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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
2
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
3
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.
4
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.
5
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
6
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.
7
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
8
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
9
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).
10
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.
12
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)
13
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
14
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
15
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
16
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.
18
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
19
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.
20
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
21
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:
22
(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
23
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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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