1

A trending walk rather than a random walk?

Time-series momentum in Australia

Faisal Mahboob

Supervised by1: Vinod Mishra

An Honours Research Essay submitted in partial fulfilment of the requirements

for the Honours degree of Bachelor of Economics, 2013

Department of Economics

Faculty of Business and Economics

Monash University

October, 2013

1 I would like to thank my supervisor Vinod Mishra for his guidance and support in my research

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Table of Contents Abstract ......................................................................................................................................................... 3

1. Introduction .............................................................................................................................................. 4

2. Literature Review ...................................................................................................................................... 7

2.1 Cross-sectional vs Time-series ............................................................................................................ 8

2.2 Momentum and Autocorrelation ........................................................................................................ 9

2.3 A Trending Walk? .............................................................................................................................. 11

2.4 Momentum Explanations .................................................................................................................. 12

2.5 Australian Evidence ........................................................................................................................... 12

3. Data and Preliminaries ............................................................................................................................ 14

3.1 Returns .............................................................................................................................................. 14

3.2 Ex Ante Volatility Estimates .............................................................................................................. 15

4. Time-series Momentum .......................................................................................................................... 17

4.1 Time-series Predictability .................................................................................................................. 17

4.2 Time-series Momentum Trading Strategy ........................................................................................ 20

5. Performance Evaluation of Time-series Momentum .............................................................................. 21

5.1 Sharpe Ratios .................................................................................................................................... 21

5.2 Factor Models ................................................................................................................................... 22

6. Conclusion ............................................................................................................................................... 25

Reference List .............................................................................................................................................. 27

Appendix (figures and tables) ..................................................................................................................... 36

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Abstract

Following the methodology proposed by Moskowitz, Ooi & Pederson (2012), we report the

presence of time-series momentum in ASX 100 stocks. We note that time-series momentum is

present only if a large cross-section of the stocks are examined. Even with the presence of time-

series momentum, the random walk hypothesis cannot be dismissed if too few stocks are

analysed. Using a sample of 69 stocks for the period January 2000 to January 2013, we find that

the most profitable strategy is a “look-back” period of 3 months and a holding period of 12

months. This strategy was found to be profitable even after adjusting for exposures to the three

Fama-French and additional Carhart risk factors. The strategy is not available to retail investors

and we caution that future research should be conducted with as many assets as possible, before

a market wide conclusion can be drawn.

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1. Introduction

One of the puzzling questions in the asset pricing literature is the existence of momentum in

asset returns. This study examines the existence of this anomaly in an Australian context through

a novel perspective. The financial economics literature usually refers to cross-sectional

momentum (CSM) when referring to the phenomenon of momentum. This study looks at time-

series momentum (TSM).

In finance, a phenomenon is considered an anomaly when an asset’s returns follow a regular

pattern, which is reliable, widely known yet it cannot be explained by an asset pricing model.

Said differently, the higher average returns observed cannot be attributed to higher risk

undertaken and vice versa. Under the framework of the Efficient Market Hypothesis (EMH),

investors and traders will eliminate the anomaly by adjusting their trading patterns so as to

exploit the widely known anomaly. Moreover, EMH proponents argue that these anomalies

cannot be exploited, even if they are present, due to real world aspects related to trading and

investing such as transaction costs, credit constraints and institutional/regulatory framework. Due

to the complexity involved with incorporating different forms of trading costs, asymmetric

information and funding constraints, we will not be concerned with such issues and whether or

not the strategy is profitable in the real world. Instead, we will be looking for systematic patterns

that persist in the Australian stock market assuming that the investor can buy or short a stock at a

moment’s notice and the investor has only these two choices.

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Momentum is defined as the systematic pattern where high (low) asset returns are followed by

additional periods of higher (lower) asset returns. Even after accounting for risk measured

through exposures to different factors, assets seem to exhibit abnormal returns if a momentum

strategy is followed. Empirical evidence suggests that momentum strategies exhibit abnormal

performance in almost all markets and asset classes around the world (see Asness, Moskowitz &

Pedersen, 2013). It is to be noted, however limited, that there is also some evidence of the

reverse (see DeBondt & Thaler, 1985). The pattern of higher (lower) asset returns followed by

lower (higher) asset returns is referred to as reversals or contrarian strategies in the literature.

Fama & French (2008) consider momentum, specifically CSM, to be the premier anomaly. They

do not find much support for reversal or contrarian strategies or other anomalies that have been

documented. Despite the enormous research into momentum, there has not been any consensus

on the underlying reasons and causes of the phenomenon. Moskowitz, Ooi & Pederson (2012)

find TSM across different asset classes and markets around the world. Rather than observing a

security’s returns relative to each other as done in the case of CSM, only the security’s past

returns are of concern. There have been many studies looking at the effects of CSM in Australia.

However, no detailed work on TSM in Australian markets has been carried out to date. We hope

to fill this gap through this study.

This study follows a slightly different route from that taken by Moskowitz, Ooi & Pederson

(2012). They show TSM in futures and forwards of different asset classes whereas this research

is focused only on individual stocks. There are three reasons for this departure. First, the

Australian stock market is much more liquid with more readily available data than the Australian

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futures market. Second, we want to check whether this anomaly can persist in one particular

asset class. Third, undertaking research into TSM at a more micro level can provide insights into

whether this pattern only exists in markets where the majority of market participants are

institutional (futures and forward markets) or it exists also in the stock market where the majority

of market participants are retail investors. This can shed light on whether the strategy should be

considered by retail traders and investors.

Most importantly, finding TSM will pose a significant challenge to the Random Walk Hypothesis

(RWH), which is a statistical description of price changes being unforecastable given past

information. RWH is usually used as a model to test for market efficiency. However, a rejection

of the RWH does not imply a rejection of the sophisticated notions of market efficiency. Under

RWH, past prices should not be indicative of whether future prices will go up or down regardless

of whether the price represents true underlying value of the asset. Undertaking this research can

contribute to the debate on asset pricing and whether the RWH should be dismissed as a model

or not.

The rest of this research essay is organised as follows. Section 2 presents an overview of the

development of momentum as an anomaly and related relevant literature. Section 3 describes the

data used in the study and the various data related issues. Section 4 describes the methodology

used in the study and documents time-series momentum in the ASX 100. Section 5 tests for

whether time-series momentum strategy leads to abnormal performance. Section 6 concludes.

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2. Literature Review

Empirically, it has been found that momentum strategies are profitable in almost all markets and

asset classes around the world. It has been found as early as in the Victorian Age by Chabot,

Ghysels & Jagannathan (2009) using a new hand-collected data set on 1,808 stocks listed on the

London Stock Exchange between 1866 and 1907.

Jegadeesh & Titman (1993) were the first to document momentum. They found momentum in

individual securities in the stock market through the formation of portfolios of stocks on the

NYSE and AMEX from 1965 to 1989. Fama & French (1998), Rouwenhorst (1998), Liew &

Vassalou (2000), Griffin, Ji & Martin (2003) and Chui, Wei & Titman (2000) extended the

analysis and detected momentum in 40 other international stock markets such as Argentina,

Australia and India. Asness, Liew & Stevens (1997) and Bhojraj & Swaminathan (2006)

detected the phenomenon in 38 country equity indices such as Morocco and Venezuela.

Shleifer & Summers (1990), Kho (1996), and LeBaron (1999) identified momentum in

currencies such as the Japanese yen and German Mark and Gorton, Hayashi & Rouwenhorst

(2008) found momentum in 31 commodity futures between 1969 and 2006. Finally, Asness,

Moskowitz & Pedersen (2013) observed the effect within and across asset classes such as bonds,

stocks, currencies, equity indices and commodity markets around the world.

Instead of focusing on returns on individual stocks, Moskowitz & Grinblatt (1999) and Grundy

& Martin (2001) added other variables such as industry factors to explain abnormal returns from

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momentum. They concluded industry momentum and not momentum in the firm-specific

component of returns helped explain momentum strategy’s abnormal performance. It is mostly

industry momentum that contributes to the momentum effect in stock returns. Finally, Asness,

Porter & Stevens (2000) found both inter- and intra-industry momentum in industries such as

Tobacco Products and Candy & Soda when they analysed all the firms listed on the NYSE,

AMEX, and Nasdaq stock exchanges from July 1963 (1973 for Nasdaq firms) through December

1998.

2.1 Cross-sectional vs Time-series

A distinction has to be made between cross-sectional momentum (CSM) and time-series

momentum (TSM). They may be related but are not the same thing. CSM refers to when a

security with higher (lower) returns in previous periods relative to its peers tend to have higher

(lower) future returns for future periods where the periods are not necessarily equal. In the

current finance literature, momentum is usually inferred to be CSM rather than TSM. The

literature covered so far has been on CSM.

On the other side of the spectrum is time-series momentum (TSM). TSM focuses purely on a

security’s own return rather than its returns relative to other securities. If a security has exhibited

high (low) returns in previous periods, it will continue to exhibit high (low) returns in the future

but not necessarily for the same number of periods forecasted.

TSM has not been widely studied yet. Only one published paper, that of Moskowitz, Ooi &

Pederson (2012), so far has explicitly mentioned TSM. There have been other papers in currency

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markets such as Lustig et al. (2011) and Rafferty (2010) following quite close, but not exactly,

the TSM methodology. They use an indicator or a trading rule as the signal to buy or sell a

security rather than constructing a trading strategy around sign predictability (discussed in

section 4.1). However, working papers by Burnside, Eichenbaum & Rebelo (2011) and Baltas &

Kosowski (2012) do follow the TSM methodology exactly. Burnside, Eichenbaum & Rebelo

(2011) use the methodology to try and explain why the strategy works whereas Baltas &

Kosowski (2012) document TSM in futures markets and their relationships with funds

employing similar strategies such as commodity trading advisors (CTAs) in the USA.

2.2 Momentum and Autocorrelation

Research into return predictability is very broad. It involves conducting various econometric

tests such as the Wald test. Economic variables are investigated and economic relationships over

various time horizons are examined. Our research agenda is only concerned with past returns

since we want to investigate the anomaly of momentum.

It is possible that momentum could just be an alternate way to exploit the pattern of small,

statistically significant, high frequency predictability in past returns documented since Fama

(1965). There is empirical evidence of positive and negative return autocorrelations (i.e. the

correlation of last period’s return with this period’s) at different time horizons in different asset

classes and various countries. Notable literature would be Fama & French (1988), Lo &

Mackinlay (1988), Poterba & Summers (1988), Cutler, Poterba & Summers (1991).

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Momentum incorporates more than just autocorrelation. Lo & Mackinley (1990) and Lewellen

(2002) have argued that CSM might be caused by autocorrelation in returns, lead-lag relations

among stocks (cross-serial correlation) or due to the cross-sectional variance in the unconditional

means of each stock’s returns. Therefore, not only does the stock’s own past returns predict

higher returns but so does the stock’s past returns being negatively correlated with the lagged

returns of other stocks and the stock’s long run average of returns being higher relative to other

stocks.

Moskowitz, Ooi & Pederson (2012) extend this line of reasoning and argue that TSM is

composed only of an autocorrelation component and a mean-squared component. A stock’s past

return and the average squared mean returns of each asset predict higher returns. Notice that

CSM and TSM are related through the common component of autocorrelation in returns. They

go on to show empirically that the primary driver of TSM, for a given strategy of “look-back”

and holding periods, is the autocorrelation in returns. We refer the reader to their paper for

further details and supporting empirical work.

Previous discussion might lead to the belief that TSM is analogous in structure to

autocorrelation. However, the superior predictability of TSM over autocorrelation is extracted

from looking at a different number of periods of past returns to those forecasted. This flexibility

is not possible with the autocorrelation studies which masked a lot of the predictability since the

number of periods of past returns and forecasted were exactly the same. Autocorrelation is a

necessary precondition for momentum, whether cross-sectional or time-series, to exist but it is

not a conclusive proof of momentum.

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2.3 A Trending Walk?

The Random Walk Hypothesis (RWH) asserts that stock prices follow a random walk, that prices

are unpredictable. Although used as a test for a particular form of market efficiency, a random

walk is a precise mathematical formulation of a stochastic process formed by successive

summation of independent, identically distributed random variables. Market efficiency, on the

other hand, is more arbitrary and harder to test as there are various models and interpretations2. A

layman definition of market efficiency asserts that investors/traders cannot earn returns above the

average stock market return on a consistent basis. An implication of the rejection of RWH is to

also discredit a form of market efficiency. Even so, this study will only scrutinize the RWH and

not consider any implications for market efficiency.

The anomaly of TSM is a direct challenge to RWH since the anomaly suggests past returns can

be used to forecast future returns while not being captured by standard risk factors such as Fama-

French factors SMB and HML and Carhart factor UMD. Furthermore, there are no constructions

of portfolios of securities. Lo & MacKinlay (1990) and Conrad & Kaul (1998) argue that the

cross-sectional variation in returns rather than the time-series component lead to momentum

profits. They initially assumed that individual stocks followed a random walk, yet returns from a

momentum strategy were positive. Thus, they concluded that time-series predictability is

negligible. However in TSM, it is only the security’s own past returns that determine its

abnormal returns. It is through examining correlations from different time horizons that gives

rise to the time-series predictability. Moskowitz, Ooi & Pederson (2012) document significant

TSM in various asset classes (except individual stocks) and across 58 liquid instruments with

2 Guerrien & Gun (2011) provide a neat discussion of the issues involved in their article "Efficient Market

Hypothesis: What are we talking about?" real-world economics review 56 (2011): 19-30.

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over 25 years of data. To date, this paper poses the most significant challenge to the RWH.

2.4 Momentum Explanations

Despite the enormous research into momentum, there has not been any consensus on the

underlying reasons and causes of the phenomenon. There are behavioural explanations (See

Tversky and Kahneman (1974), Barberis, Shleifer & Vishny (1998), Daniel, Hirshleifer &

Subrahmanyam (1998), Hong & Stein (1999), Frazzini (2006)). An interesting paper by Chui,

Titman, & Wei (2010) suggested that momentum returns could be due to cross-country cultural

differences which were measured using an individualism index developed by Hofstede (2001).

Chordia & Shivakumar (2002) attempt to link momentum returns to macroeconomic factors

which are the dividend yield, default spread, bond yields and term structure spread. On the other

hand, Cooper, Gutierrez & Hameed (2004) do not find any such relationship. Momentum returns

have been shown to be linked to firm-specific factors such as size (Hong, Lim & Stein (2000)),

credit rating(Avramov, Chordia, Jostova & Philipov (2007)), revenue growth volatility (Sagi &

Seascholes (2007)) and likelihood of bankruptcy (Eisdorfer, (2008)).

There is also evidence of a link between momentum returns with trading volume (Lee &

Swaminathan (2000)), transaction costs (Lesmond, Schill & Zhou (2004) and Korajczyk &

Sadka (2004)) and information such as analyst coverage (Hong, Lim & Stein (2000)).

2.5 Australian Evidence

In Australia, there has been considerable research into CSM. Drew, Veeraraghavan & Ye (2007)

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find CSM by examining its relationship with “trading volume” while observing periods between

1988 to 2002 in stocks. Brailsford and O’Brien (2008) find CSM in the top 500 stocks by market

capitalisation and relate it to firm size observing periods from 1979 to 2005. Hurn and Pavlov

(2003), Demir et al. (2004) and Bettman et al. (2009) all reported the profitability of CSM

strategies in the ASX stocks. Galariotis (2010) provides comprehensive and robust tests of CSM

on the ASX 100 stocks and all market securities for different time periods and market states.

Indeed, the author claims that the momentum effect is greater in Australia than most of the other

developed markets such as European countries. O'Brien, Brailsford and Gaunt (2010) suggest

that the momentum effect may just be due to the small size effect which is a stock return

characteristic where firms with low market capitalization have higher returns due to inherent

higher risk. Nonetheless, they find CSM for portfolios of firms with large market capitalization.

CSM, in general, has been found and has been suggested to be strong enough for earning above

average risk-adjusted returns in Australia.

As mentioned before, examining TSM is closely related to the calculation of autocorrelations of

stock returns. Autocorrelation studies in Australia have been documented since 1969. Praetz

(1969), Officer (1975), Brown, Keim, Kleidon & Marsh (1983) and Groenewold & Kang (1993)

have found autocorrelations with differing lags. More recently, Gaunt & Gray (2003) examined

autocorrelations structures of returns on the top and bottom 200 Australian stocks by market

capitalisation. They report statistically significant autocorrelations only for the bottom 200

Australian stocks. Any autocorrelation found was due to either illiquidity or the small firm effect

as Australian companies are generally smaller than other developed nations’ stock markets.

There is a general consensus among the authors of these studies of not being able to trade

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profitably based on returns exhibiting autocorrelation if transaction costs were taken into

account.

3. Data and Preliminaries

We will discuss here the data source, the benchmarks used and the construction of the ex ante

volatility estimates in our analysis.

3.1 Returns

Monthly returns index on the one hundred stocks part of the ASX 100 index from the period July

1988 to January 2013 was retrieved from Datastream. The ASX 100 index was chosen due to its

liquidity and companies with high market capitalisation. It covers firms with large and medium

market capitalisation. Only stocks from the ASX 100 were analysed to avoid market

microstructure effects such as stale prices and illiquidity from contaminating the results.

Monthly returns were used rather than returns of a higher or lower frequency due to the issue of

sign predictability. This will be covered in more detail in section 4.2.

The stocks were split into two samples, “Sample 1” and “Sample 2”, and a subsample of Sample

1 called “Sample 3”. This allowed us to have samples where the time-dimensions of the returns

matched up with each other for each stock. For the analysis there was a natural trade-off between

the number of stocks and stocks with the longest history of data. Sample 1 has a higher number

of past returns ranging from July 1988 to January 2013 with 18 stocks whereas Sample 2 has a

higher number of stocks with 69 stocks with historical data ranging from January 2000 to

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January 2013. Sample 3 consisted of all the stocks in Sample 1 except with returns from January

2000 to January 2013. This allowed us to carry out a comparison between Sample 2 and Sample

3. Fundamentally, the rationale between such sample selections was data availability as many

stocks did not exist from 1988. Note that there is an overlap of the same stock between the

samples.

Sample 1 started out with twenty stocks whereas Sample 2 had 71 stocks. Some stocks’ returns

were zero for a significant period of time due to non-trading months where a period is one

month. A data cleaning rule was put in place to deal with this issue. Any stock with more than

ten periods of returns of 0% was deleted. Two stocks needed to be deleted from both the samples

according to this rule. This was followed with linear interpolation to fill the returns of those

periods for those stocks which had less than ten periods of returns of 0%.

3.2 Ex Ante Volatility Estimates

To allow comparison, each stock’s return will need to be scaled by its volatility since different

stocks exhibit different levels of volatility. Every stock’s return will be divided by its volatility

. The use of the volatility last period, called ex ante volatility, is used to avoid any look-

ahead bias.

Look-ahead bias occurs when historical data is used in testing a strategy that would not have

been known or available during the period being analysed. Similarly, more sophisticated

volatility models such as GARCH were avoided since they require parameter estimates over the

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whole sample to generate the volatility estimates. These parameter estimates would have

incorporated the look-ahead bias.

The ex ante volatility was estimated using a close-to-close estimator as a proxy for realized

volatility (Shu, Jinghong, and Jin E. Zhang, 2003). These volatility estimates will be used in

section 4.1 to scale the coefficients in our regressions. Realized volatility is a specific measure of

historical volatility. Data is sampled at a very high-frequency to compute ex-post volatility at a

lower frequency. In our analysis, daily returns have been used to calculate monthly volatility.

It is calculated as follows:

(1)

where is 21 since an average month is assumed to have 21 trading days, is the daily return

on date .

3.3 Asset Pricing Benchmarks

To check the performance of the strategies, we evaluate the returns of a given strategy relative to

excess market index return, Fama-French factors SMB (‘Small Minus Big’) and HML (‘High

Minus Low’) and Carhart factor UMD (‘Up Minus Down’)3. The SMB factor is constructed by

sorting the difference in returns of stocks with low market capitalisation (called ‘Small’) to those

with high market capitalisation (called ‘Big’). Similarly, HML factor is constructed by sorting

3 We are extremely grateful to Stefano Marmi for making this data freely available at:

http://homepage.sns.it/marmi/Data_Library.html#Australia

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the difference in returns of stocks with high book value to market value ratio (called ‘High’) to

those with low book value to market value ratio (called ‘Low’).

The specifications of the factors were made by Stefano Marmi and Flavia Poma as including all

stocks for July of year to June of which have market equity data for the last fiscal year

end before March and June of time , and positive book equity data for the last fiscal year end

before March . These factors capture particular characteristics of stock returns which have been

considered as risk in empirical finance research.

4. Time-series Momentum

We first inspect time-series predictability of a stock’s return across different time horizons at the

monthly level. Then, we construct a trading strategy to exploit the time-series predictability to

test for TSM similar to Moskowitz, Ooi & Pederson (2012). To determine whether there is time-

series predictability, we search for a pattern in the t-statistics of our regressions. The pattern

should exhibit positive and significant t-statistics for the first couple of months followed by

insignificant t-statistics. This pattern suggests a return continuation for the first couple of months

with no trend after.

4.1 Time-series Predictability

There are two possible ways to discern time-series predictability. One way is through a

regression on lags and the other is to look at direction-of-change. Please note robustness checks

have not been made in this study which is a major limitation.

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Current literature suggests that there is a direct connection between asset return volatility

dependence and asset return sign dependence. If that is the case, then the sign of next period’s

return can be predicted. There is considerable literature showing asset return sign forecasting can

done successfully: Breen et al. (1989), Leitch & Tanner (1991), Wagner et al. (1992), Pesaran &

Timmermann (1995), Kuan & Liu (1995), Larsen & Wozniak (1995), Womack (1996), Gencay

(1998), Leung et al. (2000), Elliott & Ito (1999), White (2000), Pesaran & Timmermann (2004),

and Cheung et al. (2003). These studies usually analysed stocks in the USA. An important point

to note is that direction-of-change can only be forecasted at the monthly level (Christoffersen &

Diebold, 2006). Hence, our analysis is based only on monthly returns data.

We regress the stock’s excess return at month on its return lagged by months. The

regression is a pooled time-series regression. Both the returns and their lagged values are scaled

by their ex ante volatility as defined in section 3.2:

(2)

We stack all the scaled returns of all the dates and compute the t-statistics using lags from

Figure 2 and figure 3, shown in the appendix, plots t-statistics from the pooled time-series

regressions of Sample – 1 and Sample – 2 by month lag h. The positive t-statistics indicate

significant upward trend of returns and negative t-statistics indicate trend reversals. The first

sample seems to continue exhibiting TSM at higher lags such as 43 and 47 with statistically

significant correlation which is an unlikely event. The second sample is a bit more reasonable as

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it displays the pattern we are trying to detect. Nevertheless, it still displays statistically

significant negative correlation between the return at lagged month 60 with the return at month .

The second way to look at time-series predictability is by observing only the sign of the past

excess return. It is this concept that underlies the trading strategy as it exploits the fact that, at

least in sample, there is information in the sign of about the sign of (Burnside,

Eichenbaum and Rebelo, 2011):

(3)

The left-hand side of the regression is scaled again and the right-hand side doesn’t require so

since it only take the value of +1 or -1. The use of dummy variables was avoided as they would

not be able to capture the ability to short the stock which is captured by -1. The results are

similar to the first regression and we obtain almost the same pattern. Sample – 1 does not seem

to show any sort of pattern for time-series predictability whereas Sample – 2 does. This is visible

from the pattern of return continuation exhibited by Sample – 2 only through positive, significant

t-statistics for the first couple of months followed by insignificant and smaller t-statistics.

There is considerable possibility of data snooping bias due to our small samples which occurs

when a model is fit to random historical patterns that makes performance of a strategy look

superior. The continuing significance of t-statistics and lack of negative signs for the t-statistics

at higher lags is worrisome. However, the point of the regression is to only look for the pattern of

return continuation mentioned before. We see that it has been the case where the first few lags

are more significant than the rest. This is especially true and clear for Sample – 2 as shown in the

appendix.

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4.2 Time-series Momentum Trading Strategy

We explore different trading strategies based on TSM. There are three steps involved: a) we lag

the returns for different number of months which is called the “look-back” period b) we consider

whether the past excess return over the “look-back” period is positive or negative which will

determine whether the investor goes long or short c) we then calculate the returns based on the

number of months the stock is held for called the “holding-period”. We have avoided setting the

position size to be inversely proportional to the asset class’s ex ante volatility like Moskowitz,

Ooi & Pederson (2012) since we are only analysing one asset class which is stocks.

Each trading strategy of a particular “look-back” period and holding period gives a single time-

series of monthly returns. This single series is derived following the methodology used by

Jegadeesh and Titman (1993) such that the return at time represents the average return of every

stock that is being held from the past. As an example, suppose an investor has decided to hold for

two periods and bought a stock on January and another on February. There is an overlap of

returns as the return on March is dependent on the stock bought on January’s second holding

period return and the stock bought on February’s first holding period. Said differently, the stock

bought on January is still “active” in March. We take this into account by taking the arithmetic

mean of both the returns in March after having incorporated sign predictability.

To determine whether the investor/trader goes long or short we multiply the returns found earlier

by either +1 (long) or -1 (short). This is how we integrate sign predictability into our trading

strategy. For a given stock, the time- return is based on the sign of the past return from

to . We then compute the time- return based on the sign of the past return from

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to , and we continue until we get to a point where the sign of the past return is the final

return that is still being used from to . For each trading strategy of a particular

(k,h), we get a monthly time-series of positive and negative returns by taking the average of all

of the currently “active” stocks.

Finally, the returns are converted to excess returns, defined as returns minus the risk free rate.

This will remove any predictability there may be in returns from the inclusion of the risk free

rate and allow us to run factor models. We use the risk free rate used by Stefano Marmi and

Flavia Poma of yields on 90 days Australian bank-accepted bills This gives us the TSM excess

returns,

.

5. Performance Evaluation of Time-series Momentum

To determine whether the strategy can lead to abnormal returns after adjusting for risk, we look

at Sharpe ratios and alphas from the Fama-French three-factor model (Fama & French, 1993)

and the Carhart four-factor model (Carhart, 1997). Sharpe ratio (Sharpe, 1966) is the ratio of

expected returns less the risk-free rate over the standard deviation of the return. The factor

models are regressions of the returns against patterns of stock market return’s related to specific

characteristics which are considered risk.

5.1 Sharpe Ratios

Assuming that the returns are independently and identically distributed, we compute the

annualized Sharpe ratio from monthly TSM returns using:

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(4)

Figure 1 shows the Sharpe ratios for Sample – 1, Sample – 2 and Sample – 3 for only one

strategy. Only the strategy of a “look-back” and holding periods of 3 months and 12 months

respectively is regarded as it is the most profitable in our analysis using factor models discussed

in section 5.2.

The Sharpe ratios were calculated by first creating equally-weighted portfolios from a (3,12)

strategy and taking arithmetic means of the returns, risk free rates and standard deviations over

the whole sample. As expected, the ratios are above 1 suggesting that average differential return

is greater than per unit of historic variability and the strategy is reasonably profitable. Sample – 2

has higher Sharpe ratio than either of the samples suggesting a possibility of improvement in the

returns of the strategy from expanding investments into further assets.

5.2 Factor Models

We compute the alphas from the following time-series regression, the Fama-French three-factor

model:

(5)

is the overall risk of the stock market. This controls for risk the investor could have taken

by just buying an index fund rather than investing in a stock. controls for risk attributed to

investing in ‘small’ firms. These are usually stocks which have low market capitalization.

controls for risk taken on by the investor through investments in ‘value’ companies. These are

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usually stocks with high ratios of book value to market equity value. The exact details of the

construction of the factors are available on Stefano Marmi’s website4.

Table 1 and table 2 in the appendix show the t-statistics of the estimated alphas for different

combinations of the strategy for Sample – 1 and Sample – 2 respectively. The strategy backtested

on Sample – 1 exhibit extraordinary levels of abnormal performance as some of the t-statistics

are significant and extremely high. We conjecture that errors are arising from this sample as a

result of too few stocks. The overall average returns from TSM are not being captured. Note that

the strategy seems to try and extend the “look-back” and holding periods as far as possible. We

infer from this that the strategy is merely trying to capture the largest historic trends which is

unreasonable as a trading strategy. On the other hand, Sample – 2, shows results consistent with

Moskowitz, Ooi & Pederson (2012) and also show above average risk-adjusted returns. There is

abnormal performance for many grids of “look-back” and holding period combinations. Most

notably is the strategy with a “look-back” period of 3 months and a holding period of 12 months

which yields a positive, significant t-statistic of 6.48. We find that Sample – 2 exhibits abnormal

performance due to the presence of many assets. The strategy is most profitable when applied

across a spectrum of many assets, especially across different asset classes as done by Moskowitz,

Ooi & Pederson (2012).

We also compute the alphas from the following Carhart four-factor model:

(6)

4 http://homepage.sns.it/marmi/Data_Library.html#Australia

24

This regression is similar to the Fama-French three-factor model with the addition of .

This factor controls for risk taken on by the investor by investing in stocks that exhibit cross-

sectional momentum.

Table 3 and 4 of the appendix show the t-statistics of alphas from the Carhart four-factor model

on Sample – 1 and Sample – 2 respectively. Sample – 1’s t-statistics have decreased but they are

still not valid results. Sample – 2 has more interesting results since the t-statistics of the alphas

are more reasonable. The t-statistics have decreased but the significant ones have not become

insignificant with the addition of the factor. The strategy with a “look-back” period of 3

months and a holding period of 12 months again yields the highest positive and significant t-

statistic of 5.82. We can deduce that TSM excess returns, though related, are quite different from

CSM’s excess returns as the factor has not fully captured the abnormal performance of

TSM.

For Sample – 3, only the sign predictability regression and Fama-French three-factor model were

done for the analysis. From the sign predictability regression results in figure 6 in the appendix

we can see that most of the t-statistics are not statistically significant. Only lags 7, 9 and 19 are

significant which does not allow for much inference. Similar to Sample – 1, the results from the

Fama-French three-factor model shown in table 3 also suggest that the strategy is trying to

capture the largest trends as the t-statistics of alphas increase with “holding” period. However,

there is a disparity. The t-statistics of alphas do not seem to increase with “look-back” periods.

We conjecture that the strategy cannot trace back too far in history for the largest trends due to

the reduced sample size.

25

The study’s results show that a sample with a small number of stocks such as Sample – 1 and

Sample – 3 do not exhibit TSM whereas a sample with many stocks such as Sample – 2 does

exhibit TSM. Our conjecture was made more convincing when the strategy was applied to one

stock. There were no correlations of past returns to this period’s returns. TSM was not present

and it exhibited a random walk. Although the strategy is implementable in the Australian stock

market where most retail investors invest, it is still not available to them. The strategy is only

profitable when the investor is extremely diversified with investments in many stocks.

The inconsistency seems to arise from the role of the mean squared term. Moskowitz, Ooi &

Pederson (2012) find the term to be insignificant to have any effect. We believe that the term

possibly has a negative impact on the strategy if invested in too few assets and becomes

insignificant when implemented across many assets. The latter case is where the autocorrelation

in returns can be exploited fully to implement a TSM strategy.

6. Conclusion

Significant time-series momentum was found in a larger sample of stocks of the ASX 100 than in

a smaller sample. The larger sample consists of 69 stocks from the ASX 100 with returns from

the year 2000 to 2013 whereas the small sample consists of only 18 stocks from the ASX 100

with returns from the year 1988 to 2013.

Similar to Moskowitz, Ooi & Pederson (2012), the larger sample exhibits time-series momentum

for the first 12 months. However, the “look-back” period is different. The most profitable

26

strategy is a “look-back” of 3 months and a holding period of 12 months as it results in an alpha

that is the most significant and has the highest t-statistic. Time-series momentum is different

from cross-sectional momentum as the cross-sectional momentum factor doesn’t fully capture

time-series momentum’s excess returns.

We note that time-series momentum can persist in one asset class such as stocks and not only in

the futures and forward markets. Although the majority of market participants are retail

investors, the strategy is not optimal for retail investors as it requires access to many liquid

assets.

Time-series momentum is a direct challenge to the random walk hypothesis. Even in the

presence of time-series momentum, the random walk hypothesis cannot be completely dismissed.

When a smaller sample size was used, no time-series momentum was found which supported the

random walk hypothesis. Caution needs to be taken to ensure a wide range of assets are

considered when implementing the strategy or undertaking future research.

27

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36

Appendix

Figure 1

Regression 1 – Sample 1

Figure 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sample - 1 Sample - 2 Sample - 3

An

nu

aliz

ed

Sh

arp

e R

atio

Sharpe ratios of (k,h) = (3,12) strategy

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

t-st

atis

tic

Month lag

37

Regression 2 – Sample 1

Figure 3

Regression 1 – Sample 2

Figure 4

-4

-3

-2

-1

0

1

2

3

4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 t-st

atis

tic

Month lag

-6

-4

-2

0

2

4

6

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

t-st

atis

tic

Month lag

38

Regression 2 – Sample 2

Figure 5

Regression 2 – Sample 3

Figure 6

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

t-st

atis

tic

Month lag

-3

-2

-1

0

1

2

3

4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

t-st

atis

tic

Month lag

39

Fama-French three-factor model

Table 1

Table 2

Table 3

40

Carhart four-factor model

Table 4

Table 5