The Rodney L. White Center for Financial Research
Variable Selection for Portfolio Choice
Yacine Ait-Sahalia Michael W. Brandt
03-01
The Rodney L. White Center for Financial Research
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Variable Selection for Portfolio Choice
Yacine At-Sahalia
Department of Economics
Princeton University y
and NBER
Michael W. Brandt
The Wharton School
University of Pennsylvania z
and NBER
First Draft: February 2000
This Draft: February 2001 x
Abstract
We study asset allocation when the conditional moments of returns are partly
predictable. Rather than rst model the return distribution and subsequently
characterize the portfolio choice, we determine directly the dependence of the
optimalportfolioweightsonthe predictivevariables. Wecombinethe predictors
intoasingleindexthatbestcapturestime-variationsininvestmentopportunities.
Thisindex helpsinvestors determinewhicheconomic variablesthey shouldtrack
and, more importantly, in what combination. We consider investors with both
expectedutility(mean-varianceandCRRA)andnon-expectedutility(ambiguity
aversion and prospect theory) objectives and characterize their market-timing,
horizone ects, and hedgingdemands.
We thank John Campbell, George Constantinides, David Chapman, Frank Diebold, Han Hong, Paul
P eiderer, Jessica Wachter, and especially Anthony Lynch for their comments and suggestions. We also
thank seminar participantsat the 2001 Annual Meeting of the American Finance Association, Columbia
University,HarvardUniversity,MIT,NYU,PrincetonUniversity,StanfordUniversity,andtheUniversityof
Chicago. Thisresearchwasconductedduringthe rstauthor'stenureasanAlfredP.SloanResearchFellow.
Financial supportfrom theNSFundergrantSBR-9996023,theBendheim Center forFinanceat Princeton
University,andtheRodneyWhiteCenterat theWhartonSchoolisgratefully acknowledged.
y
Princeton,NJ08544-1021. Phone: (609)258-4015. E-mail: [email protected].
z
Philadelphia,PA19104-6367. Phone: (215)898-3609. E-mail: [email protected].
x
Thelatestrevisionofthispaperisavailableat: http://brandt.wharton.upenn.edu
There is by now ample evidence in the literature that the means, variances, covariances,
and higher order moments of stock and bond returns are time-varying and predictable.
However, it has proven diÆcult to translate this evidence of predictability into practical
portfolio advice because the di erent moments of returns, which in turn determine the
optimal portfolio weights, are typically predicted by di erent sets of economic variables.
Perhaps because of this diÆculty with modeling the conditional return distribution, most
professionalinvestmentadviceisgivensolelyonthebasis ofvariablesthatforecast expected
returns, such as the dividend yieldor the slopeof the term structure.
1
Lookingbeyond expected returns, it isdiÆculttodecide which selectionorcombination
ofpredictivevariablestheinvestorshouldfocuson.
2
Thisistrueeveninthefewspecialcases
where we have an explicit asset allocationformula, such as for mean-variance utility where
the optimalallocationisproportionaltothe ratioofthe conditionalmeantotheconditional
varianceof returns. Inthis mean-variancecase itisclearthatwewantto nd variablesthat
best predict the ratio of the rst two conditional moments. Choosing variables that best
predict the mean and variance separately is likely to be counter-productive. Indeed, what
shouldwedoif avariable has apositivee ect onboth means (whichthe investor likes) and
variances(whicharedetrimentaltotheinvestor)? Whatshouldwedoifthisvariableishighly
signi cantfor oneof the moments butless soforthe other? Howdowecapture the relative
importancethat the investor'spreferences placeonthe di erent moments? Thesequestions
all suggest that in a portfolio choice context we should select variables to directly predict
optimalportfolioweights,ratherthan rstselectvariablestopredictseparate featuresofthe
return distributionand then explore later their implications forasset allocation.
Itisalsointuitivelyclearthatdi erentobjectivefunctionsplacedi erentemphasesonthe
variousfeaturesoftheconditionalreturndistribution. Forexample,amean-varianceinvestor
1
It is quite natural, of course, to focus on the rst moment of the return distribution when makinga
conditionalportfoliochoice. First,expectedreturnsarethemostintuitive,andarguablythemostimportant,
inputtotheinvestor'sobjectivefunction;second,thedependenceoftheoptimalportfoliochoiceonthe rst
momentofthereturndistributionismonotonicformostpreferences,unlikethedependenceonhigherorder
moments;andthird,evenwithrelativelysimplepreferences,thedependence oftheoptimalportfoliochoice
onthewhole returndistributionissocomplexthatitcanusuallyonlybesolvednumerically.
2
Asaresult,theempiricalliteratureonconditionalportfoliochoicehasreliedonapredeterminedchoice
ofoneor atmosttwoconcurrentstatevariables. Forexample,in asingle-periodcontext: Avramov(1999),
dividend yield, book to market ratio, earnings yield, Treasury bill yield, term spread; and Kandel and
Stambaugh (1996), dividend yield. In a multiperiod setting: Balduzzi and Lynch (1999), dividend yield;
Barberis(2000), dividendyield; Brandt (1999),dividendyield, default spread,termspread,laggedreturn;
Brennan, Schwartz, and Lagnado (1997), dividend yield; bond yield, Treasury bill yield; Campbell and
Viceira(1998),Treasurybillyield;CampbellandViceira(1999),dividendyield;ChackoandViceira(1999),
observedreturnsvariance;andLynch(200),dividendyield, termspread.
aboutforecastingthesize ofthelefttailofthereturndistribution.
3
Since,again,themeans,
variances, and size of the tails are not always predicted by the same variables, these two
investorsmay choose di erentpredictorsfor theirconditionalportfoliochoice. Furthermore,
investorsmay alsodisagreeabout thevariableselectionbecause, atthe optimalchoice,they
are holding di erent portfolios of risky securities.
4
In this paper, weshow howto select and combine variablesto best predict aninvestor's
optimal portfolio weights, both in single-period and multiperiod contexts. Rather than
rst model the various features of the conditional return distribution and subsequently
characterizetheportfoliochoice,wefocusdirectlyonthe dependenceoftheportfolioweights
onthe predictors. We dosoby solving sampleanalogues of the conditionalEuler equations
that characterize the portfolio choice, as originally suggested by Brandt (1999). However,
unliketheexistingliterature,wedetermineendogenously,foragivensetofutilitypreferences,
whichofthecandidatepredictorsareimportantfortheoptimalportfolioweights(ratherthan
importantfor separate moments of the return distribution).
The advantage offocusingdirectlyonthe optimalportfolioweightsisthat webypass the
estimation of the conditional return distribution. This intermediate estimation step is the
Achilles' heel of conditional portfoliochoice, because although the moments of returns are
predictable, this predictability is for some moments actually quite tenuous. In particular,
in the literature on predicting returns, an R 2
of 10 percent is hailed, rightly so, as a great
success. Our approach is based on the hope that the relationship between the portfolio
weights, which are complicated functions of the return distribution, and the predictors is
actuallyless noisythanthe relationshipbetween theindividualmomentsandthe predictors.
Evenifitisnot,weavoidintroducingadditionalnoiseandpotentialmisspeci cationsthrough
the intermediate, but unnecessary, estimation of the return distribution.
We form a linear combination or index of the conditioning variables that best predicts
the investor's optimal portfolio weights and then judge the importance of each individual
variable by the role it plays in this index. We make no further assumptions about the
relationshipbetween the optimalportfolioweightsand the predictorsfortworeasons. First,
3
Ifreturnsarenormallydistributedalossaverseinvestorcarese ectivelyabouttheratioofthemeanto
thestandarddeviationofreturns,whichmeasuresthesize ofthetailofaGaussian density,ratherthanthe
ratioofthemeanto thevariancethatamean-varianceinvestorcaresabout.
4
Considertwoinvestorsinthesameclassofpreferences,onewhoisrelativelyriskaverseandholdsmostly
bonds,andanotherwhois lessriskaverseandholdsprimarilystocks. Sincedi erentvariableshelppredict
the momentsof bondand stock returns, these twoinvestors may also choose di erent predictorsfor their
conditional portfolio choice. Naturally, these e ects are further compounded when we compare investors
with di erent objective functions and di erent portfolio holdings, such as a mean-variance investor who
holdsstocksandalossaverseinvestorwhoholdsbonds.
the conditional moments are approximately linear; and second, the particular form of the
nonlinearitiesnotonlyvariesgreatlywiththeinvestor'spreferencesbutalsocannotgenerally
be determined explicitly. This leads us to a semiparametric approach, where the optimal
portfolioweights depend nonparametricallyona parametricindex of the predictors.
We study investors with both expected utility (mean-variance and CRRA) and non-
expected utility (ambiguity aversion and prospect theory) objectives in order to see how
the optimal index composition depends on the characteristics of the investor's preferences.
Fromanormativeperspective,ourresultscanhelpinvestorswithanyoneofthesepreferences
determinewhicheconomicvariablestheyshouldtrackand,moreimportantly,inwhatsingle
combination. Ourindex isaparsimoniousway todescribethecurrentstateofthe investor's
investment opportunities, just as in di erent economic contexts indices summarize high-
dimensionalstatevectors(theindexofleadingeconomicindicators,thebusinesscycle index,
the consumer con dence index, etc.). Macroeconomic indices are country-speci c, since
di erent countries have di erent characteristics, and for the same reason our investment
opportunitiesindex is investor-speci cbecause di erentinvestors have di erent preferences.
Forthe purpose ofgivingportfolioadvice,one advantageofour indexapproachisthatit
helpsinvestorsunderstandtheir conditionalassetallocationinamoreintuitivemanner. For
instance, itdelivers simple rules like \if the index increases, the allocation to stocks should
increase."Bycontrast, itisgenerallydiÆculttorepresentgraphicallyvariablesinmorethan
two dimensions, letalone develop economic intuitionabout their interactions.
We characterize the market-timing, horizon e ects, and hedging demands of di erent
typesofinvestors,wherewedisentanglethee ectsduetothetime-varyingreturndistribution
from those due to the speci c preference structure. Speci cally, we show how the portfolio
choice of both expected and non-expected utility investors varies as a function of the
predictors, investment horizon,and rebalancing frequency. We explain alsohow the source
of these variations di ers across the preference speci cations.
The remainder of the paper is organized as follows. We motivate the variable selection
probleminSection2withindividualmomentregressions. Section3explainsoureconometric
approachofpredictingoptimalportfolioweightswithanindexthatcapturesthecurrentstate
ofinvestment opportunities. In Section4wediscuss fourparameterizationsofthe investor's
preferences, which we then use in Section 5 for our empirical work. There we characterize
the optimal index composition and portfolio rules for di erent types of investors, di erent
horizons, and di erent rebalancingfrequencies. We conclude in Section6.
2.1 Data
Wecollect monthly, quarterly, semi-annual, and annual returns on the Standard and Poors
(S&P) 500 index, an equal-weighted portfolio of non-callable and non- ower government
bonds with more than ten years to maturity, and a maturity-matched Treasury bill from
the Center for Research in Security Prices (CRSP). The returns are sampled monthlyfrom
January 1954 through December 1997. The sample consists of 528 observations.
An ever growing set of economic variables has been shown to partly predict the means,
variances, and covariancesof returns.
5
Wecollect monthlydata onfour popular predictors:
the default spread, the log dividend to price ratio of the S&P index, the term spread, and
an S&P index trend (or momentum) variable. The default spread is the yield di erence
between Moody's Baa and Aaa rated corporate bonds. The dividend yield is the sum of
dividends payed on the S&P index over the past 12 months divided by the current level of
theindex. Thetermspreadistheyielddi erencebetweenthe ten-andone-yeargovernment
bonds. Thetrendisthedi erencebetweenthelogofthecurrentS&Pindex leveland thelog
of the average index level over the previous twelve months. Fama and French (1988,1989)
showthatthe rst threepredictorscapture cyclicaltime-variationsinexcessstockandbond
returns, and Keimand Stambaugh (1986)usea variableverysimilartoour trendtopredict
returns. The datafor the predictors isobtained from the DRI/Citibase database.
Tables1andFigure1describethedata. PanelAofTable1presentsunivariatedescriptive
statisticsforthemonthlyreturns,annualreturns, andpredictors. Weomitthequarterlyand
semi-annualreturnstopreservespace. Panel B shows pairwisecorrelations ofthe predictors
with each other and with excess stock and bond returns, their squares, and their cross-
products. Figure1 plots the time-series and autocorrelationsof the predictors.
5
Thefollowingisapartiallistofacademicpapersthat documentvariousdegreesofmeanpredictability
and the variables theyuse: Campbell (1987),term spread; Campbell and Shiller(1988a,1988b), dividend
yield; Cochrane (1991), investment to capital ratio; Fama and Schwert (1977), Treasury bill yield; Fama
and French (1988,1989), default spread, dividend yield, term spread; Ferson and Harvey (1991), default
spread,dividendyield,laggedreturns,termspread,Treasurybillyield;KeimandStambaugh(1986),default
spread, trend; Lamont (1998), dividend to earnings ratio; Lettau and Ludvigson (2000), consumption to
wealth ratio;andPonti andShall(1998),booktomarketratio. Studiesonvariancepredictabilityinclude:
Bollerslev(1986),laggedsquaredreturn,laggedvariance;Campbell(1987),termspread;Engle(1982),lagged
squaredreturn;French,Schwert,Stambaugh(1987),laggedsquaredreturn,laggedvariance;Harvey(1991),
default spread, dividend yield, lagged squared return, lagged variance, term spread, Treasury bill yield;
Schwert(1989),debttoequityratio,defaultspread,laggedvariance,volume;andWhitelaw(1994),default
spread, lagged variance, paper spread, Treasury bill yield. Finally, representative papers on predicting
covariances are: Bollerslev, Engle, and Wooldridge (1988), lagged covariances, lagged cross-products of
returns;Campbell(1987),termspread;and Harvey(1989),defaultspread,dividendyield, termspread.
We rst verify that the variables we identi ed as potentialpredictors indeed capture time-
variations in at least the rst and second moments of excess bond and stock returns. For
that purpose, we set up the following regressions:
E
t
"
r b
t+1
r s
t+1
#
=
"
Z 0
t
b
Z 0
t
s
#
and Var
t
"
r b
t+1
r s
t+1
#
=
"
Z 0
t Æ
bb Z
0
t Æ
bs
Z 0
t Æ
ss
#
; (2.1)
wherer b
t+
and r s
t+
denotebond and stock returnsin excess ofthe Treasurybillreturn and
the vector Z
t
contains subsets of the fourpredictors. Wedemean and standardize the data
to eliminate the intercepts and then estimate and Æ using Hansen's (1982) generalized
method of moments (GMM) for all one-, two-, three-, and four-dimensional subsets of the
fourpredictors. Whenever weuse overlapping returns we compute autocorrelationadjusted
asymptoticstandard errorsusing the procedureof Hodrick (1992).
Table 2 presents the regression results. For each security (bonds and stocks), moment
(mean, variance, and covariance), and return horizon (monthly and annual), the table
presents the best one-, two-, and three-variable partial regressions. It also shows the full
regression with allfour predictors. The best partial regressions are chosen according to the
Akaikeinformationcriterion(AIC).
6
One, two, orthree stars indicatethat the coeÆcientis
statisticallysigni cant atthe ten, ve, orone percent level, respectively.
The following factsemerge fromTable 2:
The default spread relates positively to the variances and covariance of both monthly
and annual stock and bond returns. The regression coeÆcients are both statistically
and economically signi cant (recall the data is demeaned and standardized, so the
magnitude of the coeÆcients is meaningful), except for the stock return variance at
the annual horizons. The default premium also relates positively, but not always
signi cantly, to expected bond and stock returns.
Thelogdividendtopriceratiorelatespositivelyandsigni cantly(at10percentlevels)
toexpected stock returnsatthe annualhorizonbut not atthe monthlyhorizon.
7
The
6
Sincethedataisdemeanedandstandardized,theAIC criterionandtheadjusted R 2
oftheregressions
arevirtuallyidentical. Tosavespace,weonlyreporttheadjustedR 2
oftheregressionsinTable2.
7
Because the dividend to priceratio does notappear to predict dividendgrowth, it must on thebasis
of the present value formula forecastreturns. In fact, aspointed out by Cochrane(1999), \price divided
byanything"sensiblehasthisforecastingpower. Sincemostpriceratiosarevariableswithveryslowmean
reversion(seeFigure1) theyforecastlong-termreturnsbetterthanshort-termreturns.
tothe covariance between stock and bond returns atthe annual horizon.
Thetermspreadisbyfarthemostimportantpredictorforexpectedreturns. Itrelates
positively to expected bond and stock returns at both horizons.
8
The coeÆcients are
statisticallyand economically signi cant, especiallyforstock returns. In addition,the
term spreadrelates negatively tostock and bond variances.
Thetrendvariablerelatesnegativelyandsigni cantlytoexpectedbondreturnsatboth
horizons. It also relates negatively to the variance of stock returns, with regression
coeÆcients that increase in magnitude and statisticalsigni cance with the horizon.
Exceptforthecovariance,theadjustedR 2
softheregressionsincreasewiththehorizon,
meaningthatreturnsandsquaredreturnsaremorepredictableatlonghorizonsthanat
short horizons. This patternisdue the slowly mean-reverting natureof the predictors
(see Figure1) and ismore pronounced forexpected returnsthan for returnvariances.
Stock andbondreturnsare about equallypredictableatboth horizons. Squaredbond
returns, however, are substantiallymore predictable than squared stock returns.
The most important nding for motivating our approach is the fact that if we had to
restrict attention to only one or two predictors, the variable selection would depend on
the conditional moments of returns that we most wanted to predict (bonds vs. stocks and
rst vs. second moments) as well as on the return horizon. This result is best illustrated
in Table 3, which lists for all four return horizons the best one or two predictors for each
moment. The problemwith selectingvariable inthe portfoliochoice contextlies inthe fact
thatthemomentsofreturns(orfunctionsofthem)thatwewanttopredictareendogenousto
the investor's preferences. Forexample,aninvestorwho isveryriskaverse and holdsmostly
bonds maywanttofocus onpredictingthe varianceofbond returnswith thedefaultspread,
while another investor who isless risk averse and holds mostly stocks may wantto focus on
predictingexpectedstock returnswith the logdividendtopriceratio and termspread. Itis
this endogeneity of the variable selectionin the portfoliochoice context that motivates our
emphasis on predicting optimalportfolioweights, ratherthan individual moments.
8
FamaandFrench(1989)documentthat theslopeoftheyieldcurvemovesintandemwiththebusiness
cycle. The yield curve is inverted at the peak of the cycle, where expected returns are low, and upward
slopingwhenarecessionturnsintoarecovery,whereexpectedreturnsarehigh.
3.1 Investor's Problem
Consider asingle-periodinvestor whomaximizesthe conditionalexpectationofanobjective
function v(W
t+1
) of next period's wealth W
t+1 .
9
The expectationis conditional ona vector
of state variables Z
t
. The maximization is over the portfolio weights
t
under the budget
constraint W
t+1
=W
t (
t 0
R
t+1
), where R
t+1
is a vector of gross returns onthe securities the
investor can buy and sell. Formally,the portfoliochoice problemis:
max
t E
h
v W
t (
t 0
R
t+1 )
Z
t i
; (3.1)
subject tothe adding-up constraint
t 0
=1, where denotes a vectorof ones. Although the
portfolioweightssum toone, notallwealth must beinvestedinrisky securities because one
of the securities may berisk-less. Furthermore, the portfoliochoice maybesubjectto aset
constraintsa c(
t
)b, such as short-sale orborrowing constraints.
The solution to the investor's problem is the mapping from the state vector Z
t
to the
portfolioweights
t
. Assuming that this mappingis time-invariant,we denote it:
10
t
(Z
t
) (3.2)
and refer toit asthe investor's portfoliochoice, policy, weight, or rule.
The relation between the portfolio policy and the predictability of individual moments
of the returns R
t+1
given the predictors Z
t
obviously depends on the speci cation of the
objective function v(W
t+1
). To illustrate this point, consider an investor with standard
mean-variance preferences. The investor's objective function is:
E h
v(W
t+1 )
Z
t i
=E h
W
t+1
Z
t i
2 Var
h
W 2
t+1
Z
t i
(3.3)
with the coeÆcient of absolute riskaversion 0. The investor's portfoliopolicyis:
t
= 1
t
W
t
0
1
t
t
W
t
0
1
t
+
1
t
t
W
t
; (3.4)
9
We describe our econometric approach for a single-period portfolio choice to keep the notation
simple. However, our estimatorextends readily to a multiperiod portfolio choice by replacing the single-
period objective function and its derivative with a multiperiod \value function" and its derivative [see
Brandt(1999)]. Infact,weapplyourapproachtomultiperiodportfoliochoicein Section5.3.3.
10
With time-invariant objective function v(W
t+1
) this assumption only requires that the conditional
distributionofthereturnsR
t+1
giventhepredictorsZ
t
istime-homogenous.
t t+1 t
t
t+1 t
weight illustrates two facts. First, even with this most simple preference speci cation
there is no straightforward link between predictability in rst and second moments and
the portfolio policy that allows the investor to identify which predictors are important for
theportfoliochoice. Second,evenifthe conditionalmomentsareapproximatelylinearinthe
statevariables,theratioformoftheportfoliopolicyimpliesthatitcanbeahighlynonlinear
and even nonmonotonicfunction.
If the portfoliochoice includes a risk-freerate, it simpli estoallocatingafraction:
tgc
t
= 1
W
t E
r tgc
t+1
Z
t
Var
r tgc
t+1
(3.5)
of wealth to the mean-variance eÆcient tangency portfolio, with excess return r tgc
t+1
, and to
investtheremainderintherisk-freeasset. Inotherwords,themean-varianceportfoliochoice
is proportional to the conditional mean-variance ratio of the tangency portfolio. Therefore,
in selectingvariablesfor the mean-variance portfolio choice, which isa normative issue, we
equivalentlyselectvariablesforpredictingthemean-varianceratioofthe tangencyportfolio,
whichis ageneric descriptive statisticof the conditionalreturn distribution.
3.2 Indices for the Conditional Portfolio Choice
Brandt (1999) shows how to estimate the optimal portfolio policy (Z
t
) without further
assumptions about the return dynamicsor functionalform of the decisionrule by replacing
theconditionalexpectationintheinvestor'sproblem(3.1)withaconsistentestimator. Fora
givenrealization ofthe statevectorZ
t
, wede nethe fullynonparametric estimator (Z^
t )of
thetrue (Z
t
)astheportfolioweightsthatsolvetheinvestor'sproblemwhentheconditional
expectation E[jZ
t
] isreplaced with a consistent estimator
^
E[jZ
t
], such that:
^
E h
v W
t (
t 0
R
t+1 )
Z
t i
! E
h
v W
t (
t 0
R
t+1 )
Z
t i
as T !1; (3.6)
for all portfolio weights
t
and state vector realizations Z
t
. In particular, Brandt suggests
estimating the conditional expectation witha standard nonparametric regression.
Unfortunately, this fully nonparametric approach does not allowus toaddress the issue
of whichpredictors are importantforthe portfoliochoice becausenonparametric estimators
typicallycannothandlealarge numberofregressors. As thenumberofpredictors increases,
the convergence rate of most nonparametric estimators to their asymptotic distribution
deteriorates exponentially. This feature of the estimators is commonly referred to as the
estimatethe conditionalexpectationfor more than two predictors.
To overcome this econometric problem, we adopt a semiparametric approach, which
explicitlyrecognizes theendogeneity ofthevariableselection. Weassume thattheinvestor's
optimal portfolioweights depend on the predictors Z
t
onlythrough a single linear index or
factor Z
t 0
with unknown parameters . 11
The dependence of the portfolioweights on this
index, however, is left completelyunrestricted. We classifyour approachas semiparametric
because the index isparametric but the portfoliopolicyis not.
Formally, werewrite the investor's problem(3.1) as:
max
t E
h
v W
t (
t 0
R
t+1 )
Z
t 0
i
; (3.7)
whichimplies that the optimal
t
depend onZ
t
onlythrough the index Z
t 0
: 12
t
(Z
t 0
; ): (3.8)
From astatisticalperspective,the index avoidsthe curseof dimensionalitybecause itallows
us to reduce the multivariate problem to one were we can implement the nonparametric
approachdescribedaboveinaunivariatesetting(sinceZ 0
t
isunivariate). Fromaneconomic
standpoint, the index o ers a convenient univariate summary statistic that describes the
current state of the time-varying investment opportunities.
The economic cost of collapsingthe multidimensionalinformationcontained inZ
t intoa
linear index Z
t 0
is that the expected utility from the unconstrained optimization (3.1)
exceeds that from the constrained optimization (3.7), unless our assumption about the
index structure of the investor's problem is true (as opposed to just an approximation for
econometricpurposes). The magnitudeofthe expectedutilitylossduetotheindex depends
ontheapplicationand canultimatelyonlybemeasuredempirically. Forour application,we
present evidence inSection 5.3.1 that this expected utility lossis minor.
We estimate the index coeÆcients through the conditional Euler equations of the
investor's unrestricted problem. Speci cally, we substitute the parametric restriction (3.8)
intothe rst-order conditions of the problem(3.1) toobtainthe followingset of conditional
11
WecanrelaxtheassumptionofalinearindexbyintroducingnonlinearitiesinZ
t
. Ourexperimentations
withnonlinearindicessuggest,however,thatintheportfoliochoicecontexttheincrementalexpectedutility
lossfrom combiningthepredictorslinearly,asopposed tononlinearly,isminimal.
12
Theoptimalportfoliochoice
t
dependsontheindexcoeÆcients notonlythroughtheindexrealization
Z
t 0
but alsothroughthefunctional form ofthepolicy function (). Forexample,consider twoindices
and
= . Inthiscase, (x; )6= (x;
)butinstead (x; )= ( x;
)for allindex realizationsx.
E
m
t+1 ( )
Z
t
E h
v 0
W
t (Z
0
t ; )
0
R
t+1
R
t+1
Z
t i
=0; (3.9)
where (Z 0
t
; ) solves the investor's restricted problem (3.7). Multiplying these moment
conditionsbypredeterminedfunctionsg(Z
t
)oftheforecastingvariables,takingunconditional
expectations, and then applying the law of iterated expectations yields a standard GMM
inference problem[see Hansen (1982)]:
13;14
min
E
m
t+1
( )g(Z
t )
0
WE
m
t+1
( )g(Z
t )
(3.10)
with optimalweighting matrix W=Cov [m
t+1 g(Z
t )]
1
. The nal step inthe construction
ofour estimatoristoreplaceboththe unconditionalexpectations andtheoptimalweighting
matrix with consistent sampleanalogues.
The main di erence between our estimator and standard GMM is that the portfolio
weights in the conditional moments (3.9) are only de ned implicitly through the investor's
portfoliooptimization. ToevaluatetheGMMcriterionforacandidate ,we rstestimatethe
sequenceofoptimalportfolioweightsf (Z 0
t ; )g
T
t=1
byreplacingtheconditionalexpectation
in the investor's problem(3.7) with nonparametric regressions, asin equation(3.6).
Appendix A presents a more detailed description of our estimator and its asymptotic
distribution. The main results are that the estimatoris consistent, asymptotically normal,
and, although it has a nonparametric component, achieves the parametric convergence rate
of p
T irrespective of the number of predictors.
15
The GMM estimator (3.10) treats the choice of as an inference problem under the
null thatthe unconstrained portfoliopolicy (Z
t
)has the index form (Z
t 0
; ). Under the
alternative that the index form is suboptimal, a more natural way to choose is through
13
Noticethatweneedtheinstrumentsg(Z
t
)toidentify . TheindexcoeÆcientsareunconditionallynot
identi edbecause (Z 0
t
; )satis esE[m
t+1 ( )jZ
0
t
]=0and henceE[m
t+1
( )]=0forany .
14
Thissetup canbeused notonlyforestimation, but alsofortesting whetherasecond set ofpredictors
should be included in the index. With the instruments g(Z
t
;Y
t
), the minimized GMM objective is an
asymptotically 2
distributedtestforthehypothesis thattheindexhaszeroloadingsonthepredictorsY
t .
15
Because the index composition is estimated and the investor's portfolio choice depends upon the
estimated index, oursubsequentestimates ofthe portfolio weights automaticallyincorporate the fact that
ex-ante predictability is uncertain. The standarderrorsof theestimated indices and portfolio policies are
largerinsmallsamplebutnotasymptotically,sinceinsuÆcientlylargesamplesthereisnouncertaintyabout
predictability. Ofcourse, theappropriateapproachto fullyaddresstheissueof parameteruncertaintyisa
Bayesianframework[seeBarberis(2000)andKandelandStambaugh(1996)].
max
E
max
t E
h
v W
t (
t 0
R
t+1 )
Z
t 0
i
=max
E
h
v W
t ( (Z
t 0
; ) 0
R
t+1 )
i
; (3.11)
wherethe equality follows fromthe lawof iteratedexpectations and the restriction(3.8). In
words, the index de ned by this maximizationgenerates asequence of conditional portfolio
choices that is unconditionally optimal or, equivalently, that minimizes the unconditional
expected utility loss from solving the constrained optimization (3.7) as opposed to the
unconstrained optimization (3.1).
Theexpectedutilitymaximization(3.11)isnestedbytheGMMestimator(3.10)through
anoptimal (inanexpectedutilitynotstatisticalsense)setofinstrumentsg(Z
t
). Speci cally,
the rst-order conditions of the maximization:
E
v W
t ( (Z
t 0
; ) 0
R
t+1 )
R
t+1
@ (Z
t 0
; )
@
=0 (3.12)
show that the problem(3.11) isequivalent tothe problem(3.10) with instruments:
g(Z
t )=
@ (Z 0
t ; )
@
(3.13)
and anarbitrary weighting matrix W (since the parmeters are exactly identi ed).
However, just because these instruments are optimal in theory, they do not necessarily
resultinmorereliableestimates oftheindexcoeÆcientsinpractice fortworeasons. First,to
construct the instruments weneed consistentestimates of the derivatives@ (Z 0
t
; )=@ . A
nonparametricestimatorofthesederivativesconvergesataslowerratethenanonparametric
estimatorof the function (Z 0
t
; ) [see Hardle (1990)] and thusmay introduce substantial
noise. Second, since the functional form of the portfolio policy may depend on the index
compositioninahighlynonlinearandirregularway(even ifthe policyiswell-behavedinthe
index for a given index composition), the derivatives @ (Z 0
t
; )=@ may cause the GMM
objective function to be less well-behaved. This makes the numerical minimization more
diÆcultand increases the risk of endingup with alocalinsteadof global minimum.
Given estimates of the optimal index composition, we judge the importance of each
predictorintheconditionalportfoliochoicethroughtherelativeweightthepredictorreceives
in the index. In other words, the relative weights the estimated index coeÆcients
^
place
onthe di erentvariables,andtheir statisticalsigni cance, telluswhichvariables,and more
importantlyinwhatcombination,arerelevantfortheinvestor'sconditionalportfoliochoice.
Toseehowthevariableselectionvariesacrossinvestorswithdi erentpreferences,weconsider
four parameterizationsof the objective functionv(W
t+1
). The rst two, mean-variance and
CRRApreferences,arestandardexpectedutilityobjectivesandresultinfairlysimilarindices
fortheconditionalportfoliochoice. Thesecondtwo,ambiguityaversionandprospect theory
preferences, are generalized or non-expected utility objectives. They produce indices that
are quite di erent from those of expected utility investors, demonstrating the endogeneity
ofthe variableselectionprobleminthe portfoliochoicecontext. In e ect,di erentinvestors
focus ondi erent aspects of the returns distribution,whichdi erent variableshelp predict.
4.1 Expected Utility
4.1.1 Mean-Variance Preferences
Wealready introducedthe objective functionof an investor with mean-variance preferences
inequation(3.3), where measures theinvestor's absoluteriskaversion
@ 2
v(W)
@W 2
Æ
@v(W)
@W .
16
An
appealingfeatureofmean-variancepreferences,andthereasonweconsiderthemhere,isthat
the optimal portfolio weights depend exclusively and analytically on the rst two moments
of returns[see equation(3.4)]. Thus, wecan directlycompare our semiparametricestimates
of the portfolio policyto parametricestimates based onindividualmomentforecasts.
4.1.2 Constant Relative Risk Aversion Preferences
Wealso consider aninvestor with constant relativerisk aversion (CRRA) or power utility:
v(W
t+1 )=
8
>
<
>
: W
1
t+1
1
if >1
lnW
t+1
if =1
; (4.1)
where now measures relative risk aversion W
t
@ 2
v(W)
@W 2
Æ
@v(W)
@W
. CRRA preferences are by
far the most popular objective function in the portfolio choice literature. This is largely
because the investor's portfolio(and consumption)policyis proportionaltowealth and the
value function is homothetic in wealth. In a multiperiod setting, these features of CRRA
preferences imply that wealth isnot astate variable in the investor's problem.
16
FortheempiricalworkwenormalizeW
t
=1,sorelativeriskaversionW
t
@ 2
v(W)
@W 2
Æ
@v(W)
@W
alsoequals .
4.2.1 AmbiguityAversion
Expected utility theory assumes that the investor can compute expectations with respect
to the return distribution, whichrequires that the agent knows the parametric structure of
the returndistribution and either knows its parameters or can form Bayesian beliefs about
them. The investor is only exposed to the \risk" inherent inthe returnsand trades o this
riskagainstexpectedrewardsthroughtheexpectedutilitymaximization. Knight(1921)and
Ellsberg(1961)argue,however, thattheinvestormaynothavealloftheinformationrequired
to form such expectations. For example, an agent may not be able or willing to assign
probabilities to a set of alternative parameterizations of the return distribution. Thus, the
investorfacesadditional\ambiguity"thatisnot capturedinthe expectedutilityframework.
Ambiguityaversionpreferences formalizethe ideathatthe investordislikesnotonlyriskbut
alsothis more vague uncertainty about the world (called Knightian uncertainty).
17
ConsideragainaninvestorwithCRRApreferences,exceptthatnowtheagentisuncertain
about whether the return distributionisp(the empiricaldistribution, forexample) orsome
other distributionp2P in the neighborhoodof p. The crucial di erencebetween ambiguity
aversionand expected utilitytheorywith modeluncertainty isthat withambiguityaversion
the investor cannot ordoes not want to assign probabilities to the set of alternative return
distributions. Following Gilboa and Schmeidler (1989) and Dow and Werlang (1992), the
investor's portfoliochoice problemin this case is given by:
max
t min
p2P E
p h
v W
t (
t 0
R
t+1 )
Z
t i
; (4.2)
were v() is the CRRA utility function in equation (4.1). This maxmin criterion is quite
intuitive. Giventhecompleteambiguityaboutthereturndistribution,theinvestorconsiders
the worst case outcome (inthe neighborhood of p) through the interior minimization. The
exterior maximizationthen achievesthe usual riskvs. expectedreward trade-o .
Toimplementthisformofambiguityaversion,weneedtocharacterizethe set ofpossible
return distributionsP. We adopt the following "-contamination parameterization:
18
P =
(1 ")p+"p : p2P
; (4.3)
17
Thereisanextensiveexperimentalliteraturecon rmingthatindividualsindeeddislikeambiguity,both
in gambling settings [e.g.,Beckerand Brownson(1964) or Curley and Yates (1985,1989)] and in nancial
markets [e.g.,CamererandKunreuther(1989)orSarinand Weber(1993)].
18
Theassumptionof"-contaminationisinformallyintroducedbyEllsberg(1961)andisnowcommoninthe
literatureonambiguityaversion. Itisused,forexample,byEpsteinandWang(1994)andLiu(1998,1999).
parameter " re ects the investor's degreeof ambiguity. 19
With"=0 the investor'sobjective
function reduces to that with standard CRRA preferences and return distributionp.
The advantage of this parameterizationis that the investor's problemsimpli es to:
max
t
(1 ")E
p h
v W
t (
t 0
R
t+1 )
Z
t i
+inf
P v W
t (
t 0
R
t+1 )
; (4.4)
whereweassumethatthesupportofthereturndistributionisindependentofthepredictors.
Thus, toimplementthe notion ofambiguity aversionwe onlyneed tochoose a value for the
parameter " and specify the support of the returndistribution to evaluatethe in mum.
Ambiguity aversion relatesclosely to the recent literature onrobustness [e.g.,Anderson,
Hansen, and Sargent (1999), Hansen, Sargent, and Tallarini (1999), and Maenhout (1999)].
Although the formalizationsdi er slightly, the behavioral motivation of the two theories is
the same. Agents are uncertain about the true modeland are unableor unwilling toassign
probabilitiestotheset of alternativemodels. Nottoosurprisingly,robustness alsoresultsin
maxmin policies. We focus on the -contamination version of ambiguityaversion becauseit
captures the essence of the theory and ismore tractable forour empirical application.
4.2.2 Prospect Theory and Loss Aversion
In another literature on decisions under uncertainty, Kahneman and Tversky (1979) argue
that humans systematically violate the axioms of expected utility theory in two important
ways. First,experimentalsubjectstend tooverweight outcomesthat are consideredcertain,
relativeto outcomes that are merely probable, which isreferred to asthe \certainty e ect".
In nancial markets, this certainty e ect makes an investor risk averse in the case of gains,
as a smallcertain gain is preferred to a probable risky gain, but risk seeking in the case of
losses, asaprobablerisky lossispreferredtoasmallcertainloss. In addition,subjects tend
to simplify decisions by disregarding components common to the alternative choices and
focusing oncomponents that di erentiate the choices, which iscalled the \isolatione ect".
Since the decomposition of alternativechoices intocommonand di erentiatingcomponents
is non-unique, however, the outcome of the decision problem depends on the investor's
perspective inthe simpli cationprocess.
Motivated by this experimental evidence, Kahneman and Tversky formulate prospect
theory, which consists of an editing stage, where alternatives are put into perspective, and
19
Alternatively, onecan interpret the portfolio choice asthe investorplaying a two-stage game against
nature. In the rst stage, nature replaces with probability " the return distribution pwith an arbitrary
distributionp2P. Inthesecondstage,naturethendrawsasetofreturnsfromthereturndistribution.
a choice stage. Utility is de ned over gains and losses relative to a reference point (such
asthe resultof anall-cashinvestmentstrategy,orlastperiod's wealth) ratherthanover the
level ofwealth asinexpected utilitytheory. Tocapture the di erentialriskpreferences over
gains and losses generated by the certainty e ect, Tversky and Kahneman (1992) propose
the followingobjective function for the choice stage:
v(W
t+1 )=
8
<
: l
W W
t+1
b
if W
t+1
<
W
W
t+1
W
b
otherwise
; (4.5)
where
W isa reference wealth level determined inthe editing stage. For example,
W could
be theinitialwealthW
t
oritsfuture value R tb
t+1 W
t
,dependingonthe investor'sperspective.
The parameter l measures the investor's loss aversion and the parameter b captures the
degree of risk seeking over losses and risk aversion over gains.
21
The kink at the origin
introduced by l>1 makes losses (relatively)more painful than gains are pleasurable.
In addition, in Tversky and Kahneman's formalization of prospect theory the investor
doesnotevaluateoutcomesonthebasis oftrueprobabilities,butrather,aspredictedby the
certainty e ect, on the basis of distorted probabilities. That is, instead of maximizing the
true expectation of the objective function,the investor maximizes:
E
v(W
t+1 )
p(W
t+1 jZ
t )
p(W
t+1 jZ
t )
Z
t
= Z
+1
1 v(W
t+1
) p(W
t+1 jZ
t )
dW
t+1
;
where () represents a subjective distortion of the objective probabilities p(). Tversky
and Kahneman suggest parameterizing this probability distortion as [see also Tversky and
Wakker (1995)]:
p(W
t+1 jZ
t )
=
p(W
t+1 jZ
t )
c
n
p(W
t+1 jZ
t )
c
+ 1 p(W
t+1 jZ
t )
c o
1=c
(4.6)
where c determines the degree of \irrationality". When c =1, the decision weights ()
reduce to the objective probabilities p(). Notice also that when 0<c<1, the weights are
notaproperprobabilitymeasure(hencetheyarecalledweights,notsubjectiveprobabilities)
because they sum toless than one.
A special case of prospect theory is loss aversion, when b=1, c=1, and l>1. In this
20
FinanceapplicationsofprospecttheorypreferencesincludeBarberis,Huang,andSantos(2000),Barberis
andHuang(2001),BenartziandThaler(1995),ShefrinandStatman(1994),andShumway(1997).
21
TverskyandKahnemanciteexperimentalevidence thatsuggestsb=0:88andl=2:25.
greater incremental utility penalty for a loss than for an equally large gain. This results
in unconditional risk aversion. Furthermore, since with c=1 the decision weights reduce
tothe objectiveprobabilities, this investorsimplymaximizes expected utility. Interestingly,
BenartziandThaler(1995) ndthatthemainaspectofprospecttheoryrelevantforportfolio
choice is loss aversion and that the concavity (resp.convexity) of the value function on the
upside (resp. downside), as well as the subjectivity of the probability distortions, are only
of second-order importance. Sharpe (1998) argues, however, that the local risk-neutrality
property of lossaversion results inportfoliochoices that are too extreme.
22
A literature that is somewhat related to loss aversion involves Value-at-Risk (VaR)
constraintswhichensurethat withprobabilityofatleast qthe investor'swealthnext period
(orsomeothertarget horizon)doesnot fallbelowsome speci edlevel.
23
The extremesq=0
andq=1correspondtoanunconstrainedinvestorand aportfolioinsurer[e.g.,Basak(1995)
and Grossman and Zhou (1996)]. Unfortunately, VaR preferences have two faults. First,
Artzner etal. (1998) show that VaRmeasures have diÆculties aggregating individualrisks,
even risks that are cross-sectionally independent, and sometimes discourage diversi cation.
Second, Basak and Shapiro (1998) nd that in a multiperiod setting a VaR-constrained
investor frequently chooses, quite paradoxically, a larger exposure to risky assets than an
otherwise equivalent unconstrained investor.
24
As a x to this problem they propose an
alternative risk management constraint that incorporates the expected value of the loss.
Similarinspirit tothis extended VaRconstraint, lossaversionpreferences penalizeforboth
the probability and magnitude of losses.
5 Empirical Results
5.1 Unconditional Portfolio Choice
Webeginourempiricalworkbycharacterizingtheunconditionalportfoliochoiceofinvestors
with expected and non-expected utility preferences. The unconditional portfolio choices
are useful for understanding the optimal index compositions in Section 5.2 and serve as
22
The problem with loss-aversion is that the iso-expected utility curves are straight lines in mean vs.
standarddeviationof returnsspace,whichmeans that inastylizedportfoliochoicebetweenasinglestock
andcashtheoptimalportfoliochoiceiseither100percentcashor100percentstock,dependingonwhether
theiso-expectedutilitycurvesaremoreorlesssteep thanthemean-variancefrontier[seeSharpe(1998)].
23
VaRisthedefactostandardmeasureofriskbecauseofitssimplicityanditspopularitywithregulators.
24
TheintuitionisthattheVaR-constrainedinvestor ndsitoptimaltoinsureagainstlossesinstateswhere
insuranceisrelativelycheap(becauselossesarerelativelysmall),butacceptsthepossibilityoflosses(up to
probability1 q)in stateswhereinsuranceisexpensive(becauselossesarepotentiallylarge).
5.1.1 Expected UtilityPreferences
Mean-Variance Investors
Panel A of Table4 presents estimatesof the unconditionalportfoliochoice ofinvestors with
mean-variancespreferencesandabsoluteriskaversion(andrelativeriskaversionsinceW
t
=1)
of two, ve, 10, and 20.
25
The investment horizon is one month or one year. The entries
inthe Without Risk-Free Ratesection of thepanel are for aportfoliochoice between stocks
and bond. Theentries inthe WithRisk-FreeRate sectionare fora portfoliochoicebetween
stocks, bond, and Treasury bills. We assume that Treasury bills are risk-free and x the
Treasury bill rate at its historical average. We impose the short-sale constraints 0x1
to prohibit unrealistic leveraging and short-selling. In brackets below each estimate are
autocorrelationadjusted asymptotic standard errors.
26
Panel A of Figure 2 helps visualize the mean-variance portfoliochoice. The two graphs
plottheexpectedreturnonwealthagainstthestandarddeviationofwealthfortheestimated
portfolio weights in Panel A of Table 4. The two lines in each graph represent the mean-
variancefrontierwith arisk-freerate(straightline)andwithoutarisk-freerate(hyperbola).
The stars and circles represent the corresponding optimal portfolios. As a reference point,
wealsoplotaportfolioof60percentstocks,20percentbonds, and20percentTreasurybills,
whichhappens to resemblethe optimalportfolioof a mean-varianceinvestor with =5.
Several well-known but nevertheless interesting features of the mean-variance optimal
portfolios emerge. Consider the portfolio choice with a risk-free rate. Except when =2,
in which case the short-sale constraints are binding, all mean-variance investors hold the
same risky position of 84 or 90 percent stocks and 16 or 10 percent bonds, depending on
the horizon but irrespective of risk aversion. Risk aversion only a ects how much wealth
the investor allocatestoriskysecurities insteadoftorisk-freeTreasurybills. Thisallocation
ranges from100 percent for =2 to about 20percent for =20.
Graphically,the factthatallmean-varianceinvestorsholdthesame riskyposition,which
is the portfolio at the tangency of the two mean-variance frontiers, but allocate di erent
fractions of wealth to it, implies that the optimal portfolios are all arranged on a straight
line in the expected return vs. standard deviation space. Also, we notice from Figure 2
25
Wecheckedthat ourmethodofmomentsestimatesarevirtuallyidenticalto theresultsofpluggingthe
unconditionalmomentsfromTable1into theanalyticexpression(3.4)fortheoptimalportfolioweights.
26
Whenevertheshort-saleconstraintsarebindingwecomputetheasymptotic standarderrors usingthe
results of Moran (1971)and Andrews (1999), whoderivethe asymptotics of an extremum estimatorof a
parameterthat islocatedattheboundaryofaclosedparameterspace.
becausethedecisionofhowmuchwealthtoinvestintheriskytangencyportfolioisinversely
proportionaltothe investors'absolute risk aversion [see equation(3.5)].
These portfolio choice patterns are the direct consequence of two-fund separation and
haveimportantimplicationsforthe variableselection.
27
Recallour exampleoftwoinvestors,
onewho isveryriskaverse, holdsmostlybonds, andwantstopredictvariances,and another
who is less risk averse, holds mostly stocks, and wants topredict means. Weargued, based
on this example, that the variable selection may di er across investors for two reasons:
the investors' preferences for predicting the various moments of returns and their portfolio
holdings. The implication of two-fund separation is that all mean-variance investors hold
the same riskyposition,unless theborrowingconstraintsare bindingorthere isnorisk-free
rate, which means their variable selection can only di er due to di erent preferences for
predicting means,variances, covariances, and higher-order moments.
The conclusion that di erent investors hold the same risky position does not apply to
theportfoliochoicewithoutarisk-freerate,althoughtwo-fundseparationholdsnevertheless
(only nowwith two risky portfolios). Withouta risk-free rate, the investors' stock holdings
decrease and the bond holdings increase with the level of risk aversion. Also, the standard
deviation of wealth decreases less than proportionallywith , which means that relativeto
the portfoliochoice with arisk-free rate the investors are taking on more risk.
Constant Relative Risk Averse Investors
Panels B of Table 4 and of Figure 2 present estimates of the unconditionalportfolio choice
of investors with CRRA preferences and relativerisk aversion of two, ve, 10, and 20. The
resultsare similar tothose formean-variancepreferences, exceptthat CRRAinvestors tend
to hold less stocks (up to seven percent less) and more bonds (up to ve percent more),
relative to equally risk averse mean-variance investors. These di erences in the portfolio
choices areattributedtothe negativeskewness instockreturnsand thepositiveskewness in
bond returns that we document inTable 1.
28
Given these di erences in the optimal portfolio weights, it is interesting that the
implications of two-fund separation for the mean-variance portfolio choice with a risk-free
27
Weusethenotionoftwo-fundseparationtorefertothefactthattwomean-varianceeÆcientportfolios
spanthemean-variancefrontier. Witharisk-freerate,thesetwoportfoliosaretheTreasurybillandtangency
portfolio. It is important, however, to realize that in the literature two-fund separation is typically an
equilibriumstatementthatreferstothe\marketportfolio"beingmean-varianceeÆcient.
28
Weobservethegreatestdi erencesbetweentheoptimalportfolioweightsofmean-varianceandCRRA
investorsat thethree-month horizon,where stockreturns exhibit themostnegativeskewness. The three-
andsix-monthestimatesarenotreportedin Table4andFigure 2topreservespace.
theory,the riskypositionof aCRRAinvestorcan depend onrelativeriskaversion,since the
investor's preferences for higher order moments, which di erentiate a CRRA investor from
anequallyriskaverse mean-varianceinvestor, are afunction ofrelativeriskaversion. In the
data,however, the e ect of thehigher-ordermoments isapparently not strongenoughtobe
noticeable in the stock holdings of CRRA investors with di erent degrees of risk aversion,
althoughitisclearly afactorinexplainingthe di erentstock holdingsofequallyrisk averse
CRRA and mean-varianceinvestors.
2930
5.1.2 Generalized or Non-Expected Utility Preferences
Ambiguity Averse Investors
The results for investors with ambiguityaversion preferences are in Panel A of Table 5 and
inFigure3. Weconsiderthe cases =f5;10gand"=f0:001;0:005;0:010g.
31
Recallthat the
case"=0correspondstotheCRRAportfoliochoicesinPanelBofTable4. Weparameterize
the worst-case returns on stocks and bonds, which we need to evaluate the in mum in the
objective function(4.4), as the empiricalunivariate minimums fromTable 1.
Ambiguityaversionhastwoe ectsontheportfoliochoice. First,anincreaseinambiguity
aversion leads investors to substitute Treasury billsfor risky securities or bonds for stocks,
dependingonwhetherornotarisk-freesecurityisavailable.
32
Second,inthecasewitharisk-
free rate, the investor doesnot take a position (positive ornegative)in bonds for 0:005,
even if we relax the short-sale constraints.
33
For the variable selection, the tendency of
investors with ambiguity aversion to not hold a position in some securities has the same
e ect as two-fund separation. It causes di erentinvestors to hold similarportfolios.
29
Tobeprecise,thee ectsofhigher-ordermomentsontherelativestockholdingsofCRRAinvestorswith
di erent risk aversion are not strong enough to be noticeable after rounding the estimates to the second
decimal. Theyareactuallynoticeable,butofcoursenotstatisticallysigni cant,atthethirddecimal.
30
Lynch(2000)reportsanalogousresultsforthemultiperiodportfoliochoiceofCRRAandmean-variance
investorsundertheassumptionofjointlog-normalityofreturns.
31
The choice of " is admittedly ad hoc. Camerer (1995) cites attempts to calibrate ambiguity aversion
preferencestogamblingexperiments. Since, itis unclear,however,howtheseexperimental resultsrelateto
ambiguityaversionin nancialmarkets,wepresentestimatesforarelativelygenerousrangeof".
32
Thise ectofanincreaseinambiguityaversionparallelsthatofanincreaseinriskaversion,whichisan
observationalequivalenceformalizedrecentlybyLiu(1999)andMaenhout(1999).
33
This tendency to nothold a position in some securities, which happens becausethe expected returns
arenotsuÆcientlypositiveornegativetojustify takingontheassociatedambiguity,isthesubjectofDow
and Werlang's(1992)paper. It alsomotivatesLiu's (1998)attemptto explainthe limitedparticipationof
U.S.householdsin nancialmarkets[see MankiwandZeldes(1991)]withambiguityaversionpreferences.
Panels B of Table 5 and Figure 3 present estimates of the unconditional portfolio choice
of prospect theory investors.
34
We set c=1 and, guided by the experimental calibrations
of Tversky and Kahneman (1992), consider the parameter values b = f0:8;0:9;1:0g and
l=f2:0;2:5;3:0g,whereb=1correspondstopurelossaversion.
35
Wesetthewealthreference
level
W,whichischosen intheediting stageof prospect theory,tothe initialwealth W
t
=1.
By far the most striking feature of the prospect theory results are the strong horizon
e ects (which is why we report results for allfour investment horizons here).
36;37
Consider
the portfolio choice with a risk-free rate. At the one-month horizon the optimal allocation
consistsofmorethan90percentTreasurybills,whileattheone-yearhorizonitis100percent
stocks, irrespective of the preference parameters. This is in obvious contrast to the results
for the other three sets of preferences, which exhibit onlyminimalhorizon e ects.
Benartzi and Thaler (1995) explain that the more often a loss averse investor evaluates
hisorherportfoliothelessattractivearehighexpectedreturnbuthighvarianceinvestments
because losses of these investments are realized more often at short horizons than at long
horizons. Loss aversion e ectively causes short-term investors to be extremely risk averse,
sincethereturndistributionstraddlesthekinkoftheutilityfunction,butlong-terminvestors
tobealmostriskneutral,asthemassofthereturndistributionmovesawayfromthekink. In
contrast,Merton(1969)andSamuelson(1969)showthataslongasreturnsformamartingale
theportfoliochoicesofmean-varianceandCRRAinvestorsare independentofthehorizon.
38
Even ifwe relaxthe martingaleassumption,the mean-varianceand CRRAportfoliochoices
donot exhibit nearly the magnitude of horizone ects that weobserve with lossaversion.
Besides the horizon e ects, the portfolio choice with loss aversion preferences is
34
To compute the standarderrors, which requires rst and second derivatives of the objective function
with respectto , wesmooththekink in theobjective function at zerowith apolynomial in-betweenthe
contactpoints-0.005and+0.005withcontinuous rstandsecondderivativesatthecontactpoints.
35
Like Barberis, Huang, and Santos (2000), who use c=1, b=1, and l=2:25, we abstract from the
probabilitydistortions. Whenc=1,theweights()reduce totheobjectiveprobabilitiesp(). Wechecked,
however,thatourresultsarequalitativelyrobusttousingTverskyandKahneman'svalueofc=0:65.
36
In asingle-period context we de ne horizon e ects as di erencesin the portfolio policies of investors
with di erent buy-and-hold horizons. An alternative de nition,and one that we employin Section 5.3.3,
refersto di erencesin the portfolio policies of multiperiod investorswith the same rebalancingfrequency
butdi erentnumbersofrebalancingperiods.
37
Thebehavioral nanceliteraturedistinguishesbetweentheinvestor'srebalancingperiodwhensecurities
aretradedandthe\evaluationperiod"whengainsandlossesarerealizedmentally(asopposedto nancially)
[BenartziandThaler(1995)orBarberisandHuang(2001)]. Forsimplicity,weassumethattherebalancing
andevaluationperiodsarethesame.
38
Thishorizonirrelevanceresultbreaksdownifthereturnsaremean-revertinginsteadofmartingalesor
inamultiperiod settingifthereturnsarecontemporaneouslycorrelatedwithinnovationstotheinvestment
opportunityset[seeBarberis(2000)andCampbellandViceira(1999)].
di erent positions in the risky securities (except at the annual horizon, where the short-
sale constraints are binding). In particular,for the portfoliochoice with a risk-free rate the
fractionof stocks inthe risky part ofthe allocationranges from60to100 percent. Itis also
intriguing that prospect theory investors construct approximately mean-variance eÆcient
portfolios (see Panel B of Figure 3)althoughthey donot all hold the tangency portfolio.
5.2 Optimal Index Composition
Turning nowtothe conditional portfoliochoice, weestimatethe optimal index composition
as described in Section 3.2 and Appendix A. We use linear instruments g(Z
t )=Z
t
, which
may be suboptimal in the sense of problem (3.11) but result in numerically more reliable
estimates.
39;40
We demean and standardize the variables Z
t
to be able to interpret the
magnitude of the index coeÆcients. We also normalize the index coeÆcients to sum to
one in absolute values, meaning j j 0
=1, since they are only identi ed up to scale. This
normalizationmeans that we can read the index coeÆcients assigned percentage loadings.
5.2.1 Expected UtilityPreferences
Mean-Variance Investors
Panel A of Table 6 presents estimates of the index coeÆcients for mean-variance investors
withaone-monthorone-yearhorizon. One,two,orthreestarsindicatethatthecoeÆcientis
statisticallysigni cant atthe ten, ve, orone percent level,respectively. Tables 7 describes
the estimated indices, where Panel A shows univariate descriptive statistics and Panel B
shows pairwisecorrelations with the fourpredictors, with the returnson bonds, stocks, and
wealth generated by the unconditional portfolio choice, and with the squared returns on
bonds, stocks, and wealth.
41
39
The instruments (3.13) tend to result in estimates that are sensitive to the starting values and are
only locally optimal. The problem is that, due to the derivative of (Z
t 0
; ) with respect to it second
argument, the optimal instruments cause the GMM objectivefunction to be less well-behaved than with
linearinstruments(see alsothediscussionin Section3.2).
40
Wecarefullyveri ed,however,thattheestimateswith linearinstrumentsarevirtuallyidentical tothe
estimatesthat correspondto theglobal minimumof theGMM criterionwith theoptimal instruments. In
particular, wesolvedforeightindices(twofor each setof preferences)usingarandomsearch optimization
algorithmthatisrobusttoill-behavedobjectivefunctions. Intheworstcase(ambiguityaversionpreferences
with =10and"=0:01),thedi erencebetweentheestimateswithlinearandoptimalinstrumentsimplies
an expected utility lossof 0.09percentin certaintyequivalent terms. The corresponding estimatesof the
indexare[0:249; 0:124;0:597;0:030]withlinearand[0:312; 0:119;0:518;0:051]withoptimalinstruments.
41
We use the return on wealth generated by the unconditional portfolio choice as ameaningful way to
collapsethe bivariatereturn series(bondsand stocks)into aunivariatevariable. Inparticular, thereturn
onwealth weightsthebondandstockreturnsrelativetotheirunconditionalimportancetotheinvestor.
feature of the results is that the index composition is remarkably insensitive to the level
of risk aversion. This is again an implication of two-fund separation because with a risk-
free rate and without short-sale constraints the optimal portfoliochoice of a mean-variance
investoristoallocateafraction tgc
t
[fromequation(3.5)]ofwealthtothetangencyportfolio
and to hold the remainingwealth in risk-free Treasury bills. Two-fund separation therefore
implies that the portfolio policies of all mean-variance investors are proportional to each
other and that, as a result, the optimal index of conditional variables must be the same
for all investors (since the index is identi ed only up to scale). In linewith this theoretical
result, the estimated indices are in fact identical across di erent levels of risk aversions if
we relax the short-sale constraints. If we impose the short-sale constraints, as we do for
Table 6, the estimates are slightly di erent because the constraints are binding more often
for aninvestor with =2than for an investor with =20.
At the monthly horizonthe index loads positively on the term spread, positively on the
defaultspread,negativelyonthelogdividendyield,andpositivelyonthetrend,withrelative
weights of about 55, 16, 13, and 15 percent, respectively. At the one-year horizon, it loads
positively on the term spread and the dividend yield, with relative weights of 57 and 29
percent. The default spread and trend each accountfor less than 11percentof the index.
To better understand the index composition, given that the predictors are correlated,
consider thepairwisecorrelationsinPanelB ofTable 7. Atthemonthlyhorizonthe indexis
most correlated with the term spread and is about equally correlatedwith the logdividend
yieldandtrend(correlationsof0.95,-0.43, and0.48,respectively). Somewhatunexpectedly,
it is nearly orthogonal to the default spread, although the loading on the default spread
exceed those on the log dividend yield and trend. At the annual horizon the index is
positively correlated with the term spread and dividend yield (correlations of 0.89, 0.08,
respectively) and isvirtually orthogonalto the default spread and trend.
Atboth horizonsthe indicesrelate positively tothe returnson wealth,with correlations
of 0.18 and 0.40,and negativelyto the squared returnsonwealth, with correlationsof -0.11
and -0.05.
42
The signs and magnitudesof these correlations suggest the following:
An increase in the index represents an unambiguous improvement in investment
opportunities,since it increases the meanand decreasesthe varianceof future wealth.
42
At the monthly horizon the positive correlation of the index with returns on wealth is attributed to
thepositiveloadingon theterm premium. Thenegativecorrelationwith squaredreturnscomes from the
positive loading on the trend. At the annual horizon both the positive correlation with returns and the
negativecorrelationwithsquaredreturnsonwealthareattributed tothetermspread.