On Market Timing and Investment Performance. II. Statistical Procedures for Evaluating
Forecasting Skills
Author(s): Roy D. Henriksson and Robert C. Merton
Source: The Journal of Business, Vol. 54, No. 4, (Oct., 1981), pp. 513-533
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/2352722
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Roy D. Henriksson and
Robert C. Merton
Massachusetts Institute of Technology
On Market Timing and
Investment Performance.
IL. Statistical Procedures for
Evaluating Forecasting Skills*
I. Introduction
In Merton(1981;hereafterreferredto as Part I),
one of us developed a basic model of markettiming forecasts where the forecaster predicts
when stocks will outperformbonds and when
bonds will outperformstocks but does not predict the magnitudeof the superiorperformance.
In that analysis, it was shown that the patternof
returns from successful market timing has an
isomorphiccorrespondenceto the patternof returns from following certain option investment
strategieswhere the implicit prices paid for the
options are less than their "fair" or marketvalues. This isomorphiccorrespondencewas used
to drive an equilibrium theory of value for
market-timingforecasting skills. By analyzing
how investorswould use the markettimer'sforecast to modify their probability beliefs about
stock returns,it was shown that the conditional
probabilitiesof a correctforecast (conditionalon
the returnon the market)provideboth necessary
and sufficient conditions for such forecasts to
have a positive value.
In the analysis presented here, we use the
* Earlierversions of the paper were presentedin seminars at Berkeley, Carnegie-Mellon,Universityof Chicago,
Dartmouth,Harvard,Universityof SouthernCalifornia,and
Vanderbilt;we thank the participantsfor their comments.
Aid from the National Science Foundationis gratefully
acknowledged.
(Journal of Business, 1981, vol. 54, no. 4)
? 1981 by The University of Chicago
0021-9398/81/5404-0005$01.50
513
The evaluationof the
performanceof investment managersis a
much studiedproblem
in finance. Based upon
the model developed in
PartI of this paper, the
statisticalframeworkis
derivedfor both
parametricand nonparametrictests of
market-timingability. If
the manager'sforecasts
are observable,then the
nonparametrictest can
be used withoutfurther
assumptionsabout the
distributionof security
returns.If the manager's forecasts are not
observable,then the
parametrictest can be
used underthe assumption of either a capital
asset pricingmodel or
a multifactorreturn
structure.The tests
differfrom earlierwork
because they permit
identificationand separationof the gains of
market-timingskills
from the gains of micro
stock-selectionskills.
514
Journal of Business
basicmodelof markettimingderivedin PartI to developbothparametric and nonparametricstatisticalproceduresto test for superiorforecasting skills.
The evaluationof the performanceof investmentmanagersis a topic
of considerableinterestto both practitionersand academics.To the
former,such evaluationsprovidea usefulaidforthe efficientallocation
of investmentfunds among managers.To the latter, significantevidence of superiorforecastingskillswouldviolatethe EfficientMarkets
Hypothesis.'Such violations,if found,wouldhave far-reachingimplications for the theory of finance with respect to optimal portfolio
holdings of investors, the equilibriumvaluation of securities, and
manydecisionsin corporatefinance.With so much at stake, it is not
surprisingthatmuchhas been writtenon this subject.Indeed,a major
applicationof modem capital markettheory has been to provide a
structuralspecificationto measureinvestmentperformance.Within
this structure,it is the practiceto partitionforecastingskills into two
components(see Fama 1972):(1) "microforecasting,"whichforecasts
pricemovementsof individualstocks relativeto stocks generally,and
(2) "macroforecasting,"whichforecastsprice movementsof the general stock marketrelative to fixed income securities. The formeris
frequentlycalled "securityanalysis" and the latter is referredto as
"markettiming."Moreover,this partitioningof forecastingskillstakes
on addedsignificancethroughthe work of Treynorand Black (1973),
who have shown that investmentmanagerscan effectively separate
actionsrelatedto securityanalysisfromthoserelatedto markettiming.
Most of the recent empiricalstudies of investment performance
focus on microforecastingand are based on a mean-variancecapital
asset pricingmodel (CAPM)framework'where the 1-periodexcess
returnon securityi can be writtenas
Zi(t) - R(t) = ai + fJZM(t)
-
R(t)] + Ei(t),
(1)
where Zi(t) is the 1-periodreturnper dollar on securityi, ai is the
expectedexcess returnfrommicroforecasting,
f3iis the ratioof the covarianceof the returnon securityi with the marketdivided by the
varianceof the returnon the market,andEi(t)has the propertythatits
expectation,conditionalon knowingthe outcomefor the marketreturn
thatis not
1. As a tautology,superiorforecastingskillsmustbe basedon information
is obtainable,then security
reflectedin securityprices.Therefore,if such information
andthemarketwillnot be efficient.Fama
priceswillnotreflectall availableinformation
(1970)providesan excellentdiscussionof boththe EfficientMarketsTheoryandvarious
attemptsto test it.
2. The capitalasset pricingmodel (CAPM)refersto the equilibriumrelationships
amongsecuritypriceswhichresultwheninvestorshavehomgeneousbeliefsandchoose
criterionfunction.Fortheoriginalderivations,
theirportfoliosbasedon a mean-variance
reviewof the
see Sharpe(1964),Lintner(1965),andMossin(1966).Fora comprehensive
model,see Jensen(1972a).
On Market Timing and InvestmentPerformance
515
ZM(t),is equal to its unconditionalexpectation which is zero. That is,
E [Ei(t) IZM(t)1
=
E [Ei(t)]
=
0.
Using this specification, both Fama (1972) and Jensen (1972b) develop theoretical structuresfor the evaluation of micro- and macroforecastingperformanceof investment managerswhere the basis for
the evaluation is a comparison of the ex post performanceof the
manager'sfund with the returnson the market.In the Jensen analysis,
the markettimeris assumedto forecast the actualreturnon the market
portfolio,and the forecastedreturnand the actualreturnon the market
are assumed to have a joint normal distribution.Jensen shows that
under these assumptions, a markettimer's forecasting ability can be
measuredby the correlationbetween the markettimer's forecast and
the realized returnon the market.3However, Jensen also shows that
the separate contributionsof micro- and macroforecastingcannot be
identifiedusing the structureof (1) unless for each period, the market-timing forecast, the portfolio adjustment correspondingto that
forecast, and the expected returnon the market are known.
Grant(1977) explains how market-timingactions will affect the results of empiricaltests that focus only on microforecastingskills. He
shows that market-timingabilitywill cause the regressionestimateof ai
in (1) to be a downward-biasedmeasureof the excess returnsresulting
from microforecastingability.
Treynor and Mazuy (1966) add a quadraticterm to (1) to test for
market-timingability. In the standardCAPM regression equation, a
portfolio's return is a linear function of the return on the market
portfolio. However, they argue that if the investment managercan
forecast marketreturns,he will hold a greaterproportionof the market
portfolio when the returnon the marketis high and a smallerproportion when the marketreturnis low. Thus, the portfolioreturnwill be a
nonlinearfunction of the marketreturn. Using annual returnsfor 57
open-endmutualfunds, they findthat for only one of the funds can the
hypothesis of no market-timingability be rejected with 95% confidence.
Kon and Jen (1979) use the Quandt (1972) switching regression
techniquein a CAPMframeworkto examinethe possibilityof changing
levels of market-relatedrisk over time for mutual fund portfolios.
Using a maximumlikelihoodtest, they separatetheir data sampleinto
differentrisk regimesand then runthe standardregressionequationfor
each such regime. They find evidence that many mutualfunds do have
discrete changes in the level of market-relatedrisk they choose which
is consistent with the view that managersof such funds do attemptto
incorporatemarket timing into their investment strategies.
3. See Jensen(1972b),pp. 317-18. This result is also derivedin Treynorand Black
(1973).
Journal of Business
516
The model of market-timingforecasts presented here differs from
those of these earlier studies in that we assume that our forecasters
follow a more qualitative approach to market timing. Namely, we
assumethat they eitherforecast thatZM(t) > R (t) or forecast thatZM(t)
< R (t). The forecastersin our model are less sophisticatedthan those
hypothesized in, for example, the Jensen (1972b) formulationwhere
they do forecast how muchbetterthe forecast superiorinvestmentwill
perform. However, as is shown in Part I, when this simple forecast
informationis combined with a prior distributionfor returns on the
market, a posteriordistributionis derived which would permitprobability statements about the magnitudesof the superior investment's
performance.
A brief formaldescriptionof our forecast model is as follows: Let
y(t) be the market timer's forecast variable where y(t) = 1 if the
forecast, made at time t - 1, for time period t is that ZM(t) > R (t) and
y(t) = 0 if the forecast is that ZM(t) - R (t). We definethe probabilities
for y(t) conditionalupon the realized returnon the marketby
prob [y(t) = 0 ZM(t) R (t)]
1 - p,(t) = prob [y(t) = 1 IZM(t) 6 R(t)]
pl(t)
(2a)
and
p2(t)
prob [y(t) = 1 ZA(t) > R(t)]
1 - p2(t) = prob [y(t)
=
0 Zm(t)
>
(2b)
R(t)].
Therefore,p,(t) is the conditionalprobabilityof a correct forecast
given that ZM(t) - R(t), and p2(t) is the conditionalprobabilityof a
correct forecast given that ZM(t) > R (t). It is assumed thatp ,(t) and
p2(t) do not dependupon the magnitudeof IZM(t) - R (t) I. Hence, the
conditionalprobabilityof a correct forecast depends only on whether
or not ZM(t) > R (t). Under this assumption, it was shown in Part I that
the sum of the conditionalprobabilitiesof a correct forecast, p ,(t) +
p 2(t), is a sufficient statistic for the evaluationof forecastingability.
Unlike the earlier studies of markettiming, this formulationof the
problempermitsus to study markettimingwithout assuminga CAPM
framework. Indeed, provided that the market timer's forecasts are
observable, we derive in Section II of this paper a nonparametrictest
of forecasting ability which does not require any assumptions about
either the distributionof returns on the market or the way in which
individualsecurityprices are formed.Althoughthe substantivecontext
of the test presentedthere is markettiming,the same test couldbe used
to evaluate forecastingability between any two securities.
If the markettimer's forecasts are not directly observable, then to
test markettimingrequiresfurtherassumptionsabout the structureof
equilibriumsecurityprices. In Section III, we derive such a test using
the assumption that the CAPM holds. However, in contrast to the
On Market Timing and InvestmentPerformance
517
Jensen formulation, our parametrictest permits us to identify the
separate contributionsfrom micro- and macroforecastingto a portfolio's return using as our only data set the realized excess returns
on the portfolio and on the market. Althoughthe test specificationin
Section III assumes a CAPMframework,it can easily be adoptedto a
multifactorpricing model as described in Merton (1973) and Ross
(1976).
II. A NonparametricTest of MarketTiming
In PartI, it was shown that a necessary and sufficientcondition for a
forecaster's predictions to have no value is that p1(t) + p2(t) = 1.
Under this condition, an investor would not modify his priorestimate
of the distributionof returns on the market portfolio as a result of
receiving the predictionand therefore would pay nothingfor the prediction. It follows that a necessary condition for market-timingforecasts to have a positive value is thatp 1(t) + p 2(t) 7 1. As shownin Part
I, a sufficientconditionfor a positive value is thatp 1(t) + p 2(t) > 1. For
example, a perfect forecaster who is always correct will have 1(t) = 1
andp2(t) = 1; therefore,pl(t) + P2(t) = 2 > 1. Formally,forecasts with
p1(t) + p2(t) < 1 can be shown to have a negative value because such
forecasts are systematically incorrect. However, such forecasts are
perverse in the sense that the contrary forecasts with p 1(t) 1 - p1(t)
andp'(t) 1 - p2(t) would satisfyp (t) + p'(t) > 1 and thereforehave
positive value. For example, a markettimer who is always wrong will
have p 1(t) + p 2(t) = 0. However, such forecasts have all the informational content of a forecasterwho is always rightbecause by following
a strategyof always doing the opposite of the forecasts that are always
wrong, one will always be right. Thus, one can reasonablyarguethat
forecasts with p1(t) + p2(t) < 1 have positive value as well, provided
one is aware that the forecasts are perverse.
Therefore, a test of a forecaster's market-timingability is to determine whetheror notp l(t) + p2(t) = 1. Of course, ifp l(t) andp2(t) were
known, then such a test is trivial. However, p1(t), p2(t), or their sum,
are rarely, if ever, observable. Generally,it will be necessary to estimatep 1(t) + p2(t), and then use these estimates to determinewhether
one can reject the naturalnull hypothesisof no forecastingskills. That
is, Ho: p1(t) + p2(t) = 1 where the conditionalprobabilitiesof a correct
forecast are not known. Essentially, this is a test of independence
between the markettimer's forecast and whether or not the returnon
the marketportfoliois greaterthan the returnfrom riskless securities.
The nonparametrictest constructed around this null hypothesis
takes advantageof the fact that conditionalprobabilitiesof a correct
forecast are sufficientstatistics to measureforecastingability and yet
they do not depend on the distributionof returnson the marketor on
any particularmodel for security price valuation. The essence of the
Journal of Business
518
test is to determine the probabilitythat a given outcome from our
sample came from a populationthat satisfies the null hypothesis. To
determinethis probability,we proceed as follows. First, we definethe
followingvariables:N1 numberof observationswhereZM S R; N2
number of observations where ZM > R; N
N1 + N2
=
total number of
observations;n1 numberof successful predictions,givenZm - R; n2
numberof unsuccessful predictions,given Zm > R; n n1 + n2
numberof times forecast that ZM - R.
By definition,E(n1/N,) = Pi and E(n2/N2) =1 - P2 where E is the
expected value operator. From the null hypothesis, we have that
E(n1/N1) = pi =
Ho
= E(n2/N2),
- P2
(3)
and from (3), it follows that
E [(n1 + n2)/(N1 + N2)1 = E (n/N)
P
p.
(4)
Ho
Both n1/N1 and n2/N2 have the same expected value under our null
hypothesis, namely,p, and both are drawnfrom independentsubsamples. Hence, only one or the other need be estimated.
Both n1 andn2 are sums of independentlyand identicallydistributed
randomvariableswith binomialdistributions.Therefore,the probability that ni - x from a subsampleof Ni drawingscan be written as
P(ni= x |Nip)
( Ni)PX(I
_
P)Nx;
1,2.
i=
(5)
Giventhe nullhypothesis,we can use Bayes's Theoremto determine
the probabilitythat n1 = x given N1, N2, and n, that is, P(n1 =
x IN1,N2,n). Denote the event that our markettimerforecasts m times
that Zm s R (i.e., n = m) as A and the event that, of the times he
forecaststhatZM- R, he is correctx times and incorrectm - x times
(i.e., n1 = x and n2 =m - x) as B. Then P(n 1 = x IN1,N2,m) =P(B |A),
and by Bayes's Theorem, we have that
P(B+ A)
P (A)
P(B IA)
(N1)(
-
P(B)
P (A)
N2 j)px(1
()m(l
tN,8
_
N2
~j)mA?x
{N
_
)Njl-xpl-x(j
-
p)N-rn
-
p)N2-tn+x
519
On MarketTiming and InvestmentPerformance
Hence, under the null hypothesis, the probabilitydistributionfor
n 1-the numberof correct forecasts, given thatZM- R-has the form
of a hypergeometricdistributionand is independentof bothp, andp 2.
Thereforeto test the null hypothesis, it is unnecessary to estimate
either of the conditionalprobabilities.So, providedthat the forecasts
are known, all the variablesnecessary for the test are directlyobservable. GivenN1, N2, andn, the distributionof n1underthe null hypothesis is determinedby (6) where the feasible range for n, is given by:
min(N,,n) =En.
n, _ max(O,n - N2) n,
(7)
Equations (6) and (7) can be used in a straightforwardfashion to
establish confidence intervals for testing the null hypothesis of no
forecasting ability. For a standard two-tail test with a probability
confidencelevel of c, one would reject the null hypothesis if n,1 , (c)
or if n1, x(c), wherei and x are defined to be the solutions to the
equations4
L- (NX )(N -)x )/(N ) = (1
-
c)12
(8a)
and
1~f(n-
c)12.
(8b)
However, we would arguethat a one-tail test (or at least one which
weights the right-handtail much more heavily than the left) is more
appropriatein this case. If forecastersare rational,then it will never
be true thatp l(t) + p 2(t) < 1, and a very smalln1 would simplybe the
"luck of the draw" no matterhow unlikely. It seems most unlikelyto
us that a "real world" forecaster who had the talents to generate
significantforecastinginformationwould not have the talent to recognize that his forecastswere systematicallyperverse, while at the same
time, we as outside observers of those forecasts can clearly see the
errors of his ways. For such a one-tail test with a probabilityconfidence level of c, one would reject the null hypothesis if n1 ? x*(c)
where x*(c) is defined as the solution to
x*(IN1) (N2x)
)(AT)
-
(N) = 1 - c.
(9)
By inspection of (8a) and (9), x*(c) < xY(c),and therefore, given an
observationin the righttail, a one-tailtest is, of-course, more likely to
distributionis discrete,the strictequalitiesof eqq. (8a)
4. Becausethe hypergeometric
and(8b)will not, in general,be attainable.Therefore,in (8a),Y shouldbe interpretedas
thelowestvalueof x for whichthe summationdoes not exceed (1 - c)12.In (8b),x should
be interpretedas the highestvalue of x for whichthe summationdoes not exceed (1 c)12.
Journal of Business
520
rejectthe null hypothesisthan a two-tailone for any fixed confidence
level c. However, this fact in no way impliesa greaterlikelihoodof
rejectingthe null hypothesiswhen it is true by using a one-tailtest.
Computationof the confidenceintervalsfor either the two-tail or
one-tailtest using(8) or (9) is straightforward
when the samplesize is
small. However, for large samples, the factorialor gammafunction
computationscan become quite cumbersome.Fortunately,for those
large samples where such computationsbecome a problem, the
hypergeometricdistributioncan be accuratelyapproximatedby the
normaldistributionsThe parametersused for this normalapproximation are the mean and variancefor the hypergeometricdistribution
given in (6), which can be writtenas
E(n)-
nN
N
(1Oa)
and
r2(n1) = [n1N1(N - N1)(N - n)]I[N2(N - 1)].
(lOb)
Tables1, 2, and 3 give valuesof n1 for a one-tailtest that rejectthe
null hypothesisat the 99%confidencelevel for differentvalues of N1,
N2, andn. As wouldbe expected,the requiredestimatedvalueof p l(t)
+ p2(t)decreasesas the size of the total sampleincreases.Tables 1-3
also demonstratethat the normal distributioncan be an excellent
approximationfor determining the confidence intervals for the
hypergeometric
distribution,even for observationsamplesas smallas
50.6
By focusingon the conditionalfrequenciesof correctforecasts,the
test proceduredescribedin (6)-(10) takes into accountthe possibility
that the markettimer may not have the same skill in forecastingup
marketsas down markets.That is, pQ(t) need not be equal to p2(t).
However,if one knowsthatthe forecasterwhose predictionsarebeing
tested has equalabilitywith respectto bothtypes of markets,then the
conditionalprobabilitiesof a correctforecast,p(t) andp2(t),are equal
to each other, andtherefore,each is equalto the unconditionalprobability of a correctforecast,p (t). That is, p1(t) = p2(t) = p (t). In that
case, one need only measurethe unconditionalfrequencyof a correct
forecastto test for market-timing
abilitywhere the null hypothesisof
5. Thelarge-sample
cases wheredirectcomputation
of the confidenceintervalsusing
(8)or (9)aremostcumbersomearewhenN1 N2orn N12.Inthesecases, thenormal
approximation
will be quite good for even moderatelylarge samples.See Lehmann
(1975,theorem19)for a generalproof.Thenormalapproximation
willnotbe a goodone
even for quite large samplesin those cases where there are substantialdifferences
betweenN1 andN2 or betweenn andN/2. However,it is preciselyin these lattercases
wheredirectcomputation
using(8)or (9)is notcumbersome
evenforverylargesamples.
6. As discussedin 5, this excellentapproximation
shouldonlybe expectedto obtain
whenN1 N2 andn N/2.
521
On Market Timing and InvestmentPerformance
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Journal of Business
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Journal of Business
524
no forecasting skills is that p (t) = .5. The distributionof outcomes
drawnfrom a populationthat satisfies this null hypothesis is the binomial distributionwhich can be written as
P(k IN,p)
= (NP k (
=
p)N(k
(N (.5)N,
k
wherek is the numberof correctpredictionsandN is the total number
of observations. One can use (11) in an analogous fashion to (6) to
construct either one-tail or two-tail confidence intervals for rejecting
the null hypothesis. Whilethe simplicityof this test may be attractive,
the readershouldbe warnedthat a test which uses (11) insteadof (6) is
only appropriateif there is strongreason to believe thatp 1(t) = p2(t).
As is discussed at lengthin PartI, the unconditionalprobabilityof a
correct forecast cannot, in general, be used as a measure of markettimingability.Specifically,it is shown thatan unconditionalprobability
of a correct forecast greater than one-half, p(t) > .5, is neither a
necessary nor a sufficient condition for a forecaster's market-timing
abilityto have positive value. To see why it is not sufficient,consider
the case of a forecaster who always predicts that the return on the
marketwill exceed the returnon riskless securities. Such completely
predictable forecasts, like a stopped clock, clearly have no value.
However, if the historical frequency with which the returns on the
marketexceeded the returns on riskless securities were significantly
greaterthan one-half, then this forecaster's unconditionalprobability
of a correct forecast would exceed one-half, and the null hypothesis
would be rejected. Indeed, if a two-tail test were used, then all that
would be requiredto reject the null hypothesis is that the historical
frequencyof up marketsversus down marketsbe significantlydifferent
than one-half.However, if this "stopped clock" forecasterwere evaluated by the test proceduredescribedin (6)-(10), then, independentof
the relativefrequenciesof up and down markets,the null hypothesisof
no forecastingability would not be rejected because for any sampleof
observations,p1(t) = 0 andp2(t) = 1, and hence, p1(t) + p2(t) = 1.
Therefore, by using the unconditionalprobabilityprocedure in (11),
one is actually testing the joint null hypothesis of no market-timing
ability andp 1(t) = p2(t).
In summary,we have deriveda nonparametricprocedurefor testing
market-timingability which takes into account the possibility that
forecastingskills are differentfor up marketsthan for down markets.
Because the critical statistic for the test is p1(t) + p2(t), it is not
essential that the individual conditional probabilities be stationary
throughtime. Rather,the critical stationaritypropertyfor the validity
of equation(6) is that their sum,p 1(t) + p2(t), be stationary,which is,
On MarketTiming and InvestmentPerformance
525
of course, true under the null hypothesis of no market-forecasting
ability. Moreover, it is straightforwardto show that this same procedurecan be used to test forecasterswho have moreconfidencein their
predictionsduring some periods than they do in other periods. One
such application can be found in Lessard, Henriksson, and Majd
(1981),where the predictionsof some foreign-exchangeforecastersare
tested using our procedure.However, it is essential to our test procedure that the forecasts of market-timingbe observable.We will therefore turn to the development of a procedure to test market-timing
ability when such forecasts cannot be observed.
III. Parametric Tests of Market Timing
To use the nonparametricproceduresto test investmentperformance,
the predictionsof the forecaster must be observable. However, it is
frequentlythe case when measuringmanagedportfolio performance
that the examineronly has access to the time series of realizedreturns
on the portfolioand does not have the investmentmanager'smarkettimingforecaststhemselves. Whileundercertainconditionsit is possible to infer from the portfolioreturnseries alone what the manager's
forecasts were, such inferences will, in general, provide noisy estimates of the forecasts. These estimates will be especially noisy if the
manager'sportfoliopositions are influencedby his microforecastsfor
individual securities. In this section, we derive procedures which
permitthe testing of timingabilityusing returndata alone. Of course,
there is a "cost" of not having the time series of forecasts, which is
that these test proceduresrequirethe assumptionof a specific generatingprocess for returnson securities. Thus, these proceduresare
parametrictests of thejoint hypothesisof no market-timingabilityand
the assumed process for the returnson securities.
As noted earlier,most of the recent empiricalstudies of investment
performanceassume a patternof equilibriumsecurityreturnswhich is
consistent with the SecurityMarketLine of the CAPMin additionto
some assumptions about the market-timingbehavior. The standardregression-equationspecificationfor portfolio returns used in these
studies can be written as
Zp(t)
R(t)
-
= a+ ?X
(t) +
E(t),
(12)
whereZp(t) is the realizedreturnon the portfolio,x(t) ZM(t) - R (t) is
the realizedexcess returnon the market,and E(t) is a residualrandom
term which is assumed to satisfy the conditions
E[E(t)] = 0
E[E(t) x(t)] = 0
E[E(t)
jE(t
- i)] =
(13)
0, i = 1,2,3.
Journal of Business
526
Providedthat the investmentmanagerdoes not attempt(or, at least,
is unsuccessful at) forecasting market returns, the standard leastsquares estimation of (12) can be used to test for microforecasting
skills. However, Jensen (1972b)shows that it is impossibleto use this
structuralspecificationto separatethe incrementalperformancedue to
stock selection from the increment due to market timing when the
return data alone are used. The tests derived here do permit such a
separation.
As in the earlier studies, we also assume that securities are priced
accordingto the CAPM, althoughthe tests can easily be adaptedto
accommodatea multifactormodelprovidedthat the factors are known.
We furtherassume that as a function of his forecast, discretely different systematicrisk levels for the portfolioare chosen by the forecaster.
For example, in the case we analyze in detail here, it is assumed that
there are two target risk levels which depend on whether or not the
return on the market portfolio is forecast to exceed the return on
riskless securities.That is, the investmentmanageris assumedto have
one targetbeta when he predictsZM(t)> R (t) and anothertargetbeta
when he predicts that ZM(t)- R(t). In Section IV, we indicate how
the test procedurescan be adaptedto the moregeneralcase of multiple
target risk levels.
Let vj, denote the targetbeta chosen for the portfolioby the manager
when his forecast is thatZM(t) - R (t) and let j2 denote the targetbeta
chosen when his forecastis thatZM(t)>R(t). If/3(t) denotes the beta of
the portfolio at time t, then p3(t)= vjl for a down-market forecast and
,8(t) = -r,2for an up-marketforecast. If the forecaster is rational,then
> vl. Of course, if/3(t) were observableat each point in time, then,
as discussed in Fama (1972),the market-timingforecast is observable,
and one could simply apply the nonparametrictests of the previous
section. However, if beta is not observable, then /(t) is a random
variable.Underthe assumptionthat beta is not observable,let b denote
the unconditional(on the forecast) expected value of /3(t). Then
7)2
b = q[p 11 + (1
-
P 1)q2]
+ (1 -
q)[P27)2 + (1
- P2)>1],
(14)
where q is equal to the unconditional(on the forecast) probabilitythat
ZM(t)- R (t). In Part I, the distributionfromwhich q is computed was
called the prior distribution.If we define the randomvariable0(t) as
equal to [/3(t) - b], then 0(t) is the unanticipatedcomponentof beta,
and its distribution,conditionalon the realized excess returnon the
market,x(t), can be written as:
conditionalon x(t) - 0:
= 8 where
=1
=
(7)i
= (q2
-
2)[1
-i1)[qpI
-
qp1 - (1 - q)(1 - P2)] with prob = pi
+ (1 -q)(1 -p2)]withprob
=
1 -p1
(l5a)
527
On Market Timing and InvestmentPerformance
and
conditionalon x(t) > 0:
_= _02where
02
+ (1
-
q)(1
-
with prob =
(15b)
=
(62
-
-
(n1
- 70)L1 - qpI - (1 - q)(l - pa)] with prob = 1 -P 2
7,l)[qP
1
P2)]
P2
From(15), it follows that the conditional(onx[t]) expected value of 0
can be written as
E(0 Ix)
(16a)
01
=
(1
-
q)(p1
+P2
-
1)Qq1
72),
forx(t)
S
0
and
E(0 X) =
=
(16b)
02
q(p1
+ P2
-
1)(72
-
1), forx(t) > 0.
The per period returnon the forecaster'sportfoliocan be writtenas
Zp(t) = R(t) + [b + 0(t)]x (t) + X + Ev(t),
(17)
where A is the expected incrementto the returnon the portfoliofrom
microforecastingor securityanalysisand E.(t) is assumedto satisfy the
standardCAPM conditions given in (13).
Under the posited returnprocess for the portfolio given in (17), a
least-squaresregression analysis can be used to identify the separate
incrementsto performancefrom microforecastingand macroforecasting. The regression specificationcan be written as
Zp(t) - R(t)
=
a
+
3Jx(t) +
2y(t) + e(t)
(18)
where y(t) max[OR(t) - ZM(t)] = max[0,-x(t)].
The motivationbehindthe specificationgiven in (18) comes from the
analysis of the value of markettimingpresentedin Section IV of PartI.
There it was shown that up to an additive noise term, the returnsper
dollarinvested in a portfoliousing the market-timingstrategydescribed
here will be the same as those that would be generatedby pursuinga
partial "protective put" option investment strategy where for each
dollarinvested in this strategy, [P2-q2 + (1 - P2)7)1] dollarsare invested
in the market; (p1 + P2 - 1)(n2 - -rp)put options on the market
portfolioare purchasedwith an exercise price (per dollarof the market)
equal to R(t); and the balance is invested in riskless securities. The
value of the markettiming(per dollarof assets managed)is that the (p1
+ P2 - 1)(X2 - ijl) puts are obtained, in effect, for no cost. Note that
y(t) as defined in (18) is exactly the returnon one such put option.
From(17), the expected returnon the portfolio,conditionalonx > 0,
can be written as
Journal of Business
528
X,
(19a)
E(Zp x>O)=R+(b+02)E(xIx>0)+
and the expected return, conditionalon x - 0, can be written as
(19b)
E(Zp Ix - O) = R + (b + 01)E(xIx - O) + X,
where bars over randomvariables denote expected values.
For the analysis of the regressioncoefficientsand errortermin (18),
it will be convenientto express the relevantvariancesand covariances
of the regression variables in terms of the expected values and variances of the randomvariablesxl(t) and x2(t) which are defined by
x1(t)
min[Ox(t)]
x2(t)
max[O,x(t)].
(20)
If var[xi(t)] o-4, i = 1,2, then we can write the variances and
covariances of the regression variables as
var[y(t)] =
U2 = U2
2 = r2o + o-2 -
var[x(t)]
cov[x (t) ,y (t)]0
cov[Zp(t),x(t)]
, = (b +
(21)
2x-lx-2
= X1X2-cr
1)(o2
cov[Zp(t),y]- ary= (b
+
02)X1X2
(b + 02)(o2
+
X2
-X
-
(b
-
X-1X2)
+ 6l)o-~.
From (18) and (21), it follows that the large sample least-squares
estimates of ,1 and /32 can be written as
plim /a
PYOXY
2
PX0Y
=
-r
2 O2...
(22)
= b+ 02
= P2172 +
(1
-P2)nl
and
plim
,
-
"-pY-
=02
x
rPX XY
-
(23)
-01
= (P1 + P2
-
1)()2-
7)
=E
From (22), plim
E[,B(t)Ix(t) > 0], and it is also equal to the
in
the
market
invested
portfolioin the option strategyused in
fraction
Section IV of Part I to replicate the market-timingstrategy. The investment manager'smarket-timingabilityis expressed as /2' The true
/2 will equal zero if either the forecasterhas no timingability-that is,
ifp1(t) + p2(t) = 1-or he does not act on his forecasts-that is, if q2 =
r%.With referenceto the replicatingoption investment strategy, from
On MarketTiming and InvestmentPerformance
529
(23), plim82 is equalto the numberof "free" put optionson the market
providedby the manager'smarket-timingskills. Indeed, as shown in
PartI, the value of market-timingskills per dollarof assets managedis
equal to (p1 + P2 - 1)(-q2 - Tj1)g(t)where g(t) is the market price of
such a put. Thus, ,/2g(t) is an estimateof the value of the market-timing
ability of the manager.
Formallyinterpreted,a negativevalue for the regressionestimate/2
would imply a negative value for market timing. However, a true
negative value for /32 would violate the rationality assumptions of p,(t)
1 andq2 : -r1.Hence, as was discussedfor the nonparametric
+p2(t)
W
tests in Section II, the reader should consider the relative meritsof a
one-tailversus the standardtwo-tailtest of significancewith respect to
rejectingthe null hypothesis that /2 = 0.
The incrementto portfolioperformancefrom microforecastingcan
also be measured using regression equation (18). The large sample
least-squaresestimate of a can be written as7
plim a = E(Z,) - R - plim f1x - plim A7=
A.
(24)
Hence, from (23) and (24), regression equation (18) can be used to
identify and estimate the separate contributionsof microforecasting
and macroforecastingto overall portfolio performance.
To completethe analysis of (18), we now investigatethe properties
of the errorterm E(t)being carefulto take into accountthe differences
between the actualbetas and their estimates. That is, if the forecaster
strictlyfollows the posited behaviorof two discrete risk levels 7rnand
7q2,then the actual,B(t)of the fundwill never be equal to the estimated
,8 unless, of course, he is a perfect forecaster.
Let us definethe followingvariables:A1 = 1 if x - 0 and the market
timer's forecast is incorrect,A1 = 0 otherwise; V1 = 1 if x - 0 and
the markettimer's forecast is correct, V1 = 0 otherwise; A2 = 1 if
x > 0 and the markettimer's forecast is incorrect,A2 = 0 otherwise;
V2 = 1 if x > 0 and the markettimer's forecast is correct, V2 = 0
otherwise. It follows immediatelythat:
E(Al) =PI
E(A2) = 1
E(V1)=
P2
1 -Pi
(25)
E(V2) =P 2
From (18), each period's estimationerrorE can be written as:
E=
A1(-P
1)(q1 -
-2)X1-V
-P
-
11712)X1 + A2P2(71 V2(1
-
P2)(71l
-
2)X2
v2)X2 + E1,
(26)
7. Whentestingfor forecastingability,the relevantportfolioreturnsare thoseearned
before any deductionfor managementfees. If one uses the returnsearnedafter the
deductionof managementfees and if the fees chargedcan be expressedas a fixed
percentage of assets, m, then plim a' = A - m.
Journal of Business
530
where E, includes any error term resulting from microforecasting.
Since, by definition,microforecastingis independentof x, Ep is independent of x. It follows by the Law of Large Numbers that as the
numberof observations, N, gets large,
V
Y. 2, and
Y.
Y.
N
will all approachtheir expectation. Therefore, from (26),
lim
N-~
Nl
-
Nj
-
I[Pi(l
P I)(l
-
-
+ [(1-
P2)P72(71-
+ lim
lEpIN
lim
N-oo
[ NP]
(1 - Pi)P1(fl
O2) -
-
P2)-P2(0
P2)(fll
XI
-
- 72)]X2
(27)
= 0.
Thus, for large samples, the coefficient from least-squaresestimation
of (18), plus the realized excess returnon the market, will give us an
unbiased estimate of the portfolio return.8
As discussed, the motivationbehindthe regressionspecification(18)
was the analysis in Part I which showed the correspondencebetween
market-timinginvestment strategies and certain option investment
strategies. However, there is alternative,but equivalent, specification
which some may find to be more intuitive. Namely, by a linear transformationof (18), we can write this alternativeregressionequationas
Zp(t)
-
R(t) = a' + fI1XA(t)+ 3x2(t) +
E,
(28)
wherex1(t)andx2(t)are as definedin (20). Becausex1(t) = 0 andx2(t) =
x(t) if x(t) > 0 f32has a rather intuitive interpretationas the "upmarket"beta of the portfolio. Similarly,because x1(t) = x (t) and x2(t)
= 0 if x(t) - 0, p8can be interpretedas the "down-market"beta of the
portfolio. Indeed, the large sample properties of the regression estimators/ and f3"fit these intuitiveinterpretations(at least in the sense
of expected values). That is,
plim ,8 = E[f3(t) Ix(t)
-P 171
+
S
0]
(1 -P1)f2
(29a)
8. Althoughunbiased,ordinaryleast squaresestimationis not efficientbecausep(t) is
not stationary,and thereforethe standarddeviationof the errorterm is an increasing
functionof Ix(t) l. To improvethe efficiencyof the estimates,one coulduse generalized
least squaresestimationto correctfor this heteroscedasticity.
531
On Market Timing and InvestmentPerformance
and
plim /2 = E [,3(t) | x(t) > ]
=P272 +2(1
(29b)
P2)77l,
- 0] = b + 1 The
where E[E3(t) x(t) > 0] =b + 02Iand E[/(t)x(t)
test for market-timingabilityusing this specificationwould be to show
that /2 is significantlygreaterthan ,. That is, show that the expected
"up-market" beta of the portfolio is greater than the expected
"down-market"beta of the portfolio.The large samplepropertiesof oi'
are the same as for a^in (18): namely, plim ^'
IV.
=
A.
Summary and Extensions
Providedthatthe forecasteronly attemptsto predictthe sign of ZM(t)R (t) but not its magnitude,and providedthat his forecasts are observable, a procedurefor testing market timing has been derived which
does not dependon any distributionalassumptionsaboutthe returnson
securities. The test includes the possibility that the forecaster's
confidencein his forecasts as measuredby (P 1,P2) can vary over time,
and indeed, if such variations are observable, then the test can be
refined to measure his forecastingability for each such variation.
In the case where the forecaster's predictionsare not observable, a
parametrictest procedure was derived which permits separate measurementsof the contributionsto portfolio performancefrom market
timing and security analysis. As is apparentfrom the analysis of the
errortermin Section III, this test will accommodatethe case where the
two-targetrisk levels chosen by the managervary over time provided
that these variationsare randomarounda stationarymean.
The test is also applicableto the case where the forecaster selects
from morethantwo discretesystematicrisktargetlevels, as long as the
differentlevels are based on differinglevels of confidence in the forecasts and not differingexpectations of the level of the return. In this
case, the large sampleleast-squaresestimates of)/1 and 32 representa
weighted average of the differentrisk levels.
The test procedurespresentedhere can be extended to evaluate the
performanceof a markettimerwho segmentshis predictionof x(t) into
more than two discrete regions. For example, a forecastermighthave
that - 10%o- x(t) < 0, that
fourpossible predictions:thatx(t) < - 10%F,
0 - x(t) < 10%, and that 10%lo x(t). We briefly illustrate how the
analysis would be applied to such multipleregions for the parametric
test case. As in the two-regioncase, it is assumed that the probability
of a particularforecast will only depend on the region in which x(t)
falls. However, there are now more than two possible forecasts.
Journalof Business
532
Specifically,we assumethat there are n differentregionsthat the
forecastermightpredictand definepij as the probabilitythat the
forecaster'spredictionwas thatx(t) wouldbe in thejth region,given
thatx(t) actuallyendedup in theith region.Theonlyconstrainton the
conditionalprobabilitiesis that j=1pij = 1, i = 1,2 . . . , n. The return
on theforecaster'sportfoliocanbe definedas in SectionIIIexceptthat
now
n
b = E
i=1
n
> jpj,
i
j-1
thatx(t) willendupin regioni and
where6iis definedas theprobability
,qj is the chosenlevel of systematicriskwhenthe forecastis thatx(t)
will end up in regionj.
to (28)in the two regioncase
Theregressionequationcorresponding
x + E,wherexi = x if x is in
can be writtenas Z -R = a
region i, and xi = 0 otherwise.
A
Thelargesampleleast-squaresestimatesof /pianda are:plim =
j- pijqj and plim a' = X. Fromthis analysis,it follows that for
regions(thatis, forlargeenoughvaluesof
finelypartitioned
sufficiently
returns
to separatetheincremental
in
is
least
possible
n), it at
principle,
withoutany restrictionson the disfrommicro-andmacroforecasting
tributionof forecasts.All that is requiredare the actualreturnsfrom
the market,the portfolio,and risklesssecurities.
References
Fama, E. 1970.Efficientcapitalmarkets:a review of theoryandempiricalwork.Journal
of Finance 25, no. 2:383-417.
Fama, E. 1972. Componentsof investmentperformance.Journal of Finance 27, no.
2:551-67.
Grant, D. 1977. Portfolioperformanceand the "cost" of timingdecisions. Journal of
Finance 32, no. 2:837-46.
Jensen, M. C. 1972a. Capitalmarkets:theory and evidence. Bell Journalof Economics
and ManagementScience 3, no. 2:357-98.
Jensen, M. C. 1972b. Optimalutilization of market forecasts and the evaluation of
investmentperformance.In G. P. Szego andK. Shell (eds.), MathematicalMethodsof
Investmentand Finance. Amsterdam:North-Holland.
Kon, S. J., and Jen, F. C. 1978. The investment performanceof mutualfunds: an
empiricalinvestigationof timing, selectivity, and marketefficiency.Journalof Business 52, no. 2:263-89.
Lehmann,E. 1975.Nonparametrics:Statistical Methods Based on Ranks. San Francisco: Holden-Day.
Lessard, D.; Henriksson,R.; and Majd, S. 1981. Risk and returnon foreign currency
assets: an evaluationof forecastingability.Workingdraft. Cambridge,Mass.: Massachusetts Instituteof Technology.
Lintner,J. 1965.Securityprices, risk, andmaximalgainsfromdiversification.Journalof
Finance 20 (December):587-615.
Merton,R. C. 1973.An intertemporalcapitalasset pricingmodel.Econometrica41, no.
5:867-87.
Merton, R. C. 1981. On markettimingand investmentperformance.I. An equilibrium
theory of value for marketforecasts.Journal of Business 54 (July): 363-406.
On MarketTiming and InvestmentPerformance
533
in a capitalasset market.Econometrica34, no. 4:768-83.
Mossin,J. 1966.Equilibrium
Quandt,R. E. 1972.A new approachto estimatingswitchingregressions.Journalof the
American Statistical Association 67, no. 338:306-10.
Ross, S. 1976.The arbitragetheoryof capitalassetpricing.Journalof EconomicTheory
3 (December):343-62.
Sharpe,W. F. 1964.Capitalassetprices:a theoryof marketequilibrium
underconditions
of risk. Journal of Finance 19, no. 4:425-42.
Treynor,J., and Black, F. 1973. How to use securityanalysis to improveportfolio
selection. Journal of Business 46, no. 1:66-86.
Treynor,J., and Mazuy, F. 1966. Can mutualfunds outguessthe market?Harvard
BusinessReview 44 (July-August):131-36.