf Et f ’1 (Ut+1 ) .
Ut = Ct
If f (x) = x this expression reduces to power utility. These utility functions are not state-
separable. As with habits, these utility functions distinguish risk aversion from intertemporal
substitution“one coef¬cient can be set to capture the consumption-interest rate facts, and a
completely separate coef¬cient can be set to capture the equity premium. So far, this style
of model as in Epstein and Zin (1989), Weil (1989), Kandel and Stambaugh (1991), and
Campbell (1996) does not generate time-varying risk aversion, but that modi¬cation should
not be too dif¬cult, and could lead to a model that works very much like the habit model I
surveyed here.
Habit persistence is the opposite of durability. If you buy a durable good yesterday, that
lowers your marginal utility of an additional purchase today, while buying a habit-forming
good raises your marginal utility of an additional purchase today. Thus the durability of goods

438
SECTION 21.3 BIBLIOGRAPHY


should introduce a non-time-separability of the form u(ct + θxt ), xt = f(ct’1 , ct’2 , ...)
rather than the habit persistence form u(ct ’ θxt ). Since goods are durable, and we have
a lot of data on durables purchases, it would be good to include both durability and habit
persistence in our models. (In fact, even “nondurables” contain items like clothing; the truly
nondurable purchases are such a small fraction of total consumption that we rely on very little
data.) One must be careful with the time horizon in such a speci¬cation. At a suf¬ciently
small time horizon, all goods are durable. A pizza eaten at noon lowers marginal utility
of more pizza at 12:05. Thus, our common continuous time, time separable assumption
really cannot be taken literally. Hindy and Huang (1992) argue that consumption should
be “locally substitutable” in continuous time models. Heaton (1993) found that at monthly
horizons, consumption growth displays the negative autocorrelation suggestive of durability
with constant interest rates, while at longer horizons consumption is nearly unforecastable
after accounting for time-aggregation.
There is also a production ¬rst order condition that must be solved, relating asset prices
to marginal rates of transformation. The standard here is the q theory of investment, which is
based on an adjustment cost. If the stock market is really high, you issue stock and make new
investments. The trouble with this view is that f 0 (K) declines very slowly, so the observed
price volatility implies huge investment volatility. The q theory adds adjustment costs to
damp the investment volatility. The q theory has had as much trouble ¬tting the data as
the consumption-based model. Cochrane (1991d) reports one success when you transform
the data to returns “ high stock returns are associated with high investment growth. The
more recent investment literature has focused on specifying the adjustment cost problem
with asymmetries and irreversibilities, for example Abel and Eberly (1996).
There is an important literature that puts new utility functions together with production
functions, to construct complete explicit economic models that replicate the asset pricing
facts. Such efforts should also at least preserve if not enhance our ability to understand the
broad range of dynamic microeconomic, macroeconomic, international and growth facts that
the standard models were constructed around. Jermann (1998) tried putting habit persistence
consumers in a model with a standard technology Y = θf (K, L) from real business cycle
models. The easy opportunities for intertemporal transformation provided by that technol-
ogy meant that the consumers used it to dramatically smooth consumption, destroying the
prediction of a high equity premium. To generate the equity premium, Jermann added an
adjustment cost technology, as the production-side literature had found necessary. This mod-
i¬cation resulted in a high equity premium, but also large variation in riskfree rates.
Boldrin, Christiano and Fisher (1997) also added habit-persistence preferences to real
business cycle models with frictions in the allocation of resources to two sectors. They gen-
erate about 1/2 the historical Sharpe ratio. They ¬nd some quantity dynamics are improved
over the standard model. However, they still predict highly volatile interest rates and persis-
tent consumption growth.
To avoid the implications of highly volatile interest rates, I suspect we will need repre-
sentations of technology that allow easy transformation across time but not across states of


439
CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS


nature, analogous to the need for easy intertemporal substitution but high risk aversion in
preferences. Alternatively, the Campbell-Cochrane model above already produces the equity
premium with constant interest rates, which can be interpreted as a linear production func-
tion f(K). Models with this kind of precautionary savings motive may not be as severely
affected.
Tallarini (1999) uses non-state separable preferences similar to those of Epstein and Zin in
a general equilibrium model with production. He shows a beautiful observational equivalence
result: A model with standard preferences and a model with non-state-separable preferences
can predict the same path of quantity variables (output, investment, consumption, etc.) but
differ dramatically on asset prices. This result offers one explanation of how the real busi-
ness cycle and growth literature could go on for 25 years examining quantity data in detail
and miss all the modi¬cations to preferences that we seem to need to explain asset pricing
data. It also means that asset price information is crucial to identifying preferences and cal-
culating welfare costs of policy experiments. Finally, it offers hope that adding the deep
modi¬cations necessary to explain asset pricing phenomena will not demolish the success of
standard models at describing the movements of quantities.
The Constantinides and Duf¬e model has roots in a calculation by Mankiw (1986) that
idiosyncratic risk could make the representative consumer seem more risk averse than the
individuals. Work on evaluating the mechanisms in this model in microeconomic data is
starting. Heaton and Lucas (1996) calibrate idiosyncratic risk from the PSID, but their model
explains at best 1/2 of the sample average stock return, and less still if they allow a net supply
of bonds with which people can smooth transitory shocks. More direct tests of these features
in microeconomic consumption data are underway, for example Brav, Constantinides and
Geczy (1999), Storesletten, Telmer and Yaron (1999).
Kandel and Stambaugh (1986) present a model in which a small amount of time-varying
consumption volatility and a high risk aversion coef¬cient generate the large time-varying
discount factor volatility we need to generate returns predictability.
Aiyagari and Gertler (1991), though aimed at the point that the equity premium might be
explained by a “too low” riskless rate, nonetheless was an important paper in specifying and
solving models with uninsured individual risks and transactions costs to keep people from
trading them away.


21.4 Problems

1. Derive the analogue to the Hansen-Jagannathan bound in continuous time for an “excess
return,” i.e. considering a self-¬nancing portfolio, rather than a single return less the risk
free rate.
2. Suppose habit accumulation is linear, and there is a constant riskfree rate or linear



440
SECTION 21.4 PROBLEMS


technology equal to the discount rate, Rf = 1/δ. The consumer™s problem is then
∞ ∞
X X X X
’ Xt )1’γ
t (Ct t t
φj Ct’j
max δ s.t. δ Ct = δ et + W0 ; Xt = θ
1’γ t t
t=0 j=1

where et is a stochastic endowment. In an internal habit speci¬cation, the consumer
considers all the effects that current consumption has on future utility through Xt+j .
In an external habit speci¬cation, the consumer ignores such terms. Show that the two
speci¬cations give identical asset pricing predictions in this simple model, by showing
that internal-habit marginal utility is proportional to external-habit marginal utility, state
by state.
3. Suppose a consumer has quadratic utility with a constant interest rate equal to the
subjective discount rate, but a habit or durable consumption good, so that utility is
1
u(ct ’ θct’1 ) = ’ (c— ’ ct + θct’1 ).
2
Show that external habit persistence θ > 0 implies positive serial correlation in
consumption changes. Show that the same solution holds for internal habits, or
durability. Show that durability leads to negative serial correlation in consumption
changes.
4. Many models predict too much variation in the conditional mean discount factor, or
too much interest rate variation. This problem guides you through a simple example.
Introduce a simple form of external habit formation,
u = (Ct ’ θCt’1 )1’γ
and suppose consumption growth Ct+1 /Ct is i.i.d. Show that interest rates still vary
despite i.i.d. consumption growth.
5. We showed that if m satis¬es the Hansen-Jagannathan bound, then proj(m|X) should
also do so. Hansen and Jagannathan also compute bounds with positivity, solutions to
min σ(m) s.t. p = E(mx), m ≥ 0, E(m) = µ.
Does proj(m|X) also lie in the same bound?
6. One most often compares consumption-based models to Hansen-Jagannathan bounds.
Can you compare the CAPM discount factor m = a ’ bRem to the bound? To the bound
with positivity?




441
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454
PART V
Appendix




455
Chapter 23. Continuous time
This chapter is a brief introduction to the mechanics of continuous time stochastic processes;
i.e. how to use dz and dt. I presume the reader is familiar with discrete time ARMA models,
i.e. models of the sort xt = ρxt’1 + µt , and draw analogies of continuous time constructs to
those models.
The formal mathematics of continuous time processes are a bit imposing. For example,
the basic random walk zt is not time-differentiable, so one needs to rethink the de¬nition
Rt
of an integral and differential to write obvious things like zt = s=0 dzt . Also, since zt is
a random variable one has to specify not only the usual measure-theoretic foundations of
random variables, but their evolution over a continuous time index. However, with a few
basic, intuitive rules like dz 2 = dt, one can use continuous time processes quite quickly, and
that™s the aim of this chapter.


23.1 Brownian Motion


zt , dzt are de¬ned by zt+∆ ’ zt ∼ N (0, ∆).


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