Asymmetric Momentum Effect Under Uncertainty

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Review of Finance (2010), 129doi: 10.1093/rof/rfq021

Asymmetric Momentum Effects Under Uncertainty

DAVID KELSEY1, ROMAN KOZHAN2 and WEI PANG3

1University of Exeter; 2University of Warwick; 3Kingston University

Abstract. This paper studies asymmetric profitability of the momentum trading strategy. When

investors face Knightian uncertainty, they react differently to past winners and losers, which createsasymmetric patterns in price continuations. This asymmetry increases with the level of market and

idiosyncratic uncertainty relating to the fundamental value of stocks. We provide a model explaining

this phenomenon and empirical evidence supporting the hypothesis. Our results also imply that

momentum is more likely to continue for downward trends in a highly uncertain market.

JEL Classification: D81, G11, G12

1. Introduction

A large body of literature has documented the existence of momentum strategies in

stock trading: buying past winners and selling past losers generates significant aver-

age profit. Most of the evidence concerns the predictability of time-series variation

in stock returns (Jegadeesh and Titman (1993), Chan et al. (1996), Rouwenhorst

(1998), Moskowitz and Grinblatt (1999), Lee and Swaminathan (2000)). The in-

terpretation of the evidence on return momentum is mainly based on behavioural

biases such as underreaction or overreaction to new information (see Barberis et al.

(1998), Daniel et al. (1998), Hong and Stein (1999)). Specifically, Hong and Stein

(1999) consider the interaction of newswatchers who under-react to the news and

trend followers who create overreaction to the news. De Long et al. (1990) model

the overreaction to the signal through the behaviour of positive feedback traders

who despite the presence of arbitragers in the market create momentum effects.

The mechanism in these models works as follows. Investors who trade on the fun-

damental value of the asset react to signals concerning the fundamental value andcause the price to change accordingly. Momentum traders who trade on the price

We thank participants of the workshop Market Complexity at the University of Warwick and

the research seminar at Kingston University for helpful comments. Special thanks go to Vince Daly,

Subrata Ghatakfor, Cars Hommes, Mark Salmon, Author Shaw, Willem Spanjers, Wing Wah Tham,

the editor and an anonymous referee for their helpful comments. We gratefully acknowledge financial

support from EU NEST STREP, FP6 Grant on Complex Markets, no. 516446, which made the

project possible.

C The Authors 2010. Published by Oxford University Press [on behalf of the European Finance Association].All rights reserved. For Permissions, please email: [email protected]

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2 D. KELSEY ET AL.

trend observe the price change and enter the market pushing the price further, thus

creating the overreaction pattern.

In these models, one of the key factors determining the strength and magnitude

of the momentum effect in stock returns is investors reaction to information.

Furthermore, there are empirical studies showing the difference in price drifting

patterns caused by bad news and by good news (Andersen et al. (2007) found such

asymmetry in the FX market, Hong et al. (2000), Chan (2003) found it in equity

markets, and Beber and Brandt (2009) found it in US bond markets). A recent study

by Epstein and Schneider (2008) argued that such patterns can arise due to the

presence of Knightian uncertainty (hereafter, we use terms Knightian uncertainty

and uncertainty interchangeably). They show that in face of Knightian uncertainty

investors overvalue negative information and undervalue positive information. This

paper aims to model and test the effect of Knightian uncertainty on momentum

returns and the presence of asymmetric patterns between subsequent returns on

winner and loser portfolios.

Knightian uncertainty refers to situations where objective probabilities are un-

known or imperfectly known. It can arise because of missing information, dis-

agreement of expert opinions or lack of confidence in the quality of information.

Consequently, it is impossible to describe such ambiguous situations using precise,

objective probabilities. One may argue that financial markets are a good place to

study the effects of such uncertainty. Facing uncertainty, the majority of people

are found to be uncertainty averse, i.e. they prefer choices with better informationto similar choices with poor information.1 With uncertainty aversion, people tend

to overweight possible worse outcomes and underweight better outcomes. We use

maxmin expected utility, introduced by Gilboa and Schmeidler (1989), to model

uncertainty-aversion. In this model, people make decisions based on a cautious

maxmin criteria: each possible course of action is evaluated with respect to the least

favourable probability distribution from a given set of priors. The chosen action

is the one for which the minimum expected utility is the highest. The degree of

uncertainty-aversion depends on how large the set of priors is. For a small set of pri-

ors the resulting preferences only exhibit a small degree of uncertainty-aversion. As

a result they are close to the expected utility. A large set of priors would correspond

to more extreme types of uncertainty-aversion2.

Our main theoretical contribution is the demonstration that the presence of un-

certainty can be a reason for a pronounced asymmetry in momentum returns. We

establish a positive relationship among asymmetry, level of uncertainty and the pro-

portion of uncertainty averse investors in the market. We build a model with three

types of traders: arbitragers, uncertainty averse traders and momentum traders.

1 For the experimental evidence on uncertainty averse behaviour, see Camerer and Weber (1992).2 Thus an uncertainty averse preference overweighting unfavourable outcomes (not necessarily the

worst) is equivalent to maxmin preferences with a smaller set of priors. It is easy to see that our results

are held for general uncertainty averse preferences as long as the priors are not unique.

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ASYMMETRIC MOMENTUM UNDER UNCERTAINTY 3

Arbitragers are uncertainty neutral and behave with von Neumann-Morgenstern

utility preferences.3 Uncertainty averse traders lack confidence on the distribution

of the fundamental value of a stock and hence consider a set of possible priors

for the fundamental value. When choosing the distribution of fundamental value,

uncertainty averse traders act as if a good signal is unreliable and a bad signal is

reliable. Consequently, the aggregated price change triggered by a good signal is

less than the aggregated price change triggered by a bad signal. Given the differ-

ent magnitudes of price change and their demand functions, momentum traders

would accordingly create asymmetry between positive and negative price continu-

ation. Finally, we show that such asymmetry would disappear with the absence of

uncertainty and uncertainty averse traders in our model.

We firstly derive the equilibrium where investors use static optimisation to form

their demands. Arbitragers and uncertainty averse traders do not consider the dy-

namic aspect of a trading strategy while forming their demand in each trading

period.4 At the price of this limitation, we are able to obtain illustrative results

showing asymmetry in price continuation in relation to the level of uncertainty. To

make the arguments general, we further impose full rationality on arbitragers, so that

they choose their periodic demand with the strategic consideration of the future path

of price and other traders trading behaviour. Thus the trading price of the asset in

each period not only clears the market but also satisfies dynamic consistency of ar-

bitragers behaviour. We solve this generalised model numerically and confirm sim-

ilar findings that asymmetric momentum patterns arise with a positive relationshipbetween the level of uncertainty and the proportion of uncertainty averse traders.

In our model, we derive asymmetric responses to good and bad signals through

modelling uncertainty concerning the fundamental value rather than the signal

quality as suggested in Epstein and Schneider (2008). In doing so, we are able to

construct empirical testing for the predictions of the model. We adopt the empirical

measurements of stock uncertainty in Zhang (2006) and use a similar data set

with a prolonged period. Unlike most previous studies, we investigate separately

winner and loser momentum returns realised on firms with different levels of

uncertainty. We find that the momentum effect is stronger and more likely to last

for losers with a greater level of uncertainty. That is, for stocks with a high level

of uncertainty, selling the loser attributes to the greater part of momentum profit.

Short-selling stocks with greater uncertainty will realise greater profits following

3 When the degree of uncertainty aversion tends to zero, multiple prior utility converges to von

Neumann-Morgenstern utility.4 This is to assume that different groups of arbitragers and uncertainty averse traders enter the market

in each period, and buy and hold until receiving the dividend. An alternative interpretation is that

arbitragers behave in a boundedly rational way, as argued in Hong and Stein ( 1999). They are not

time-consistent and do not condition on other traders actions. However, as shown in our general

analysis, imposing time-consistency of trading behaviour on arbitrager does not change our results

on asymmetry momentum effects of uncertainty.

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4 D. KELSEY ET AL.

negative momentum than buying stocks with greater uncertainty following positive

momentum. However, the difference between post-formation returns of losers and

winners is not significant for stocks with less uncertainty. The inequality of profit

distribution following negative and positive momentum is increasing and only

becomes significant with increased uncertainty. To investigate the effect of market

uncertainty, we define the market crashes in 1987 and 2000 to be a crisis period

and assume a high market uncertainty during this period. We find that asymmetry

between negative and positive momentum returns are more profound during the

crisis period in comparison to the rest of sample period, named as the non-crisis

period in the paper.

There are several studies showing the difference in price drifting patterns caused

by bad news and by good news (Chan (2003), Hong et al. (2000)). Hong et al.

(2000) have observed strong asymmetry in profitability of momentum strategies

following past losers and past winners, categorised by market capitalisation and

analyst coverage. They attribute the explanation to the gradual-information-flow

model in Hong and Stein (1999) and interpret that lower analyst coverage implies

slower response to bad news than to good news. In this paper, we use six sorting

variables including market capitalisation and analyst coverage, and observe asym-

metric patterns with all variables. In contrast to Hong et al. (2000), we argue these

sorting variables as proxies for the level of uncertainty rather than the rate of in-

formation flow. As our model shows, the dislike of uncertainty by investors causes

asymmetry in momentum return. The asymmetric patterns become more prevalentas uncertainty level associated with stocks increases. Our empirical findings are

consistent with these predictions. Moreover, in our view, the comparison between

the crisis and the non-crisis periods provides exclusive support for our argument

that uncertainty accounts for asymmetric momentum returns. It is hard to argue

that bad news travels slower than good news during a crisis. Hence we believe that

our model and empirical evidence presented in this paper provide an alternative

explanation of asymmetric momentum returns.

The rest of the paper is organised as follows. In Section 2 we develop the

model and establish our testable hypotheses. Section 3 introduces the data and

testing methodology. Empirical findings are reported in Section 4. We conclude in

Section 5.

2. The Model

We consider a model featuring three types of investors: uncertainty neutral arbi-

tragers (denoted R henceforth), present in the market in a measure , uncertainty

averse traders (denotedUhenceforth), present in the market in a measure 1 and

momentum traders (denotedM), present in the market in a measure . Momentum

traders disregard any information about the fundamental value of the stock and

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form their demands by extrapolating past prices. Both arbitragers and uncertainty

averse traders form their demand based on the difference between todays price and

their expectation of the future fundamental value of the stock. The principal differ-

ence between those two types is that uncertainty averse traders perceive uncertainty

about the fundamental value which is reflected in their demand functions while

arbitragers are uncertainty neutral.5

There are three trading periods 0, 1, 2 and 3 and two trading assets, bond and

stock. We assume that bond is in perfectly elastic supply and pays zero risk-free

rate of return while stock is in a zero net supply. Stock pays a risky dividend

d= d+ m + i at the end of trading period 3, where d is the mean dividend, mandi are market and idiosyncratic shocks respectively.

2.1 INVESTORS BELIEFS

At time 0, arbitragers and uncertainty averse traders start trading in the market.

Arbitragers observe an objective distribution of the dividend

d Nd, 2d

,

where variance 2d = 2m +

2i is equal to the sum of variances of market and

idiosyncratic components.

In contrast, uncertainty averse traders perceive uncertainty about the future div-

idend payment, namely fundamental uncertainty, which is caused by either themarket shock m or the idiosyncratic shock i . We model preferences of un-

certainty averse traders using the multi-prior approach developed in Gilboa and

Schmeidler (1989). The set of priors for the market shock consists of normal dis-

tribution with the set of variance values [2m m, 2m +m] and the set of mean

values [m, m], 2m > m > 0 andm > 0. Priors for the idiosyncratic com-

ponent of the stock are also described by a family of normal distributions with

the variance interval [2i i , 2i +i ] and means [i ,i ],

2i > i > 0 and

i > 0. Since m andi are uncorrelated, the fundamental value d has variance

in the interval [2d, 2d+], where = m +i . The set of mean values

is [d, d+], where d is the mean dividend and = m +i .6 In sum,

priors which uncertainty averse traders hold about the future dividend payment are

described by a family of normal distributions

d Nd+ , 2d+

, [, ] and [, ].

This captures the idea that uncertainty averse traders have a lack of confidence

about the true distribution of the future dividend payments. Note that if = 0 and

5 All traders are risk averse.6 Garlappi et al. (2007) provide an interpretation of parametric sets of multiple priors as confidence

intervals around expected returns.

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6 D. KELSEY ET AL.

= 0, there is no fundamental uncertainty perceived in the model and uncertainty

averse traders prior beliefs are reduced to a single distribution and coincide with

beliefs of uncertainty-neutral arbitragers.

The uncertainty averse traders form their expectations according to the multi-

prior model:

EU(d) = min[,]

E(d) = d, (1)

where the subscript in the expectation operator denotes the prior with respect to

which the expectation is calculated.

The intuition behind this specification of uncertainty averse preferences is thatthey do not know precisely the distribution of the true fundamental value of stocks.

As a result, instead of using a unique probability distribution, they consider set of

priors about the future payoff. Since the traders are uncertainty averse, they choose

their investment strategies by maximising the expected utility in the worst-case

scenario. Equation (1) indicates that at time 0 uncertainty averse traders discount

their expectations about the true price acquiring an ambiguity premium, which is

related to the width of the mean interval [, ].

The uncertainty-neutral arbitragers expectations of the fundamental value is

ER(d) = d.

2.2 DEMAND FUNCTIONS

The demands of both arbitragers and uncertainty averse traders are proportional to

the difference between their expectation of future dividend payment and the current

price,

DI0 = I(EI(d) p0), and,

I > 0,

where Iis eitherR andU. Coefficients I reflect risk tolerance of traders which is

allowed to be different for arbitragers and uncertainty averse traders.

Momentum traders base their trade at time tonly on the price change from t 2

to t 1.7 Their demand for stock at time tis given by

D

M

t =

M

(pt1 pt2), and M

> 0.

2.3 PERIOD 0

Since there is no price changes prior to time 0, DM0 = 0, i.e., period 0 provides

a benchmark against which the momentum traders can measure changes in stock

7 The model is an illustration to a more general case where momentum trading is based on price

changes over a particular interval from t k 1 to tk.

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prices. The market clearing condition DR0 + (1 )DU0 + D

M0 = 0 implies

p0 =RER(d) + (1 )UEU(d)

R + (1 )U= d (1 ), (2)

where = R

R+(1)U. Since at time 0 there are uncertainty averse traders in

the market who demand an uncertainty premium for holding the stock, there is

a price discount at time 0.8 The more fundamental uncertainty is perceived, the

wider the interval of the fundamental value of the stock, d, and the greater the price

discount is at time 0. At the same time, the price discount depends on the proportion

of uncertainty averse traders relative to uncertainty-neutral arbitragers. The feweruncertainty averse traders there are in the market, the smaller the price discount

that is observed.

2.4 PERIOD 1

At time 1, both arbitragers and uncertainty averse traders receive a noisy signal s

about the fundamental value of the stockd. Specif ically, traders observe s = d+

with N(0, 2 ).9

Since uncertainty averse traders perceive uncertainty about the future dividend

payoffdvariable, they also perceive uncertainty about the distribution of the signals.

Upon receiving the signal s, uncertainty-neutral arbitragers update their proba-

bilistic beliefs in a Bayesian way, so that

ER(d|s) = d+2d

2d+ 2

(s d) = d+ R(s d) = d+ R, (3)

where R =2d

2d+2

and = s d is the magnitude of the signal relative to the

expected value of the future dividend payoff for arbitragers.

Uncertainty averse traders update their beliefs about the fundamental value of

the stock by applying Bayes rule to all the likelihoods of the priors to obtain a

family of posteriors (see Epstein and Schneider(2008))

d|s N d+ +

2d+

2d+ + 2 (s (d+ )),

2d+ 22d+ + 2

, (4)

with [, ] and [, ]. The slope coefficient2d+

2d++2

indicates the

extent to which the uncertainty averse traders trust the precision of signal s.

8 Epstein and Schneider(2008) made the same argument about the expectation of uncertainty averse

investors.9 We can extend the model to the case of uncertain noise, as it is done in Epstein andSchneider(2008)

but this will not change the main conclusion of the model, so we skip it for the sake of simplicity.

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8 D. KELSEY ET AL.

If the variance of the signal noise 2 is high relative to the variance of the

fundamental value 2d, uncertainty averse traders take the information contained in

the signal less seriously. If the variance of the fundamental value is high compared

to the variance of the noise, the information content in the signal is high and the

slope coefficient is higher, thus uncertainty averse traders rely more on the signal.

The traders expectation is

EU(d|s)= min [, ] [,]

E,(d|s)= d+Us(d)

(s (d))= d+U ,

where = s (d) and

U =

+ =

2d

2d+2

, if 0,

=2d+

2d++

2

, if < 0.

Thus, uncertainty averse traders respond to the signal in an asymmetric manner and

we have < R < +. The intuition is similar to that in Epstein and Schneider

(2008). If the news about d is good (that is, the signal is higher than the lowest

expected value of the fundamental price s d), uncertainty averse traders

treat the signal less seriously and react less by using the lower fundamental variance

2d (and by doing so increasing the impact of the noise in the signal). If the

news is bad(s < d), uncertainty averse traders view the signal as more reliableand react more by using the higher value of the fundamental variance 2d+. This

causes an asymmetric effect in the reaction to news they react stronger to bad

news and weaker to good news.10

Arbitragers and uncertainty averse traders demand at time 1 is

DI1 = I(EI(d|s) p1), I= R,U

and again momentum traders do not trade at time 1. The market clearing condition

at time 1, DR1 + (1 )DU1 + D

M1 = 0 implies

p1 =RER(d|s) + (1 )UEU(d|s)

R + (1 )U= ER(d|s) + (1 )EU(d|s)

= d(1 )+

()R + (1 )U

= d(1 )+ AR, (5)

10 Note that arbitragers measure the magnitude of a signal relative to the expected value of dividend

d, and uncertainty averse traders measure it relative to the maximin expected value of dividendd.

Since only uncertainty averse traders have different reaction to bad and good signals, the measurement

of signal strength is what matters in our analysis.

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where A = R + (1 )U is an average response of both uncertainty averse

traders and uncertainty neutral arbitragers to the signal s.

At time 1, the price consists of four components. The first one is a rational

expectation component equal to the mean of the future dividend payoff. The

second component reflects the price discount due to the presence of uncertainty

averse traders who require a premium for holding the uncertain stock. The

third component is the price revision due to the arrival of new information. The

sensitivity to new information is a weighted combination of R and U which

reflects a consensus sensitivity between arbitragers and uncertainty averse traders.

The price will adjust up if there is good news 0, and adjust down if there is bad

news < 0. The coefficient U is greater for bad news than for good news, thus,

ceteris paribus, the up-movement of price will be less than the down-movement

of price. The fourth component is the price adjustment due to the difference in

the valuations of signal magnitude by arbitragers (measured as ) and uncertainty

averse traders (measured as ). Since we use the measurement of the signal strength

in the equation forA, arbitragers responses to new information are adjusted

down by in the formation of the market clearing price.

Without the presence of the uncertainty averse traders, i.e., = 1, the price p1becomes d+ R(s d) which is revised solely on the expectation of arbitragers

and the asymmetric price movements disappear. If there is no uncertainty, i.e. = 0

and = 0, asymmetry in the reactions to bad and good signals also disappears.

2.5 PERIOD 2

At time 2, there is no new information and both arbitragers and uncertainty averse

traders do not revise their beliefs, while momentum traders enter the game. They

interpolate prices p0 andp1 to form their demand

DM2 = M(p1 p0).

Demands of two other traders types are

DI2 = I(E(d|s) p2), I= R,U.

We obtain the equilibrium price of the stock at time 2 as a solution of the market

clearing condition DR2 + (1 )DU2 + DM2 = 0,

p2 = p1 +M

R + (1 )U(p1 p0)

= d(1 ) + (1+ )

()R + (1 )U

= d(1 ) + (1+ )[AR], (6)

where =M

R+(1)U.

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10 D. KELSEY ET AL.

Period 2 price is a continuation of the price at period 1 due to the demand pressure

from momentum traders, whose market proportion is indicated by . Momentum

traders buy if they observe a past positive price change and sell on the observed

negative price change. Their trades simply amplify the asymmetric price movements

created in period 1.

At time 3, a dividend d is realised and paid to all traders. Arbitragers make

profit on average while the average performance of uncertainty averse and mo-

mentum traders depends on their risk appetites and severity of uncertainty in

the market. This might suggest that uncertainty neutral arbitragers should dom-

inate the market over time. However, this is only true if attitudes towards un-

certainty are independent of other constraints such as wealth and gender. Mean-

while, De Long et al. (1990) provided a general discussion on why momentum

traders persist in the long run even though the strategy of chasing the trend

might not be profitable. Since this aspect of the model is not closely relevant

to the purpose of the paper, we restrict our attention to factors affecting price

continuations.

The mechanism of the momentum effect is close to Hong and Stein (1999).

Momentum traders demands respond to the price change in the past period.

They submit a market order to buy when price rises and to sell when price de-

clines. In comparison, arbitragers and uncertainty averse traders demands are

driven by current market price and their beliefs about the future fundamental value

of the stock. There is a single signal arriving at time 1 which causes arbitragersand uncertainty averse traders to create a price change from time 0 to 1. This

price change is picked up and continued by momentum traders at time 2 through

a round of momentum trading. The asymmetry appears in our model due to the

existence of uncertainty averse traders. While being uncertain about the funda-

mental value of stock, the traders create different magnitudes of price changes by

reacting less on the arrival of good signals than on bad signals. Momentum traders

who simply trade on the previous price changes will continue the difference in

past price changes under good/bad signals and generate asymmetric momentum

effects.

2.6 MOMENTUM AND ITS ASYMMETRIC EFFECT

Note that in the benchmark period with no momentum traders ( = 0), the price

continuation is equal to 0. Momentum traders enter the market at time 1 and generate

the price continuation during the third period of time as

Cont(, ,)= p2 p1 = (p1 p0)= R + (1)U

R, (7)

which increases with the strength of the relative signal .

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Our aim is to emphasise the asymmetric momentum between winners and losers,

so for a relative signal > 0 we define

Asym(, ,) = Cont(, ,) + Cont(, ,)

= [R + (1 )+] R

+ ()[R + (1 )] R

= [2R + (1 )(+ )],

where,

+ =

2

d

2d + 2

2

d

+

2d+ + 2=

22

(m +i )2d+

2

2 (m +i )2

.

As analysed before, in the presence of uncertainty, the asymmetric effect arises

due to the difference in responses to good (+) and bad signals () by uncertainty

averse traders. We have Asym < 0 meaning that, ceteris paribus, the reaction to a

bad signal dominates the reaction to a good signal. The magnitude of the asym-

metry is positively related to fundamental uncertainty including both the market

uncertainty and idiosyncratic uncertainty. Specifically, an increasing market un-

certainty (higherm andm) would increase the asymmetric momentum effects

while keeping all other parameters of the model fixed. Similarly, stocks with a

higher level of idiosyncratic uncertainty (higher i andi ) would also have a

higher level of asymmetry. If there is no uncertainty ( = 0 and = 0), the valuesof slope coefficients will coincide (+ = ), and we have Asym(, ,) = 0.

Also, asymmetry in momentum depends on the proportion of uncertainty aversion

traders in the market. The more uncertainty-averse traders there are in the market,

the larger the magnitude of the asymmetry. If there are no uncertainty averse traders

in the market ( = 1), the asymmetric response to new information disappears.

We summarise the implications of the model as the following testable hypotheses.

Hypothesis 1. Due to fundamental uncertainty, negative momentum effects are

greater than positive momentum effects.

Hypothesis 2. The more idiosyncratic uncertainty is, the more profound the asym-

metry between realised negative and positive momentum profits.

Hypothesis 3. An increase in market uncertainty leads to an increase in cross-

sectional asymmetry of momentum return.

2.7 FULLY RATIONAL ARBITRAGERS

Following Hong and Stein (1999), we do not consider the dynamic nature of the

arbitragers trading strategy in the above model. However, it is natural to ask

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12 D. KELSEY ET AL.

Figure 1. Equilibrium Prices and Momentum Asymmetry with Fully Rational Traders.The figure presents equilibrium prices when fully rational traders are present in the model. The left

panel presents price paths with good news (upper lines) and bad news (lower lines) for different levelof uncertainty. The right panel presents the measure of momentum asymmetry for different levels of

uncertainty. Parameters are = 0.5, = 0.4, 2d = 1, 2 = 0.2 andd= 0. We set d= 0.5 and =

0.5 to represent positive news and = 0.5 andd= 0.5 to represent negative news. We computeprices for a set of 5 uncertainty levels, U1(, ) = U(0, 0); U2(, ) = U(0.01, 0.2); U3(, ) =U(0.02, 0.4); U4(, ) = U(0.03, 0.6) and U5(, ) = U (0.04, 0.8). In the left panel, the slopeof the price path following bad news is much steeper than the one following good news for higheruncertainty level such as U5. In the right panel, the measure of asymmetry has a clear downwardslope indicating larger asymmetry for higher levels of uncertainty and asymmetry disappears in U1when we have = 0 and = 0.

whether our basic results presented in the previous section will continue to hold ifarbitragers are fully rational. In this section, we solve the model by assuming that

uncertainty neutral arbitragers trade strategically in each period with consideration

of the future price path in addition to the expected value of liquidated dividend

and current market price. We use backward induction to derive arbitragers optimal

demand in each period so that their f inal wealth is maximised. Unfortunately the

equilibrium results in this case become mathematically cumbersome and lose their

intuitive interpretation. Thus we present here a graphic illustration of our numerical

solutions.

As shown in Figure 1, our main results and conclusions regarding the momentum

asymmetry remain similar in the context of fully rational arbitragers. Firstly, we

see that the slope of the price path following bad news is much steeper than the

one following good news for higher uncertainty level such as U5. Such asymmetry

is originated in period 0 to 1 and continued in the momentum effects appeared in

period 1 to 2; second, in the benchmark case U1 of no uncertainty ( = 0 and

= 0), the momentum effect still exists, however, the price reactions to good and

bad signals are perfectly symmetric. The right panel of the f igure summarised the

findings. The measure of asymmetry has a clear downward slope indicating larger

asymmetry for higher levels of uncertainty and asymmetry disappears in U1 when

= 0 and = 0.

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3. Data and Methodology

Our sample consists of all firms listed on the NYSE, the AMEX and the NASDAQ

during the period January 1983 through December 2005. The stock return data are

from the Center for Research in Security Prices (CRSP) Monthly Stocks Combined

File. In our sample, we exclude ADRs, REITs, closed end funds, and primes and

scores from the analysis. We also exclude stocks with a share price below $5 and

higher than $1000 at the portfolio formation date and as well as firms with less

than 12 months of past return data on CRSP.

Book value and other financial data are from Compustat. Analyst forecast re-

visions and analysts coverage data are from I/B/E/S. Following Zhang (2006) wedelete observations for which the absolute value of earnings forecast revision ex-

ceeds 100% of the prior year-end stock price.

There have been few studies on the empirical measurement of ambiguity in

the literature. Zhang (2006) proposed six proxies for uncertainty and argued their

suitability. These proxies are mainly variables describing firms characteristics.

We adopt this approach and use the same proxies to measure the uncertainty

level relating to a firm. The first variable is firm size measured as the market

capitalisation (MV) at the beginning of each period for ex ante returns. The sec-

ond variable is firm age (AGE) measured as the number of years since the firm

was first covered by CRSP. The third variable is analyst coverage (COV) mea-

sured as the number of analysts following the firm in the ex ante period. Thefourth proxy is stock volatility (SIGMA), which is measured as the standard de-

viation of weekly excess return over the period for calculating ex ante return.

The fifth variable is dispersion in analyst earnings forecasts (EFD) measured as

the standard deviation of analyst forecasts scaled by the prior ex ante period-end

stock price to mitigate heteroscedasticity. The sixth variable is cash flow volatility

(CVOL) which is the standard deviation of cash flow from operations in the past

5 years.

In order to implement the momentum strategy, we calculate cumulative return

Rt12,t1 for the past 12-month period from t 12 through t 1 months prior

to the portfolio formation. The performance of the strategy is based on the future

one-month returns Rt+1 for the periodtto t+ 1. Following Jegadeesh and Titman

(1993), we also form the portfolios and calculate the returns for 6 6 momentumstrategies to increase the power of our tests.

Table I presents descriptive statistics and the correlation matrix for variables

of interest and the general patterns are similar to those that are found in Zhang

(2006). The mean monthly return has a slight right skewness in the distribution, for

instance, forRt+1, we have 1.10% as the mean and 0.52% as the median. There is

a large variation in firm size and age in the sample. Firm age has the median of 12

indicating that young firms account for a big portion of the sample. In contrast to

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14 D. KELSEY ET AL.

Table I. Descriptive Statistics and Correlation Matrix

This table provides descriptive statistics for variables used in the analysis. Rt+1 denotes returns from

month tto t+ 1, Rt+6 is accumulated returns from month tto t+ 6, Rt12,t1 is accumulated returns

from month t 12 to t 1, Rt6,t1 is accumulated returns from month t 6 to t 1. Firm size

(MV) is the market capitalisation (in millions of dollars) at the end of month t. Firm age (AGE) is the

number of years since the firm was first covered by CRSP. Analyst coverage (COV) is the number of

analysts following the firm in the previous year. Earnings forecast dispersion (EFD) is the standard

deviation of analyst forecasts in month t scaled by the prior year-end stock price. Stock volatility

(SIGMA) is the standard deviation of weekly market excess returns over the year ending at the end

of month t. Cash flow volatility (CVOL) is the standard deviation of cash flow from operations in

the past 5 years (with a minimum of 3 years), where cash flow from operations is earnings before

extraordinary items minus total accruals estimated from the balancesheet approach, scaled by averagetotal assets. Stocks with a price less than $5 are excluded from the sample. The sample period is from

January 1983 to December 2005.

Panel A: Descriptive Statistic

N Mean St. Dev Min Q1 Med. Q3 Max

Rt+1 928614 1.10% 13.45% 98.13% 5.36% 0.52% 6.94% 1034%

Rt+6 918742 6.61% 36.09% 99.50% 12.29% 4.17% 21.25% 3110%

Rt12,t1 918702 25.53% 76.48% 97.56% 9.43% 13.15% 40.89% 4998%

Rt6,t1 926766 13.06% 46.73% 96.89% 8.84% 6.97% 25.43% 3753%

MV 923995 4048.6 26915.2 0 68.49 265.82 1240.02 1.106

AGE 928614 16.56 15.06 2 6 12 22 81

SIG 918702 1.88% 6.71% 0 0.43% 0.91% 1.92% 1376%

CVOL 647103 0.16 5.73 4.6106 5.9103 0.02 0.08 1871.4

COV 638244 8.36 8.39 1 2 5 11 63

EFD 398966 0.67% 2.51% 0 0.072% 0.21% 0.57% 141.9%

Panel B: Correlation Matrix

Rt+6 Rt12,t1 Rt6,t1 MV AGE SIG CVOL COV EFD

Rt+1 0.374 0.006 0.011 0.005 0.007 0.017 0.004 0.000 0.019

Rt+6 1 0.002 0.024 0.011 0.014 0.042 0.010 0.001 0.028

Rt12,t1 1 0.695 0.003 0.057 0.265 0.066 0.029 0.030

Rt6,t1 1 0.002 0.045 0.238 0.051 0.033 0.014

MV 1 0.189 0.020 0.002 0.316 0.041

AGE 1 0.088 0.017 0.329 0.012

SIG 1 0.097 0.044 0.044

CVOL 1 0.083 0.032COV 1 0.031

Zhang (2006), stock returns are not very volatile, as suggested by a mean SIGMA

of 1.88% per week and a median of 0.91% per week.

Panel B shows that firm size (MV), firm age (AGE) and analyst coverage (COV)

are positively correlated with each other but negatively correlated with stock volatil-

ity (SIG), cash flow volatility (CVOL) and earnings forecast dispersion (EFD). This

supports the idea that these proxies reflect the same property of a firm.

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Based on these measurements, we tested our hypotheses by assigning stocks to

portfolios based on past returns and the uncertainty level associated with the stock.

The following section provides the results of the comparison of the portfolio returns

by momentum and uncertainty proxy.

4. Empirical Findings

4.1 PORTFOLIO RETURNS BY MOMENTUM AND UNCERTAINTY PROXY

We follow the treatments used in Jegadeesh and Titman (1993) and calculate the

stock returns of JK trading strategy, where stocks are selected on the basis of

returns over the past J months and held for K months. In this paper, we test our

hypotheses respectively for two trading strategies, 12 1 and 6 6 strategies.

We rank the stocks in descending order on the basis of RtJ,t1 returns and

construct five equally weighted portfolios. We consider the upper and the lower

quintile portfolios as winners and losers portfolios representing positive and nega-

tive momentum respectively. Stocks in each of these portfolios are ranked according

to their uncertainty proxy and five sub-portfolios are formed. These portfolios are

rebalanced every month and their following monthly return is calculated.

The stocks in uncertainty quintile 1 (denoted U1) are regarded as the least

uncertain and the stocks in uncertainty quintile 5 (denoted U5) are regarded as the

most uncertain. This results in 10 equal weighted portfolios for each uncertaintyproxy for each trading strategy.

Table II shows the main results for momentum returns for 12 1 trading strategy

based on momentum and uncertainty proxies.

We firstly examine variation in portfolio returns for winners and losers respec-

tively by each uncertainty level and verify the existence of momentum effects.

Portfolios at all uncertainty levels have higher returns following winners than fol-

lowing losers. The momentum effects are more significant for portfolios with higher

uncertainty levels. The results are applied to all uncertainty proxies and confirm an

implication suggested by Zhang (2006) that momentum trading strategies should

work better for portfolios with more uncertainty. For example, for AGE proxy and

the portfolio at uncertainty quintile 1, the post momentum return is 0.67%; however,

the post momentum return of U5 portfolio by buying the winners and selling the

losers is 2.41%.

The focus of the paper is on the separate comparison of positive and negative

momentum returns across uncertainty levels. We observe that, for all uncertainty

proxies, portfolios with more uncertainty have lower returns following past losers

than portfolios with less uncertainty. The results are less unified for the returns

following winners. Portfolios with more uncertainty have higher returns following

winners than portfolios with less uncertainty for 4 proxies except proxies MV and

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16 D. KELSEY ET AL.

Table II. Positive and Negative Momentum Returns and Asymmetric Effect (12 1)

The table reports average monthly portfolio returns sorted by price momentum and uncertainty proxy.

Each month we first sort stocks into five quintiles based on returns from months t 12 to t 1.

For each momentum quintile, we further sort stocks into five groups based on uncertainty proxy.

Firm size (MV) is the market capitalisation (in millions of dollars) at the end of month t. Firm age

(AGE) is the number of years since the firm was first covered by CRSP. Analyst coverage (COV) is

the number of analysts following the firm in the previous year. Earnings forecast dispersion (EFD)

is the standard deviation of analyst forecasts in month t scaled by the prior year-end stock price.

Stock volatility (SIGMA) is the standard deviation of weekly market excess returns over the year

ending at the end of month t. Cash flow volatility (CVOL) is the standard deviation of cash flow from

operations in the past 5 years. Stocks are held for 1 month, and portfolio returns are equally weighted.

The sample period is from January 1983 to December 2005; t-statistics in parentheses are adjustedfor autocorrelation.

Uncertainty Proxy: MV Uncertainty Proxy: AGE

Neg. Pos. Momentum Asym. Eff Neg. Pos. Momentum Asym. Eff

U1 0.90% 2.03% 1.13% (1.98) 2.92 (5.10) 0.95% 1.63% 0.67% (1.39) 2.58 (5.31)

U2 0.58% 1.82% 1.24% (2.10) 2.40 (4.08) 0.73% 1.93% 1.20% (2.26) 2.65 (4.99)

U3 0.20% 1.79% 1.59% (2.67) 2.00 (3.35) 0.42% 1.89% 1.47% (2.58) 2.31 (4.05)

U4 0.10% 1.69% 1.79% (3.16) 1.60 (2.82) 0.00% 1.94% 1.94% (3.24) 1.94 (3.24)

U5 0.54% 1.46% 2.00% (3.79) 0.92 (1.75) 0.67% 1.75% 2.41% (3.84) 1.08 (1.72)

Uncertainty Proxy: SIGMA Uncertainty Proxy: CVOL

Neg. Pos. Momentum Asym. Efft Neg. Pos. Momentum Asym. Eff

U1 0.71% 1.73% 1.02% (2.79) 2.44 (6.68) 0.73% 1.55% 0.83% (1.74) 2.28 (4.82)

U2 0.59% 1.73% 1.14% (2.40) 2.33 (4.88) 0.68% 1.76% 1.08% (2.04) 2.45 (4.64)

U3 0.30% 1.93% 1.63% (2.85) 2.24 (3.91) 0.35% 1.87% 1.52% (2.59) 2.23 (3.80)

U4 0.08% 1.84% 1.93% (2.97) 1.76 (2.71) 0.08% 1.91% 1.83% (2.71) 1.99 (2.94)

U5 0.50% 1.55% 2.06% (2.64) 1.05 (1.35) 0.40% 1.80% 2.20% (2.95) 1.40 (1.88)

Uncertainty Proxy: COV Uncertainty Proxy: EFD

Neg. Pos. Momentum Asym. Eff Neg Pos. Momentum Asym. Eff

U1 0.56% 1.80% 1.24% (1.98) 2.36 (3.77) 0.62% 2.35% 1.73% (2.80) 2.96 (4.79)

U2 0.68% 1.86% 1.18% (1.91) 2.54 (4.10) 0.54% 1.96% 1.42% (2.33) 2.51 (4.10)

U3 0.37% 2.01% 1.64% (2.70) 2.39 (3.92) 0.35% 1.72% 1.37% (2.22) 2.07 (3.36)U4 0.03% 2.08% 2.06% (3.56) 2.11 (3.64) 0.17% 1.65% 1.48% (2.29) 1.83 (2.82)

U5 0.26% 1.85% 2.12% (3.83) 1.59 (2.88) 0.44% 1.59% 2.02% (3.11) 1.15 (1.77)

EFD. The magnitude of returns following negative momentum is greater compared

to returns following positive momentum. As shown in Table II, taking firm age

proxy for example, following negative momentum, the return of the U5 portfolio

is 1.62% lower than the return of the U1 portfolio; following positive momen-

tum, the return of the U5 portfolio is only 0.12% higher than the return of the U1

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Figure 2. Positive and Negative Momentum Returns Under Uncertainty (12 1).The six panels show the post-formation 1-month returns across 5 uncertainty levels measured re-spectively by six proxies. The black lines in each panel represent post-formation 1-month returnsfollowing winner (upper quintile) pre-formation returns, which have mostly flat slopes. The grey linesrepresent post-formation 1-month returns following loser (lower quintile) pre-formation returns. Thegrey lines decline obviously across the uncertainty axis for all uncertainty proxies. This suggestsa greater effect of negative momentum compared to positive momentum for portfolios with moreuncertainty. The sample period is from January 1983 to December 2005.

portfolio. Furthermore, we compare the divergence of the returns following nega-

tive and positive momentum. Using the return of the U3 portfolio as a benchmark,

we can see that the portfolio returns under negative momentum for each uncer-

tainty level diverge more than under positive momentum. For example, as shown

in Table II, for MV uncertainty proxy, the return of the U3 portfolio is 0.7%

lower than the return of the U1 portfolio under negative momentum; under posi-

tive momentum, the difference is only 0.24%. This observation is mostly true for

each uncertainty level and for all proxies. We obtain similar results for portfolios

using 6 6 trading strategies (see Table III). Figure 2 illustrates the asymmetric

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18 D. KELSEY ET AL.

Table III. Positive and Negative Momentum Returns and Asymmetric Effect (6 6)

The table reports average annual portfolio returns from month t+ 1 to t+ 6 sorted by price momen-

tum and uncertainty proxy. Each month we first sort stocks into five quintiles based on returns from

months t 6 to t 1. For each momentum quintile, we further sort stocks into five groups based on

uncertainty proxy. Firm size (MV) is the market capitalisation (in millions of dollars) at the end of

month t. Firm age (AGE) is the number of years since the firm was first covered by CRSP. Analyst

coverage (COV) is the number of analysts following the firm in the previous year. Earnings forecast

dispersion (EFD) is the standard deviation of analyst forecasts in month tscaled by the prior year-end

stock price. Stock volatility (SIGMA) is the standard deviation of weekly market excess returns over

the year ending at the end of month t. Cash flow volatility (CVOL) is the standard deviation of cash

flow from operations in the past 5 years (with a minimum of 3 years). Stocks are held for 6 months,

and portfolio returns are equally weighted. The sample period is from January 1983 to December2005; t-statistics in parentheses are adjusted for autocorrelation.

Uncertainty Proxy: MV Uncertainty Proxy: AGE

Neg. Pos. Moment. Asym. Eff Neg. Pos. Moment. Asym. Eff

U1 5.96% 12.29% 6.33% (4.81) 18.25 (13.8) 5.34% 8.84% 3.50% (3.15) 14.18 (12.7)

U2 3.67% 10.45% 6.78% (4.99) 14.12 (10.4) 4.21% 10.00% 5.80% (4.63) 14.21 (11.3)

U3 2.15% 9.99% 7.84% (5.59) 12.14 (8.65) 2.72% 10.83% 8.12% (5.98) 13.55 (9.98)

U4 0.73% 9.41% 8.68% (6.02) 10.14 (7.04) 1.15% 10.39% 9.24% (6.04) 11.54 (7.55)

U5 1.01% 7.38% 8.39% (6.26) 6.38 (4.76) 1.96% 9.33% 11.29% (7.09) 7.36 (4.63)

Uncertainty Proxy: SIGMA Uncertainty Proxy: CVOL

Neg. Pos. Moment. Asym. Eff Neg. Pos. Moment. Asym. Eff

U1 5.17% 10.11% 4.94% (5.21) 15.29 (16.1) 4.88% 8.90% 4.02% (3.72) 13.78 (12.7)

U2 4.08% 10.61% 6.53% (5.74) 14.70 (12.9) 4.48% 9.66% 5.18% (4.20) 14.14 (11.4)

U3 2.82% 10.62% 7.80% (5.63) 13.44 (9.70) 2.99% 10.58% 7.60% (5.60) 13.57 (10.0)

U4 0.52% 9.89% 9.37% (5.85) 10.41 (6.50) 1.92% 10.09% 8.17% (5.09) 12.00 (7.48)

U5 0.68% 8.19% 8.87% (4.60) 7.52 (3.90) 1.12% 9.84% 10.96% (5.62) 8.71 (4.47)

Uncertainty Proxy: COV Uncertainty Proxy: EFD

Neg. Pos. Moment. Asym. Eff Neg Pos. Moment. Asym. Eff

U1 4.19% 10.38% 6.19% (4.33) 14.57 (10.1) 5.02% 14.86% 9.84% (6.71) 19.88 (13.5)

U2 3.57% 9.95% 6.38% (4.45) 13.52 (9.43) 3.77% 11.26% 7.49% (5.30) 15.03 (10.6)

U3 2.72% 10.74% 8.02% (5.70) 13.46 (9.56) 3.36% 9.65% 6.29% (4.48) 13.02 (9.26)U4 2.02% 11.08% 9.06% (6.28) 13.10 (9.08) 1.86% 8.47% 6.61% (4.46) 10.33 (6.98)

U5 0.61% 10.03% 9.43% (6.83) 10.64 (7.71) 0.54% 7.61% 7.08% (4.53) 8.15 (5.21)

momentum effects. For most uncertainty proxies, the dark line representing post

formation returns following positive momentum is flatter than the grey line rep-

resenting post formation returns following negative momentum across uncertainty

levels. This implies greater price continuation following negative momentum than

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Figure 3. Asymmetry of the Momentum Returns Under Uncertainty.The figure shows the asymmetric effect of momentum returns, the sum of loser and winner post-formation 1 month returns(leftgraph) and 6 months returns(right graph),acrossuncertainty levels foreach uncertainty proxy. The downward sloped lines suggest that the reaction to negative momentumdominates the reaction to positive momentum for portfolios with higher level of uncertainty. Thesample period is from January 1983 to December 2005.

following positive momentum for stocks with uncertainty. This effectively com-

pletes the test of Hypothesis 1.

In order to test Hypothesis 2, we sum the returns following positive and nega-tive momentum for each uncertainty level. If an increased uncertainty level has no

impact on positive and negative momentum effects on portfolio returns, the sum

of the returns should be similar across U1 to U5. In fact, we find that such a sum

decreases with an increase in uncertainty associated with portfolios. As shown in

Table II, following the two types of momentum, the portfolio with more uncertainty

has lower sum of post returns than the portfolio with less uncertainty for all un-

certainty proxies. For example, for the COV proxy, the sum of U5 portfolio return

is 0.77% lower than the sum of U1 portfolio return. This implies that the effect of

negative momentum dominates the effect of positive momentum more significantly

with increasing uncertainty level associated with stocks. We obtain similar results

for portfolios using 6-6 trading strategies (see Table III). Figure 3 plots the main

results. The downward sloped curves suggest that asymmetric momentum effects

become more profound with increasing uncertainty.

To test Hypothesis 3, we split the sample period into two, crisis period and non-

crisis period. We construct the crisis period as the sum of time length for two well-

known financial crises in the US stock markets, from September 1987 to January

1988 and from January 2000 to January 2003. The rest of the sample period between

1983 to 2005 forms the non-crisis period. We assume that there is higher market

uncertainty concerning stocks in the crisis period than in the non-crisis period.

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20 D. KELSEY ET AL.

The formation of the portfolio and the calculation of portfolio returns for two

trading strategies are the same as before for each subperiod. Since the crisis period

is relatively short, we test for 6 6 months strategy to obtain the results. We found

that the portfolio returns are overall lower in the crisis period under both positive

and negative momentum. Except for proxies MV and COV, positive momentum

disappeared for portfolios with more uncertainty and negative momentum deepened

with the increased uncertainty associated with the portfolio. Using proxy AGE for

example, as shown in Table IV, while U5 portfolio return is 18.7% lower than U1

portfolio following negative momentum during crisis period, U5 portfolio return is

4.4% lower than the U1 portfolio following negative momentum during the non-

crisis period. Following positive momentum, U5 portfolio return is negative and

2.3% lower than the U1 portfolio during the crisis period, U5 portfolio return is

almost 1% higher than the U1 portfolio during the non-crisis period.

Figure 4 illustrates that asymmetric patterns are more significant during the crisis

period compared to the non-crisis period. Except proxies COV and MV, a steeper

downward sloped curve for all other uncertainty proxies across uncertainty level

for the measurement of asymmetry during crisis suggests that asymmetry between

negative and positive momentum is more significant during the crisis compared to

the non-crisis period. This confirms our Hypothesis 3.11

4.2 LIKELIHOOD OF PRICE CONTINUATION

In order to make sure that the results are not driven by extreme outliers, we perform

a robustness check and test the hypothesis in a different way. We look for patterns

by comparing the likelihood of price continuation for each individual stock by

momentum and uncertainty proxy. If negative momentum has stronger effects on

stock returns than positive momentum and asymmetry increases with an increase

in uncertainty, we observe that price continuation is more likely following negative

momentum than following positive momentum, especially for stocks with a high

uncertainty level.

In order to implement the approach, we define negative past average monthly

returns rather than relative past return losers as negative momentum. We construct

our hypothesis on the relationship between past returns (momentum) and post

returns as the following 2 2 contingency table,

Nnn = N(RtJ,t1 < 0, Rt+K < 0), Nnp = N(RtJ,t1 0, Rt+K 0),

Npn = N(RtJ,t1 0, Rt+K 0), Npp = N(RtJ,t1 > 0, Rt+K > 0),

11 We found that both positive and negative momentum returns have downward slopes across uncer-

tainty level during the crisis period. Specifically, the slope for portfolio returns following negative

momentum during crisis period is steeper for all uncertainty proxies compared to the non-crisis

period. This implies a even stronger reaction to negative momentum during the crisis period when

markets general uncertainty level is higher.

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Table IV. Crisis vs. Non-Crisis: Momentum Returns and Asymmetric Effect (6 6)

The table compares momentum returns during the crisis and the non-crisis periods. We define the

crisis periods as combined periods from September 1987 to January 1988 and from January 2000

to January 2003. Non-crisis periods consist of all other complimentary months. For each month we

first sort stocks into five quintiles based on returns from months t 6 to t 1. For each momentum

quintile, we further sort stocks into five groups based on uncertainty proxy. Firm size (MV) is the

market capitalisation (in millions of dollars) at the end of month t. Firm age (AGE) is the number of

years since the firm was first covered by CRSP. Analyst coverage (COV) is the number of analysts

following the firm in the previous year. Earnings forecast dispersion (EFD) is the standard deviation

of analyst forecasts in month t scaled by the prior year-end stock price. Stock volatility (SIGMA)

is the standard deviation of weekly market excess returns over the year ending at the end of month

t. Cash flow volatility (CVOL) is the standard deviation of cash flow from operations in the past 5years (with a minimum of 3 years). Stocks are held for 6 months, and portfolio returns are equally

weighted. The sample period is from January 1983 to December 2005; t-statistics in parentheses are

adjusted for autocorrelation.

Crisis Period

Un ce rt ai nt y Pr ox y: MV Unc er ta int y P roxy : AGE Un ce rt ai nt y Pr ox y: S IGMA

Neg. Pos. Asym. Effect Neg. Pos. Asym. Effect Neg. Pos. Asym. Effect

U1 4.35% 1.44% 5.79 (1.60) 2.89% 0.65% 3.54% (1.14) 5.88% 9.07% 14.95 (5.68)

U2 5.13% 0.52% 4.61 (1.22) 1.40% 2.70% 1.30% (0.39) 1.17% 4.71% 5.88 (1.82)

U3 3.85% 1.60% 2.26 (0.60) 2.53% 3.05% 0.52% (0.15) 3.50% 2.16% 1.34 (0.36)

U4 4.77% 2.92% 1.85 (0.50) 6.05% 0.23% 6.28% (1.54) 9.60% 1.83% 11.4 (2.79)

U5 4.98% 0.97% 4.01 (1.16) 15.8% 1.84% 17.7% (4.14) 15.3% 9.46% 24.8 (4.82)

Un ce rt ain ty P roxy: C VOL Un ce rt ai nt y Pr ox y: COV Un ce rt ai nt y Pr ox y: EF D


U1 0.67% 1.48% 4.44 (1.30) 8.63% 3.61% 12.2% (2.96) 5.91% 2.19% 3.72 (1.00)

U2 1.14% 2.65% 0.71 (0.19) 3.49% 0.03% 3.52% (0.91) 5.00% 0.20% 4.80 (1.30)

U3 0.37% 0.94% 1.24 (0.32) 3.28% 2.52% 0.76% (0.20) 2.70% 0.32% 2.38 (0.63)

U4 0.13% 0.31% 6.40 (1.57) 3.43% 2.91% 0.52% (0.15) 5.76% 2.51% 8.27 (1.98)

U5 2.16% 6.53% 19.6 (4.10) 4.30% 3.50% 0.80% (0.21) 7.27% 3.20% 10.4 (2.37)

Non-Crisis Period

Un ce rt ai nt y Pr ox y: MV Unc er ta int y P roxy : AGE Un ce rt ai nt y Pr ox y: S IGMA

Neg. Pos. Asym.Effect Neg. Pos. Asym. Effect Neg. Pos. Asym. Effect

U1 7.7 1% 14.61% 22.32 (16.8) 5.75% 10.23% 15.98% (13.6) 5.0 5% 1 0.29% 15.34 (15.0)

U2 5.1 6% 12.14% 17.30 (12.3) 5.16% 11.24% 16.40% (12.3) 4.5 8% 1 1.61% 16.19 (13.4)

U3 3.1 6% 11.41% 14.58 (9.82) 3.61% 12.15% 15.76% (10.8) 3.89 % 1 2.05% 15.94 (10.8)

U4 1.6 7% 10.51% 12.18 (7.87) 2.37% 12.19% 14.56% (9.04) 2.24 % 1 1.88% 14.12 (8.38)

U5 0.33% 8.47% 8.14 (5.65) 0.39% 11.22% 11.61% (7.09) 1.8 2% 1 1.18% 13.00 (6.59)

(Continued)

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22 D. KELSEY ET AL.

Table IV. Continued

Unc er ta int y P roxy : CVOL Un ce rt ai nt y Pr ox y: COV U nce rt ain ty P roxy: E FD


U1 5.21% 10.15% 15.36 (13.9) 6.36% 12.75% 19.11% (13.4) 6.97% 17.12% 24.10 (15.9)

U2 5.57% 10.85% 16.42 (12.8) 4.76% 11.64% 16.40% (10.9) 5.34% 13.23% 18.57 (12.6)

U3 3.86% 12.22% 16.08 (11.3) 3.73% 12.13% 15.86% (10.6) 4.44% 11.32% 15.77 (10.6)

U4 3.38% 11.75% 15.13 (8.85) 2.94% 12.46% 15.41% (9.87) 3.22% 10.43% 13.66 (8.93)

U5 0.91% 12.61% 13.52 (6.59) 1.44% 11.14% 12.58% (8.60) 1.93% 9.45% 11.38 (7.01)

Figure 4. Crisis vs. Non-Crisis: Asymmetry of Momentum Returns under Uncertainty (6 6).The figure compares the asymmetric effect of momentum returns during the crisis and the non-crisis

periods. We define crisis periods as combined periods from September 1987 to January 1988 and fromJanuary2000to January2003.Non-crisisperiods consist of allothercomplimentary months. Themea-sure of asymmetry is the sum of loser and winner post-formation 6 month returns across uncertaintylevels for each uncertainty proxy. The downward sloped lines suggest that the reaction to negative mo-mentum dominates the reaction to positive momentum for portfolios with higher level of uncertainty.

where N() stands for the number of stocks satisfying the corresponding condition.

For example, the number of stocks with realisation that post return is negative fol-

lowing negative momentum (negative past return) is denoted by Nnn , the notations

in the other three cells follow similarly.

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We firstly sort stocks into uncertainty quintiles for each uncertainty proxy. In

each uncertainty quintile, we sort stocks into 2 groups by positive and negative

average past J-month returns. We further classify stocks in each group into one of

two categories based on positive and negative post K-month returns. This results

in 20 groups of stocks for each uncertainty proxy. The number of stocks (firms) in

each group is then counted and compared as shown in the contingency table.

We define the ratio of positive momentum, rpp =NppNp

, as the number of firms

which had positive post formation returns following positive pre-formationJ-monthreturn divided by the number of firms which had positive pre-formation returns.

The ratio of negative momentum, rnn =Nnn

Nn, is similarly defined.

Table V presents the main results for 12-1 trading strategy. Firstly, we found thatthe majority have positive returns in the past 12 months. This is consistent with clas-

sic asset pricing theories. However, following positive momentum, the number of

firms having price continuation is decreasing with increased uncertainty across un-

certainty quintile for all proxies. In comparison, following negative mo

Asymmetric Momentum Effect Under Uncertainty

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