Timothy Crack Paper

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Price momentum in the New Zealand stock market:a proper accounting for transactions costs and risk*

Sam Tretheweya, Timothy Falcon Crackb

aPricewaterhouseCoopers, Auckland, New ZealandbDepartment of Finance and Quantitative Analysis, University of Otago, Dunedin, New Zealand

Abstract

We test for recently reported momentum profits in New Zealand using a practi-

tioner technique that we have not yet seen in the academic literature. This tech-nique simultaneously weighs returns, risk and transactions costs at each

portfolio rebalance, rather than blindly chasing returns and then accounting for

risk and transactions costs after the fact. We reverse the findings of the earlier lit-

erature because our gross profits are more than fully consumed once transactions

costs are properly accounted for. Although we focus on momentum trading in

New Zealand, our practitioner technique is broadly applicable to investigations

of trading anomalies.

Key words: Price momentum; New Zealand; Price impact; Market efficiency;Equity trading

JEL classification: G11, G14

doi: 10.1111/j.1467-629X.2010.00355.x

1. Introduction

We test for recently reported profits from price momentum trading strategies

in the New Zealand stock market (Gunasekarage and Kot, 2007; Stork, 2008). A

particular strength of the paper is the use of a practitioner technique for optimal

portfolio rebalancing subject to risk and transactions costs; we have not seen this

technique used before in the academic literature. Our simplest unconstrained

* The opinions expressed in this paper are those of the authors and do not necessarily

represent those of PricewaterhouseCoopers. We thank Simon Benninga, Robin Grieves,an anonymous asset manager working for a bulge bracket investment bank and ananonymous referee for helpful comments Any errors are ours

Accounting and Finance 50 (2010) 941965


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momentum portfolio generates gross returns of 185 basis points (bps) per month

over the July 1992 to September 2006 period in line with earlier literature. This

compares very favourably with an NZSE40/NZX50 (i.e. New Zealand Stock

Exchange) benchmark return of only 78 bps per month over the same period.After accounting for transactions costs, risk and other practical considerations,

however, our realized net return falls to only 52 bps per month more than eras-

ing the profits.

The paper proceeds as follows. Section 2 provides a review of selected litera-

ture on price momentum. Section 3 discusses the data and method. Section 4

presents our empirical results. Section 5 concludes.

2. Literature review

2.1. The price momentum anomaly

Levy (1967) concludes that superior profits can be achieved by investing in

securities which have historically been relatively strong in price movement.

Jegadeesh and Titman (1993) demonstrate that strategies that buy past winner

stocks and sell past loser stocks generate significant excess returns. Jegadeesh

and Titman measure past performance over the prior three to 12 months and

allow for subsequent holding periods of three to 12 months. Price momentum of

this form has now been found by researchers in most markets: Rouwenhorst

(1998) reports significant momentum effects in 11 out of 12 European countries

(Sweden is the exception); Leippold and Lohre (2008) report significant momen-

tum effects in the US and 14 out of 16 European countries (Ireland and Austria

are exceptions); Chuiet al. (2000) report significant momentum effects in seven

out of eight Asian countries (Japan is the exception); Hurn and Pavlov (2003),

Demiret al.(2004) and Stork (2008) report momentum profits in Australia; and

Gunasekarage and Kot (2007) and Stork (2008) report momentum profits in

New Zealand.

Fama and French (1996) suggest that momentum profits may be due to

data snooping. Jegadeesh and Titman (2001) respond to this with furtherout-of-sample evidence, dismissing the Fama and French data snooping argu-

ment. Fama and French (1996) argue that, although many of the CAPM anoma-

lies can be explained by their three-factor model, the momentum profits of

Jegadeesh and Titman (1993) are an exception. Fama and French (1996) suggest

that investors underreaction to recent news produces momentum effects, but

their overreaction to less-recent news causes a longer-term reversal. Danielet al.

(1998) suggest that investors are overconfident about their own abilities and the

accuracy of their private information and that this leads them to push up the

prices of past winners and push down the prices of past losers. Barberiset al.(1998) suggest sentiment-driven explanations for underreaction and momentum

fit H d St i (1999) th t t b d d ti l Th

942 S. Trethewey, T. F. Crack/Accounting and Finance 50 (2010) 941965


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information, ignoring the rest. With heterogeneous agents and slow diffusion of

information through the economy, this leads to underreaction in the short term.

Hong et al. (2000) follow up and conclude that this underreaction is especially

noticeable for negative news, and that momentum profits are stronger in smallstocks and stocks with low analyst coverage. Grinblatt and Moskowitz (2004)

find that being a consistent winner can double the subsequent return associated

with being in the top momentum decile. Grinblatt and Han (2005) suggest that

the disposition effect (Shefrin and Statman, 1985) could be driving momentum

profits, because selling winners too soon and delaying the sale of losers would

generate price underreaction consistent with momentum. Sadka (2006) finds that

part of the return from momentum trading is compensation for bearing liquidity

risk. Given this brief review, it is fair to say that momentum profits are both

widely recognized and difficult to explain.In New Zealand, two recent studies identify a strong price momentum effect

(Gunasekarage and Kot, 2007; Stork, 2008). Stork (2008) reports momentum

profits in New Zealand, but he focuses on very concentrated large capitalization

portfolios that are not suitable for institutional asset managers and he does not

account for transactions costs. Gunasekarage and Kot (2007) look at the perfor-

mance of portfolios formed on the basis of recent three- to 12-month formation

periods. They form three equally weighted portfolios: relative winners, a middle

group and relative losers. They then look at subsequent performance of these

portfolios over three- to 12-month holding periods. They report momentum

strategy outperformance of an NZX index by 12.63 per cent per annum before

transactions costs and 8.80 per cent per annum after transactions costs (Gun-

asekarage and Kot, 2007, p. 114). They subtract only an arbitrary slice (one-fifth)

of gross returns as an ad hoc transactions cost and do not account for actual

spreads or price impact. We argue below that our method is a significant

advance on Gunasekarage and Kot (2007) and Stork (2008).

3. Data and method

3.1. Data

Our trading strategy uses monthly rebalancing of a portfolio of individual

New Zealand stocks, but some parts of the implementation require daily data.

We use securities that are members of the NZSE40 Capital Index and its replace-

ment, the NZX50 Free Float Gross Index. Index membership, index member

weights and closing prices are obtained on a daily basis from the NZX for the

NZSE40 from June 1991 to March 2004 and for the NZX50 from March 2003

to September 2006. Further daily data from the New Zealand Stock Exchange

Database at the University of Otago are collected to identify bid-ask spreads,dividend and stock split price adjustments and sector classifications. Data are

h k d i Y h ! Fi d th Ot U i it Bl b

S. Trethewey, T. F. Crack/Accounting and Finance 50 (2010) 941965 943


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Terminal. The New Zealand Government three-month Treasury bill yield is used

as the risk-free rate of return.

Table 1 provides descriptive statistics for the stocks in the benchmark portfo-

lio. Comparing 2006 with 1991, we see that liquidity has steadily improved overthe time series: The average market capitalization has roughly doubled; average

daily dollar turnover has more than tripled; and average relative spreads have

roughly halved.

Although not shown in Table 1, the corresponding time series improvement

in liquidity is even more dramatic for the median stock in each of the less

liquid turnover quartiles. Even so, significant differences in liquidity remain in

the cross-section: For example, by the end of the sample, the median stock in

the least liquid turnover quartile still has one-ninth the market capitalization,

two times the relative spread, and one twenty-sixth the daily dollar turnover ofthe median stock in the most liquid turnover quartile. Any nave momentum

trading strategy that fails to account for these cross-sectional differences in

liquidity will bias us towards overly optimistic profits because, as we shall see,

it is the less liquid stocks with higher transactions costs that possess the most

attractive momentum characteristics. Sections 3.3 and 3.4 discuss how our

momentum strategy accounts properly for these liquidity and transactions

costs issues.

3.2. The benchmark portfolio

The performance of our momentum portfolio is measured against either the

NZSE40 or NZX50, depending on the time period. The NZSE40 Capital Index

consisted of the 40 largest publicly traded companies in NZ. All listed securities

from these companies were included in the index; therefore, the index regularly

had more than 40 constituents. Without loss of generality, we refer to the index

members as stocks because the non-stock index members (e.g. warrants and

convertible notes) were of very small capitalization. The NZSE40 was discontin-

ued in March 2004, and the NZX50 was introduced. At the end of the NZSE40

period, we expand our portfolios universe of benchmark stocks to the NZX50.The NZX50 comprises the 50 largest companies listed issues, subject to liquid-

ity constraints. As of 18 November 2009, the 112 members of the NZSE All

Share had a total market capitalization of NZD46.1 billion, whereas the NZX50

securities had a total market capitalization of just over two-thirds this, at

NZD32.7 billion (Bloomberg Terminal, 2009).

To simplify various technical differences in construction, we use the official

index weights obtained from the NZX for each index, but we record the dollar

growth in the benchmark portfolio using the same split- and dividend-adjusted

database returns that we use to calculate dollar growth in our active portfolios.This creates a level playing field for the competition between the benchmark and

th ti tf li O i d th i th f li htl diff t f th t



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e1

pledesc

riptivestatistics

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

lA:No.offirms

ean*

n/a

46

48

49

47

51

52

52

52

52

50

45

46

51

51

51

lB:Mo

nthlystockreturns(%)

ean

2.65

1.65

2.44

)2.23

0.89

0.98

)0.65

)0.01

0.23

)0.16

0.3

8

0.21

1.84

1.75

0

.14

1.24

edian

1.62

0.00

0.18

)1.80

0.80

0.46

0.00

0.00

0.00

0.00

0.8

2

0.00

1.75

1.60

0

.00

0.47

andard

Deviatio

n

12.36

10.99

15.28

17.23

8.05

7.35

8.85

12.32

9.40

10.17

10.2

3

12.01

9.48

7.58

7

.15

6.16

werQu

artile

)3.17

)3.76

)3.49

)6.71

)2.20

)2.21

)4.43

)6.21

)3.90

)4.73

)2.4

3

)3.89

)1.68

)0.97

)2

.97

)2.83

pperQu

artile

8.49

7.65

7.37

3.00

4.57

4.43

4.00

6.20

3.86

4.40

4.6

6

3.66

5.81

4.63

3

.84

4.17

lC:Ma

rketcapitalization($NZDmillion

)

ean

614

629

728

909

948

942

966

884

938

898

864

942

839

1251

1296

1305

edian

162

172

218

310

302

353

387

395

516

512

507

517

387

447

536

593

andard

Deviatio

n

1208

1182

1476

1796

1935

1933

1997

2023

2075

1836

1402

1446

1378

1957

1847

1607

werQu

artile

64

85

103

184

171

182

199

226

235

228

210

220

184

217

289

291

pperQu

artile

431

396

483

715

728

841

917

758

905

880

1010

1059

967

1418

1769

2024

lD:Turnover($NZDthousandtradedperday)

ean

687

657

1185

1233

1351

1024

1152

2091

2101

2006

1851

1660

1638

1906

2066

2502

edian

85

144

249

234

222

273

259

363

501

458

514

479

519

490

556

520

andard

Deviatio

n

1549

1108

2629

2782

4194

2178

2836

6047

6137

5991

5233

5864

4508

5934

6320

9386

werQu

artile

30

54

105

107

94

123

125

146

143

89

159

157

162

182

231

184

pperQu

artile

390

547

721

571

574

633

625

1203

1886

1547

1520

1457

1536

1366

1482

1698



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e1(con

tinued)

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

200

1

2002

2003

2004

2005

2006

lE:RelativeBid-AskSpread(%)

ean

2.33

2.04

1.47

1.73

1.60

1.52

1.58

2.07

1.55

1.91

1.50

1.33

1.43

0.91

0.95

1.05

edian

1.53

1.41

1.12

1.17

1.07

0.99

1.06

1.21

0.96

1.12

0.91

0.87

0.83

0.68

0.77

0.80

andard

Deviatio

n

2.55

2.71

1.26

2.05

2.06

1.94

2.19

2.58

2.14

2.60

2.26

1.91

2.87

0.89

0.91

0.91

werQu

artile

0.87

0.76

0.73

0.70

0.65

0.68

0.69

0.85

0.57

0.63

0.56

0.51

0.55

0.44

0.46

0.48

pperQu

artile

2.60

2.06

1.71

1.92

1.72

1.57

1.68

2.37

1.83

2.30

1.61

1.46

1.35

1.12

1.09

1.21

meannumberoffirmsrepresentstheyearlyaveragenumberoffirmsthatwereconstituentsoftheNZSE

40orNZX50andweretherefore

partofthe

entum

portfolio.Nomomentumportfolioswereformedin1991;thesedatawereusedonlytoconstructa

historicalvariancecovariancema

trix.Panels

andE

useapooledsamplewhereeachstocksdataareobservedonthefirsttradingdayofthemonth(whentheportfoliorebalancetakes

place),with

urnoverobservationbeingthe40-daymo

vingaverageusedinthepriceimpactcalculation.



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3.3. Price momentum portfolio construction

We execute a quantitative active equity alpha optimization, but with only one

signal: price momentum. This widely used practitioner technique is described indetail in the practitioner book by Grinold and Kahn (2000a).1 A similar tech-

nique appears in Chincarini and Kim (2006, Chapter 9). We begin by construct-

ing ex-ante alphas (also called signals) for each stock, each month using a

series of steps involving scaling and neutralization of raw alphas. The raw alphas

are measures of relative strength. In our case, our raw alphas are simply the log

of the ratio of price one month ago to price seven months ago. That is, they are

six-month log price relatives (or six-month formation period returns) calculated

using a one-month gap. The six-month period is consistent with most prior liter-

ature; the one-month gap is included to reduce the impact on profits of possibleshort-term price reversals not caused by bid-ask bounce (we use mid-spread

prices). The literature suggests there was a short-term reversal in New Zealand

stocks at the weekly horizon in early data (Bowman and Iverson, 1998, using

19671986 data), but that it was absent at that horizon in later data (Boebel and

Carson, 2001, using 19911999 data).

These alphas are built to be benchmark neutral. In other words, holding the

benchmark exposes you to no ex-ante alpha and no active bets, but actively chas-

ing these alphas goes hand-in-hand with actively stepping away from the bench-

mark. We then run an optimization routine each month to rebalance our

portfolio weights (the choice variables) by tilting them towards positive ex-ante

alphas and away from negative ex-ante alphas. We retard this alpha chasing by

including a penalty in our objective function that quantifies the exposure to

active risk associated with actively stepping away from the benchmark. This

active risk is moderated using a client risk aversion coefficient. We also include

penalties in our objective function for the transactions costs incurred by chasing

alpha (the objective function appears below in equation (1)). Our transactions

costs include the explicit cost associated with buying stocks at the ask and selling

them at the bid and also the implicit price impact incurred when we need to

walk up or down the centralized limit order book (CLOB) to fill a trade thereby pushing prices against us (see Section 3.4, for details of our model of

price impact).

Although addressing the same question and using similar data, our approach

is in stark contrast to the approach of Gunasekarage and Kot (2007). They form

equally weighted long-short portfolios of winners and losers and then ex-post

use an ad hoc estimate of transactions costs. We, however, form optimally

1 An extended version of this paper is available from the authors upon request. It containsa deeper discussion of the optimization and its constraints, the steps in the alpha construc-tion, the variance-covariance matrix estimation, and the results. It also contains an expli-



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weighted portfolios, allow realistic underweighting that avoids shorting and use

realistic transactions costs using actual spreads and a practitioner model of price

impact.

Our approach also contrasts with Korajczyk and Sadka (2004), who use USstocks. They form equal- or value-weighted portfolios of recent winners and then

account ex-post for the transactions costs needed to rebalance the portfolios each

month. They subsequently compute Sharpe ratios and (Jensen alpha type)

abnormal returns relative to the Fama-French three-factor model.

Although relatively standard, the Gunasekarage and Kot (2007) and

Korajczyk and Sadka (2004) techniques just described are both nave implemen-

tations of an active trading strategy. Practitioners do not blindly chase antici-

pated returns and then passively account ex-post for the transactions costs and

risk; doing so is sub-optimal. Rather, practitioners weigh all three simulta-neously: A portfolio manager may, for example, avoid overweighting a small

capitalization stock that has attractive momentum characteristics if it has high

transactions costs or unfavourable risk characteristics. Korajczyk and Sadka do

attempt to address this deficiency by also using liquidity conscious portfolios

with weights that are related to the liquidity of the stock (Korajczyk and Sadka,

2004, pp. 1054, 1075), but they acknowledge that this approach to portfolio for-

mation is optimal only under fairly restrictive conditions (Korajczyk and Sadka,

2004, p. 1054).

Like most quantitative active equity strategies, we try to avoid benchmark tim-

ing2 by constraining the ex-ante portfolio beta each month to equal 1 (in prac-

tice, we are rebalancing only once a month, so we incur some unintended

benchmark timing as the beta slips away from 1; we account for this in our per-

formance measurement). We also constrain portfolio turnover each month, limit

the size of the active bets in any stock and require the portfolio to be fully

invested in equities. For the relatively unconstrained strategies, we allow short

selling, but ultimately our feasible strategies are all long only.

Korajczyk and Sadka choose to use long-only portfolios of winners. They

argue that this avoids the asymmetric costs associated with the short side of a

long-short strategy (Korajczyk and Sadka, 2004, p. 1045). Many other research-ers (e.g. Chenet al., 2002; Gunasekarage and Kot, 2007) use long-short arbitrage

strategies. All these papers, however, overlook our tilts approach, where a pas-

sive benchmark fund is actively tilted towards over- and underweights, but with-

out breaching the long-only constraint, and where the optimization takes care of

the transactions costs. Although not a true long-short fund, and although the

long-only constraint may carry a considerable impact (Grinold and Kahn,

2000a, Chapter 15; Grinold and Kahn, 2000b), our resulting long-only

2 Benchmark neutrality also reduces the likelihood that abnormal returns are generatedby any firm characteristics common to the firms in this small market. This is because, by



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implementations allow consideration of both winners and losers and look like

standard institutional practice for a quantitative fund.

The choice variables in the optimization are the vectorhP of holdings in the

portfolio of stocks that are members of the benchmark. The optimization is amaximization of a value added (VA) objective function (Grinold and Kahn,

2000a, p. 119) modified for transactions costs. The objective function has the fol-

lowing form:

VA aP kx2PTC 1

whereaP hP0aP is the portfolio ex-ante alpha calculated as the inner product

of the portfolio holdings vector and the vectoraP of ex-ante alphas for the indi-

vidual stocks; k is the client risk aversion coefficient; x2P r

2Pr

2B

hP0VhP-hB

0VhB is the forecast active risk of the portfolio (where B denotes

benchmark), whereV is a variance-covariance matrix of returns; andTC is the

estimated transactions costs (including both actual spreads and estimated price

impact) associated with rebalancing the portfolio. Everything in this objective

function is scaled to be in annual return terms (see Section 3.4).

The portfolio is assumed to be launched at the end of June 1992, with bench-

mark weights (as if an active manager had just taken over a passive fund) and

with an initial investment ranging from $1 in the relatively unconstrained case

up to $150 million. At the end of each subsequent month, we examine ourexisting holdings, and we rebalance by running the optimization to chase the

just-calculated ex-ante alphas by choosing new portfolio weights subject to

the above-mentioned penalties and constraints. We calculate gross returns over

the following month, calculate the new end-of-month dollar balance of the fund

and then we recalculate the ex-ante alphas and rebalance again (in all, we rebal-

ance 171 times over our sample period). For each strategy, monthly turnover

was reassuringly credible and in line with our intuition (averages are reported

later in the paper). All strategies are self-financing; no new money enters the

portfolios. For each gross return simulation, we also perform a separate net

return simulation, where monthly transactions costs are subtracted from the

gross return before calculating the new end-of-month dollar balance of the fund.

At each step, we ensure that all information used for the optimization was avail-

able to a portfolio manager at that date. When stocks enter or exit the index, we

temporarily increase the turnover allowance (as a function of the index weight of

the stock that is entering or exiting) to allow the manager to trade into or out of

the position quickly without breaching the turnover constraint. We run the

momentum trading strategy using different fund sizes, different ex-ante alpha

formation periods, with and without the one-month gap, different client risk

aversion coefficients, different allowances for turnover, different allowances foractive weight and with and without short selling. The results are discussed in



10/26

3.4. Measurement of transactions costs

Our optimization uses a transactions cost penalty in the objective function

given by TC= 12 hP0

(RS+ PI) calculated as 12 times the inner product ofthe portfolio holdings vectorhPand the sum of the vectorsRSandPI, which are

described below. In practice, fund managers using this technique rebalance more

frequently than monthly, but otherwise our model of transactions costs is used

by practitioners in essentially the same way that we use it here (Grinold and

Kahn, 2000a).

The termsRSandPIare vectors whose elements contain the relative spread,

RSi, and price impact, PIi, functions for each stock i. These stock-specific

functions are estimated as follows (Grinold and Kahn, 2000a, p. 452):

RSi1

2 hi;thi;t

askbidbidask=2

2

PIi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivolumetrade

volumedaily

s rdaily 3

whereh*i,tis the new optimal portfolio holding of stockiat timet, andhi,tis the

existing holding of stock i at time t immediately before the rebalance,bid and

ask represent the current bid and ask prices for stock i (subscript suppressed),

volumetradedenotes the number of shares required to be traded to reach the new

portfolio holding of stock i, volumedaily represents the average daily volume of

stockifor the past 40 trading days (split adjusted) andrdailyis the past 250-day

standard deviation of daily returns to stocki(subscript suppressed on the right-

hand side of equation (3)). To avoid confusion, note thathi,t, the existing holding

in stock i just prior to the rebalance, is whath*i,t)1 (the most recent rebalanced

optimal holding) has evolved into over the months time periodt)

1 tot.Korajczyk and Sadka (2004) calibrate their price impact models explicitly

using US TAQ data. Without intraday data we have instead used, in equation

(3), an implicit scaling based on the traders rule of thumb that it costs approxi-

mately one days volatility to trade one days volume (Grinold and Kahn, 2000a,

p. 452). Note that Korajczyk and Sadka use academic models of price impact

(e.g. Glosten and Harris, 1988; Breen et al., 2002), whereas we use a model of

price impact taken directly from the practitioner literature (Grinold and Kahn,

2000a, p. 452).

Like Glosten and Harris (1988), our model of transactions costs for a givenstock is composed of a cost that is fixed as a function of volume traded by the

t t (i th l ti d) d t th t i i bl f ti f l



11/26

modelling the percentage price impact (i.e. as a percentage of initial stock price)

as a square root function of the number of shares traded by the strategy (in that

stock during that month). This square root form has its foundation in Barra

research that analysed Loeb (1983) and found the results consistent with asquare root pattern (see Grinold and Kahn, 2000a, p. 452). This square root

form is clearly a concave functional form, and it gives immediately a concave

functional form for percentage price impact as a function of dollar volume of the

strategy in that stock.

Loeb (1983), Glosten and Harris (1988), Hausmanet al. (1992), Keim and

Madhavan (1996) and Breenet al. (2002) all present theoretical models and/or

empirical results that are consistent with a percentage price impact function that

is, like ours, concave when price impact as a percentage (of original price or of

portfolio value) is expressed in terms of dollar volume. Hasbrouck (1991) pre-sents a price impact model but the functional form for percentage price impact

as a function of dollar volume is not clear.

Although concave when expressed as percentage price impact as a function of

dollar volume, if we multiply both sides of (3) by dollar volume, to look at total

absolute price impact cost (in dollars) as a function of dollar volume, the result-

ing convex functional form involves dollar volume to the power of 3/2 (Grinold

and Kahn, 2000a, p. 452).

Finally, the annualized transactions cost is the cost of a trade divided by the

rebalance period in years. Therefore, we scale the transactions costs by a factor

of 12 in the objective function.

3.5. Testing for the existence of momentum profits

Our simulated portfolio strategy generates a time series of 171 months of port-

folio gross returns, portfolio returns net of transactions costs and benchmark

returns. Momentum profits can be tested for with the following regression:

rP;trf;t aPbPrB;trf;t et 4

whererP,tdenotes the realized monthly return on the active portfolio at time t,

rf,tis the monthly risk-free rate of return,rB,tis the realized monthly benchmark

return, aP is the ex-post realized monthly alpha, bP is the realized portfolio beta

and et rP;trf;t aP bPrB;trf;t is the realized residual return. Simplereturns are used for the regression and appear for all reported results. When

portfolio gross returns (i.e. ignoring any transactions costs) are used in (4),aP is

a risk-adjusted measure of exceptional performance, and we can measure its sta-

tistical significance with the standard t-statistic. When portfolio returns net of

transactions costs are used in (4), aP is a risk-adjusted and transactions cost-adjusted measure of exceptional performance. Risk-adjusted here means that



12/26

using an objective function that included an explicit penalty for risk, and also

that the regression equation removes the portion of portfolio return associated

with the benchmark. The realized ex-post alpha thus accounts for client risk

aversion and benchmark riskwhich are, ultimately, what matter to clients ofinstitutional asset managers.

4. Results

4.1. The profitability of price momentum

Our relatively unconstrained portfolio is designed to be analogous to the

unconstrained strategies reported in the literature, such as Gunasekarage and

Kot (2007). It has an initial investment of $1 and is allowed to short sell, takeactive weights of 20 per cent (this is the maximum allowed value of any element

of the vector hPhBj j in any month) and have a maximum two-sided turnoverof 50 per cent per month. We include penalties for the bid-ask spread and price

impact in the objective function, but the tiny $1 portfolio size means that the

price impact is effectively zero.

Figure 1 and Table 2 provide strong evidence of a momentum effect with

mean monthly returns well in excess of the mean monthly benchmark return and

significant levels of alpha (i.e. realized abnormal return as in equation (4)) at the

0.1 per cent level for the gross returns and the 1 per cent level for the net returns.

The relatively unconstrained portfolio produces a gross alpha of 112 bps per

month before subtracting transactions costscomparable with Gunasekarage

and Kot (2007). This strategy is, however, not realistic because a larger fund

would push prices against it as it walks up or down the CLOB. Also, although

short selling is allowed in New Zealand, it is not widely used. Finally, the strat-

egy pushes the two-sided turnover constraint of 50 per cent to the limit every

time the portfolio is rebalanced. This indicates that the returns will be severely

decreased once price impact is considered and turnover is properly constrained.

Net of transactions costs (effectively the relative spread only in this case), the rel-

atively unconstrained portfolio still produces a monthly alpha of 87 bps permonth and a good information ratio (i.e. Sharpe ratio calculated using residual

returns) ofIR = 0.67.

The last two rows of Table 2 show that removing the ability to short and tight-

ening the turnover and active weight constraints immediately kills off more than

three-quarters of the realized alpha; theIR on the constrained $1 portfolio is still

good, however, and is 0.47 after transactions costs.

In all portfolios reported in Table 2, unintentional benchmark timing caused a

slight decrease in portfolio return. We constrained our ex-ante portfolio beta to

equal 1 when we rebalanced each month, but in a real-world trading strategy,the portfolio managers would rebalance something like 10 times per month

( h t th l l t d t l h f ll f h t t



13/26

enough that our portfolio betas slip away from 1, which introduces unintentional

benchmark timing. We focus on the ex-post alpha aP rather than on the activereturn rP rB because the alpha represents the intentional return to stock selec-tion, whereas the active return includes the unintentional benchmark timing

return.

Table 3 reports the results for trading strategies where we have estimated a

more realistic portfolio than in Table 2. Each of these strategies was imple-

mented with a $100 million initial portfolio, no short selling, an active weight

constraint of 5 per cent and a two-sided turnover constraint of 20 per cent per

month. We also report strategies with different ex-ante alpha constructions (with

and without a one-month gap and using three or six months for the formationperiod).

Although the mean gross return to the realistic portfolios in Panel A and Panel

B of Table 3 exceeds the mean return to the benchmark, we see that without

exception the gross alphas are economically small and statistically insignificant,

and the net alphas are statistically significantly negative at the 0.1 per cent level.

The benefits of stepping away from the benchmark have therefore failed to

exceed the cost of doing so. Real-world constraints mean that we have not been

able to capture the promising momentum profits we saw in the relatively uncon-

strained portfolios in Table 2.An asterisk in Table 3 marks our base portfolio. In the remainder of the

paper we vary its characteristics/constraints to deduce which are pivotal in

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

Jun-1992 Jun-1994 May-1996 May-1998 May-2000 May-2002 May-2004 May-2006

Cumulativelevel

$1 Portfolio, relatively unconstrained, gross return

$1 Portfolio, relatively unconstrained, net return

$1 Portfolio, constrained, gross return

$1 Portfolio, constrained, net return

Benchmark (NZSE40/NZX50)

All portfolios are momentum strategies with an initial fund value of $1. The relatively unconstrained portfolios are allowed to short sell andhave a maximum turnover of 50 per cent, and a maximum active weight of 20 per cent. The constrained portfolios are not allowed to short sell and have amaximum turnover of 20 per cent and a maximum active weight of 5 per cent. Gross return refers to the estimated portfolio return prior to transaction costs.Net return refers to the estimated portfolio return after transaction costs (i.e. spreads and price impact).

Figure 1 Profitability of relatively unconstrained return momentum strategies.



14/26

e2

tivelyunconstrainedreturnmomentumstrategies

folio

N

Mean

Return

StdDevof

Returns

Alpha

Alpha

t-value

Beta

ActiveBeta

Bmark

Timing

Return

Resid.

IR

R2

F Stat

hmark

171

0.78%

0.04460

Unconstrained,GrossReturn

171

1.85%

0.05601

1.12%

3.29

0.770

)0.230

)0.05%

0.87

0.376

101.7

Unconstrained,NetReturn

171

1.60%

0.05662

0.87%

2.54

0.781

)0.219

)0.05%

0.67

0.378

102.8

trained

,GrossReturn

171

1.00%

0.04305

0.24%

2.33

0.918

)0.082

)0.02%

0.62

0.905

1611.0

trained

,NetReturn

171

0.94%

0.04297

0.18%

1.76

0.918

)0.082

)0.02%

0.47

0.908

1666.8

stimatesaremonthly.

Nisthenumberofmonthlyrebalances.Allportfolioshaveasix-monthformation

period,withaone-monthgappriortothe

ngperiod.Therelativelyunconstrainedportfoliosareallowedtoshortsellandhaveamaximumturnover

of50percentandamaximumactiveweight

percent.Theconstrainedportfoliosare

notallowedtoshortsellandhaveamaximumturnoverof20percentandamaximumactiveweightof5per

Allportfolioshaveaninitialvalueof$1.Grossreturnreferstotheestim

atedportfolioreturnpriortotransactioncosts.Netreturnrefers

totheesti-

dportfolioreturnaftertransactioncosts

(i.e.spreadsandpriceimpact).A

llportfolioswereconstrainedtohaveex-antebetaof1ateachmo

nthlyrebal-

AllF-Statsaresignificantatbetterthan

1percent.



15/26

e3

sactionscostsandtheprofitabilityofreturnmomentumstrategies

mation-Return

bination

N

Mean

Return

StdDevof

Returns

Alpha

Alpha

t-value

Beta

Active

Beta

Bmark

Timing

Return

Resid.

IR

R2

F Stat

hmark

171

0.78%

0.04460

lA:No

gap

Month,

GrossReturn

171

0.82%

0.04309

0.05%

0.87

0.951

)0.049

)0.01%

0.23

0.969

5299.4

Month,

NetReturn

171

0.49%

0.04441

)0.29%

)4.46

0.977

)0.023

)0.01%

)1.18

0.964

4541.2

Month,

GrossReturn

171

0.84%

0.04362

0.07%

1.20

0.962

)0.038

)0.01%

0.32

0.967

5021.0

Month,

NetReturn

171

0.50%

0.04363

)0.27%

)4.25

0.960

)0.040

)0.01%

)1.13

0.964

4565.1

lB:One-monthgap

Month,

GrossReturn

171

0.80%

0.04360

0.03%

0.53

0.962

)0.038

)0.01%

0.14

0.969

5224.1

Month,

NetReturn

171

0.45%

0.04342

)0.32%

)5.37

0.958

)0.042

)0.01%

)1.42

0.969

5222.1

Month,

GrossReturn*

171

0.80%

0.04298

0.03%

0.60

0.949

)0.051

)0.01%

0.16

0.970

5539.8

Month,

NetReturn

171

0.52%

0.04369

)0.25%

)3.84

0.961

)0.039

)0.01%

)1.02

0.962

4270.3

lC:Theeffectoftransactionscosts

Month,

GrossReturn*

171

0.80%

0.04298

0.03%

0.60

0.949

)0.051

)0.01%

0.16

0.970

5539.8

Month,

GrossReturn

essRelativeSpread

171

0.78%

0.04383

0.00%

0.05

0.969

)0.031

)0.01%

0.01

0.972

5807.4

Month,

NetReturn

171

0.52%

0.04369

)0.25%

)3.84

0.961

)0.039

)0.01%

)1.02

0.962

4270.3

stimatesaremonthly.

Nisthenumbero

fmonthlyrebalances.Nogapreferstoimmediateinvestmentaftertheformationperiod.Allportfolioshave

itialvalueof$100million,maximumturnoverof20percentandmaximumactiveweightof5percent.O

ne-monthgapreferstoleavinga

one-month

between

theformationandholdingperiodtoremoveanypotentialreversa

leffects.Thenumberofrebalancesindicatesthelengthinmonthsofthestrat-

ormatio

nperiod.Grossreturnreferstotheestimatedportfolioreturnprio

rtotransactionscosts.Netreturn

referstotheestimatedportfolio

returnafter

actioncosts.Grossreturnlessrelativesp

readreferstotheestimatedportfolioreturnaftertherelativespreadtransactionscostshavebeenre

moved,but

rethepriceimpactcostshavebeenremoved.AllF-Statsaresignificantatbetterthan1percent.

basep

ortfolio:initialvalueof$100million,20percentmaximumturnover,5percentmaximumactivew

eight,fullobjectivefunction,six-monthfor-

onperiodandone-monthgapuntilholdingperiod.Thisportfolioisusedforcomparisonthroughouttheresults.



16/26

Figure 2 and Panel C of Table 3 demonstrate that deducting the spread com-

ponent of transactions costs from the gross $100 million base portfolio returncauses the strategy to match the returns of the benchmark within rounding error.

Then, subsequently removing the price impact transaction cost causes the strat-

egy to significantly underperform the benchmark.

Momentum trading strategies are designed to exploit short-lived effects. It

is not surprising, therefore, that they are reported to have high turnover

(Keim, 2003; Lesmond et al., 2004; Sadka, 2006). For the $100 million base

portfolio, the mean two-sided turnover was 127 per cent per annum, but it

was 697 per cent per annum in the relatively unconstrained portfolio. These

turnovers correspond to average stock holding periods of 9.4 and 1.7 months,respectively. The shorter holding period brings with it greater momentum

profits (as seen here and in Gunasekarage and Kot, 2007), but the portfolio

turnover required to achieve a short holding period in the base portfolio

involves an infeasible amount of price impact. Although the mean transaction

cost attributable to the spread component was only 5 bps per month, the

mean transaction cost attributable to price impact was 28 bps per month.

Price impact thus accounts for 85 per cent of the transactions costs of the

strategy and cannot be ignored.

Korajczyk and Sadka (2004) also discuss the size of spread and price impactcomponents of transactions costs for stand-alone momentum strategies, but

they do not explicitly identify the relative sizes of these components We

75

100

125

150

175

200

225

250

275

300

325

350

375

Jun-1992 Jun-1994 May-1996 May-1998 May-2000 May-2002 May-2004 May-2006

Cumulativelevel

$100m base portfolio, gross return

$100m base portfolio, gross return less relative spreads

$100m base portfolio, net return

Benchmark (NZSE40/NZX50)

All portfolios are momentum strategies that are variations, by transaction costs only, of the base portfolio*. Gross return refers to the estimated portfolio returnprior to transaction costs. Net return refers to the estimated portfolio return after transaction costs (i.e. spreads and price impact). Gross return less relative spreadrefers to the estimated portfolio return after the relative spread transaction costs have been removed. *The base portfolio: initial value of $100 million, 20 per centmaximum turnover, 5 per cent maximum active weight, full objective function, six-month formation period and one-month gap until holding period.

Figure 2 Transactions costs and the profitability of return momentum strategies.



17/26

1245 bps per month, depending upon portfolio formation strategy (Korajczyk

and Sadka, 2004, p. 1058). We deduce that their price impact transactions

costs in a USD 5 billion portfolio are in the range 40100 bps per month for

the value- and liquidity-weighted strategies; the particular values in theseranges depend upon the portfolio formation strategy and the model of price

impact (Korajczyk and Sadka, 2004, Figures 4(a), 5(a), 6(a) and 7(a)). We con-

clude that their price impact component of transactions costs is, like ours, the

largest part of the transactions costs for any reasonably sized strategy. It is

hardly surprising that their transactions costs are noticeably larger than ours

in absolute magnitude (twice or more), because our optimization includes an

explicit transactions costs penalty in the objective, which means that, like a

practitioners, our portfolios are chosen specifically so as to minimize exposure

to stocks with higher transactions costs.In our implementation, we trade only once per month. We may, therefore, be

overestimating practitioner price impact as a function of dollar volume. Figure 2

and Panel C of Table 3 show, however, that even if price impact were zero, the

base portfolios performance still only matches that of the benchmark. On the

returns side, however, our infrequent rebalancing may lose exposure to ex-ante

alphas, and we may therefore underestimate the ability of the model to gain trac-

tion with our alphas.

4.2. The impact of relative spreads and fund size

Table 4 reports estimates of the effects of different assumed relative spreads

(Panel A) and initial fund sizes (Panel B) on the performance of the momentum

strategy. All portfolios are compared with our base portfolio and the full form

of the objective function is used.

The results in Panel A of Table 4 show that the mean returns per month of the

momentum strategy are very stable across the variations of relative spread.

Using a blanket 80 bps relative spread for all stocks generates a higher mean

return on a gross and net basis. The performance using the 80 bps spread is only

marginally higher than the performance using a minimum 50 bps spread andslightly higher again than using actual spreads.

In Panel B of Table 4, all portfolios considered are variations of the base port-

folio, with only the initial fund value changing. We can see the alpha and theIR

dropping almost monotonically as the fund size increases and the price impact

begins to bite. The $1 portfolio, with effectively no price impact, predictably pro-

duces the highest mean return and a gross alpha of 24 bps per month (the net

alpha for this strategy appeared in Table 2 and was a respectable 18 bps per

month). The effect of the price impact function is not noticeable until the fund

size reaches $10 million. In fund sizes above $50 million, we observe that theprice impact function significantly retards the strategy from trading in stocks

th t d d l h



18/26

e4

effectofrelativespreadsandfundsizeon

theprofitabilityofreturnmomentumstrategies

folio

N

Mean

Return

StdDevof

Returns

Alpha

Alpha

t-value

Beta

Active

Beta

Bmark

Timing

Return

Resid.

IR

R2

F Stat

hmark

171

0.78%

0.04460

lA:Relativespreads

bpsspread,GrossReturn

171

0.86%

0.04343

0.09%

1.53

0.959

)0.041

)0.01%

0.41

0.971

5645.8

bpsspread,NetReturn

171

0.52%

0.04297

)0.25%

)4.00

0.946

)0.054

)0.01%

)1.06

0.964

4528.4

inimum

50bpsspread,

ossReturn

171

0.82%

0.04440

0.04%

0.69

0.982

)0.018

0.00%

0.18

0.973

6034.9

inimum

50bpsspread,

etReturn

171

0.52%

0.04345

)0.25%

)3.97

0.956

)0.044

)0.01%

)1.05

0.964

4461.8

tualspread,GrossReturn*

171

0.80%

0.04298

0.03%

0.60

0.949

)0.051

)0.01%

0.16

0.970

5539.8

tualspread,NetReturn

171

0.52%

0.04369

)0.25%

)3.84

0.961

)0.039

)0.01%

)1.02

0.962

4270.3

lB:Fundsize(allgrossreturns)171

1.00%

0.04305

0.24%

2.33

0.918

)0.082

)0.02%

0.62

0.905

1611.0

million

171

0.97%

0.04283

0.21%

2.48

0.928

)0.072

)0.02%

0.66

0.934

2377.6

0millio

n

171

0.90%

0.04265

0.14%

2.02

0.936

)0.064

)0.02%

0.54

0.958

3827.4

0millio

n

171

0.82%

0.04311

0.05%

0.80

0.952

)0.048

)0.01%

0.21

0.970

5457.2

00million*

171

0.80%

0.04298

0.03%

0.60

0.949

)0.051

)0.01%

0.16

0.970

5539.8

50million

171

0.84%

0.04438

0.07%

1.13

0.980

)0.020

0.00%

0.30

0.970

5500.0

stimatesaremonthly.

Nisthenumberofmonthlyrebalances.Allportfolioshaveamaximumturnoverof

20percentandmaximumactive

weightof5

ent.PanelAcontainsvariationsontherelativespread.80bpsspreadreferstoallstockshavingtheirrelativespreadsettoablanket80bps.Minimum

pssprea

dreferstoallstockshavingactualspreadsoverlaidwithaminimu

mrelativespreadof50bps.Actualspreadindicatesthatthereportedbid-ask

adhasb

eenusedtocalculatetherelative

spread.AllportfoliosinPanelA

hadaninitialvalueof$100million.PanelBcontainsvariationsoftheportfo-

zebygrossreturns.Grossreturnreferstotheestimatedportfolioreturn

priortotransactioncosts.Netreturnreferstotheestimatedportfolioreturn

transac

tioncosts(i.e.spreadsandpriceimpact).AllF-Statsaresignificantatbetterthan1percent.

basep



onperiodandone-monthgapuntilholdingperiod.



19/26

4.3. The effect of optimization parameters

Table 5 reports the impact on our base portfolios performance when we vary

the components of the objective function (Panel A and Panel B) or the tightnessof the turnover or active weight constraints (Panels C and Panel D).

Grinold and Kahn (2000a, p. 119) quote high (k 15), moderate (k 10) andlow (k 5) values for client risk aversion. Panel A of Table 5 indicates that,other things being equal, the tight limits on active weights and turnover and the

presence of the price impact penalty term in the objective function, combined

with the inability to short sell, retard portfolio trade to the extent that the risk

aversion is simply not biting in their presence.

Panel B of Table 5 explores dropping various terms from the full objective

function for the base portfolio. We see that dropping the price impact penaltyin the objective function immediately adds 21 bps per month to the gross

alpha (but unsurprisingly this change destroys approximately 100 bps of net

alpha per month; not reported in the tables). Then, dropping the spread com-

ponent from the objective function adds an additional 9 bps per month to the

gross alpha. Then, dropping the risk aversion component adds just one more

basis point to the gross alpha but hurts theIR because of the additional active

risk taken on.

Looking at Panels C and D of Table 5, we can see that relaxing the turnover

and active weight constraints has only a slight effect on gross returns unless the

price impact penalty term in the objective is removed, in which case the effect on

gross alpha is significant, jumping by 27 bps per month (but again unsurprisingly

this change destroys approximately 200 bps of net alpha per month; not reported

in the tables). Then also dropping the risk aversion down tok 5 has only amarginal impact on alpha but again hurts theIR through additional active risk

taken on. The implication is that the price impact and risk aversion penalty

terms are important for retarding alpha chasing that would otherwise be blind to

transactions costs or active risk, respectively.

4.4. Market capitalization, winners, losers and shorts

Ignoring transactions costs, the small capitalization holdings of our base

portfolio generate 50 bps of alpha per month (compared with 30 bps of alpha

generated per month by their nave sub-index portfolio).3 The large capitaliza-

tion holdings of the base portfolio lose 24 bps of alpha per month (roughly

matching their sub-index portfolio). These results are consistent with Lesmond

et al. (2004), who find that the stocks that generate large momentum returns

are precisely those stocks with high trading costs. No wonder we found that

3



20/26

e5

effectoftheoptimisationparametersontheprofitabilityofreturnmomentumstrategies(allgrossreturns)

folio

N

Mean

Return

StdDevof

Returns

Alpha

Alpha

t-value

Beta

Active

Beta

Bmark

Timing

Return

Resid.

IR

R2

F Stat

hmark

17

1

0.78%

0.04460

lA:Ris

kaversion

mbda=

5

17

1

0.85%

0.04344

0.08%

1.36

0.958

)0.0

42

)0.01%

0.36

0.968

5140.7

mbda=

10*

17

1

0.80%

0.04298

0.03%

0.60

0.949

)0.0

51

)0.01%

0.16

0.970

5539.8

mbda=15

17

1

0.83%

0.04389

0.05%

0.95

0.971

)0.0

29

)0.01%

0.25

0.974

6455.2

lB:Objectivefunction

ctiveRe

turn

17

1

1.09%

0.04301

0.34%

2.69

0.892

)0.1

08

)0.03%

0.71

0.856

1004.9

ctiveRe

turnandActiveRisk

17

1

1.09%

0.04316

0.33%

3.13

0.919

)0.0

81

)0.02%

0.83

0.902

1547.5

ctiveRe

turn,ActiveRisk,and

Relative

Spreads

17

1

1.00%

0.04306

0.24%

2.35

0.918

)0.0

82

)0.02%

0.62

0.905

1607.3

ctiveRe

turn,ActiveRisk,andFull

ransactionsCosts*

17

1

0.80%

0.04298

0.03%

0.60

0.949

)0.0

51

)0.01%

0.16

0.970

5539.8

lC:Constraintlevels

urnover

50%,ActiveWeight5%

17

1

0.80%

0.04317

0.03%

0.58

0.954

)0.0

46

)0.01%

0.15

0.972

5871.4

urnover

20%,ActiveWeight5%*

17

1

0.80%

0.04298

0.03%

0.60

0.949

)0.0

51

)0.01%

0.16

0.970

5539.8

urnover

10%,ActiveWeight5%

17

1

0.84%

0.04351

0.07%

1.08

0.958

)0.0

42

)0.01%

0.29

0.964

4512.8

urnover

20%,ActiveWeight10%

17

1

0.83%

0.04282

0.06%

0.96

0.943

)0.0

57

)0.01%

0.25

0.965

4722.5

urnover

20%,ActiveWeight20%

17

1

0.87%

0.04374

0.10%

1.46

0.961

)0.0

39

)0.01%

0.39

0.961

4124.4

lD:Variationsonbaseportfolioconstrain

tsandpenalties

urnover

20%,ActiveWeight5%*

17

1

0.80%

0.04298

0.03%

0.60

0.949

)0.0

51

)0.01%

0.16

0.970

5539.8

urnover

50%,ActiveWeight20%

17

1

0.82%

0.04349

0.04%

0.70

0.957

)0.0

43

)0.01%

0.19

0.964

4551.8

urnover

50%,ActiveWeight20%,

NoPIin

Obj.Fn.

17

1

1.06%

0.04327

0.31%

2.14

0.873

)0.1

27

)0.03%

0.57

0.808

710.9



21/26

e5(con

tinued)

folio

N

Mean

Return

StdDevof

Returns

Alpha

Alpha

t-value

Beta

Active

Beta

Bmark

Timing

Return

Resid.

IR

R2

F Stat

urnover

50%,ActiveWeight20%,

NoPIin

Obj.Fn.,

ambda

=

5

171

1.07%

0.04486

0.32%

1.75

0.850

)0.150

)0.04%

0.47

0.713

420.5

stimatesaremonthly.

Nisthenumberofmonthlyrebalances.Allportfo

lioshaveaninitialvalueof$100

millionandaregrossreturnvar

iantsofthe

portfolio*withthespecifiedparameters

altered.WithinPanelA,lambda

istheclientriskaversioncoefficie

ntusedintheobjectivefunction.

Higherlev-

flambd

arepresentgreaterriskaversion.

ThelevelsofriskaversionwherechosenfollowingGrinoldandKa

hn(2000a,p.119).InPanelB,th

eportfolios

tovariationsontheformoftheobjectivefunction.Fulltransactionscosts

isthesumofrelativespreadsand

priceimpactcosts.WithinPanelC,thecon-

ntlevelsrepresentthemaximumlevelof

activeweightorturnoverthattheportfoliomayhaveeachtimeit

isrebalanced.TheportfoliosinPanelDare

tionsonthebaseportfoliowithchanges

intheconstraintsandtheobjectivefunctionpenalties(PIreferstopriceimpact).AllF-Statsaresignificantat

rthan1percent.

basep



onperiodandone-monthgapuntilholdingperiod.



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the price impact term is so biting, given that price impact is greater in small

stocks and that it is the small stocks that are providing the ex-post alpha per-

formance.

Again, ignoring transactions costs, the winner (i.e. overweight) sub-portfolioof our base portfolio generates 20 bps of alpha per month (compared with

14 bps of alpha generated by its nave sub-index portfolio), but the loser (i.e.

underweight) sub-portfolio loses 32 bps of alpha per month (compared with

13 bps of alpha lost by its nave sub-index portfolio). In the relatively uncon-

strained portfolio, however, winners generate 36 bps of alpha per month (com-

pared with 8 bps of alpha generated by their sub-index, losers (underweight but

not short) roughly match their sub-index, and shorts generate 46 bps of alpha

more per month than the)3 bps provided by their sub-index portfolio. The

short constraint in the base portfolio thus confounds the ability of the optimizerto correctly weight the underachievers, and, by doing so, confounds the ability

for the optimizer to correctly exploit the winners (see related discussion in

Grinold and Kahn, 2000a, p. 421 and Grinold and Kahn, 2000b).

The importance of the short position here is also consistent with earlier liter-

ature that finds that stock prices are more likely to underreact to bad news

than to good news (Hong et al., 2000, p. 277), and, as such, the ability to

short the losers is very valuable. Similarly, Lee and Swaminathan (2000, Table

I) and Lesmondet al. (2004, Table 1) find that portfolios of loser stocks subse-

quently underperform average stocks by more than winner stocks outperform

them.

Gunasekarage and Kot (2007, p. 120) consider long-only portfolios, and they

also find that the winners are driving the momentum profits. In contrast to us,

however, they say that it is the larger capitalization stocks that generate alpha.

They have a much broader sample of stocks than ours, and our small stocks

probably account for half of their large stocks, while our large stocks have no

impact at all.

If we were to trade only the highly liquid stocks, we could reduce the liquidity

problems and minimize price impact. This is what Stork (2008) does, with

reported profitability before transactions costs. Unfortunately, concentratedportfolios of only a few stocks are not attractive to institutional asset managers.

The active risk is simply too high, and, if implemented in any size, price impact

would again become an issue. On top of these problems, our analysis indicates

that the momentum profits are predominantly sourced in the smaller capitaliza-

tion stocks. For all these reasons, we believe a concentrated high-liquidity strat-

egy is not feasible.

Although the net returns to all of our realistic portfolios are negative, the

momentum strategy may be useful as one of a group of alpha signals imple-

mented simultaneously, or as a trade timing indicatorwhen you have to geta trade completed and want to know whether you should wait to execute it or

t



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5. Conclusion

We test whether recently reported profits from price momentum trading strate-

gies in the New Zealand stock market (Gunasekarage and Kot, 2007; Stork,2008) are able to be captured using a simulated portfolio trading strategy. Within

the NZSE40/NZX50 index, the smaller stocks generate gross momentum profits,

but have high spreads and low turnover; the low turnover, in turn, implies high

price impact. In a tiny unconstrained portfolio, smaller stocks, winner stocks

and shorts combine to generate gross performance so exceptional that perfor-

mance net of spreads is excellent (and price impact is negligible). In a portfolio

of any size and with short sale constraints, however, transactions cost avoidance

and risk aversion necessarily retard trade in the smaller stocks, winner stocks

provide no profits and shorts are unavailable. In this case, the trading strategystill steps away from the benchmark to chase anticipated momentum profits, but

gross returns are less than anticipated and only just cover bid-ask spread costs;

portfolio size combined with high turnover in small stocks means that once price

impact is accounted for, performance lags the index. A proper accounting

for transactions costs and risk has therefore reversed the finding of the earlier lit-

erature.

A particular strength of our analysis is the introduction of a practitioner tech-

nique (quantitative active equity alpha optimization) for optimal portfolio rebal-

ancing subject to risk and transactions costs. This technique is broadly

applicable to simulated trading strategies, but we have not seen it used elsewhere

in the academic literature.

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