This document presents a model of market momentum. The model shows that: [1] Momentum is more pronounced in a confident market where investors incorporate new information into prices more slowly. [2] Only idiosyncratic shocks, not systematic shocks, can produce momentum, as systematic shocks do not affect cross-sectional stock returns. [3] Empirical evidence supports the predictions, finding momentum is greater when volatility (and uncertainty) is lower, and when stocks experience larger idiosyncratic shocks.

1. Market confidence and momentum
Kevin Q. Wang and Jianguo Xu
Abstract
We develop a model in which equity fundamentals are subject to random
shocks. Investors learn about the shocks through noisy information. The model
shows that momentum is more pronounced in a more confident market. We
conduct tests of the prediction and find supportive evidence. Specifically, we find
that market volatility negatively predicts momentum profits. This evidence
supports the prediction since a more volatile market is likely to be less confident.
The model also predicts that idiosyncratic shocks, not systematic shocks, produce
momentum. This is consistent with empirical findings from a number of studies.
1

2. Momentum refers to the phenomenon that an arbitrage portfolio comprised of a long
position on stocks that perform better and a short position on stocks that perform worse during
the past 3-12 months (the ranking period) earns a positive profit during the next 3-12 moths (the
holding period). This phenomenon, first documented by Jagadeesh and Titman (1993), has been
confirmed in many later studies. It has been found that momentum extends to later time periods
(Jagadeesh and Titman (2001)), exists in industry portfolios (Moskowitz and Grinblatt (1999)),
value and size portfolios (Lewellen (2002)), and extends to markets other than the United States
(Rouwenhorst (1998)). It has also been found that equity indices also exhibit momentum
(Richards (1995, 1997), Chan, Hameed, and Tong (2000)). The momentum profit is “abnormal”
because it cannot be explained by known risk factors. Fama and French (1996) find that
momentum is the only anomaly that cannot be explained by their three-factor model. Momentum
is particularly annoying because the finding that historical returns help predict future returns
implies that financial markets is not efficient even in the weak form.
Time-varying return patterns of winners and losers in a momentum strategy are
impressive. Figure 1 shows the performance of a 6-6 momentum strategy for the 24 months after
ranking.1 Figure 2 depicts the momentum strategy in a slightly longer time horizon, including
pre-ranking, ranking, and post-holding periods. Together figures 1 and 2 suggest that the winner
(loser) stocks experience positive (negative) shocks. The price changes in the ranking period are
most impressive, suggesting that most of the shocks are incorporated into prices during the
ranking period. For the 6-6 momentum strategy and for the full sample of 1926-2007, the
ranking period winner-loser return spread is about 84% while the holding period return spread is
about 4%. Thus, about 95% of the price adjustments (run-up or run-down) have occurred during
1
A J-K momentum strategy refers to a strategy that ranks past J month returns and hold the portfolio for K months.
Throughout this paper we focus on a 6-6 strategy unless otherwise stated.
2

3. the ranking period, with about 5% left for the holding period which gives rise to the appearance
of momentum profits. It is clear that in terms of the longer time window, momentum profits over
the holding period are of minor magnitude, relative to the huge winner-loser performance
difference over the ranking period. Nonetheless, monthly profit of about 0.7% is economically
significant and commands an explanation.
In this paper, we develop a model that is motivated by the evidence. We aim to
understand the roles of random shocks and investor learning in generating momentum profits.
We organize the thoughts into a model with two risky assets whose payoffs are subject to
random shocks. The representative investor learns about the shocks via noisy information. The
learning is not immediate due to noises in the information, which leads to gradual adjustment of
asset prices and appearance of underreaction. The model produces predictions about sources of
momentum and variations of momentum profits in different market conditions.
First, only idiosyncratic shocks lead to momentum. Systematic shocks that affect all
stocks do not produce momentum. Intuitively, systematic shocks are shared by all stocks and
thus do not affect cross sectional stock returns beyond risk loadings. Since momentum cannot be
explain by risk factors, it cannot be due to systematic shocks. This prediction is consistent with
earlier findings (e.g., Grundy and Martin (2001)) that momentum profit is stronger when stocks
are sorted on idiosyncratic past returns. Sorting on idiosyncratic returns better captures
idiosyncratic shocks than sorting on gross returns. This prediction is also consistent with the
finding of Hou, Peng, and Xiong (2005) that momentum is more pronounced among high R-
square stocks. Stocks with higher R-squares are likely to have experienced larger idiosyncratic
shocks which generate a larger momentum payoff.
3

4. Second, the model predicts that momentum should be more pronounced when market
confidence is higher. The intuition is that in a more confident market, investors react less to new
information, including information about random shocks. Therefore, shocks are incorporated into
price slower, which implies larger momentum profit. We empirically test this prediction and find
supportive evidence. Specifically, we find that momentum is more pronounced when volatility is
lower. Since lower volatility implies higher market confidence, this evidence supports our
prediction that a more confident market exhibits less momentum.
This finding that market confidence negatively predicts momentum can be related to the
finding of Cooper, Gutierrez and Hammed (2004) that momentum depends on market states.
They find that momentum only exists in “UP” markets. We control for market states and find
that the effect of volatility persists after controlling for market status. The explanation that
Cooper, Gutierrez and Hammed propose for their finding is that investors become more
overconfident after good market performance. Although our evidence does not contradict the
behavioral explanations, we argue that it is not necessary to introduce overconfidence to explain
momentum. After random shocks, especially as large as shown above, it is rational that investors
gradually update their opinions. As shown by Leroy (1973) and Lucas (1978), unforecastability
of asset returns is neither a necessary nor a sufficient condition of economic equilibrium.
We emphasize that the appearance of momentum exists from the stand point of
econometricians who have information that is not available to investors. From the perspective of
real time investors who have to base their decisions on information available to them, they
cannot predict momentum and contrarian in equity prices. Thus, there is no tradable strategy
available for them (Lewellen and Shanken (2002)). Investors who trade on beliefs about
fundamentals cannot exploit momentum. In addition, momentum is by nature a statistical
4

5. strategy which is not available to individual investors who usually only hold a small number of
stocks in their portfolios. The remaining question is whether portfolio managers can benefit from
a momentum strategy. Even for professional traders or money managers, there are at least three
unfavorable features of a momentum strategy: 1) high turnover and transaction costs, 2) negative
skewness, and 3) costly and risky short selling. Furthermore, it is useful to emphasize that
momentum trading is not a risk-free arbitrage opportunity. For example, it is not profitable
during the 1990-1995 and 2001-2003 periods.
Our model differs from existing rational explanations for momentum. Berk, Green and
Naik (1999) propose that momentum arises from the persistence in expected returns. Johnson
(2002) argues that since the growth rate risk carries a positive price, high growth firms tend to
have high expected returns. Sorting on past returns tends to sort firms by recent growth rates.
Momentum arises because winners have higher expected returns than losers. In theory our model
does not deny expected return as an alternative explanation for momentum. However, it is
difficult to attribute the extremely high (low) returns of the winner (loser) portfolio during the
ranking period to expected returns. Even a casual look at Figure 2 suggests that non-expected
shocks are at work. We explore unexpected shocks as a source of momentum in this study.
The model also differs from existing behavioral explanations for momentum. Daniel,
Hirschleifer, and Subrahmanyam (1998) develop a model in which investors are overconfident in
private information and this underreact to public information which produces momentum.
Barberis, Shleifer, and Vishny (1998) allow investors to suffer from the cognitive biases of
representativeness and conservatism. Investors in their model initially underreact and then
overreact when a pattern is observed in data. Hong and Stein (1999) allow investors to focus on a
subset of information. In their model, “newswatchers” focus on private information about future
5

6. fundamentals and ignore price history. “Momentum traders” look at price history only. Both
types of investors in their models have bounded rationality in the sense that they fail to take all
information into consideration. In comparison, we introduce random shocks to asset fundamental.
We assume investors are rational Bayesian learners with limited and imperfect information to
learn the true payoffs.
Our model and evidence is in spirit consistent with the arguments of Chan, Jegadeesh,
and Lakonishok (1996) that is momentum is due to slow travel of information. We contend that
instead of “slow travel of information”, momentum may be explained by “slow adjustment of
opinions”. Although seemingly identical, we contend that slow adjust of opinions is more
appealing because of two reasons. First, slow travel of information only applies to private
information. Public information reaches all information receivers immediately in today’s
financial markets. At the same time, whether momentum is due solely to private information is
unclear. In contrast, slow adjustment of opinions applies to both private and public information.
Second, our model can explain a set of accumulated evidence about momentum as discussed
above. More importantly, our model produces a new prediction that is empirically confirmed.
Therefore, our model is better in the sense of being able to explain more evidence and producing
new empirically testable predictions.
A comment on the difference between opinion and information is at demand. Varian
(1989) vividly asks: when someone conveys a probability belief to another agent, what should
the other agent respond? If he updates his posterior belief just as my probability belief, he has
interpreted my probability belief as information, or credible. If he does not update his posterior
at all, then he has interpreted my belief as opinion, or incredible. Very likely, he will partially
adjust his posterior based on my probability belief. In that case, he interprets my belief as
6

7. partially information and partially opinion. So the critical question is: when one makes a
pronouncement, is he conveying information or just conveying his opinion. Or more precisely,
how much of the pronouncement is information and how much is opinion? We can ask the same
question for any announcement. For any announcement, by a company or by a statistics bureau,
one can ask how much he can trust this announcement and how much he should update his
beliefs. Obviously all announcements, no matter how objective or numerical they seem to be, are
based on one set of methodology. The methodology itself is a set of opinions. Besides, even if
the methodology may seem to be very objective, there are always space for subjective
interpretations and discretionary judgments. At the end, we are reaching the sense that there
exists no “pure” information.
The rest of the paper is structured as follows. In Section I, we construct and solve the
model. Implications for price underreaction and momentum are derived. Testable predictions are
discussed. In Section II, we present empirical findings from our tests. Section III concludes.
I. The Model
A. Model structure
Consider a market with one safe asset and two risky assets. The safe asset pays a fixed
interest income at the end of each period. Investors can buy or sell the safe asset infinitely. For
simplicity, the interest rate is assumed to be zero. The risky assets, A and B, pay stochastic
payoffs, θ = (θ A ,θ B )T .
There are two periods and three times, t=0, 1, 2. At time 0 risk neutral investors enter the
market endowed unit of each of the risky assets. At time 1 one signal is observed for each of
7

8. risky assets about its payoff. Investors trade to reach a new equilibrium. At time 2 the asset is
liquidated after realizing the payoff.
The risky assets are subject to a random shock at time 1, after which the payoff is
x = θ + η , η = (η A ,η B )T . At time 1 a signal about each asset, s = ( s A , s B )T , is observed
s = x+ε , (1)
where ε represents a vector of noises. Plugging x into (1) give
s = θ +η + ε . (2)
Random variables, θ , η , and ε , are independent. The correlation between these two asset
payoffs is ρ = corr (θ A ,θ B ) = corr (η A ,η B ) . The noises are independent. All variables are
⎛µ ⎞
normally distributed: θ ~ N ( µθ , Σθ ) , η ~ N (0, Ση ) , ε ~ N (0, Σ ε ) , where µθ = ⎜ A ⎟ ,
⎜µ ⎟
⎝ B⎠
⎛ σ θ2 ρσ θ2 ⎞ ⎛ σ2 ρσ η2 ⎞ ⎛σ 2 0 ⎞
Σθ = ⎜ ⎟ , Ση = ⎜ η 2 ⎟ , Σε = ⎜ ε ⎟
⎜ ρσ 2 σ θ2 ⎟ ⎜ ρσ σ η2 ⎟ ⎜ 0 σ 2 ⎟ . The precisions of these variables are
⎝ θ ⎠ ⎝ η ⎠ ⎝ ε ⎠
denoted by τ θ = Σθ 1 , τ η = Ση 1 , and τ ε = Σ ε 1 , respectively.
− − −
Remark 1: The model is set up in its simplest form. Investors can buy and sell the safe
asset infinitely thus there is no wealth effect on prices of the risky assets. Two is the minimum
number of risky assets required to study cross sectional variations in risky returns such as
momentum. Two is also the minimum number of periods to study time series price behavior.
One way to interpret the liquidation at time 2 is that investors receive a decisive signal without
noise about the payoff. Extending the model into multiple periods with multiple assets and
assume investors observe a noisy signal each period does not add new insights. Investors are
8

9. assumed to be risk neutral because in this paper we do not consider risk aversion as an
explanation for price momentum or reversal. We do not model asset heterogeneity, therefore we
assume θ A and θ B , η A andη B , ε A and ε B have identical variances, respectively.
Remark 2: The timing of the model is as below. At time 0 the nature makes its first move
to pick values for θ A and θ B . At time 1 the nature makes its second move to pick values for η A ,
η B , ε A , and ε B . Investors know the joint distribution but not the value of these variables.
B. Equilibrium
For normal distribution, the posterior belief after observing the signals is given by the
standard formula (see, e.g., DeGroot (1970)).
E ( x | s ) = (τ x + τ ε ) −1 (τ x µθ + τ ε s )
Plug in (1) and rearrange, we have:
E ( x | s ) = µθ + (τ x + τ ε ) −1τ ε ~ ,
s (3)
where ~ = θ + η + ε − µθ is the information “surprises”.
s
From equation (3) it is clear that the error in initial expectation, θ − µθ , and the shock to
the payoff, η , are equivalent in the belief updating formula. This is not surprising since an error
in initial beliefs constitute a “shock” when revealed. We could have assumed that investor’s
beliefs are rational in the sense that this difference equals zero. This assumption does not change
the model.
Expanding (3) gives:
⎛ E ( x A | s) ⎞ ⎛ µ A ⎞ ⎛ ks A + (1 − k ) sB ⎞
⎜ E ( x | s ) ⎟ = ⎜ µ ⎟ + δ ⎜ (1 − k ) s + ks ⎟ ,
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (4)
⎝ B ⎠ ⎝ B⎠ ⎝ A B⎠
9

10. (1 + ρ )σ x
2
(1 − ρ 2 )σ x + σ ε2
2
where δ = , k= are constants.
(1 + ρ )σ x + σ ε
2 2
(1 − ρ )σ x + σ ε2 + ρσ ε2
2 2
Risk neutrality implies that equilibrium prices equal the expected payoff at times 0 and 1.
⎛ PA ⎞ ⎛ µ A ⎞
0
⎜ 0 ⎟ = ⎜ ⎟. (5)
⎜P ⎟ ⎜µ ⎟
⎝ B ⎠ ⎝ B⎠
⎛ PA ⎞ ⎛ µ A ⎞
1
⎛ ks + (1 − k ) s B ⎞
⎜ 1⎟ = ⎜ ⎟+δ⎜ A⎜ (1 − k ) s + ks ⎟ . (6)
⎜P ⎟ ⎜µ ⎟ ⎟
⎝ B⎠ ⎝ B⎠ ⎝ A B⎠
At time 2, the payoffs are realized and there is no uncertainty,
⎛ PA2 ⎞ ⎛θ A + η A ⎞
⎜ 2⎟=⎜
⎜ P ⎟ ⎜θ + η ⎟ . ⎟ (7)
⎝ B⎠ ⎝ B B⎠
We define return as the dollar price change during a time period, rather than the
percentage price change. This allows us to avoid dealing with the division operation in
calculating percentage returns. Equations (5), (6), and (7) jointly give
⎛ rA ⎞
1
⎛ ks + (1 − k ) sB ⎞
⎜ 1⎟ =δ⎜ A
⎜ (1 − k ) s + ks ⎟ ,
⎟ (8)
⎜r ⎟
⎝ B⎠ ⎝ A B⎠
⎛ rA ⎞ ⎛θ A + η A − µ A ⎞ ⎛ ks A + (1 − k ) sB ⎞
2
⎜ 2⎟=⎜
⎜ r ⎟ ⎜θ + η − µ ⎟ − δ ⎜ (1 − k ) s + ks ⎟ .
⎟ ⎜ ⎟ (9)
⎝ B⎠ ⎝ B B B⎠ ⎝ A B⎠
Notice worthy is that in this model all returns are unexpected. Risk neutrality implies
zero risk premium and thus zero expected returns. Returns are driven by changes in expected
payoff which constitutes surprises to the market.
10

11. C. Price underreaction
Equation (4) describes the revision in expectations as a scaled weighted average of the
information surprises. The weights on the signal of the asset and the other asset are k and 1-k,
(1 − ρ 2 )σ x + σ ε2
2
k= . Because (1 − ρ 2 )σ x + σ ε2 ≥ σ ε2 ≥ ρσ ε2 , with the equalities hold when
2
(1 − ρ )σ x + σ ε + ρσ ε
2 2 2 2
ρ = 1 , we have k ∈ [0.5, 1] . So the weight on an asset’s own signal is at least as large as the
weight on the other asset’s signal. In the special case of perfectly correlated payoffs ( ρ = 1 ), the
weights on both signals are equally split into one half. Another special case is when the assets are
independent ( ρ = 0 ). In this case k=1, which means that the beliefs about the two assets are
updated independently on its own signal without taking into consideration the other signal. In
this case the model degenerates into a single risky asset model. We assume that ρ ∈ [0,1] .
Although it is theoretically possible for ρ to be negative, equity returns are usually positively
correlated. We summarize this discussion in the following observation.
OBSERVATION 1: The weight investors put on asset’s own signal is at least one half. It
decreases with asset correlation.
(1 + ρ )σ x
2
The scaling factor, δ = , is between 0 and 1. The implication is that if the
(1 + ρ )σ x + σ ε2
2
initial expectation does not equal the true payoff at time 1, investors only partially adjust their
expectations after observing the signal. This is a natural result of Bayesian updating. It is also
straightforward that δ increases with ρ and decreases with σ ε2 . The intuition is simple. If the
assets are more correlated, essentially there is less variation in the payoff and the signals provide
11

12. better information about the assets. And when the signals are less noisier, investors put more
weight on the signal and less weight on the prior expectation. In the special case that the signals
are completely revealing ( σ ε2 = 0) , the scaling factor reaches unity.
OBSERVATION 2: If the initial expectations do not equal the true payoff, investors partially
adjust their expectations toward the true payoff. The speed of adjustment increases with asset
correlation and information quality.
An obvious yet important message from equation (4) is that if there is no difference
between investors’ initial expectation and the true payoff, on average investors’ posterior beliefs
will not be biased. That is, erroneous initial expectations or shocks are necessary for price
underreaction in this model. This partial adjustment of expectations toward the true value is
rational in the Bayesian sense. The deviation in posterior expectation from true asset value
comes from the initial error in expectations, which is partially corrected by the signal, or from
the random shock to the fundamental, which investors partially learn from the signal.
It is useful to compare this model to behavioral models. In this model we allow errors in
initial beliefs or random shocks. We assume investors are rational Bayesian learners with limited
and imperfect information to learn the true payoffs. In comparison, Daniel, Hirschleifer, and
Subrahmanyam (1998) allow investors to be overconfident in their own private information and
give themselves too much credit (too less blame) in case of success (failure). Barberis, Shleifer,
and Vishny (1998) allow the representative investor to suffer from the cognitive biases of
representativeness and conservatism. Hong and Stein (1999) allow investors to focus on a subset
12

13. of information. In their model, “newswatchers” focus on private information about future
fundamentals and ignore price history. “Momentum traders” look at price history only. Both
types of investors in their models have bounded rationality in the sense that they fail to take all
information into consideration.
Essentially, we allow investors to have errors in their expectations but do not allow
investors to make mistakes when using the Bayesian method. I also allow investors have random
errors in their expectations. But it is not easy to take advantage of such possible existence of
errors. This is completely consistent with the argument of Grossman and Stiglitz (1980). In the
end, market efficiency is not that price is correct, but no free lunch. Market efficiency does not
preclude profit from unique insight and hard work.
D. Momentum
Price underreaction is not equivalent to momentum. The former concerns the time series
autocorrelation of asset or portfolio returns. The later is a cross sectional phenomenon: past
winners continue to outperform past losers for some period of time. To have momentum, we
introduce cross sectional differences in the assets. Consider a shock to one of the stock, A.
Without loss of generality, assume η A > 0 . The expected shock to stock B is E (η B ) = ρη A .
Without loss of generality, assume that initial expectation is not biased thus the shock is the only
source for different payoff. On average stock A will be the winner and stock B will be the loser
during period 1. A momentum strategy of buying A and selling B earns expected profit of
M = [ E ( PA2 ) − E ( PA )] − [ E ( PB2 ) − E ( PB )] .
1 1
Substitute in equations (9) and organize, we have
13

14. σ ε2
M = (1 − ρ )η A . (10)
(1 − ρ )σ x + σ ε2
2
Equation (10) says that momentum profit is jointly determined by the shock, (1 − ρ )η A ,
σ ε2
and the underreaction to shock, . To understand the first term in equation (10),
(1 − ρ )σ x + σ ε2
2
(1 − ρ )η A , we decompose the total shock η A into ρη A and (1 − ρ )η A . The former represents the
part that is shared by asset B because of the inter-asset correlation, and the latter represents the
part that is idiosyncratic to asset A. From now on we label the former as systematic shock and the
latter idiosyncratic shock. Equation (10) says that only the idiosyncratic shock help generates
momentum. Intuitively, systematic shock affects both assets thus does not help generate
momentum, which is a cross sectional phenomenon. This discussion is supported by the finding
of Grundy and Martin (2001) that sorting on idiosyncratic returns produces larger momentum
profit than sorting on raw returns. This is because idiosyncratic returns better capture
idiosyncratic shocks.
σ ε2
The second term determines the size of momentum for given
(1 − ρ )σ x + σ ε2
2
idiosyncratic shock. More intuition can be obtained by decomposing the variation of asset payoff
σ x2 into ρσ x2 and (1 − ρ )σ x2 . The former captures the variation that is shared by both assets,
while the latter captures the variation that is idiosyncratic to individual assets. Therefore, we can
label ρσ x as the “systematic” variance while (1 − ρ )σ x the “idiosyncratic” variance. The second
2 2
term in equation (10) says that given an idiosyncratic shock, the strength of momentum is
determined by the ratio of the variance of noise to that of noise plus idiosyncratic asset payoff.
14

15. Several observations follow from equations (10), which makes the model subject to
empirical scrutiny. First, expected momentum profit increases with the idiosyncratic shock,
(1 − ρ )η A . If shocks to asset fundamentals can be directly identified, we should expect
momentum to be related to idiosyncratic shocks. However, this might not be possible for most
cases due to 1) shocks may simply be unobservable, and 2) the value implication of shocks may
be difficult to calculate. When shocks cannot be identified, we can infer shocks from price
changes. To the extent that more idiosyncratic shocks lead to more dispersed stock returns, we
expect momentum to be more pronounced when individual stock returns are more dispersed. In
equation (10) dispersion of stock returns is measured by the parameter ρ . It is obvious from
equation (10) that momentum decreases with ρ . This is consistent with the finding of Hou, Peng
and Xiong (2005) that momentum is more pronounced for stocks with low R-square, the returns
of which is more dispersed.
Second, momentum decreases with the variance of asset payoff σ x . Since σ x inversely
2 2
measures the confidence of prior expectations, momentum increases with market confidence.
Intuitively, when investors are more confident in their initial beliefs, they adjust their
expectations less to new information, leaving more space for momentum profit. This argument
share some similarity with the one based on overconfidence. Investors may become too confident
in their initial beliefs if they are overconfident in private information. However, overconfidence
is not necessary for this to happen.
Third, momentum profit decreases with information quality. When information is more
precise, investors learn faster and the shock is incorporated into prices faster. In the extreme case
of perfectly revealing information, σ ε2 = 0 , there will be no momentum. This implication
15

16. predicts that momentum should be more pronounced in a market when information quality is
lower. However, a caution needs to be exerted when drawing this prediction. In such a market
investors also tend to be less confident in their prior expectations because they do not have high
quality information. Since weak prior confidence lead to less momentum, the overall effect is
unclear.
So for a market in which stocks move more synchronously, momentum should be less
pronounced. If a bear market is more synchronous than a boom market, momentum should be
more pronounced in a boom market. If synchronicity displays a U shape with market returns,
momentum profit should display an inverse U shape with market returns. The evidence that
Japan does not have momentum and that in Japan stocks are very synchronous is consistent with
this argument.
II. Empirical evidence
The model produces the novel prediction that momentum should be more pronounced
when the market is more confident. In this section we empirically test this prediction in two steps.
We split the stock market of the United States from 1926 to 2007, for which period we have data,
into 5-year periods. We calculate the volatility and momentum for each subperiods and examine
whether there exists a relationship between volatility and momentum. Essentially, we consider
the subperiods as “markets” and look for volatility-momentum correlation among these
“markets”. Second, for each month, I calculate the 6-month momentum profit and correlated this
profit to the volatility prior to the formation of the momentum portfolio. The model predicts that
higher volatility lead to lower momentum profits.
16

17. A. Data and method
The data for the study are all NYSE and AMEX stocks listed on the CRSP monthly file.
Our sample period covers January 1926 to December 2007. Stocks are sorted at the end of each
month t into deciles based on their prior six month, t-5 to t, returns. The test-period profit is
calculated for t+2 to t+6. Because we need 6 months to calculate past and future returns, our first
momentum portfolio is for June 2006 and last momentum portfolio is for June 2007. We follow
the usual practice to one month between the ranking and holding period. We define each
momentum portfolio as long in the prior six month winners (highest decile) and short in the prior
six-month losers (lowest decile). We exclude stocks with a price at the end of the formation
period below $1 to mitigate microstructure effects associated with low-price stocks.
B. Raw momentum
Table I reported the momentum profit for the whole sample period of 1927-2006. To
compare with other studies, I also report momentum for three subsamples: 1927-1964, 1965-
1989, 1990-2006. The subsamples are selected as before, after, and the same as the 1965-1989
sample in the original Jagadeesh and Titman (1993) study.
For the whole sample period of 1926-2007, the average monthly profit is 0.65% per
month. This profit is highly significant. Momentum is more pronounced during the 1965-1989
period (1% per month) than during the earlier 1926-1964 period (0.43% per month) and the later
1990-2007 period (0.66% per month).
It is notice worthy that a momentum strategy has negative skewness for the whole sample
and all three subsamples. Generally, losing portfolios are more positively skewed while winning
portfolios are more negatively skewed. The momentum portfolio is significantly negatively
17

18. skewed for the whole sample and all three subsample (insert evidence about significance of
skewness). This finding is consistent with the finding of Harvey and Siddique (2000), The
momentum strategy is especially negatively skewed during the early period of 1926-1964. To the
extent that investors like positive skweness and do not like negative skewness for their portfolios,
the negative skewness of the momentum strategy helps explain why momentum profit is not
arbitraged away.
Figure 3 plots the accumulative momentum profit from July 1926. Upward slopes suggest
positive momentum profit while downward slopes suggest negative momentum profit. As can be
seen from this picture, momentum is positive for most of the times. However, momentum is
negative for the periods of 1930-1940, 1991-1994, 2000-2004, etc.
Figure 3 also plots market volatility during the 6-month portfolio ranking periods. The
clear pattern is that when volatility shoots up, momentum profit attenuates or even become
negative. For example, during the 1930s, market volatility is very high and momentum profit is
negative. The same pattern shows up in early 1990s and 2000s. This pattern provides a primitive
support to our prediction that momentum increases with market confidence.
C. Momentum across decade
Table II reports momentum within decades together with average market return and
volatility during each decade. The idea is to consider the market in different decades as different
markets. We aim to identify a correlation between volatility and momentum between these
“different markets”. Because Cooper et al (2004) find that market states help predict momentum,
we also calculate the average market return during each decade. Momentum profit is strong in
1940s, 1950s, 1960s, 1980s, and 1990s. It does not exist during 1930s and 2000s. For 1920s and
18

19. 1970s, momentum profit is significant at the 5% but not the 1% level. Table II also report the
simple average of momentum and non-momentum periods. The cutting is based on both 5% and
1% significance levels. Either case, the evidence suggests that momentum periods have higher
market returns and lower volatility. The former is consistent with Cooper et al (2004). The latter
supports our prediction.
D. Momentum conditional on market volatility
Encouraged by primitive support of our prediction, this section we conduct a more
systematic test of our prediction that market volatility negatively predicts momentum. For each
month, market volatility is calculated using daily returns during the 6-month ranking period. We
rank months on this volatility measure into quintiles. Quintile 1 includes the lowest volatility
months and quintile 5 includes the highest volatility months. Table III reports the average
momentum profits conditional on this ranking. The results are reported for the full sample of
1926-2007 and for the three subsamples of 1926-1964, 1965-1989, and 1990-2007.
The clear pattern is that momentum profit is positive and significant for low volatility
months, quintiles 1, 2, and 3. For quintiles 4 and 5, the momentum profit is insignificant, positive
or negative. The difference between quintiles 1 and 5 is highly significant. In other words,
momentum is positive and significant for 60% of months with lower volatility. Momentum profit
does not exists for the other 40% months with higher volatility. The decreasing of momentum
from low volatility to high volatility months is nearly monotonic. The result is robust across
subsamples.
Cooper et al (2004) finds that market states, defined as aggregate market returns during
the past 36 months, help determine the existence of momentum. They define “UP” markets as
19

20. months for which the past 36 month market return is positive and “DOWN” markets as months
with negative past 36-month market returns. To disentangle the effects of market states and
volatility on momentum, we sort months based on market volatility into low, medium, and high
volatility months. We independently sort months based on markets states into bad, medium and
good. Then we calculate momentum profits for the 9 groups jointly determined by market state-
volatility ranks. Table IV reports the results.
Several observations emerge from Table IV. First, after controlling for market states,
higher volatility still leads to lower momentum profit. Second, in the highest volatility group,
momentum does not exist no matter the market state is bad, medium or good. Third, after
controlling for volatility, better market state does not lead to better momentum profit. In fact, in
low volatility months higher market return leads to lower momentum. In comparison, in high
volatility months higher market returns lead to higher momentum. This reverse of pattern can
possibly due to the possibility that market state is a partial proxy for market confidence. In a
confident market (low volatility), market states does not capture additional variation in
confidence. In contrary, in a unconfident market (high volatility), market states capture further
variation in confidence.
Table IV is not conclusive due to the opposite effect of market state on momentum when
volatility is low versus when volatility is high. Table V conduct a regression analysis of
momentum on market state and volatility. Individually, market state positively predict
momentum while volatility negatively predicts momentum. The former is consistent with the
result of Cooper et al (2004). The latter is consistent with the evidence in Tables II, III and IV.
In the full sample of 1927, market state is not significant after controlling for market volatility.
On the other hand, market volatility continue to be significant after controlling for markets states.
20

21. Regression on the subsamples of 1926-1964, 1965-1989, 1990-2007 suggests that the
insignificance of market state is mainly due to the early sample period of 1926-1964. Recall that
according to Table II, momentum is negative and significant at the 10% significance level.
Overall, Tables IV and V suggests that market volatility is a better proxy for market confidence
than market state.
21

22. III. Concluding Remarks
We develop a model that aims at idiosyncratic shocks and investor learning. We
emphasize that these are two important factors, which jointly contribute to the momentum effect.
On the one hand, there exist random shocks to asset fundamentals. The shocks have an
idiosyncratic component. Alternatively, errors in investors’ expectation serve the same function.
On the other hand, investors learn about fundamentals via noisy signals. The noise prevents the
shocks from being incorporated into price quickly, giving rise to momentum.
The model implies that in the presence of the random shocks and information noise,
investor under-reaction should be observed. This implication is consistent with the documented
evidence on post event price continuation. The model further predicts that such continuation
should be more pronounced when investors are more confident in their prior beliefs. We have
documented some preliminary evidence that is consistent with this prediction.
We plan to consider two issues in future work. First, it is important to have a detailed
analysis of volatility-based forecasts of the momentum payoff. We are in the process of
performing robustness checks and conducting further tests. Second, an important direction is to
consider feasibility to extend beyond the representative agent framework. Intuitively, the
momentum profits should be intimately linked to investor trading, and hence may be associated
with interesting trading patterns. How to introduce heterogeneity among investors into the model,
which can produce trading predictions, is an inviting direction for further research.
22

23. References:
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coverage, and the profitability of momentum strategies, Journal of Finance 55, 265–295.
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losers: Implications for stock market efficiency, Journal of Finance 48, 65–91.
Jegadeesh, Narasimhan, and Sheridan Titman, 2000, Cross-sectional and time-series
determinants of momentum profits, Working paper, University of Illinois.
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24

25. Richards, Anthony J., 1995, Comovements in national stock market returns: Evidence of predict-
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284.
25

26. Table I. Momentum profit
For each month from June 1926 to June 2007, NYSE/AMEX stocks in the CRSP database are ranked into
deciles based on their past 6-month returns, t-5 to t. A momentum strategy of buying winners (decile 10) and
selling losers (decile 1) is formed. Returns within deciles are equal weighted to calculate the portfolio return.
The average equal weighted monthly return for the next 6 month excluding the immediate following month,
t+2 to t+6, is reported. t values are adjusted for autocorrelation using the Newey-West method. Also reported is
the skewness of portfolio returns. The results are reported for the full sample and three subsamples.
Subsamples are selected as before, after, and the same as the 1965-1989 sample in the original study of
Jegadeesh and Titman (1993).
1926-2007 1926-1964
Return T Skewness Return T Skewness
Loser 0.0111 4.06 1.68 0.0130 2.91 1.99
2 0.0116 4.92 1.59 0.0124 3.18 1.83
3 0.0126 5.79 1.48 0.0128 3.57 1.74
4 0.0125 6.10 1.36 0.0125 3.69 1.61
5 0.0129 6.57 1.21 0.0129 3.96 1.43
6 0.0132 7.02 0.99 0.0131 4.20 1.19
7 0.0134 7.26 1.00 0.0133 4.37 1.23
8 0.0140 7.69 0.69 0.0140 4.73 0.91
9 0.0148 7.96 0.64 0.0146 4.91 0.94
Winner 0.0176 8.17 0.56 0.0173 5.10 0.86
W-L 0.0065 4.87 -2.45 0.0043 2.03 -2.95
1965-1989 1990-2007
Return T Skewness Return T Skewness
Loser 0.0075 1.95 0.23 0.0120 3.38 -0.06
2 0.0109 3.30 0.21 0.0108 3.92 -0.32
3 0.0125 4.05 0.08 0.0121 5.03 -0.57
4 0.0129 4.43 0.06 0.0118 5.45 -0.47
5 0.0131 4.78 -0.10 0.0124 6.13 -0.39
6 0.0139 5.21 -0.04 0.0124 6.55 -0.46
7 0.0140 5.33 -0.10 0.0126 6.78 -0.63
8 0.0148 5.49 -0.15 0.0130 7.00 -0.58
9 0.0156 5.47 -0.22 0.0142 7.24 -0.76
Winner 0.0174 5.16 -0.10 0.0186 7.58 -0.45
W-L 0.0100 5.21 -0.65 0.0066 3.08 -0.85
26

27. Table II. Momentum across decades
The 1926-2007 period is split into decades. The decade 1920 includes year 1926 to 1929. The decade 2000
include year 2000-2007. NYSE/AMEX stocks in the CRSP database are ranked into deciles based on their past
6-month returns. A momentum portfolio of buying the winners and selling the losers is formed and held for 6
months skipping the immediate next month. Within each decade the average momentum profit are reported.
The t values are adjusted for autocorrelation using the Newey-West method. Also reported are the average
monthly return and volatility for the value weighted market index. Simple average of momentum profit, t value,
market return, and market volatility across momentum and non-momentum decades based on the 1% and 5%
significance levels are also reported.
Decade Momentum t Market return Volatility
1920 0.0102 2.25 0.0130 0.055
1930 -0.0103 -1.72 0.0050 0.104
1940 0.0078 3.79 0.0087 0.044
1950 0.0093 6.64 0.0146 0.032
1960 0.0118 5.10 0.0073 0.036
1970 0.0064 1.99 0.0062 0.049
1980 0.0122 6.44 0.0139 0.048
1990 0.0098 3.63 0.0142 0.039
2000 0.0022 0.79 0.0033 0.042
Average 0.0066 3.21 0.0096 0.050
Momentum and non-momentum decades: 5% significance
Non-momentum -0.0041 -0.47 0.0042 0.073
Momentum 0.0096 4.26 0.0111 0.043
Momentum and non-momentum decades: 1% significance
Non-momentum 0.0021 0.83 0.0069 0.063
Momentum 0.0102 5.12 0.0117 0.040
27

28. Table III. Momentum on market volatility
For each month during June1926-June 2007, NYSE/AMEX stocks in the CRSP database are ranked into
deciles based on their past 6-month returns. A momentum portfolio of buying the winners and selling the
losers is formed and held for 6 months skipping the immediate next month. For each month we calculate
market volatility as the standard deviation of daily market return during the ranking period. Months are
independently ranked into quintiles based on this volatility measure. The average momentum profit and
autocorrelation adjusted t values within each quintile are reported. The results are reported for the full sample
and three subsamples: 1926-1964, 1965-1989, and 1990-2007.
Volatility Rank Loser Winner W-L T (Loser) T (Winner) T (W-L)
1926-2007
Low 0.0069 0.0189 0.0119 2.02 5.75 7.69
2 0.0091 0.0214 0.0122 3.44 7.96 9.34
3 0.0044 0.0137 0.0093 1.35 4.54 6.69
4 0.0139 0.0174 0.0033 2.65 4.03 1.43
High 0.0210 0.0166 -0.0044 2.48 3.06 -1.01
High-Low 0.0140 -0.0022 -0.0163 2.32 -0.51 -5.00
1926-1964
Low 0.0101 0.0191 0.0090 2.40 5.17 5.74
2 0.0139 0.0237 0.0098 3.60 5.46 4.43
3 0.0079 0.0186 0.0107 2.00 4.64 6.31
4 0.0130 0.0115 -0.0015 1.48 1.67 -0.42
High 0.0204 0.0128 -0.0076 1.36 1.31 -1.02
High-Low 0.01 -0.0063 -0.0166 0.93 -0.84 -2.88
1965-1989
Low -0.0021 0.0168 0.0189 -0.33 2.45 6.44
2 0.0007 0.0171 0.0164 0.16 3.85 7.65
3 -0.0015 0.0084 0.0099 -0.30 1.62 4.65
4 0.0203 0.0243 0.0040 2.74 4.20 1.07
High 0.0200 0.0207 0.0007 2.36 3.20 0.16
High-Low 0.0220 0.004 -0.0182 2.74 0.54 -4.38
1990-2007
Low 0.0122 0.0206 0.0083 3.68 6.25 3.77
2 0.0060 0.0167 0.0107 1.10 3.63 5.68
3 0.0068 0.0182 0.0113 1.31 4.30 3.74
4 0.0085 0.0171 0.0087 0.97 3.03 1.51
High 0.0263 0.0202 -0.0061 3.58 4.83 -1.29
High-Low 0.014 -0.0004 -0.0144 2.15 -0.09 -3.20
28

29. Table IV. Momentum on market states and volatility
For each month during June1926-June 2007, NYSE/AMEX stocks in the CRSP database are ranked into
deciles based on their past 6-month returns. A momentum portfolio of buying the winners and selling the
losers is formed and held for 6 months skipping the immediate next month. For each month we calculate
market volatility as the standard deviation of daily market return during the ranking period. Months are equally
ranked into 3 groups based on this volatility measure. For each month we also calculate market state as the
return on the value weighted market index during the past 36 months following Cooper et al (2004) and
independently rank months into 3 groups based on this market state measure. The average momentum profit
and t values within each market state-volatility group are reported.
Market Volatility
Market State Low Medium High High-Low
Bad 0.0157 0.0085 -0.0101 -0.0258
5.45 3.84 -2.15 6.22
Medium 0.0128 0.0127 0.0037 -0.0091
8.05 8.56 0.80 2.02
Good 0.0084 0.0090 0.0046 -0.0038
4.90 4.66 1.43 1.40
Good-Bad -0.0073 0.0005 0.015
-2.76 0.20 3.49
29

30. Table V. Regression of momentum on market state and volatility
For each month during June1926-June 2007, NYSE/AMEX stocks in the CRSP database are ranked into
deciles based on their past 6-month returns. A momentum portfolio of buying the winners and selling the
losers is formed. For each month we calculate market volatility as the standard deviation of daily market return
during the ranking period. For each month we also calculate market state as the return on the value weighted
market index during the past 36 months following Cooper et al (2004). Momentum profit is regressed on
market state and volatility. t values are adjusted for autocorrelation using the Newey-West method.
1926-2007 1926-1964 1965-1989 1990-2007
Intercept -0.00033 0.02 0.013 0.011 0.017 0.011
-0.09 5.39 3.67 2.07 3.51 1.97
Market State 0.72 0.44 0.34 0.82 0.67
2.38 1.71 0.91 2.69 2.50
Market Volatility -0.34 -0.26 -0.23 -0.40 -0.27
-3.22 -3.29 -2.28 -3.60 -2.45
# of months 984 984 984 468 300 216
Adj. R-square 0.064 0.089 0.107 0.091 0.149 0.121
30

31. Figure 1. Accumulative return of winners and losers post ranking. The upper blue dashed (lower black
solid) line is the accumulative return on the winner (loser) portfolio from month 1 to month 24 after portfolio
ranking. Winners (losers) are NYSE/AMEX stocks ranked into the top (bottom) 10% on the past 6 month
returns. Returns are demeaned by the average return of all NYSE/AMEX stocks.
31

32. Figure 1. Accumulative return of winners and losers around ranking. The upper blue dashed (lower black
solid) line is the accumulative return on the winner (loser) portfolio from month -24 to month 24 around
portfolio ranking. Winners (losers) are NYSE/AMEX stocks ranked into the top (bottom) 10% on the past 6
month returns. Returns are demeaned by the average return of all NYSE/AMEX stocks.
32

33. Figure 3. Accumulative momentum profit and market volatility. The black solid line (left axis) is the
accumulative momentum profit from June 1926 to June 2007. Momentum profit is the return on a portfolio
buying past 6 month winners and selling past 6 month losers and held for 6 month. The dashed blue line (right
axis) is the smoothed market volatility, calculated as the standard deviation of daily returns on a value-
weighted market portfolio during the 6 months ranking period.
33