Active momentum trading versus passive ‘ naive diversification

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Active momentum trading versus passive naive

diversificationANURAG N. BANERJEE

a& CHI-HSIOU D. HUNG

b

aDurham University, Durham Business School , Durham , UK

bSchool of Business, University of Dundee, Division of Accounting & Finance , Dundee ,

UK

To cite this article: ANURAG N. BANERJEE & CHI-HSIOU D. HUNG (2013): Active momentum trading versus passive naivediversification, Quantitative Finance, 13:5, 655-663

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use no information and weight risky assets randomly. Overall,

they find that, over the long run, the rewards and penalties

cancel out and that the momentum strategist is no better than

a simple naive randomizer. One interesting outcome of the

construction of this universe of naive investors is that the me-

dian naive investor follows the naive diversification strategy

or the 1/N allocation rule. Essentially, such a 1/N allocation

rule is a passive strategy that tracks the equally weighted stock

portfolio without using past information.

In this paper, we aim to understand the merits and demerits

of the active momentum trading and this passive 1/N strategy.

For this purpose we examine their raw and risk-adjusted prof-

its and idiosyncratic variances (in other words, the variance

of returns due to stock selection Sharpe (1992)). We further

investigate the relation between these strategies in terms of

their correlation and the dimensions of risk exposures.

We consider a naive diversification strategy (NDS) that is

long in theequally weighted (1/N) portfolioof stocksand short

in therisk-free asset at thebeginning of a period. This creates an

initial zero-net-worth position as in the momentum strategies.The returns on the 1/N portfolio in excess of the risk-free rate,

using the one-month Treasury bill rate as a proxy, are thus the

profits of the zero-net-worth strategies of the NDS. At the end

of the period the NDS re-balances the stock portfolio in order

to remain equally weighted. The NDS then holds the portfolio

for the next period and continues to be financed at the risk-free

rate. This strategy can be implemented practically and tracked

over time.

The momentum strategies (MS) first form portfolios of the

winner (top 10% past returns) and the loser (bottom 10% past

returns) stocks and then hold a long position on the winner

portfolio and a short position on the loser portfolio. The mo-mentum strategies further execute intensive trading in order to

establish overlapping strategy positions.

We use the monthly equity data of the NYSE, AMEX and

NASDAQ over the sample period between 1926 and 2005,

various sub-sample periods and 100 randomly selected 10-

year periods. The set of sample stocks that we use to construct

the equally weighted portfolio is the same as that for con-

structing the momentum strategies. We find that the average

profit of the NDS is around 1% per month, close to that of

the MS, in our sample period and in the sub-sample periods

ofJegadeesh and Titman (1993, 2001). The MS generates the

same level of profits as the NDS during the period of bull

markets such as the dot-com bubble and makes profits fromselling short the losers in the great depression period. Further,

we divide the sample into two groups of large and small size

stocks for the various sub-sample periods and find that, in each

size group, the difference between the average profits of the

NDS and the MS is close to zero and statistically insignificant.

Moreover, the variance of the idiosyncratic component of

the momentum profits is 20 times higher than the NDS profits.

Importantly, the risk-adjusted alpha of the difference in profits

between the momentum and the NDS is virtually zero and

statistically insignificant. The NDS and the momentum strate-

gies are orthogonal, i.e. their profits have zero correlation. The

momentum profitsshowa positive marketbeta anda significantloading of nearly 1 on the momentum factor. In contrast, the

NDS has a significantly positive market beta, but does not have

significant exposure on the momentum factor.

On a risk-adjusted basis,the winners outperform theNDS by

14 basis points per month, while the losers under-perform the

NDS by 18 basis points per month. The results, taken together,

suggest that asset managers might be able to earn higher risk-

adjusted excess returns than the NDS by either taking a long

position in the winner portfolio, or by taking a short position in

theloser portfolio. TheMS, however,are notbetter off pursuing

the combined longshort trading because a simple strategy

that equally weights the feasible set of stocks can achieve the

same level of average profit, either raw or risk-adjusted, as the

momentum strategies, but with lower idiosyncratic variance.

Our findings are robust with respect to sampling and period-

specific effects as we simulate the distributions of the aver-

age returns of the winner and the loser portfolios as well as

the momentum profits by re-sampling with replacements. In

each of the 100 randomly chosen 10-year periods we use all

stocks feasible for trading in the sample during that period,

and then construct the empirical distribution of the average

profits. At the 5% level, all of the 100 samples accept the

null hypothesis that the momentum profits are equivalent tothe profits of the NDS. We further consider the momentum

strategies based on 12-month formation and 12-month holding

periods as in Jegadeesh and Titman (1993). Overall, our results

still hold, showing that adopting naive 1/N diversification is

a good investment policy relative to the 12-month momentum

strategies.

Our findings are important for practitioners and investors.

The results illustrate that the MS requires the analysis of past

return information and then takes both sides of the extreme

return positions, thereby taking up the risk from the tails of the

return distribution. By doing so, they significantly reduce the

exposure to the market risk but face large dispersions in theprofits.

The NDS, by contrast, simply weights stocks equally and

performs as well as the MS, and even has lower idiosyncratic

variance. From a practical point of view, pursuing the active

momentum strategies would not be beneficial once further tak-

ing into account the costs of the intensive longshort trading,

the risks and regulation restrictions on short sales of stocks.

The rest of the paper is structured as follows. Section 2

defines the naive diversification strategy and the momentum

strategies. In section 3 we examine the profitability of these

strategies over the whole sample period, some interesting sub-

sample periods, and 100 randomly selected 10-year periods.

Section 4 analyses the theoretical and empirical relations be-tween the momentum and the 1/N naive diversification strate-

gies, risk factor exposures and idiosyncratic variance. Section

5 concludes.

2. The naive 1/N diversification strategy and the momen-

tum strategies

We define N stocks that are feasible for trading at a given

month as in Jegadeesh and Titman (1993). The set ofN stocks

we use for forming the NDS strategies is in fact the same

Jones and Lamont (2002), for example, describe the difficultiesof short sales, including the risks, costs, legal and institutionalrestrictions, and the need of sufficient stock supply from investorswho are willing to lend.
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as the set of stocks for forming the momentum strategies.

Specifically, we use all the monthly equity data of the New

York Stock Exchange (NYSE), the American Stock Exchange

(AMEX) and NASDAQ files from the Center for Research in

Security Price (CRSP)for theperiod between January 1926 and

December 2005. We includes all stocks classified as ordinary

common shares (CRSP share codes 10 and 11) with exchange

codes of 1, 2 and 3 as of the end of the previous year. We thus

exclude all non-common equities such as American deposit

receipts, companies incorporated outside of the U.S., shares of

beneficial interests, certificates, real estate investment trusts,

close-end funds, etc. A stock must meet the following criteria

in order to be included for analysis: first, a stock must have

observations on returns for the current month and over the past

6 or 12 months (depending on the respective analysis period),

stock price, and shares outstanding. Second, a stock must have

a price equal to or higher than $5 at portfolio formation.

Our tests focus on the representative overlapping momen-

tum strategies that form portfolios by sorting stocks on their

past 6-month compounded returns and hold portfolios for6 months. At the end of each month, the stocks within the

top 10% of past returns comprise the winner portfolio FW,

and stocks within the bottom 10% of past returns comprise the

loser portfolio FL. Portfolios are equally weighted at formation

and are held for six months without re-balancing during the

holding period. The momentum strategy is then defined as

FP = (FW FL).Denote r = [r1, r2, . . . , rN] as the vector of returns on the

feasible stocks in excess of the risk-free rate rf. We compute

monthly excess portfolio returns (rW = rFW and rL = rFL)and the profits, rP = rFP, for the momentum strategies using

single-period returns as in Liu and Strong (2008). The momen-tum strategies have six overlapping strategy positions, each

starting one month apart. The monthly portfolio returns from

the overlapping strategies are averages of the six strategies as

in Jegadeesh and Titman (1993).

We construct the 1/N naive diversification strategy over

these N feasible stocks. The excess portfolio returns from the

strategy,

1

N1 =

1

N,

1

N, . . . ,

1

N

,

are then given by r = (1/N)Ni=1 ri., where r is effectivelythe profit of the 1/N naive diversification strategy.

Banerjee and Hung (2011) demonstrate that this strategy doesnot involve information costs and is the cross-sectional mean

and the projection median of their uniformly distributed

random strategies.

3. Profitability of the strategies

The top panel of figure 1 plots the time-series of profit

differences between the momentum and the NDS strategies.

The profit differences mainly center around zero and exhibit

sporadic deviations from zero most of the time. During the

In the case where a stock is delisted during the holding period, theliquidating proceeds are reinvested in the remaining stocks in theportfolio.

periods from 1929 through 1933 and in the late 1990s, how-

ever, there exist large positive and negative deviations. These

periods correspond, respectively, to the great depression and

the dot-com bubble. We thus carefully examine these sub-

sample periods.

Panel A of table 1 presents the average excess returns of

the winner and the loser portfolios, the average momentum

profits as well as the average differences in excess returns

between these portfolios and the NDS. The winner and the

loser portfolios have average excess returns of 1.78% and

0.70% per month, respectively. The momentum strategies gen-

erate a statistically significant average profit of 1.08% per

month. Strikingly, the average profit of the NDS is as high as

1.06% permonthand is highlysignificant.The winnerportfolio

outperforms the NDS by a statistically significant 0.72% per

month, while the loser portfolio under-performs the NDS by a

statistically significant 0.36% per month. Importantly, the av-

erage difference between the profits of the momentum and the

zero-net-worth NDS strategies is virtually zero and statistically

insignificant.We next look into the question of whether size matters in

our analysis. For this purpose, we divide the sample, each

month, into two groups of large and small size stocks using

the median of the market value of all sample stocks at the

end of the previous month as the cut-off point. We then form

the momentum and the zero-net-worth NDS strategies within

each of the groups. Panel A of table 2 shows that, in each

size group, the difference between the average profits of the

momentum and the NDS strategies is close to zero and statisti-

cally insignificant. Within the sample of small size stocks, the

average monthly momentum profit is five basis points lower

(but statistically insignificant) than that of the NDS; within thesample of large size stocks, the average momentum profit is 11

basis points higher (but statistically insignificant) than that of

the NDS.

3.1. Over sub-sample periods

Next, we examine the profitability of the momentum and the

1/N naive diversification strategies over the sample periods

ofJegadeesh and Titman (1993, 2001) from 1965 to 1989 and

from 1990 to 1998, respectively. Panel B of table 1 shows that,

over the former sub-period, the momentum strategies generate

a statistically significant average profit of around 1.1% per

month.The NDS hasa statistically significant average monthlyprofit of 0.9%. The average difference in profits between the

momentum and the NDS strategies is 0.18%, but statistically

insignificant. Compared with the NDS, the winner portfolio

significantly over-performs by 0.72%, while the loser portfolio

significantly under-performs by 0.36% per month. The results

in panel C of table 1 for the latter subperiod are similar. Again,

there is virtually no difference in profits between the momen-

tum and the NDS strategies. The middle panels of figure 1 plot

the differences in profits between the momentum and the NDS

strategies over the sample periods of Jegadeesh and Titman

(1993, 2001). The profit differences mainly center around zero

and only exhibit sporadic deviations from zero.As discussed above, we observe large profit differences be-

tween the momentum and the NDS strategies in the early and

later periods of our sample. We therefore further examine the
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Figure 1. Distributions of the differences in profits between the momentum strategy and the NDS for the various sample periods.

Table 1. Profitability of the six-month momentum and the 1/N naive diversification strategies. This table presents mean excess returns ofthe momentum strategies and the 1/N naive diversification strategies (NDS) for the whole sample period (panel A), the sub-sample periodsof Jagadeesh and Titman (1993, 2001) (panels B and C), the periods during the great depression (panel D) and the dot-com bubble (panelE). We use the one-month T-bill rate as a proxy for the risk-free rate. Winner and Loser are, respectively, the returns on the winner andthe loser portfolios in excess of the one-month T-bill rate. WinnerNDS and LoserNDS are, respectively, the return differences betweenWinner, Loser and the NDS. Momentum profit-NDSis the difference between the profits of the momentum and the NDS. t-Statistics are

in parentheses. , and denote statistical significance at the 10%, 5% and 1% level, respectively.

Winner Loser Momentum profits NDS WinnerNDS LoserNDS Momentum profit- NDS

Panel A: The whole sample: 01-1927 to 07-2005Mean return (%) 1.78 0.70 1.08 1.06 0.72 0.36 0.02

(6.97) (2.4) (5.86) (4.96) (6.10) (2.99) (0.06)Panel B: JagadeeshTitman (1993): 01-1965 to 12-1989Mean return (%) 1.62 0.54 1.13 0.90 0.72 0.36 0.18

(3.90) (1.33) (4.68) (2.64) (4.40) (2.87) (0.43)Panel C: JagadeeshTitman (2001): 01-1990 to 12-1998Mean return (%) 2.22 0.79 1.60 1.34 0.88 0.54 0.09

(3.41) (1.32) (4.36) (3.02) (3.21) (2.17) (0.18)Panel D: Great depression: 08-1929 to 03-1933Mean return (%) 1.69 2.41 0.72 2.02 0.33 0.39 2.74

(1.02) (0.76) (0.34) (0.98) (0.42) (0.27) (0.72)Panel E: Dot-com bubble: 01-1995 to 02-2000Mean return (%) 3.96 1.11 2.86 1.59 2.37

0.48 1.27

(3.13) (1.21) (3.37) (2.49) (2.81) (-1.27) (1.31)

sub-sample periods between August 1929 and March 1933

during the great depression as designated by the NBER and

between January 1995 andMarch2000 for thedot-com bubble.

Interestingly, we find that during the great depression period,

asshown inpanel D of table 1, thewinner portfolio loses 1.69%

per month, but the loser portfolio incurs an even larger aver-

age monthly loss of2.41%. The average momentum profitof 0.72%, although statistically insignificant, mainly comes

from the short position on the loser portfolio. The NDS also

suffers an average monthly loss of

2.02%. The average of the

excess returns on the winner portfolio is only 33 basis points

higher than the average profit of the NDS, but statistically

insignificant. The average difference in profits between the

momentum and the NDS strategies, although as high as 2.74%,

is statistically insignificant because of the greater volatility in

this period. The right-hand panel of the lower part of figure 1

shows that the profit differences over this period display a few

very large negative outliers.

Panel E of table 1 shows that, during the dot-com bubble

period, the winner portfolio has an average monthly excess

return of 3.96%, which is 2.37% higher than that of the NDS.

The average excess return on the loser portfolio is 48 basis

points lower than the NDS, albeit statistically insignificant.

The average difference between the profits of the momentum

and the NDS strategies is 1.27%, but statistically insignificant.

The left-hand panel of the lower part of figure 1 shows that the


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660 Feature

where t = [1t, . . . , it, . . . , Ntt], Bt = [1ti t, . . . , N t]is the vector of factor loadings, xt are common risk factors,2t is the idiosyncratic variance at time t and Nt is the total

number of assets at time t.

The idiosyncratic variance of NDS is simply Var(rt | xt) =2t (1/Nt). Also note that Cov(rj t, r

1(1/Nt) | xt) = 2t Fj t1(1/N

t)

=2

t(1/N

t),

j= W,L. Therefore, it is easy to see thatthe conditional mean of the excess returns on the winner and

loser portfolios can be given by

E(rjt | xt, rt) = (t+Btxt)Fjt +

rt (t + Btxt)11

Nt

.

Since Fji t = 10/Nt, if the i th asset is in the winner portfolio,we obtain the idiosyncratic variance as

Var(rjt | x, r) = 2Fj tFj t = 2t10

Nt.

Therefore, it immediately follows that the conditional mean

and idiosyncratic variance of the momentum profits are

E[rPt (t + Btx)FP | xt, rt] = 0, (2)Var[rPt | xt] = 2t

20

Nt. (3)

The above results suggest the theoretical relations that: (1) the

momentum strategies and NDS are orthogonal to each other,

and (2) the idiosyncratic variance of profits is 20 times higher

than the idiosyncratic variance of the NDS. We test these pre-

dictions below.

4.1. Orthogonality

Under the assumptions made for the factor model (1) and

from (2), the profits of the momentum strategies are orthogonal

to the profits of the 1/N naive diversification strategies. We

examine this prediction by plotting the excess returns on the

winner portfolio, the loser portfolio and the momentum profits

against the 1/N naive diversification strategies (r) in figure 3.

The scatter plot shows that the slope coefficients for both rWand rL seem to be close to one and that the momentum profits

are randomly scattered against r.

Formally, we run a weighted (by

2t /Nt ) linear regression(WLS) of the momentum profits on the profits of the NDS.

Specifically, we examine the following time-series model to

test the null H0 : P = 0:rj t = j + j rt + u jt, t = 1, . . . , T, j {W,L,P},

Var(ujt) 2t

Nt,

where 2t is the variance oft in (1), and Nt is the total number

of assets at time t.

The WLS results show a P of0.04 (t = 0.43), which is inline with thepredictionsof(2) andsuggeststhat themomentum

profits are orthogonal to the profits of the NDS. We also test

whether the sensitivities of the winner and the loser portfolios

on the NDS are statistically different from one. We run WLS

regressions of excessportfolio returns on theprofits of theNDS

to test the null of H0 : W = L = 1. The results show thatW = 0.85 and L = 0.81 are both smaller than one witht = 1.91 and t = 2.52, respectively.

Figure 2. Distributions of the average profits of the NDS andthe excess returns of the winner and the loser portfolios and themomentum profits over 100 random sample periods. The differencein profits between the momentum strategy and the NDS is denoted byprofitNDS. Foreach runof thesimulation, we select an experimentalperiod of 120 months with the starting month randomly chosenbetween July 1926 and December 1995.

Figure3. Scatterplots of the excessreturns on thewinner (blue line)and the loser (green line) portfolios and the momentum profits (redline) versus the profits of the NDS.

4.2. Risk-adjusted profits and risk exposure

We examine the risk-adjusted profits and risk factor exposuresof the momentum and the 1/N naive diversification strategies.

Specifically, we run the time-series regression

rj t =j + jMKTt + sj SMBt + hjHMLt+ m jMOMt + vjt, j {W,L,P,NDS}, (4)

where rj t is the excess portfolio return or profit for strategy j

at time t, and MKT is the return on the CRSP value-weighted

market index at time t in excess of the one-month T-bill rate.

The Fama and French (1993) size factor, SMB, is defined as

the monthly return difference between two portfolios that con-

sist of large and small stocks. The FamaFrench value factor,

HML, is defined as the monthly return difference betweentwo portfolios with high and low book-to-market equity ratio.

The momentum factor,MOM, is the monthly return difference

between two portfolios with high and low returns over the past


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Table 3. Risk exposures of the six-month momentum and the 1/N naive diversification strategies. Panel A reports the factor loadings of theexcess returns on the winner (Winner) and the loser (Loser) portfolios, the momentum profits and the NDS for the period from January1927 to December 2005. Momentum profit is for the strategies that are long in the winner and short in the loser portfolios. Panel B reportsthe factor loadings of WinnerNDS, LoserNDS and Momentum profit-NDS, which are the differences between the excess returns of thewinner and loser portfolios, the momentum profits and the profits of the NDS, respectively. MKTis the return on the CRSP value-weightedmarket index in excess of the one-month T-bill rate. SMB is the monthly return difference between two portfolios that consist of large andsmall size stocks. HML is the monthly return difference between two portfolios with high and low book-to-market equity ratios. MOM isthe monthly return difference between two portfolios with high and low returns over the past 2 to 12 months. The t-statistics in parentheses

are calculated by applying the Newey-West heteroskedasticity-and-autocorrelation-consistent standard errors. , and denote statisticalsignificance at the 10level, respectively.

Constant MKT SMB HML MOM

Panel A: Excess returns of the momentum strategy and the NDSWinner 0.0039 1.2230 0.8001 0.0484 0.5495

(4.85) (44.14) (10.70) (1.18) (15.17)Loser 0.0007 1.1580 0.7992 0.0635 0.4471

(0.74) (35.89) (16.85) (1.29) (12.45)Momentum profit 0.0032 0.0649 0.0010 0.1119 0.9966

(2.49) (1.72) (0.01) (1.64) (21.51)NDS 0.0025 1.0502 0.4251 0.0888 0.0186

(7.10) (96.80) (14.50) (4.48) (0.88)Panel B: Return differences between the momentum strategy and the NDS

WinnerNDS 0.0014 0.1728 0.3751 0.1372 0.5680(1.72) (6.66) (4.67) (2.93) (13.46)

LoserNDS 0.0018 0.1079 0.3741 0.0253 0.4286(1.93) (3.20) (6.59) (0.45) (11.51)

Momentum profitNDS 0.0007 0.9852 0.4241 0.2007 1.0151(0.50) (26.85) (3.93) (2.89) (18.63)

2 to 12 months. The coefficients sj , hj and mj arethe loadings

on the size, value and the momentum factors, respectively.

Table 3 shows the regression results for the entire sample

period. The NeweyWest standard errors are applied to correct

for heteroskedasticity and autocorrelation of residuals. Panel A

shows that both thewinner andloser portfolios aresignificantly

exposed to the market, SMB and momentum factors. The mo-mentum strategies have an average risk-adjusted profit of 0.3%

per month and are marginally exposed to the market factor and

with a highly significant loading of nearly 1 on the momentum

factor, but with insignificant loadings on the SMB and HML

factors. The NDS has a highly significant alpha of 0.25% per

month with statistically significant loadings of 1.05, 0.42 and

0.09, respectively, on the market, SMB and HML factors. The

NDS does not have significant exposure on the MOM factor.

Thus the momentum and the NDS bear different sources of

risk. After accounting for the factor exposures, the magnitude

of the risk-adjusted momentum profit is close to that of the

NDS.

Panel B of table 3 shows that the differences between theexcess returns of the winner portfolio and the profits of the

NDS have a positive and marginally significant alpha with

positive and highly significant exposures to the market, SMB

andMOM, but witha negativeexposureto theHML factor. The

differences between the excess returns of the loser portfolio

and the profits of the NDS have a negative and marginally

significant alpha with positive and highly significantexposures

to the market and the SMB factor, but with a negative and

highly significant exposure to the MOM factor. Importantly,

the difference in profits between the momentum and the NDS

has a very small and statistically insignificant alpha of seven

basis points per month.

The returns on these factors are obtained from Ken Frenchs datalibrary.

4.3. Idiosyncratic variance ratio

We test the null hypothesis that the ratio of idiosyncratic vari-

ance (CVR = Var[rPt|xt]/Var(rt|xt)) between the momentumprofits and the NDS profits are equal to 20 as is the prediction

of (3). Specifically, we test the null hypothesis:

H0 : Var[rPt | xt]Var(rt | xt)

= 20.

Since the idiosyncratic variance (2t ) and the number of assets

(Nt) vary over time, we cannot usethe traditional variance ratio

test. We test this hypothesis by constructing a moment-based

estimator for the CVR,

CVR = 1T

Tt=1

CVRt, where CVRt =v2Ptv2NDSt ,

where

vPt and

vNDSt are the residuals from the regressions

in (4). Note that

E(CVRt) = Ev2Ptv2NDSt =

Ev2PtEv2NDSt =

Var[rPt | xt]Var(rt | xt)

.

The second equality follows from Heijmans (1999). Finally,

using (3) we obtain the expected variance ratio. Therefore,

under the null hypothesis by the central limit theorem, the

standardized statistic is

ZCVR =

T(CVR E(CVR))

stdev(CVR)

D N(0, 1). (5)

Under the null hypothesis E(CVR) = 20. We obtain a calcu-latedZCVR of 1.005 over the whole sample period, and, hence,accept the hypothesis that the idiosyncratic variance of the

momentum portfolio is 20 times higher than the idiosyncratic

variance of the NDS.
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Table 4. Profitability of the 12-month momentum and the 1/N naive diversification strategies. This table presents mean excess returns ofthe momentum strategies (12 month by 12 month) and the 1/N naive diversification strategies (NDS) for the whole sample period (panelA), the sub-sample periods of JagadeeshTitman (1993, 2001) (panels B and C), the periods during the great depression (panel D) and thedot-com bubble (panel E). We use the one-month T-bill rate as a proxy for the risk-free rate. Winner and Loser are, respectively, the returnson the winner and the loser portfolios in excess of the one-month T-bill rate. WinnerNDS and LoserNDS are, respectively, the returndifferences between Winner, Loser and the NDS. Momentum profitNDS is the difference between the profits of the momentum and the

NDS. t-Statistics are in parentheses. , and denote statistical significance at the 10%, 5% and 1% level, respectively.

Winner Loser Momentum profits NDS WinnerNDS LoserNDS Momentum profitsNDS

Panel A: The whole sample: 01-1928 to 01-2005Mean return (%) 1.38 1.11 0.28 1.04 0.35 0.07 0.76

(5.29) (3.76) (1.55) (4.79) (2.70) (0.56) (2.42)Panel B: JagadeeshTitman (1993): 01-1965 to 12-1989Mean return (%) 1.28 0.64 0.63 0.83 0.45 0.19 0.20

(2.75) (1.50) (2.45) (2.33) (2.22) (1.34) (0.46)Panel C: JagadeeshTitman (2001): 01-1990 to 12-1998Mean return (%) 2.36 1.96 0.40 1.71 0.66 0.26 1.31

(3.14) (3.42) (0.96) (4.02) (1.70) (1.07) (2.85)Panel D: Great depression: 08-1929 to 03-1933Mean return (%) 1.40 1.28 0.12 1.12 0.28 0.17 -1.00

(1.15) (1.09) (0.21) (1.34) (0.53) (0.33) (1.04)Panel E: Dot-com bubble: 01-1995 to 02-2000

Mean return (%) 3.93 5.08 1.15 5.11 1.19 0.04 6.26(1.03) (0.72) (0.33) (0.90) (0.60) (0.02) (0.70)

Table 5. Risk exposures of the 12-month momentum and the 1/N naive diversification strategies. Panel A reports the factor loadings of theexcess returns on the (12 month by 12 month) winner (Winner) and the (12 month by 12 month) loser (Loser) portfolios, the momentumprofits and the NDS for the period from January 1927 to December 2005. Momentum profitis for the (12 month by 12 month) strategies thatare long in the winner and short in the loser portfolios. Panel B reports the factor loadings of Winner-NDS, Loser-NDS and Momentumprofit-NDS, which are the differences between the excess returns of the winner and loser portfolios, the momentum profits and the profitsof the NDS, respectively. M K T is the return on the CRSP value-weighted market index in excess of the one-month T-bill rate. S M B is themonthly return difference between two portfolios that consist of large and small size stocks. H M L is the monthly return difference betweentwo portfolios with high and low book-to-market equity ratios. M O M is the monthly return difference between two portfolios with high andlow returns over the past 2 to 12 months. The t-statistics in parentheses are calculated by applying the NeweyWest heteroskedasticity-and-

autocorrelation-consistent standard errors. , and denote statistical significance at the 10level, respectively.

Constant M K T S M B H M L M O M

Panel A: Excess returns of the momentum strategy and the NDS Winner0.0010 1.2493 0.7377 0.1945 0.5280(1.07) (42.23) (9.04) (3.43) (12.28)

Loser 0.0013 1.1573 0.9209 0.2894 0.1302(1.01) (27.14) (14.19) (4.69) (2.43)

Momentum profit 0.0002 0.0920 0.1832 0.4839 0.6582(0.14) (2.26) (1.56) (5.59) (11.13)

NDS 0.0025 1.0498 0.4251 0.0894 0.0183(7.03) (95.76) (14.37) (4.46) (0.86)

Panel B: Return differences between the momentum strategy and the NDSWinnerNDS 0.0015 0.1994 0.3126 0.2839 0.5463

(1.55) (6.42) (4.08) (4.41) (11.47)LoserNDS

0.0013 0.1075 0.4958 0.2000

0.1119

(1.03) (2.68) (6.02) (3.33) (2.32)Momentum profitNDS 0.0028 0.9579 0.6084 0.5734 0.6765

(1.68) (21.03) (5.71) (6.04) (9.64)

4.4. Twelve-month momentum

For a robustness check, we further consider the momentum

strategies based on 12-month formation and 12-month holding

periods as in Jegadeesh and Titman (1993). Using exactly the

same criteria for theselection of thestockset, thestocks that are

feasible for trading exclude those whose prices are below $5

at portfolio formation. Consistent with Jegadeesh and Titman

(1993), panel B of table 4 reports the average (12-by-12) mo-mentum profit as 0.63% per month and statistically significant.

Panel A for the entire sample period shows that the winner

and the loser portfolios have average excess returns of 1.38%

and 1.11% per month, respectively. The average momentum

profit is 0.28% per month, but statistically insignificant. The

average profit of the NDS remains statistically significant at

1.04% per month. The momentum strategies under-perform

the zero-net-worth NDS strategies by 0.76% per month. The

overall pattern of the results presented in table 4 shows that

momentum strategies do not outperform the NDS strategies.Table 5 shows, for the entire sample period, the regression

results and the t-statistics using the NeweyWest standard

errors. In contrast to the results in table 3, panel A of table 5


10/10

Feature 663

shows that theHML factor becomes statistically significant for

the winner portfolio, the loser portfolio and the momentum

profit. Further, the average risk-adjusted return on the winner

portfolio is essentially zero and statistically insignificant, while

the alpha of the NDS remains statistically significant at 0.25%

per month.

Panel B of table 5 shows that the model alpha of the differ-

ence between the excess return on the winner portfolio and the

NDS profit is now negative, reflecting the insignificant model

alpha of the winner portfolio as shown in panel A, while the

pattern of risk exposures is similar to that of the 6-month-by-6-

month momentum. The model alpha of the difference between

the loser portfolio and the NDS profit is close to zero and

statistically insignificant. The model alpha of the difference

in profits between the momentum and the NDS is a negative

28 basis points per month and statistically significant. Overall,

the 12-month momentum strategies under-perform the NDS

strategies.

Overall, our results still hold, and, in fact, the 12-month mo-

mentum strategies under-perform the NDS strategies, showingthat the use of adopting the naive 1/N diversification is a

good investment policy relative to the 12-month momentum

strategies.

5. Conclusions

In this paper we examine the merits and demerits of the active

momentum strategies that use stocks of the top and bottom

10% returns and the passive 1/N naive diversification strategy

which is long in the equally weighted portfolio of stocks and

short in the risk-free asset. The momentum and the NDS strate-

gies areorthogonal.The NDSgenerates an average profit of 1%per month, close to that of the momentum strategies over the

sample periodbetween 1926 and2005, andvarious sub-sample

periods, including those examined by Jegadeesh and Titman

(1993,2001). The evidence shows that the differences in the

average profits between the momentum and the NDS strate-

gies are economically and statistically insignificant. The find-

ings are robust with respect to market capitalization effects as

well as sampling and period-specific effects in our simulations,

where we randomly select 10 years for 100 times. From the

viewpoint of investment, our findings suggest that pursuing

the active momentum trading would not be beneficial relative

to the passive 1/N naive diversification strategy.

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