Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth
Author(s): Fernando Alvarez and Urban J. Jermann
Source: Econometrica, Vol. 73, No. 6, (Nov., 2005), pp. 1977-2016
Published by: The Econometric Society
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Econometrica,Vol. 73, No. 6 (November,2005), 1977-2016
USING ASSET PRICES TO MEASURE THE PERSISTENCE
OF THE MARGINAL UTILITY OF WEALTH
BY FERNANDO ALVAREZ AND URBAN J. JERMANN1
We derive a lower bound for the volatilityof the permanentcomponentof investors'
marginalutilityof wealth or, more generally,asset pricingkernels.The bound is based
on returnpropertiesof long-termzero-couponbonds, risk-freebonds, and other risky
securities.We find the permanentcomponent of the pricingkernel to be very volatile;
its volatilityis about at least as large as the volatilityof the stochasticdiscount factor.
A related measure for the transitorycomponent suggest it to be considerablyless important. We also show that, for many cases where the pricing kernel is a function of
consumption,innovationsto consumptionneed to have permanenteffects.
KEYWORDS:Pricingkernel, stochasticdiscount factor, permanentcomponent, unit
roots.
1. INTRODUCTION
OPPORTUNITIES
THE ABSENCEOFARBITRAGE
implies the existence of apricing
kernel, that is, a stochastic process that assigns values to state-contingent payments. As is well known, asset pricing kernels can be thought of as investors'
marginal utility of wealth in frictionless markets. Since the properties of such
processes are important for asset pricing, they have been the subject of much
recent research.2 Our focus is on the persistence properties of pricing kernels;
these are key determinants of the prices of long-lived securities.
The main result of this paper is to derive and estimate a lower bound for the
volatility of the permanent component of asset pricing kernels. The bound is
based on return properties of long-term zero-coupon bonds, risk-free bonds,
and other risky securities. We find the permanent component of pricing kernels
to be very volatile; its volatility is about at least as large as the volatility of
the stochastic discount factor. A related bound that measures the volatility of
the transitory component suggests it to be considerably less important than the
permanent component.
Our results complement the seminal work by Hansen and Jagannathan
(1991). They used no-arbitrage conditions to derive bounds on the volatility
1We thank Andy Atkeson, Erzo Luttmer, Lars Hansen, Pat Kehoe, Bob King, Narayana
Kocherlakota,Stephen Leroy, Lee Ohanian,and the participantsin workshopsand conferences
at UCLA, the Universityof Chicago, the Federal Reserve Banks of Minneapolis,Chicago, and
Cleveland,and Duke, Boston, Ohio State, Georgetown,and Yale Universities,NYU, Wharton,
the SED meeting in Stockholm,SITE, the Minnesotaworkshopin macroeconomictheory, and
ESSFM for their comments and suggestions.We thank Robert Bliss for providingthe data for
U.S. zero-couponbonds. Alvarez thanksthe NSF and the Sloan Foundationfor support.Earlier
versions of this paper were circulatedunder the title "The Size of the PermanentComponentof
Asset PricingKernels."
2A few prominent examples of researchalong this line are Hansen and Jagannathan(1991),
Cochraneand Hansen (1992), and Luttmer(1996).
1977
1978
E ALVAREZAND U. J. JERMANN
of pricing kernels as a function of observed asset prices. They found that, to
be consistent with the high Sharpe ratios in the data, stochastic discount factors have to be veryvolatile. We find that, to be consistentwith the low returns
on long-term bonds relative to equity, the permanent component of pricing
kernels has to be very large. This propertyis important,because the low frequencycomponentsof pricingkernels are importantdeterminantsof the prices
of long-lived securities such as stocks. Recent work on asset pricing has highlighted the need for a better understandingof these low frequency components; see, for instance, Bansal and Yaron(2004) and Hansen, Heaton, and Li
(2004). Our results are also related to Hansen and Scheinkman(2003), where
they present a general frameworkfor linkingthe short and long run properties
of asset prices.
Asset pricingmodels link pricingkernels to the underlyingeconomic fundamentals. Thus, our analysisprovides some insights into the long-termproperties of these fundamentalsand into the functions that link pricing kernels to
the fundamentals.On this point, we have two sets of results.
First, under some assumptionsabout the function of the marginalutility of
wealth, we derive sufficientconditions on consumptionso that a pricingkernel
has no permanent innovations. We present several examples of utility functions for which the existence of an invariantdistributionof consumption implies pricingkernelswith no permanentinnovations.Thus, these examplesare
inconsistentwith our main findings.This result is useful for macroeconomics
because, for some issues, the persistencepropertiesof the processes that specify economic variablescan be very important.For instance, on the issue of the
welfare costs of economic uncertainty,see Dolmas (1998); on the issue of the
volatility of macroeconomicvariables such as consumption, investment, and
hours worked, see Hansen (1997); and on the issue of internationalbusiness
cycle comovements,see Baxterand Crucini(1995). The lesson from our analysis for these cases and many related studies of dynamic general equilibrium
models is that models should be calibratedso as to generate macroeconomic
time series with importantpermanentcomponents.
Following Nelson and Plosser (1982), a large body of literature has tested
macroeconomictime series for stationarityversus unit roots.3More recently,a
large and growingliteratureon structuralvector autoregressions(VARs) is using identifyingassumptionsbased on restrictingthe originof permanentfluctuations in macroeconomicvariablesto certain types of shocks.The relationship
between such structuralshocks and macroeconomicvariablesis then compared
to the implicationsof different classes of macroeconomicmodels. See, for instance, Shapiroand Watson(1988), Blanchardand Quah (1989), and, more recently, Gali (1999), Fisher (2002), and Christiano,Eichenbaum,and Vigfusson
(2002). The identificationstrategies used in this literature hinge criticallyon
3Assetprices have also been includedin multivariateanalysesof persistenceof grassdomestic
product(GDP) and consumption,see, for instance,Lettau and Ludvigson(2004).
MARGINAL UTILITY OF WEALTH
1979
the presence of unit roots in the key macroeconomictime series. The results
in our paper provide validation for this approachby presenting new evidence
about the importance of permanent fluctuations.We introduce new information about persistence from the prices of long-termbonds. Prices of long-term
bonds are particularlyinformativeabout the persistence of pricingkernels because they are the market's forecast of the long-term changes in the pricing
kernel.
As a second set of results,we measurethe volatilityof the permanentcomponent in consumptiondirectlyand compare it to the volatilityof the permanent
component of pricingkernels. This can provide guidance for the specification
of functional forms of the marginalutility of wealth.4Specifically,we find the
volatility of the permanent component of consumptionto be lower than that
of pricing kernels. This suggests the use of utility functions that magnify the
permanentcomponent.
The rest of the paper is structuredas follows. Section 2 contains definitions
and a preview of the main results. Section 3 presents theoretical results. Section 4 presents empiricalevidence. Section 5 relates pricingkernels and aggregate consumption.Section 6 concludes. Proofs are in AppendixA. AppendixB
describesthe data sources. Appendix C addresses a small sample bias.
2. DEFINITIONSAND PREVIEW OF THE MAIN RESULT
In this section we startwith some key definitionsand assumptions.Then, to
previewthe main theoreticalresult of the paper,we state without derivationan
expressionfor the lower bound of the permanentcomponent of the stochastic
discount factor. We compute this lower bound for two benchmarkcases: one
with only permanentmovements and one with only transitorymovements.
Let Dt+k be a state-contingent dividend to be paid at time t + k and let
Vt(Dt+k)be the currentprice of a claim to this dividend.Then, as can be seen,
for instance, in Duffie (1996), arbitrageopportunitiesare ruled out in frictionless marketsif and only if a strictlypositivepricingkernelor state-priceprocess,
{Mt},exists so that5
Et(Mt+k Dt+k)
(D
(1) (1)
t Vt
(D,+k)=
M,
4See Daniel and Marshall(2001) on the related issue of how consumptionand asset prices are
correlatedat differentfrequencies.
SAs is well known, this result does not require complete markets,but assumes that portfolio
restrictionsdo not bindfor some agents.This last conditionis sufficient,but not necessary,for the
existence of a pricingkernel. For instance, in Alvarez and Jermann(2000), portfolio restrictions
bind most of the time; nevertheless,a pricingkernel exists that satisfies(1).
1980
E ALVAREZAND U. J. JERMANN
For our results, it is importantto distinguishbetween the pricingkernel, Mr+l,
and the stochasticdiscountfactor, M,+l/Mt.6We use Rt+1for the gross return
on a generic portfolio held from t to t + 1; hence, (1) implies that
(2)
1=Et(
Mt+ ?R
-f
Rt+)
We define Rt+l,k as the gross return from holding from time t to time t + 1 a
claim to one unit of the numeraireto be delivered at time t + k:
Rt+l,k =
+1it(l+k)
Vt(lt+k)
The holding return on this discount bond is the ratio of the price at which
the bond is sold, Vt+(lt+k), to the price at which it was bought, Vt(lt+k).With
this convention, Vt(lt) = 1. Thus, for k > 2 the returnconsists solely of capital
gains; for k = 1, the return is risk-free.In this paper we focus on the limiting
long-term bond, which has return Rt+,,o
limk,o Rt+1,k.
Throughoutthe paper we maintain the assumptionthat stochastic discount
factors Mt+1/Mtand returns Rt+1are jointly stationary and ergodic. An immediate implication of the stationarityof stochastic discount factors is that
all bond returns are stationary.The assumption of stationarityof returns is
standard in the asset pricing literature. In Section 4 we review some of the
evidence on the stationarityof interest rates. Under our maintained assumption about stationarity,we find that pricingkernels M, have a large permanent
component. Alternatively, if we were to consider Mt+l/Mt as nonstationary, then Mt would not be stationary either. To use a time series analogy,
if log M+1 - logMt were to have a unit root, then log M would be integrated
at least of order 2.
Below we decompose the pricingkernel Mt into two components,
Mt = M, Mt,
where Mtf is a martingale,so it capturesthe permanentpart of M,, and MfTis
the transitorycomponent of Mt. The main resultof this paper is that the volatility of the growth rate of the permanent component, MAP+/Mf,relative to the
volatilityof the stochasticdiscount factor, Mt+l/Mr,is at least as large as
(3)Rtl
E log Rt+- - E log R,+1,
Rt+
Rt+1
R'+ +
Elog Rt+,
L(RRt+ l,1 )
6For instance, in the Lucas representativeagent model, the pricing kernel Mt is given by
ftU'(c,), where / is the preference time discount factor and U'(ct) is the marginalutility of
consumption.In this case, the stochasticdiscountfactor,Mr+l/Mr,is given by PfU'(c,+l)/U'(c,).
MARGINAL UTILITY OF WEALTH
1981
where Rt+1is the return of any asset and L(1/R,t+,1) is a measure of the
volatility of the short-term interest rate to be described in detail below. For
this preliminarydiscussion note that L = 0 if interest rates have zero variance and otherwise L > O. The numerator of this expression is the difference between two (log) excess returnsor two risk premiums.As is easily seen,
if the term premium for the bond with infinitely long maturity is positive,
> O, this expression is maximized by selecting the asset
E(log(Rt+l,,/Rt+l,1))
with the highest expected log excess returnE(log(Rt+1/Rt+,lj)).
We now compute the lower bound for two examples for which the volatility of the permanent component of the pricingkernel is obvious. Consider an
investor with time separable expected utility, and consider two consumption
processes: independent and identicallydistributed(i.i.d.) consumptiongrowth
and i.i.d. consumptionlevel. The pricingkernel is
=
=
U'(c,)
-
Mt+l1
+
ct-,
where U has CRRA y.
1: Assume that ct+l/ct is i.i.d. Clearly Mt has only permanent
EXAMPLE
shocks. In this case, it is easy to verify that interest rates Rt+l,1are constant,
which implies that L(1/Rt+1,i) = 0 and that
logRt+l,k =0,
log
Rt+l,1
so that all term premiumsare zero. With these values, expression (3) is equal
to 1, so that the volatility of the permanent component of the stochastic discount factor is, indeed, at least as large as the volatility of the stochastic discount factor.
EXAMPLE2: Assume that ct+l is i.i.d. ClearlyMt has no permanentcomponent. In this case, neither short-terminterest rates nor returns on long-term
bonds are constant in general. Indeed,
U'(ct)
E[U(ct+l)
and
Rt+l,k
=
(1+ P))
U'(ct)
Ut(Ct+l)
_M,
=
M,t+
for k>2;
that is, for k > 2, the holding return equals the inverse of the stochastic discount factor. It is now easy to show that the highest lower bound computed
1982
F ALVAREZ AND U. J. JERMANN
from expression (3) is attained by choosing the return Rt+1 = Rt+l,k for k > 2
and that this lower bound equals 0. Indeed, ruling out arbitrageimplies that
for any returnRt+1,
1.
Et(Mt+1Rt+)=
Mt
Using Jensen's inequality,
0=logEt(
MRt+)>
Etlog ( MRt+),
which implies
EtlogR+l < Et logM
Mt+l
with equality if Rt+1and Mt/Mt+l are proportional. Thus, because Rt+l,k =
Mt/Mt+l, for k > 2 no log return is higher than the log return of long-term
bonds. Setting Rt+i = Rt+1,kfor k > 2 gives the highest lower bound (3) and its
value will be zero. Hence we have verified that the bound shows that, for the
case where the level of consumptionis i.i.d., there is no permanentcomponent.
3. THEORETICAL RESULTS
In this section we first show an existence result for the multiplicativedecomposition of Mt into a transitoryand permanent component, and we derive a lower bound for the volatility of the permanent component. We then
present a related bound for the volatilityof the transitorycomponent. We also
present a proposition that guarantees the applicabilityof our bound for the
permanentcomponent to any appropriatemultiplicativedecompositionunder
some regularityassumptions. Finally, we compare our bound to a result by
Cochraneand Hansen (1992) about the conditional and unconditionalvolatility of stochasticdiscount factors.
We startwith two conditionsunderwhichwe can decompose the kernel into
permanentand transitorycomponentsproperlydefined.
ASSUMPTION1: Assume there is a number 3 such that
0< lim
kx
Vt(((l+k)
< oo
pk
for all t.
In the language of Hansen and Scheinkman (2003), the number /3 is the
dominant eigenvalue of the pricing operator. Assumption 1 can be violated
MARGINAL UTILITY OF WEALTH
1983
either if the limit does not exist or if it takes the values 0 or oc. The existence
of the limit imposes a regularitycondition on the shape of the term structure
for large k. Specifically,it requiresthat the yield -(1/k) log Vt(lt+k)converges
fast enough as k -> oc. The limit can take the value 0 or oc if bond prices are
nonstationary.For instance,consider the case where after date s there are only
two possible outcomes: either the yields of bonds of all maturities are equal
to -log/3 or they are equal to -log 8. In this case there is no /3 for which
the limit in Assumption 1 is strictlypositive and finite. Since we have assumed
throughoutthat bond prices are stationary,this possibilityis ruled out.
ASSUMPTION2: Assume that for each t + 1 there is a randomvariable xt+
such that
Mt+1 Vt+ (lt+l+k)
/3t+1
<Xt+
k
a.s.
with E,xt+l finite for all k.
Assumption 2 strengthens Assumption 1. Instead of requiring that
Vt(lt+k)/kk has a finite limit, Assumption2 requiresthat for each k its product
with the marginalvaluation is bounded by a variable that has a finite conditional expectation.
1: UnderAssumptions1 and 2, thereis a uniquedecomposition
PROPOSITION
M, = MTMP
with EtMfP+= Mf and
li
= lim
Mft MP
k-oo
MT
lim
EtMt+k
ft+k
O t+k
tk
k-oo Vt(lt+k)
Due to Assumption 1, MP is well defined, strictlypositive, and finite. Assumption 2 is used to establish that Mf as defined above is a martingale.The
decomposition obtained throughProposition 1 is unique given its constructive
nature.
The value of the permanent component is the expected value of the
process M in the long run, relative to its long-term drift /3. We call MP the
permanent component because it is unaffected by information at t that does
not lead to revisions of the expected value of M in the long run. The decomposition in Proposition 1 is analogous to the one used by Beveridge and
Nelson (1981). Beveridge and Nelson's decomposition is additive instead of
1984
F. ALVAREZAND U. J. JERMANN
multiplicativeand their permanentcomponent is a randomwalk, while in our
decompositionthe permanentcomponent is a martingale.
The component MTis a scaled long-terminterestrate. Given our stationarity
assumptionfor the stochastic discount factor, interest rates inherit this property, and interpreting MT as containing only transitorycomponents follows
naturally.This stationaritypropertyfor MT is again linked to a related property in the Beveridge and Nelson decomposition. Interest rates are a function
of the expected growth rate of the pricingkernel. Thus, assumingstationarity
for interest rates is similarto the assumptionbehind the Beveridge and Nelson
decompositionthat growthrates are stationarywhile levels are not.
Nothing in Proposition 1 rules out the possibilitythat there exist other decompositions of M into two parts, where one part is a martingale and the
other contains transitorycomponents. Such alternativedecompositionscould
exist independentlyof whether Assumptions 1 and 2 apply.With Assumptions
1 and 2 holding, it might still be possible to construct a decomposition in another way. Alternatively,Proposition 1 has nothing to say for the case where
Assumptions 1 and 2 would not hold. However, as we will show later in this
section, our volatilitybounds also apply to such decompositions more generally.
To characterizethe importance of permanent and transitorycomponents,
we use Lt(x,t+)
log Etx,+ - Et logxt+l and L(xt+) = logExt+1 - Elog x,+
as measures of the conditional and unconditionalvolatility of xt+,. Throughout the rest of the paper we refer to the expected values of different random
variableswithout statingexplicitlythe assumptionthat these randomvariables
are integrable.The following result can then be shown.
PROPOSITION
2: Assume thatAssumptions1 and 2 hold. Then (i) the conditional volatilityof thepermanentcomponentsatisfies
(4)
LtM
) >EtlogRt+l-EtlogRt+l,
for anypositive returnRt+\. Furthermore,(ii) the unconditionalvolatilityof the
permanentcomponentsatisfies
(S)+'
Mt
E (log R,+,)
Rt",l!
}
E(log R,+i,o)
Rt+
l,
for anypositiveR,t+ such that E(log(Rt+l/Rt+,11))+ L(1/Rt+,1l) > 0.
Inequality(4) bounds the conditionalvolatilityof the permanentcomponent
in the same units as L by the difference of any expected log excess returnrelative to the returnof the asymptoticdiscount bond. Inequality(5) bounds the
MARGINAL UTILITY OF WEALTH
1985
unconditional volatility of the permanent component relative to that of the
stochastic discount factor. As we furtherdiscuss below, equation (5) describes
a property of the data that is closely related to Cochrane's(1988) size of the
randomwalk component.
To better understandthe measure of volatilityL(x), note that if var(x) = 0,
then L(x) = 0; the reverse is not true, because higher order moments than the
variance also affect L(x). More specifically,the variance and L(x) are special cases of the general measure of volatility f(Ex) - Ef(x), where f(.) is
a concave function. The statistic L(x) is obtained by making f(x) = logx,
while for the variance, f(x) = -x2. It follows that if a random variable x1 is
more risky than x2 in the sense of Rothschild-Stiglitz, then L(xl) > L(x2)
and, of course, var(xl) > var(x2).7As a special case, if x is lognormal, then
L(x) = 1/2var(logx). Volatility L(x) has been used to measure income inequalityand is known as Theil's second entropymeasure (Theil (1967)). Based
on Proposition2, Luttmer(2003) has worked out a continuous-timeversion of
our volatility bound and shown its relationshipto Hansen and Jagannathan's
volatilitybound for stochasticdiscountfactors.
The following propositioncharacterizesthe transitorycomponent;an upper
bound to its relative volatility can then be easily obtained along the lines of
Proposition2.
3: UnderAssumptions 1 and 2, R,+,1, = MTI/MT and
PROPOSITION
L(R 1)
L(MT1)
\MT
L(
V
Mt /)-
Rt+1,0o
E(log
R
)+
Rt---+l,1!
L(RRt+l,l )
for anypositiveRt+lsuch that E[log(Rt+1/Rt+,1,)]+ L(1/Rt+i)1) > 0.
Our decomposition does not require the permanent and transitorycomponents to be independent. Thus, knowing the amount of transitoryvolatility
relative to the overall volatility of the stochastic discount factors adds independent information in addition to knowing the volatility of the permanent
component relativeto the volatilityof the stochasticdiscountfactor.As we will
see below, for data availabilityreasons,we will be able to learn more about the
volatility of the permanent component than about the volatility of the transitory component. Kazemi (1992), in a related result,has shownthat in a Markov
economy with a limiting stationarydistribution,Rt,t+, = Mt/Mt+i.
As we mentioned above, the decomposition derived in Proposition 1 is not
necessarilythe only one that yields a martingaleand a transitorycomponent,
and thus the bounds derived above might not necessarilyapplyto other cases.
7Recall that xl is more risky than x2 in the sense of Rothschild and Stiglitz if, for E(xl) =
E(x2), E(f(xi)) < E(f(x2)) for any concave function f.
1986
F. ALVAREZ AND U. J. JERMANN
To strengthen our results, we show here that the volatility bounds derived in
Proposition2 are valid for any decompositionof the pricingkernel into a martingale and a transitorycomponent under some regularityassumptions.To do
this, we need a definition for the transitorycomponent, which we describe as
havingno permanentinnovations.
DEFINITION:We say that a random variable indexed by time, Xt, has no
permanentinnovationsif
lim
(6)
k-oo
Et+1(Xt+k)
= 1
a.s. for all t.
Et(Xt+k)
We say that there are no permanentinnovationsbecause, as the forecasting
horizon k becomes longer, informationarrivingat t + 1 will not lead to revisions of the forecasts made with current period t information.Alternatively,
condition (6) says that innovations in the forecasts of Xt+k have limited persistence, since their effect vanishes for large k. As can easily be seen, a linear
process that is covariance-stationaryhas no permanentinnovations.
PROPOSITION
4: Assume that the kernelhas a componentwith transitoryinnovationsMtr,that is, a componentfor which (6) holds, and a componentwith
permanentinnovationsMP thatis a martingale,so that
M
Let
Vt,t+k
= MtTMP.
be definedas
covt(Mt+k, MtP+k)
Vtt+k
E(MTA
Et(Mtkj
(
)Et((Mt+k)
and assumethat
lim
lim
k-oo
(1 + Vt+l,t+k)
=1
a.s.
a.s.
(1 + Vt,t+k)
Thenthe boundsin equations(4) and (5) apply.
For an example that illustratesthis result, see the supplementarymaterialto
this article (Alvarez and Jermann(2005)).
Following Cochrane and Hansen (1992, pp. 134-137), one can derive the
lower bound for the fraction of the variance of the stochastic discount factor
accounted for by its innovations,
E[vart( M,+)]
(+l-Rt+ var(Mt)
\Mt
1
var[Vt(lt+l)]
1)2
o a(Rt+l)
(E [V(1
t+)])
MARGINAL UTILITY OF WEALTH
1987
where R,t+ stands for any return.This lower bound takes a value of about 0.99
when R,t+ is an asset with a Sharperatio of 0.5 and one-period interestvolatility is low, such as var[Vt(l1t+)]= 0.052.A naturalinterpretationof this result is
in terms of persistent and transitorycomponents, and the conclusionwould be
in line with our main result. However, such an interpretationis not necessarily
correct.Indeed, one can easily constructexamplesof pricingkernelswith oneperiod interest rates that are arbitrarilysmooth and that have no permanent
innovations.The examplewe use in Section 4.3 is of this type. Nevertheless,our
results confirm such a naturalinterpretationof the findingsof Cochrane and
Hansen. We learn from our analysisthat the reason the two results can have
a similarinterpretationis because the term premiumsfor long-termbonds are
very small.
3.1. Yieldsand ForwardRates:AlternativeMeasuresof TermSpreads
For empiricalimplementation,we want to be able to extractas much information from long-termbond data as possible. For this purpose,we show in this
section that for asymptoticzero-coupon bonds, the unconditionalexpectations
of the yields and the forwardrates are equal to the unconditionalexpectations
of the holding returns.
Consider forwardrates. The k-periodforwardrate differentialis defined as
the rate for a one-period deposit that matures k periods from now relative to
a one-period deposit now:
f,(k)
-log (-
(Vt(lt+k-)l)
log
Vt,\'
Forward rates and expected holding returns are closely related. They both
compare prices of bonds with a one-period maturitydifference: the forward
rate does it for a given t, while the holding return considers two periods in a
row. Assuming that bond prices have means that are independent of calendar
time, so that EVt(lt+k) = EVr(l7+k) for every t and k, then it is immediatethat
the log excess holding
E[ft(k)] = E[ht(k)], where ht(k) = log(Rt+1,k/Rt+l,1),
return.
We define the continuouslycompoundedyielddifferentialbetween a k-period
discountbond and a one-period risk-freebond as
yt(k) _ log Vt(k)l/k I )
(Vt(lt+k)
Concerningholding returns,for empiricalimplementation,we assume enough
regularityso that
Rt+l,k
Et logi(R+
kt+lim,k
log( (Rt+o,o).-ht().
lim (mERt
-l
Flog
F,log
k-+oo
k-#oc
Rt+1,1/
Rt+,,/
1988
F. ALVAREZAND U. J. JERMANN
The next propositionshows that under regularityconditions, these three measures of the term spreads are equal for the limitingzero-couponbonds.
PROPOSITION
5: If the limitsof h,(k), f,(k), and yt(k) exist,the unconditional
expectationsof holdingreturnsare independentof calendartime;that is,
E(logRt+l,k) = E(logR,7+,k) for all t, r, k,
and if holdingreturnsandyields are dominatedbyan integrablefunction, then
ht li =
E[lm
k-roo
E[l
kA--0o
k
o
(k)
limyt(k)].
kA--0
o
In practice,these three measuresmay not be equallyconvenient to estimate
for two reasons. One is that the term premiumis defined in terms of the conditional expectation of the holding returns.However, this will have to be estimated from ex post realized holding returns,which are very volatile. Forward
rates and yields are, accordingto the theory, conditional expectationsof bond
prices. While forwardrates and yields are more serially correlated than realized holding returns,they are substantiallyless volatile. Overall,they should be
more preciselyestimated.The other reason is that,while resultsare derivedfor
the limitingmaturity,data are availableonly for finite maturities.To the extent
that a term spread measure converges more rapidlyto the asymptoticvalue,
it will be preferred.In the cases considered here, yields are equal to averages
of forwardrates (or holding returns), and the average only equals the last element in the limit. For this reason, yield differentials,y, might be slightlyless
informativefor k finite than the term spreads estimated from forwardrates
and holding returns.
4. EMPIRICALEVIDENCE
The main objectiveof this section is to estimate a lowerbound for the volatility of the permanentcomponent of pricingkernels, as well as the related upper
bound for the transitorycomponent. We address these two points in Sections
4.1 and 4.2. We also present two sets of additionalresults that help interpret
these estimates. First,we consider a simple process for the pricingkernel that
correspondsto the specificationimplied by many studies of dynamicgeneral
equilibriummodels. We show how our main findingscan provide guidance for
the degree of persistence that such models should reasonablydisplay.Second,
we measure the part of the permanent component that is due to inflation.As
is well known,price levels are typicallynonstationary.We documentthe extent
to which our findingsprovide information about the permanent components
of real variablesover and above the permanentcomponents in price levels.
1989
MARGINAL UTILITY OF WEALTH
4.1. The Volatilityof the PermanentComponent
TablesI, II, and III present estimates of the lower bound to the volatilityof
the permanentcomponent of pricingkernels derivedin Proposition2. Specifically,we report estimates of
(7)
E(log
+ ) - E(log Rt+lO
)
R+L) + L( Rt+l,l )
E(log Rt+l,l
obtained by replacingeach expected value with its sample analog for different
data sets.
TABLEI
SIZE OF PERMANENT COMPONENT BASED ON AGGREGATE EQUITY
AND ZERO-COUPON BONDS
Maturity
Equity
Premium
Term
Premium
E[log(R/R1)]
(1)
E[log(Rk/R1)]
(2)
Adjustment
for Volatility
of Short Rate
Size of
Permanent
L(1/R1)
Component
L(P)/L
(3)
(4)
(1) - (2)
E[log(R/R1)]
-E[log(Rk/R1)]
(5)
P[(5) < 0]
(6)
A. ForwardRates
25 years
0.0664
(0.0169)
29 years
E[f(k)];
-0.0004
(0.0049)
-0.0040
(0.0070)
HoldingPeriodIs 1 Year
0.0005
0.9996
(0.0002)
(0.0700)
1.0520
(0.1041)
0.0669
(0.0193)
0.0704
(0.0256)
0.0003
B. HoldingReturns
25 years
0.0664
(0.0169)
29 years
E[h(k)];
-0.0083
(0.0340)
-0.0199
(0.0469)
HoldingPeriodIs 1 Year
0.0005
1.1164
(0.0002)
(0.5186)
1.2899
(0.7417)
0.0747
(0.0342)
0.0863
(0.0446)
0.0145
C. Yields
25 years
E[y(k)];
0.0082
(0.0033)
0.0082
(0.0035)
HoldingPeriodIs 1 Year
0.0005
0.8701
(0.0002)
(0.0534)
0.8706
(0.0602)
0.0582
(0.0196)
0.0582
(0.0226)
0.0015
E[y(k)];
0.0174
(0.0031)
0.0168
(0.0033)
HoldingPeriodIs 1 Month
0.0004
0.7673
(0.0002)
(0.0717)
0.7755
(0.0795)
0.0588
(0.0213)
0.0595
(0.0241)
0.0664
(0.0169)
29 years
D. Yields
25 years
29 years
0.0763
(0.0180)
0.0030
0.0266
0.0050
0.0028
0.0067
Note: For A, term premia (column 2) are given by 1-year forward rates for each maturity minus 1-year yields for
each month. For B, term premia (column 2) are given by overlapping holding returns minus 1-year yields for each
month. For C, term premia (column 2) are given by yields for each maturity minus 1-year yields for each month.
For A, B, and C, equity excess returns are overlapping total returns on NYSE, Amex, and Nasdaq minus 1-year
yields for each month. For D, short rates are monthly rates. Newey-West asymptotic standard errors using 36 lags are
shown in parentheses. P values in column 6 are based on asymptotic distributions. The data are monthly from 1946:12
to 1999:12. See Appendix B for more details.
TABLE II
SIZE OF PERMANENT COMPONENT BASED ON GROWTH-OPTIMAL PO
AND 25-YEAR ZERO-COUPON BONDS
Growth
Optimal
E[log(R/R )]
(1)
z
L;
.
'i
tfl
a)
N
w
P4
I4
Term
Premium
Adjustment for Volatility
of Short Rate
E[log(Rk/R1)]
(2)
L(1/R1)
(3)
Size of Perm
Compone
L(P)/L
(4)
A. Growth-OptimalLeveragedMarketPortfolio(Portfolio Weight:3.46 for Monthly Holding Period; 2.14
One-year holding period
0.9998
0.0005
Forwardrates
0.1095
-0.0004
(0.0002)
(0.0402)
(0.0049)
(0.0426
1.0708
-0.0083
Holding return
(0.3203
(0.0340)
0.9210
0.0082
Yields
(0.0381
(0.0033)
One-month holding period
0.0004
0.8946
Yields
0.1689
0.0174
(0.0031)
(0.0002)
(0.0519
(0.0686)
B. Growth-OptimalPortfolioBased on the 10 CRSP Size-DecilePortfolios
One-year holding period
a).0005
Forwardrates
0.1692
-0.0004
(0.0049)
(C
).0002)
(0.0437)
-0.0083
Holding return
(0.0340)
Yields
0.0082
(0.0033)
One-month holding period
Yields
0.0174
0.0004
0.2251
(0.0737)
(0.0031)
(0.0002)
0.9999
(0.0276
1.0459
(0.2053
0.9488
(0.0199
0.9209
(0.0320
TABLE III
SIZE OF PERMANENTCOMPONENTBASED ON AGGREGATEEQUITY AND COUPO
Equity
Premium
U.S.
1871-1997
TermPremium
E[logR/Rl]
(1)
E[y]
0.0494
(0.0142)
0.0034
(0.0028)
E[h]
(2)
Adjustment
L(1/R)
(3)
0.0003
(0.0001)
0.0043
(0.0064)
1946-1997
U.K.
1801-1998
0.0715
(0.0193)
0.0122
(0.0025)
0.0239
(0.0083)
0.0002
(0.0020)
0.0004
(0.0001)
0.006
(0.0129)
0.0003
(0.0001)
0.0036
(0.0058)
1946-1998
0.0604
(0.0198)
0.0092
(0.0038)
0.0018
(0.0143)
0.0007
(0.0002)
Size of Pe
Comp
L(P)/
(4)
0.926
(0.054
0.907
(0.123
0.824
(0.046
0.911
(0.172
0.978
(0.080
0.836
(0.222
0.837
(0.090
0.958
(0.228
Note:Column1 givesthe averageannuallog returnon equityminusthe averageshortrate for the year.Column2 givesthe av
bonds minus the average short rate for the year, or the average annual holding period return on long-term government coupon bon
Newey-West asymptotic standard errors with five lags are shown in parentheses. See Appendix B for more details.
1992
E ALVAREZAND U. J. JERMANN
In Table I, we report estimates of the lower bound given in equation (7)
and of each of the three quantities entering into it, as well as the asymptotic
normalprobabilitythat the numeratoris negative. We present estimates using
zero-coupon bonds for maturitiesat 25 and 29 years, for various measures of
the term spread (based on yields, forwardrates, and holding returns),and for
holdingperiods of 1 year and 1 month. As returnR,+1,we use the CRSPvalueweighted index that covers the NYSE, Amex, and NASDAQ. The data are
monthly,from 1946:12to 1999:12.Standarderrorsof the estimated quantities
are presented in parentheses; for the size of the permanent component, we
use the delta method. The variance-covarianceof the estimatesis computedby
using a Newey and West (1987) windowwith 36 lags to accountfor the overlap
in returnsand the persistence of the different measuresof the spreads.8
Based on the asymptotic(normal) distribution,the probabilitythat the term
spread is larger than the log equity premium is very small, in most cases well
below 1%. Hence, the hypothesis that the pricing kernel has no permanent
innovation is clearly rejected. Not only is there a permanent component, it is
very volatile. We find that the lower bound of the volatility of the permanent
component is about 100%;none of our estimates is below 75%. The estimates
are precise; standarderrorsare at or below 10%,except for holding returns.
Two points about the result in Table I are noteworthy.First, the choice of
the holding period, and hence the level of the risk-freerate, has some effects
on our estimates. For instance,by usingyields with a yearlyholding period, the
size of the permanent component is estimated to be about 87%. Instead, by
using yields and a monthly holding period, we estimate it to be 77%. This difference is due to the fact that monthlyyields are about 1%below annualyields,
affectingthe estimate of the denominatorof the lower bound.9Second, by estimating equation (7) as the ratio of sample means, our estimates are consistent
but biased in small samplesbecause the denominatorhas nonzero variance.In
Appendix C we present estimates of this bias. They are quantitativelynegligible for forwardrates and yields, on the order of about 1% in absolute value
terms. Estimates of the bias are somewhatlargerfor holding returns.
Since (7) is defined for any return Rt+1, we select portfolios with high
E(log(Rt+1/Rt+1,1))in TableII to sharpenthe bounds based on the equitypremium in Table I. Table II contains the same information as Table I, except
8For maturitieslonger than 13 years, we do not have a complete data set for zero-coupon
bonds. In particular,long-termbonds have not been consistentlyissued duringthis period. For
instance, for zero-coupon bonds that mature in 29 years, we have data for slightly more than
half of the sample period: data are missing at the beginning and in the middle of our sample.
The estimates of the variousexpectedvalues on the right-handside of (7) are based on different
numbersof observations.We take this into accountwhen computingthe variance-covarianceof
our estimators.Ourproceduregives consistentestimatesas long as the periodswith missingbond
data are not systematicallyrelated to the magnitudesof the returns.
90ur data set does not containthe informationnecessaryto presentresultsfor monthlyholding
periods for forwardrates and holding returns.
MARGINAL UTILITY OF WEALTH
1993
that Table II covers only bonds with 25-year maturity.We find estimates of
E(log(Rt+1/Rt+,,,)) of up to 22.5% comparedto 7.6% in TableI. The smallest
estimate of the lower bound in TableII is 89% as opposed to 77% in TableI.
In panel A of Table II we let Rt+l be a fixed-weightportfolio of aggregate
equity and the risk-free rate that maximizes E(log(Rt+1/Rt+l,l)), that is, we
are derivingthe so-called growth-optimalportfolio (see Bansal and Lehmann
(1997)). Depending on the choice of the holding period, E(log(Rt+,/Rt+,1)) is
up to 9% largerthan the premiumpresented in TableI, with a share of equity
of 2.14 or 3.46. In panel B of TableII, we choose a fixed-weightportfolio from
the menu of the 10 CRSP size-decile portfolios. This leads to an average log
excess returnof up to 22.5%.
TableIII extends the sample period to over 100 years and adds an additional
country, the United Kingdom. For the United States, given data availability,
we use coupon bonds with about 20-year maturity.For the United Kingdom,
we use consoles. For the United States, we estimate the size of the permanent
component between 82% and 93%, dependingon the time period andwhether
we consider the term premium or the yield differential.Estimated values for
the United Kingdomare similarto those for the United States.
A naturalconcernis whether 25- or 29-yearbonds allowfor good approximations of the limitingterm spread.FromFigure 1, whichplots term structuresfor
three definitionsof term spreads,we take the long end of the term structureto
be either flat or decreasing.Extrapolatingfrom these pictures suggests, if anything, that our estimates of the size of the permanentcomponent presented in
TablesI and II are on the low side. In this figure,the standarderrorbands are
wider for longer maturities,which is due to two effects. One is that spreadson
long-termbonds are more volatile, especiallyfor holding returns.The other is
that for longer maturities,as discussedbefore, our data set is smaller.
Note that for equation (7) to be well defined, specificallyfor L(1/Rt+,,,)
to be finite, we have assumed that interest rates are stationary.10While the
assumption of stationaryinterest rates is confirmed by many studies (for instance, Ait-Sahalia (1996)), others report the inabilityto reject unit roots (for
instance, Hall, Anderson, and Granger(1992)). Cochrane (2005, p. 199) sums
up the issue eloquently:"the level of nominal interest rates is surely a stationaryvariablein a fundamentalsense: we have observationsnear 6% as far back
as ancient Babylon, and it is about 6% again today."Also, consistent with the
idea that interest rates are stationaryand, therefore, L(1/Rt+,,,) is finite, Table III shows lower estimates for the very long samples than for the postwar
period.
10Equation(4), which defines a bound for the size of the permanentcomponent in absolute
terms, does not requirethis assumption.
1994
F ALVAREZ AND U. J. JERMANN
Average ForwardRates in Excess of One-YearRate
2
1.5
...
-
.....
:......... ...
1
,
........
..
.
...
aD
D 0.5
Q.
CI
C
0
a)
.a -0.5
-1
-1.5
5
0
10
15
20
25
30
Average HoldingReturnson Zero-CouponBonds in Excess of One-YearRate
4
2
a) O0
>1
-------lc;-
----
-----
------
a)
Q.
-2
------,,_
---
--,,,
c
a)
| -4
-6
-8
t
5
10
15
20
25
30
AverageYields on Zero-CouponBonds in Excess of One-YearRate
1.4
1.2cu 1a) 0.8 Q.
8 0.6 a)
Q-0.40.2 -
15
maturity,in years
FIGURE1.
30
1995
MARGINAL UTILITY OF WEALTH
4.2. The Volatilityof the Transitory
Component
We now report on estimates for volatility of the transitorycomponent and
the related upper bound for the volatility of the transitorycomponent relative to the volatility of the stochastic discount factor. As shown in Figure 2,
L(1/Ro) goes up to 0.04 for 29-yearmaturity,while it is about 0.015 for 20-year
maturity.The correspondingupperbound for the volatilityrelativeto the overall volatility L(1/Ro)/L(M'/M) reaches a maximumof 23% for 29-year maturity, while it is about 9% for 20-year maturity.This upper bound is based
on the CRSP decile portfolios as reported in Table II. Unfortunately, these
estimates are somewhatdifficultto interpretbecause there is no apparentconvergence for the available maturities. Moreover, the lack of a complete data
set for all maturities seems to result in a substantialupwardbias of the estimates of L(1/Rk) for maturities k > 20 years. Figure 3 shows that the data
for the longest maturities are concentrated in the part of the sample characterized by high volatility.A simple way to adjustfor this sample bias would be
to assume that the ratio of the volatilities for different maturitiesis constant
across the entire sample. We can then consider the volatility for the 13-year
bond, the longest span for which we have a complete sample, as a benchL(1/Rk) with one standard deviation band
Il
Upper bound for L(1/Rk )/L(M'IM)with one standard deviation band
.
(I "41-
1)._
0.3-
0.05 -
0.25 0.04 -
0.20.03 0.15 0.02 -
*
I
0.1 '
/
*/
?/
0.01 -
;
~ ~ ~
?/
?(
0.05 -
'''
I I
0
~~~~~
/
?/~
:.
I I'-
10
20
30
0
10
20
Maturity, k
Maturity, k
FIGURE 2.
30
1996
F ALVAREZ AND U. J. JERMANN
Log holding returs for selected discount bonds
1950
FIGURE3.-Log
1960
1970
1980
1990
holding returns for selected discount bonds.
mark. The ratio of the volatilities of the 13-yearbond for the entire sample
over that for the sample covered by the longest availablematurity,29 years, is
about 0.8, so the relative upper bound would be adjustedto about 18%, down
from 23%.
1997
MARGINAL UTILITY OF WEALTH
Concerning the measurement of the permanent component, note that the
averageterm spreadfor the 13-yearbond is actuallylargerfor the shortersample covered by the 29-year bond, although by only 20 basis points. Thus, any
adjustmentwould, if anything,further increase the estimates of the volatility
of the permanentcomponent in equation (7).
4.3. An Exampleof a PricingKernel
We present here an example that illustratesthe power of bond data to distinguishbetween similarlevels of persistence.In particular,the example shows
that even for bonds with maturitiesbetween 10 and 30 years, one can obtain
strong implicationsfor the degree of persistence. Alternatively,the example
shows that to explain the low observed term premia for long-term bonds at
finite maturitieswith a stationarypricingkernel, the largest root has to be extremely close to 1. The example is relevant,because many studies of dynamic
general equilibriummodels imply stationarypricingkernels.
Assume that
logMt+l = log 3 + p logM, + ,t+l
with 8t+i - N(0, o2). Simple algebrashows that
(8)
h(k)=
2 (-
_
p2(k-1)
This expression suggests that if the volatility of the innovation of the pricing
kernel, ao2,is large, then values of p only slightlybelow 1 may have a significant
quantitativeeffect on the term spread. In Table IV, we calculate the level of
persistence, p, required to explainvarious levels of term spreads for discount
bonds with maturitiesof 10, 20, and 30 years. As is clear from TableIV, p has
to be extremelyclose to 1.
For this calculationwe have set ,2= 0.4 for the following reasons.Based on
Proposition2 and assuminglognormality,we get
+var g M
2 E
R
Rt(log +,I),
TABLE IV
FORBONDSWITHFINITEMATURITIES
REQUIREDPERSISTENCE
Term Spread
Maturity
(Years)
10
20
30
0%
0.50%
1%
1.50%
1.0000
1.0000
1.0000
0.9986
0.9993
0.9996
0.9972
0.9987
0.9991
0.9957
0.9980
0.9987
1998
E ALVAREZAND U. J. JERMANN
where R,+1 can be any risky return. Based on our estimates in Table II,
the growth-optimal excess return should be at least 20%, so that
var(log(Mt+1/Mt))> 0.4. Finally,for p close to 1 we can write
var(log Mt+I)=
2p
2
4.4. Nominal versusReal PricingKernels
Because we have so far used bond data for nominalbonds,we have implicitly
measured the size of the permanent component of nominal pricing kernels,
that is, the processes that price future dollaramounts.We present now two sets
of evidence that show that the permanentcomponent is to a large extent real,
so thatwe have a directlinkbetween the volatilityof the permanentcomponent
of pricingkernels and real economic fundamentals.
First, assume, for the sake of this argument,that all of the permanentmovements in the (nominal) pricing kernel come from the aggregate price level.
Specifically,assume that Mt = (1/Pt)M, where Pt is the aggregateprice level.
Thus 1/Pt convertsnominal payouts into real payouts and MT prices real payouts. Because 1/P, is directly observable,we can measure the volatility of its
permanent component directly and then compare it to the estimated volatility of the permanent component of pricing kernels reported in Tables I, II,
and III. It turns out that the volatility of the permanentcomponent in 1/Pt is
estimated at up to 100 times smaller than the lower bound of the volatilityof
the permanent component in pricing kernels estimated above. This suggests
that movements in the aggregate price level have a minor importance in the
permanentcomponent of pricingkernels and, thus, permanentcomponents in
pricing kernels are primarilyreal. It should be noted that this interpretation
is only valid to the extent that the behavior of the official consumer price index accuratelyreflects the properties of the price level faced by asset market
participants.
The next proposition shows how to estimate the volatilityof the permanent
component based on the L(.) measure.
PROPOSITION
6: Assume that the process Xt satisfiesAssumptions1 and 2
and that the following regularityconditionsare satisfied:(a) Xt+l/Xt is strictly
= 0. Then
stationary and (b) limk_ooj,L(EtXt+k/Xt)
(9)
lim L (X )
L(XP)A; / = k-+oc
k
\ Xt /
The usefulness of this proposition is that L (XIP1/XP) is a naturalmeasure
for the volatilityof the permanentcomponent. However, it cannot directlybe
1999
MARGINAL UTILITY OF WEALTH
estimated if only Xt is observable, but XP and XT are not observable separately. The quantitylimk_, (1/k)L(Xt+k/X,) can be estimated with knowledge of only Xt. This result is analogous to a result in Cochrane (1988), with
the main difference that he uses the varianceas a measure of volatility.
Cochrane (1988) proposes a simple method for correctingfor small sample
bias and for computingstandarderrorswhen using the variance as a measure
of volatility.Thus,we will focus our presentationof the resultson the variance,
having established first that, without adjustingfor small sample bias, the variance equals approximatelyone-half of the L () estimates,whichwould suggest
that departuresfrom lognormalityare small. Overall,we estimate the volatility
of the permanentcomponent of inflation to be below 0.5% based on data for
1947-1999 and below 0.8%based on data for 1870-1999. This comparesto the
lower bound of the (absolute) volatility of the permanent component of the
pricingkernel,
(10)
L Mt+l ) > E[logR+1 - logRt+ ,l],
thatwe have estimated to be up to about 20% as reportedin column5 in Tables
I, II, and III.
Table V contains our estimates. The first two rows display results based on
estimating an AR1 or AR2 for inflation and then computing the volatility
TABLEV
DUE TO INFLATION
SIZE OF THEPERMANENTCOMPONENT
1947-1999
AR1
AR2
(1/2k)var(logPt+k/Pt)
1870-1999
AR1
AR2
L(Pt/Pt+k)/var(logPt+k/Pt)
AR(2)
o2
Size of Permanent Component
0.66
0.87
-0.24
0.0005
0.0004
0.0021 (0.0009)
0.0015 (0.0006)
k = 20
k = 30
L(Pt/Pt+k)/var(logPt+k/Pt)
(1/2k)var(logPt+k/Pt)
AR(1)
0.0043 (0.0031)
0.0030 (0.0027)
(k = 20)
(k = 30)
0.50
0.51
AR(1)
AR(2)
0.28
0.27
0.00
,2
0.0052
0.0052
Size of Permanent Component
0.0049 (0.0013)
0.0050 (0.0006)
k = 20
0.0077 (0.0035)
k =30
0.0067 (0.0038)
(k = 20)
(k = 30)
0.51
0.49
Note: For the AR(1) and AR(2) cases, the size of the permanent component is computed as one-half of the spectral
density at frequency zero. The numbers in parentheses are standard errors obtained through Monte Carlo simulations.
For (1/2k)var(logPt+k/Pt), we have used the methods proposed by Cochrane (1988) for small sample corrections
and standard errors. See our discussion in the text for more details.
2000
E ALVAREZAND U. J. JERMANN
of the permanent component as one-half of the (population) spectral density at frequency zero. For the postwar sample, 1947-1999, we find 0.21%
and 0.15% for AR1 and AR2, respectively. The third row presents the results using Cochrane's (1988) method that estimates var(logX+ 1/XP) using
limk,(l1/k) var(logXt+k/X). For the postwar period, the volatility of the
permanent component is 0.43% or 0.30%, depending on whether k = 20
is approxior 30.11 The table also shows that L(Xt+k/Xt)/var(logXt+k/Xt)
of
inflation
Note
that
the
roots
the
for
0.5.
mately
process
reported in TableV
are far from 1, supportingour implicit assumptionthat inflation rates are stationary.
A second view about the volatility of the permanent component can be
obtained from inflation-indexedbonds. Such bonds have been traded in the
United Kingdom since 1982. Consideringthat an inflation-indexedbond represents a claim to a fixed number of units of goods, its price provides direct
evidence about the real pricing kernel. However, because of the 8-month indexation lag for U.K. inflation-indexedbonds, it is not possible to obtain much
information about the short end of the real term structure. Specifically,an
inflation-indexedbond with outstandingmaturityof less than 8 months is effectively a nominal bond. For our estimates, this implies that we will not be
able to obtain direct evidence of E(logRt+,,1) and L(1/Rt+,1l) in the definition
of the volatilityof the permanentcomponent as given in equation (5). Because
of this, we focus on the bound for the absolutevolatilityof the pricingkernel as
given in equation (10). For the nominal kernel,we use averagenominal equity
returns for ElogRt+1, and, for ElogRt+,,,, we use forward rates and yields
for 20 and 25 years, from the Bank of England'sestimates of the zero-coupon
term structures,to obtain an estimate of the right-handside of (10). For the
real kernel,we take the averagenominal equityreturnminusthe averageinflation rate to get E log Rt+1;for Elog Rt+,,,, we use real forwardrates and yields
from a zero-coupon term structureof inflation-indexedbonds. The right-hand
side of (10) differs for nominal and real pricing kernels only if there is an inflation riskpremiumfor long-termnominalbonds. If long-termnominalbonds
have a positive inflationriskpremium,then the lowerbound for the permanent
component for real kernels will be largerthan for nominal kernels.
TableVI reportsestimatesfor nominaland real kernels.The data are further
describedin Appendix B. Consistentwith our findingthat the volatilityof the
permanentcomponent of inflation is very small, the differences in volatilityof
the permanentcomponents for nominal and real kernels are very small. Comparingcolumns 3 and 6, for one point estimate the volatilityof the permanent
component of real kernels is larger than the estimate for the corresponding
nominal kernels;for the second case, they are basicallyidentical. In any case,
is
defined as
isdefinedoX
"Cochrane's (1988) estimator
a 12 = ( T T)()
)k k[XJ -- Xjjk k - T x
estimator(XTk
=
(XT -x0)]2, with T the sample size, x
logX, and standarderrorsgiven by (5 )O ok.
TABLE VI
INFLATION-INDEXED BONDS AND THE SIZE OF THE PERMANENT COMPONENT OF PRICING KERNELS:
Nominal Kernel
(1)- (2)
Size of
Permanent
Equity
Forward Yield
Maturity
(Years)
E[log(R)]
(1)
E[log(F)]
25
0.1706
(0.0197)
0.0762
(0.0040)
E[log(Y)]
Component
L(P)
(3)
(2)
0.0815
(0.0046)
0.0944
(0.0212)
0.089
(0.0200)
Inflation Rate
E[log(7r)]
(4)
E[log(F)]
0.0422
(0.0063)
0.0342
(0.0023)
Note: Real and nominal forward rates and yields are from the Bank of England. Stock returns and inflation rates are from Globa
given in parentheses, are computed with the Newey-West method with 3 years of lags and leads. See Appendix B for more details.
2002
E ALVAREZAND U. J. JERMANN
the correspondingstandarderrorsare largerthan the differencesbetween the
resultsfor nominal and real kernels.
5. PRICING KERNELSAND AGGREGATECONSUMPTION
In many models used in the literature, the pricing kernel is a function of
currentor lagged consumption.Thus, the stochastic process for consumption
is a determinantof the process of the pricingkernel. In this section, we present
sufficient conditions on consumption and the function mapping consumption
into the pricingkernel so that pricingkernels have no permanentinnovations.
We are able to define a large class of stochasticprocessesfor consumptionthat,
combinedwith standardpreference specifications,will result in counterfactual
asset pricing implications.We also present an example of a utility function in
which the resultingpricingkernels have permanentinnovationsbecause of the
persistence introduced through the utility function. Finally, we estimate the
volatility of the permanent component in consumption directly and compare
it to our estimates of the volatility of the permanent component of pricing
kernels.
As a startingpoint, we present sufficient conditions for kernels that follow
Markovprocesses to have no permanent innovations.We then consider consumptionwithin this class of processes. Assume that
Mt=
f(t)f(st),
where f is a positive function, and that st E S is Markovwith transitionfunction Q, which has the interpretationPr(st+lE Als- = s) = Q(s, A).
We assume that Q has an invariantdistributionA*and that the process {st}
is drawn at time t = 0 from A*.In this case, st is strictly stationary and the
unconditionalexpectations are taken with respect to A*.We use the standard
notation
(Tkf)()
Jf
(s s, ds'),
where Qk is the k-step ahead transitionconstructedfrom Q.
PROPOSITION 7: Assume that there is a unique invariant measure, A*. In
addition, if either (i) limk_,oo(Tkf)(s) = f fdA* > 0 and is finite or in case
limk, (rTkf)(S) is not finite, if (ii) limk_j[(Tk 1f)(s') - (Tkf)(s)] < A(s) for
each s and s', then
klim
k-oo
Et+1(Mt+k)= 1.
Et(Mt+k)
MARGINAL UTILITY OF WEALTH
2003
We are now ready to consider consumptionexplicitly.Assume that
Ct = r(t)ct = T(t)g(st),
where g is a positive function, s, E S is Markovwith transitionfunction Q, and
r(t) represents a deterministictrend. We assume (a) that a unique invariant
measure A*exists. Furthermore,assume (b) that
lim (Tkh)(s)=
k-oo c
hdA*
for all h(.) bounded and continuous.
8: Assume that Mt = 3(t)f (ct, xt) with f () positive,bounded,
PROPOSITION
and continuous,and that (ct, x,) = Stsatisfiesproperties(a) and (b) withf () > 0
withpositiveprobability.ThenMt has no permanentinnovations.
An example covered by this proposition is CRRA utility, 1/(1 - y)c~-Y with
relative risk aversion y, where f(ct) = c 7 with c c> >E > 0. If consumption
would have a unit root, then properties (a) and (b) would not be satisfied.
For the CRRA case, even with consumptionsatisfyingproperties(a) and (b),
Proposition 8 could fail to be satisfied because ct7 is unbounded if c, gets
arbitrarilyclose to zero with large enough probability.It is possible to construct examples where this is the case, for instance, along the lines of the
model in Aiyagari (1994). This outcome is driven by the Inada condition
u'(O)= oo. Note also that the bound might not be necessary.For instance, if
logct = plog ct_ + ?t with e - N(0, o2) and IpI< 1, then, log f(c) = -ylogct,
and direct calculationsshow that condition (6), which defines the propertyof
no permanentinnovations,is satisfied.
5.1. ExampleswithAdditionalState Variables
There are manyexamplesin the literaturefor whichmarginalutilityis a function of additional state variables and for which it is straightforwardto apply
Proposition8, very much like for the CRRA utilityshown above. For instance,
the utility functions that displayvarious forms of habits such as those used by
Ferson and Constantinides(1991), Abel (1999), and Campbell and Cochrane
(1999). On the other hand, there are cases where Proposition8 does not apply;
for instance, as we show below, for the Epstein-Zin-Weil utility function. In
this case, even with consumptionsatisfyingthe conditions requiredfor Proposition 8, the additional state variable does not have an invariantdistributions.
Thus, innovationsto pricingkernels alwayshave permanenteffects.
2004
F ALVAREZ AND U. J. JERMANN
Assume the representative agent has preferences represented by nonexpected utility of the recursiveform
Ut =
(Ct, EtUt+l),
where U, is the utility starting at time t and / is an increasingconcave function. Epstein and Zin (1989) and Weil (1990) develop a parametriccase in
which the risk aversioncoefficient, y, and the reciprocalof the elasticityof intertemporalsubstitution,p, are constant.They also characterizethe stochastic
discount factor Mt+ /Mt for a representativeagent economy with an arbitrary
consumptionprocess {Ct}as
1Mt+l
(ll)
t+1-
(Ct+l
-R
M,= Ct)
with 0 = (1 - y)/(1 - p), where 3 is the time discount factor and Rc+ is the
gross return on the consumption equity, that is, the gross return on an asset
that pays a of dividendsequal to consumption{Ct}.
Inspection of (11) reveals that a pricingkernel Mt+1for this model is
(12)
where Yt+l= Rt .Y
Mt+l= 0(ft+l)Yt+llCt+CO,
and Yo= 1.
The next propositionshowsthat the nonseparabilitiesthat characterizethese
preferences for 0 : 1 are such that, even if consumption is i.i.d., the pricing
kernel has permanent innovations. More precisely, assume that consumption
satisfies
(13)
C = tct,
where ct E [c, c] is i.i.d. with cumulative distributionfunction (c.d.f.) F. Let
Vtcbe the price of the consumption equity, so that R+1 = (Vt+ + Ct+1)/1Vc.
We assume that agents discountthe future enough so as to have a well-defined
price-dividendratio. Specifically,we assume that
l/e
c' 1-ly
(14)
max Pr1- /a -
ce[c,c]
dF(c')
< 1.
C/
PROPOSITION
9: Let thepricingkernelbe givenby (12), let the detrendedconsumptionbe i.i.d. as in (13), and assumethat (14) holds. Thentheprice-dividend
ratiofor the consumptionequityis given by VtI/Ct= ict-1 for some constant
i > 0; hence, Vtc/Ct is i.i.d. Moreover,
(15)
xt+?1,k
+
((1 I
))
'
Et(Mt+k) E {(1+ 1C(-))
}
Et+,1(Mt+k)
2005
MARGINAL UTILITY OF WEALTH
thus thepricingkernelhaspermanentinnovationsif and only if 0
Cthas strictlypositivevariance.
1, y
1, and
Note that 0 = 1 correspondsto the case in which preferences are given by
time separableexpected discounted utility;hence, with i.i.d. consumption,the
pricing kernel has only temporary innovations. Expression (15) also makes
clear that for values of 0 close to 1, the volatilityof the permanentcomponent
is small.
5.2. The Volatilityof the PermanentComponentin Consumption
We present here estimates of the volatilityof the permanent component of
consumption, obtained directly from consumption data. We end up drawing
two conclusions.One is that the volatilityof the permanentcomponent in consumptionis about half the size of the overallvolatilityof the growthrate, which
is lower than our estimates of the volatility of the permanent component of
pricing kernels. This suggests that, within a representative agent asset pricing framework,preferences should be such as to magnify the importance of
the permanent component in consumption.12The other conclusion, as noted
in Cochrane (1988) for the randomwalk component in GDP, is that standard
errorsfor these direct estimates are large.
As in Section 4.4 for inflation, we use Cochrane's method based on
is close to 0.5. Specifithe variance, since L(Xt+k/Xt)/var(logXt+k/Xt)
cally, for k up to 35, it lies between 0.47 and 0.49. Our estimates for
(1/k)var(logXt+k/Xt)/var(logXt+1/Xt), with associated standard error
bands, are presented in Figures 4 and 5 for the periods 1889-1997 and
1946-1997, respectively. For the period 1889-1997, shown in Figure 4, the
estimates stabilize at around 0.5 and 0.6 for k larger than 15. For the postwar period, shown in Figure 5, standarderrorbands are too wide to drawfirm
conclusions.
6. CONCLUSIONS
The main contributionof this paper is to derive and estimate a lower bound
for the volatilityof the permanentcomponent of asset pricingkernels. We find
that the permanent component is about at least as volatile as the stochastic
discountfactor itself. This result is drivenby the historicallylow yields on longterm bonds. These yields contain the market'sforecasts for the growthrate of
the marginalutility of wealth over the period that correspondsto the maturity of the bond. A related bound that measures the volatilityof the transitory
component suggests it to be considerablyless importantthan the permanent
12Thisconclusionwould not be valid if asset marketparticipationis limited,unless the participants'consumptionexhibitsthe same persistencepropertiesas the aggregate.
.
.^.
.. ..................
. .. .. ...
.........
.
.
2006
1.4
..
.
.
F ALVAREZ AND U. J. JERMANN
..............
J ..........
...
.
.1 ..........
...
I
..
.........
.....
......I. .............
..............
. .
1.2\..
.
1
. . . . . . . . . .
.........
.....
.
- - - - - - -
,
.........
. . . . . . . . . . . . . . .
'
~ ~ ~ ~~
~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
h~~
.....
.....
-
......:'.
.
-
0.8 -
\
v
~~~~~~~~~~~~.
t~~~~~~~~~~
.
~
-'
"
-.
.
.........
.
...
.
.
..
.
....
-.......
:
4
:
0.6 -
'E
:
:..........:
:Z
e :-:
......
.................
0.4 -
0.2 -
0
-
0
?
5
10
15
20
25
30
35
FIGURE4.-l/ k times the variance of k differences of consumption divided by the variance
of the first difference: 1889-1997. Bands show one asymptotic standard error; a period is 1 year.
Id
-I
10-
8
6
.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
.. ...
2
.
.
.
.
. .... . . . . . . . . . . . . . . . ..... . . . . .
....
.
..
-~.
-
. . . . .
-
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0
5
10
15
20
25
30
35
FIGURE5.-l/k
times the variance of k differences of consumption divided by the variance
of the first difference: 1846-1997. Bands show one asymptotic standard error; a period is 1 year.
2007
MARGINAL UTILITY OF WEALTH
component. We also relate the persistence of pricingkernels to the persistence
of their determinantsin standard models, notably consumption. We present
sufficientconditions for consumptionand preference specificationsto imply a
pricing kernel with no permanent innovations. We present evidence that the
permanent component of pricing kernels is determined, to a large extent, by
real as opposed to nominal factors. Finally,we present some evidence that the
importance of the permanent component in consumption is smaller than the
permanentcomponent in pricingkernels.Withina representativeagent framework, this evidence points towardutilityfunctions that magnifythe permanent
component.
Dept. of Economics, Universityof Chicago, Chicago, IL 60637, U.S.A.;
and NBER;[email protected]
and
The
Wharton
School, Universityof Pennsylvania,3620 LoDept. of Finance,
cust Walk,Philadelphia,PA 19104-6367,U.S.A.;and NBER;jermann@wharton.
wharton.upenn.edu/ -jermann.
upenn.edu; http://finance.
ManuscriptreceivedJanuary,2002;final revisionreceivedApril,2005.
APPENDIX A: PROOFS
2: We show that (i) Rt,t+l,oo= MtT/Mt+ and (ii)
PROOFOF PROPOSITION
=L
M
MLt
tMP
M'
mt+l
+Etlog
Rt+1,'
Rt+l,cc
and then that this implies
L
P
t
Mt+1
\Mt
>Elog
Rt+lj
Etlog
-E/log
Rt,+l,o
Rt+,1
Rt+1,1
(i) Using Assumption 1,
Et+lMt+k
Rt, t+1,l,
m
-k-oc
V,t+l(1t+k)
Vt(lt+k)
)
Mt+l
limko
EtMt+kP+k
Mt
Mt+
lim
k--*o EtMt+k
Mt
Mt+i
M
Mt
Et+1Mt+k/1t+k
l
lim
,=m
k---o
MT
M+l
M,+
E
EtMt+k/3t+k
Mt
E ALVAREZ AND U. J. JERMANN
2008
(ii) By definition,
=(A.1
\
Mt+
,
Mt,+l(
-gV(lt
=
+ +
L-Elog
log
Mtl l
M
+1 L,( I
t
1,,o
EtM17
log MP
+M1t
MT
t+i )
M
Rt+,,
= E, log Rt+i-log
IR+1,1 +
Mt
Mt
Mt
Mt
Mt+
log
Et
l(A
Rt+j-- log
, =0
ogE1)(Rt > EtM
log
R,+,,I
)
because from no-arbitrageand concavityof the log,
-E,\ -E = E log(R+1)
,
M
Mt+
-Et log
t
> E, log(Rt+1).
Mt
For an unconditionalversion of the bound we use that L(x,+1) = ELt(xt+,) +
L(E,xt+1). Using this result,we take unconditionalexpectations
ELt M+ )=ELt
(
Mt )
( Mt
M
=L
L
) -EEtlogR
(M ) t
MtP )
MP
M
Mt
Mt
- Rt++')
(
R,+;
Rt+E,1
- E log
Rt+,,,
> ElogR
+L
e1r
aMt+l
M, ),E-Mt R,+,1"
Rt+mt1
V(
R,+l,u
and form the ratio
L( M+I
M,+l
L( "Mt
L( M,+)-L(R,
)-Elog
L (M,+
LMt+
Rt+
'
Rt+l,1
MARGINAL UTILITY OF WEALTH
2009
so that if [-L(1/R+1,1 ) - Elog(Rt,+1,/Rt+l,l)] < 0 and Elog(Rt+1/Rt,+,1)+
L(l/Rt+l,1) > 0,
>
L(PM?Rt
t
-L(1)-(
E log
>
R
-E log Rt+,
Rt+l,l1
+l,1
Elog++L(--t
Rt +
MRt)l
l,
1, l
)
and if [-L(1/Rt+1,1) - Elog(Rt+l,o/Rt+l,l)] > 0 and Elog(Rt+1/Rt+l,1)+
L(l/Rt+l,1) > 0,
I<.
E loE
)
1L(MP--o)
M
?
It <
(
log
L--,)\Mt
Elog
)
Rt+l,
-E log
logQ
+
R,t+l
^Rt+l,1
Rt+l,
+L
Q.E.D.
-I
Rt+l,l
4: Given the proof of Proposition 2, we only need
PROOFOF PROPOSITION
= MtT/MT+1.By definition,
to show that under the stated assumptions,Rt,t+,,oo
R,+,
=lim
Vt+l(lt+k)
k-oo
Vt(lt+k)
=
.
lim
k-o0
Et+lMt+k Mt
Mt+1
EtMt+k
and by the definition of Vt,t+k,the first term equals
Et+l[Mt+k]
Et[Mt
Et+ [Mt+k]Et+l[Mt+k] ( 1 + Vt+l,t+k)
l
Et,[Mtr+k]Et[Mt+k](1+
Vt,t+k)
= MP+l/MP, due to the as-
Taking limits gives limkoo{E,t+[Mt+k]/Et[Mt+k]}
sumption that
lim
k-o
1 + Vt+l,t+k
_ 1 + Vt,t+k
=1,
that MP+1/Mfis a martingale,and due to the definitionof no permanentinnovations. Thus Rt,t+,, = (MtP+/MP)(Mt/Mti)
= M t+/M 1.
Q.E.D.
5: By definition,
PROOFOF PROPOSITION
1
lim Et logRt+l k - lim ht(oo) - yt(oo) = k-k-+oc
oo
k
k
-
logRtk
j=1
Takingunconditionalexpectationson both sides, we have that
E{ht(oo) - yt(oo)}
1
= E lim Et logRt+1,k - E lim -k
k-->oc
k->-o
k
logRtjk-(j1)
k 10gR+-j=l
j=l
(j-1).
2010
E ALVAREZAND U. J. JERMANN
Since, by assumption, expected holding returns and yields, E logRt+l,k and
(1/k)
_1 logRt+j,k-(j_1), are dominated by an integrable random variable and
the limit of the right-handside exists, then by the Lebesgue dominatedconvergence theorem,
E lim Et log Rt+l,k = lim E log Rt+l,k,
k-o
kk
k
Elim k ylogRt+jk-(j-l)
-" EogR,. ock
k-*oc
= lim
ElogRt+j,kj1)
k-
j=l
j=1
Denote the limit
(A.2)
lim E log Rt+l,k = r,
k->oc
which we assume to be finite. Since, by hypothesis, ElogRt+j,k_(j_1)=
ElogRt+l,k-(j-_) for all j, then
k
limk
1
yt+jk-(j-1)
k
E log
k Rt+1 ,k(j)
= klim
=
r,
j=l
j=1
where the second inequalityfollows from (A.2). Thus, we have that
E{ht(oo)-
y,(oc)}
1k
= lim ElogRt+l,k - lim - ElogRt+k-(j-1)
j=1
= r- r= .
Q.E.D.
PROOF OF PROPOSITION6: Using assumption (a) that Mt+l/Mt is strictly
stationary,some algebrashows that
kL
kM,LMt
k )
logE
kl?gE(M
Mt
)+E
)+Elog(y)
logMt+
Again using the stationarityassumptionand some algebra,we have
L(Mt+k
k
Ek
Mt,
+ELt(Mt
,
kMt
_j=l
m]k 1-
+ElogR
l.
2011
MARGINAL UTILITY OF WEALTH
Going to the limit, which given Assumptions 1 and 2 exists,we get
Mt+k\
1
l
k-ock
Mt
11
= lim-L
Et
k-ook
M+k
(M_t+_
i)-E E logRt+,, + ELtL
.
/
1
Mt
logR+,t
Finally,with assumption(b) we have the postulated result, given that from the
proof of Proposition2 it is easy to see that
EL(Mt+pl
Rt+
MELMPELMttMt+ -E log Rt+
Q+iE.D.
\P)
^ /M,
)
(
Q.E.D.
gt+1,?
7: Using the Markovassumptionunder (i) and (ii),
PROOFOFPROPOSITION
we have
lim Et+l(Mt+k) = lim
k-.o
k-+o
Et(Mt+k)
(Tk-1f)(s')
fQ.E.D.
= 1.
(Tkf)(s)
PROOF OF PROPOSITION8: Properties (a) and (b) define setwise conver-
gence, and with f () bounded, expected values converge.
Q.E.D.
PROOFOFPROPOSITION
9: First, we show a lemma that consumption equity
prices and consumptionequity dividend-priceratios are i.i.d. Then we use the
lemma to show that the kernel has permanentinnovations.
A.1: Assume that ct is i.i.d. withc.d.f. F and that r < 1, where
LEMMA
X c/c
T
-max 3r1-p
cE[c,c]
/e
1-e
/ cJ \c
dF(c')
Thentheprice of consumptionequityVt/Ct = f*(ct), wherethefunction f* is the
uniquesolutionto
T*f*=f*,
f*(c) = c7-l
for some constant q > 0 and the operatorT is definedas
(Tf)(c) = 3r1-~'
Moreover, Vt
=
-'v(Ct) -f
^ )
(ct)
.
C,.
-
[f (c') + l]?dF(c')
I~~~~~~},
2012
F ALVAREZAND U. J. JERMANN
PROOF:Using the pricing kernel (12), we obtain that consumption equity
must satisfy
--
[VtC]0= Et
+
(c'+I[CVt
t+lt+
Guessing that Vt = vtrt, we obtain
vt =
Et
I
TCt+
r
_
[Vt+
Ct_
+ Ct+l]
_
-
_
and dividingby ct on both sides, we can write
1/
I (1-)
/C/
[Tf](c) =f r1-"
[f (c') + 1] dF(c')
(-
where f is the price-dividend ratio of the consumptionequity:f(c) = v(c)/c.
The operator T can be shown to be a contraction:hence, it has a unique fixed
point. Moreover, a is given by
I'~
J
,1/0
r1-P f c'(1-') [f*(')
+ 1] dF(c')
where f* satisfies Tf* = *.
Q.E.D.
Using Lemma A.1, we can write the returnon the consumptionequity as
(A.3)
R+1 =
c+
,+
v(c,A1)
(
1
Then using (12) and (15), and throughsome algebra,we get
Et+ Mt+k
xt+l,k
Et,Mt+k
/1
I =_+o-
Et+, [p0(t+l)C-P+qoY-1]
Et[ 3?(t+l)Ct.-OYte+
\e~l I
Et1
1
c+l- (cy-l)
0-1Q.E.D.
APPENDIX B: DATA
For TableI, the data on monthlyyields of zero-coupon bonds from 1946:12
to 1985:12 come from McCulloch and Kwon (1993), who use a cubic spline
MARGINAL UTILITY OF WEALTH
2013
to approximatethe discount function of zero-coupon bonds using the price of
coupon bonds. They make some adjustmentsbased on tax effects and for the
callablefeature of some of the long-termbonds. The data for 1986:1to 1999:12
are from Bliss (1997). From the four methods available,we use the method
proposed by McCullochand Kwon (1993). The second part of the sample does
not use callable bonds and does not adjust for tax effects. Forwardrates and
holding period returns are calculated from the yields of zero-coupon bonds.
The 1-month short rate is the yield on a 1-month zero-coupon bond. Yields
are availablefor bonds of maturitiesgoing from 1 month to 30 years, although
for longer maturities,yields are not availablefor all years.
For Table III, for the United States, equity returns are from Shiller (1998);
short-termrates are from Shiller (1998) before 1926 and from Ibbotson Associates (2000) after 1926;and long-termrates are from Campbell(1996) before
1926 and from IbbotsonAssociates (2000) after 1926.
Ibbotson Associates' (2000) short-term rate is based on the total monthly
holding return for the shortest bill not having less than 1-month maturity.
Shiller (1998), for equityreturns,used the Standardand Poor Composite Stock
Price Index. The short-termrate is the total return to investing for 6 months
at 4-6-month prime commercialpaper rates. To adjustfor a default premium,
we subtract0.92%from this rate. This is the averagedifferencebetween T-bills
from Ibbotson Associates (2000) and Shiller's (1998) commercialpaper rates
for 1926-1998.
The data for the United Kingdom are from the Global Financial Database: http://www.globalfindata.com.
Specifically,the bill index uses the 3-month
bills
from
1800
on
commercial
through 1899 and the yield on treasury
yield
bills from 1900 on. The stock index uses Bank of England shares exclusively
through 1917. The stock price index uses the Banker's Index from 1917 until 1932 and the Actuaries General/All-ShareIndex from 1932 on. To adjust
for a default premium, we have subtracted 0.037% from the short rate for
1801-1999. This is the average difference between the rates on commercial
bills and treasurybills for 1900-1998.
For TableV, the inflationrates are computed using a price index from Januaryto December of each year. Until 1926, the price index is the PPI; afterward,
the consumerprice index (CPI) index is from IbbotsonAssociates (2000).
For Table VI, the aggregate equity index is from Global Financial Data,
further described above. Inflation is based on the CPI, given by Global Financial Data. The Bank of England publishes estimates of nominal and real
term structures for forward rates and yields. We use the series that corresponds to the Svenssonmethod, because these are availablefor the whole samand Anderson and
ple period, 1982-2000. See http://www.bankofengland.co.uk/
Sleath (1999) for details.
2014
F. ALVAREZAND U. J. JERMANN
APPENDIX C: SMALLSAMPLEBIAS
We derive here an estimate of the size of the small sample bias in our estimates in TableI. For notational convenience, define
a
b
b-
log
-Elog
aRt+l,1I
R'++,1C
R,
Rt+l,l
Elog Rt+I- + J(Rt1)
?10^R
+l(
t1
In TableI, we estimate this ratio as the ratio of the estimates a/b - f (a, b). Using a second-orderTaylorseries approximationaround the population values
and consideringthat a is an unbiased estimatorof a, we can write
E
a -a
b+ I 1(aV
var(b) - cov(a, b) + --E(b
)Ib
bb
7
- b
a
_bb + biasl + bias2.
Ea
- b)
We estimate bias1directlyfrom the point estimates and the variance-covariance matrixof the underlyingsample means. We estimate bias2by a^ var(c),
with c the sample mean of 1/R,,t+l. For forward rates, we estimate the size
of the overall bias, biasl + bias2,as [0.0071, -0.0012] for the two maturitiesin
panel A of Table I, where a negative number means that our estimate should
be increasedby that amount. Correspondingvalues for panels B, C, and D are
[0.0591, 0.1277], [-0.0077, -0.0112], and [-0.0165, -0.0209].
REFERENCES
ABEL, A. (1999): "Risk Premia and TermPremia in General Equilibrium,"Journalof Monetary
Economics,43, 3-33.
AIT-SAHALIA,Y. (1996): "NonparametricPricingof Interest Rate DerivativeSecurities,"Econo-
metrica,64, 527-560.
Journal
AIYAGARI,R. S. (1994): "UninsuredIdiosyncraticRisk and AggregateSaving,"Quarterly
of Economics, 109, 659-684.
ALVAREZ,F, AND U. J. JERMANN (2000): "Efficiency, Equilibrium, and Asset Pricing with Risk
of Default,"Econometrica68, 775-797.
(2005): "Supplement to 'Using Asset Prices to Measure the Persistence of the
Marginal Utility of Wealth'," Econometrica, available at http://www.econometricsociety.
org/ecta/supmat.
3841exam.pdf.
ANDERSON, N., AND J. SLEATH(1999): "New Estimates of the UK Real and Nominal Yield
Curves,"QuarterlyBulletin, Bank of England,November.
BANSAL,R., AND B. N. LEHMANN(1997): "Growth-OptimalPortfolioRestrictionson Asset Pricing Models,"MacroeconomicDynamics,1, 333-354.
BANSAL,R., AND A. YARON (2004): "Risks for the Long Run: A Potential Solution of Asset
PricingPuzzles,"Journalof Finance,59, 1481-1509.
BAXTER, M., AND M. CRUCINI(1995): "Business Cycles and the Asset Structureof Foreign
Trade,"InternationalEconomicReview,36, 821-854.
BEVERIDGE,S., AND C. R. NELSON(1981): "ANew Approach to Decomposition of Economic
Time Series into Permanentand TransitoryComponentswith ParticularAttentionto Measurement of the BusinessCycle,"Journalof MonetaryEconomics,7, 151-174.
MARGINAL UTILITY OF WEALTH
2015
BLANCHARD,
O., ANDD. QUAH(1989): "TheDynamicEffects of AggregateDemand and Supply
Disturbances,"AmericanEconomicReview,79, 655-673.
BLISS,R. (1997): "TestingTermStructureEstimationMethods,"in Advancesin Futuresand OptionsResearch,Vol. 9, ed. by P Boyle, G. Pennachi,and P.Ritchken.Greenwich,CT:JAI Press,
197-231.
J. Y. (1996): "UnderstandingRisk and Return,"Journalof PoliticalEconomy, 104,
CAMPBELL,
298-345.
J. Y., ANDJ. H. COCHRANE
CAMPBELL,
(1999): "By Force of Habit:A Consumption-BasedExplanationof Aggregate Stock MarketBehavior,"Journalof PoliticalEconomy,107, 205-251.
CHRISTIANO,L., M. EICHENBAUM,AND R. VIGFUSSON(2002): "What Happens after a Technol-
ogy Shock?"UnpublishedManuscript,NorthwesternUniversity.
J. H. (1988): "HowBig Is the RandomWalkin GNP?"Journalof PoliticalEconomy,
COCHRANE,
96, 893-920.
(2005):Asset Pricing.Princeton,NJ: PrincetonUniversityPress.
COCHRANE,J. H., AND L. P. HANSEN(1992): "Asset Pricing Exploration for Macroeconomics," in
NBER MacroeconomicsAnnual, ed. by O. J. Blanchardand S. Fischer. Cambridge,MA: MIT
Press, 152-165.
DANIEL, K. D., AND D. A. MARSHALL(2001): "Consumption and Asset Returns at Short- and
Long-Horizons,"UnpublishedManuscript,NorthwesternUniversity.
J. (1998): "RiskPreferencesand the WelfareCost of Business Cycles,"Reviewof EcoDOLMAS,
nomic Dynamics,1, 646-676.
DUFFIE,D. (1996):DynamicAsset PricingTheory.Princeton,NJ: PrincetonUniversityPress.
L. G., ANDS. E. ZIN(1989): "Substitution,Risk Aversion, and the TemporalBehavior
EPSTEIN,
of Consumptionand Asset Returns:A TheoreticalFramework,"Econometrica,57, 937-969.
FERSON,W. E., AND G. M. CONSTANTINIDES
(1991): "Habit Persistence and Durability in Ag-
gregate Consumption:EmpiricalTests,"Journalof FinancialEconomics,29, 199-240.
FISHER,J. D. M. (2002): "TechnologyShocks Matter,"WorkingPaper 2002-14,Federal Reserve
Bank of Chicago.
GALI,J. (1999): "Technology,Employment,and the Business Cycle:Do TechnologyShocks Explain AggregateFluctuations?"AmericanEconomicReview,89, 249-271.
HALL, A. D., H. M. ANDERSON,AND C. W. J. GRANGER(1992): "A Cointegration Analysis of
TreasuryBill Yields,"Reviewof Economicsand Statistics,74, 116-126.
HANSEN,G. D. (1997): "TechnicalProgressand Aggregate Fluctuations,"Journalof Economic
Dynamicsand Control,21, 1005-1023.
HANSEN, L. P., J. C. HEATON,AND N. LI (2004): "Consumption Strikes Back," Unpublished
Manuscript,Universityof Chicago.
HANSEN,L. P, AND R. JAGANNATHAN
(1991): "Implications of Security Market Data for Models
of DynamicEconomies,"Journalof PoliticalEconomy,99, 225-262.
HANSEN, L. P., AND J. SCHEINKMAN(2003): "Semi Group Pricing," Unpublished Manuscript,
Universityof Chicagoand PrincetonUniversity.
IBBOTSONASSOCIATES(2000): Stocks, Bonds, Bills, and Inflation-1998
Yearbook. Chicago: Ib-
botson Associates.
H. B. (1992): 'An IntertemporalModel of Asset Prices in a Markov Economy with
KAZEMI,
LimitingStationaryDistribution,"Reviewof FinancialStudies,5, 85-104.
LETTAU,M., AND S. LUDVIGSON(2004): "Understanding Trend and Cycle in Asset Values:
Reevaluatingthe WealthEffect on Consumption,"AmericanEconomicReview,94, 276-299.
E. G. J. (1996): "Asset Pricing in Economies with Frictions,"Econometrica,64,
LUTTMER,
1439-1467.
(2003): "TwoDecompositions of the Local Varianceof State Prices,"UnpublishedManuscript,Universityof Minnesota.
MCCULLOCH,J. H., AND H.-C. KWON(1993): "U.S. Term Structure Data, 1947-1991," Working
Paper 93-6, Ohio State University.
2016
E ALVAREZ AND U. J. JERMANN
NELSON,C. R., AND C. I. PLOSSER(1982): "Trend and Random Walks in Macroeconomic TimeSeries: Some Evidence and Implications," Journal of Monetary Economics, 10, 139-162.
NEWEY,W. K., AND K. D. WEST(1987): 'A Simple, Positive Semidefinite, Heteroskedasticity and
Autocorrelation Consistent Covariance Matrix," Econometrica, 55, 703-708.
SHAPIROM., AND M. WATSON(1988): "Sources of Business Cycle Fluctuations," NBER Macroeconomics Annual, 3, 111-148.
SHILLER, R. (1998): "Annual Data on US Stock Market Prices, Dividends, Earnings,
1871-Present with Associated Interest Rate, Price Level and Consumption Data," available
at http://www.econ.yale. edul shiller/data/chapt26.html.
THEIL,H. (1967): Economics and Information Theory. Amsterdam: North-Holland.
WEIL,P. (1990): "Non-Expected Utility in Macroeconomics," QuarterlyJournal ofEconomics, CV,
29-42.