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Two centuries of trend following

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  • Capital Fund Management

Abstract and Figures

We establish the existence of anomalous excess returns based on trend following strategies across four asset classes (commodities, currencies, stock indices, bonds) and over very long time scales. We use for our studies both futures time series, that exist since 1960, and spot time series that allow us to go back to 1800 on commodities and indices. The overall t-stat of the excess returns is $\approx 5$ since 1960 and $\approx 10$ since 1800, after accounting for the overall upward drift of these markets. The effect is very stable, both across time and asset classes. It makes the existence of trends one of the most statistically significant anomalies in financial markets. When analyzing the trend following signal further, we find a clear saturation effect for large signals, suggesting that fundamentalist traders do not attempt to resist "weak trends", but step in when their own signal becomes strong enough. Finally, we study the performance of trend following in the recent period. We find no sign of a statistical degradation of long trends, whereas shorter trends have significantly withered.
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Journal of Investment Strategies 3(3), 41–61
Two centuries of trend following
Yves Lempérière
Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France;
email: yves.lemperiere@cfm.fr
Cyril Deremble
Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France;
email: cyril.deremble@cfm.fr
Philip Seager
Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France;
email: philip.seager@cfm.fr
Marc Potters
Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France;
email: marc.potters@cfm.fr
Jean-Philippe Bouchaud
Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France;
email: jean-philippe.bouchaud@cfm.fr
(Received April 14, 2014; revised May 15, 2014; accepted May 19, 2014)
We establish the existence of anomalous excess returns based on trend-following
strategies across four asset classes (commodities, currencies, stock indexes and
bonds) and over very long time scales. We use for our studies both futures time
series that have existed since 1960 and spot time series that allow us to go back to
1800 on commodities and indexes. The overall t-statistic of the excess returns is
approximately equal to five since 1960 and approximately equal to ten since 1800,
after accounting for the overall upward drift of these markets. The effect is very
stable, across both time and asset classes. It makesthe existence of trends one of the
most statistically significant anomalies in financial markets. When analyzing the
trend-following signal further, we find a clear saturation effect for large signals,
suggesting that fundamentalist traders do not attempt to resist “weak trends”,
but step in when their own signal becomes strong enough. Finally, we study the
performance of trend following in the recent period.We find no sign of a statistical
degradation of long trends, whereas shorter trends have significantly withered.
This work is the result of many years of research at Capital Fund Management (CFM). Many
colleagues must be thanked for their insights, in particular P. Aliferis, N. Bercot,A. Berd, D. Challet,
L. Dao, B. Durin, P. Horvai, L. Laloux, A. Landier, A. Matacz, D. Thesmar, T. Tu and M. Wyart.
41
42 Y. Lempérière et al
1 INTRODUCTION
Are markets efficient, in the sense that all public information is included in current
prices? If this were so, price changes would be totally unpredictable in the sense
that no systematic excess return based on public information could be achievable.
After decades of euphoria in economics departments,1serious doubts were raised by
behavioral economists, who established a long series of pricing “anomalies” (Schw-
ert 2003). The most famous of these anomalies (and arguably the most difficult to
sweep under the rug) is the so-called excess volatility puzzle, unveiled by Shiller and
others (Leroy and Porter 1981; Shiller 1981). Strangely (or wisely?) the 2013 Nobel
committee decided not to take sides, and declared that markets are indeed efficient (as
claimed by laureate Eugene Fama), but that the theory actually makes “little sense”
(as argued by Robert Shiller, who shared the same prize!).2(See also de Bondt and
Thaler (1985), Black (1986) and Summers (1986) for insightful papers on this debate.)
In the list of long-known anomalies, the existence of trends plays a special role.
First, because trending is the exact opposite of the mechanisms that should ensure
that markets are efficient, ie, reversion forces that drive prices back to the purported
fundamental value. Second, because persistent returns validate a dramatically simple
strategy, “trend following”, which amounts to buying when the price goes up, and
selling when it goes down. Simple as it may be (Covel 2009), this strategy is at the heart
of the activity of commodity trading advisors (CTAs; Bartas and Kosowski 2012), an
industry that now manages (as of 2013 Q4) no less than US$325 billion, representing
around 16% of the total assets of the hedge fund industry, and accounting for several
percent of the daily activity of futures markets (Mundt 2014).3These numbers are
by no means small, and make it hard for efficient market enthusiasts to dismiss this
anomaly as economically irrelevant.4The strategy is furthermore deployed over a wide
range of instruments (indexes, bonds, commodities, currencies, etc) with positive
reported performance over long periods, suggesting that the anomaly is to a large
extent universal, across both epochs and asset classes.5This reveals an extremely
1“There is no other proposition in economics which has more solid empirical evidence supporting
it than the efficient market hypothesis”, as M. Jensen famously wrote in 1978.
2Together with a third scientist, Lars Hansen, who had not directly taken part in the debate.
3Futures markets allow traders to go short as easily as going long. Therefore, both up-trends and
down-trends can be exploited equally.
4Jensen (1978) actually stressed the importance of trading profitability in assessing market effi-
ciency. In particular, if anomalous return behavior is not definitive enough for an efficient trader to
make money trading on it, then it is not economically significant.
5Note that the excess return of trends cannot be classified as a risk premium either (see Lempérière
et al 2014; Narasimhan and Titman 2011). On the contrary, trend following is correlated with
“long-vol” strategies.
Journal of Investment Strategies 3(3)
Two centuries of trend following 43
persistent, universal bias in the behavior of investors who appear to hold “extrapolative
expectations”, as argued in many papers coming from different strands of the academic
literature (see, for example, Bouchaud and Cont 1998; DeLong et al 1990; Greenwood
and Shleifer 2014; Hirshleifer andYu 2012; Hommes et al 2008; Hong and Stein 1999;
Kent et al 1998; Kirman 1991, 1993; Smith et al 1988 and the references therein).
Many academic studies have already investigated this trend anomaly on a wide
range of assets, and have convincingly established its statistical significance in the
last few decades (Clare et al 2012; Szakmary et al 2010). Recently, this time horizon
has been extended to 100 years by Hurst et al (2012), and the effect still exists
unabated. The aim of the present paper is to extend the time horizon even further, to
200 years, as far in the past as we have been able to go in terms of data. We find that
the amplitude of the effect has been remarkably steady over two centuries. This also
allows us to assess the recent weakening of the effect (as testified by the relatively
poor performance of CTAs over the last five years). We show that the very recent past
is fully compatible with a statistical fluctuation. Although we cannot exclude that this
recent period is a precursor of the “end of trends”, we argue theoretically that this is
an unlikely scenario. We give several mechanisms that could explain the existence
and persistence of these trends throughout history.
Note that trends exist not only for market factors such as indexes, bonds and curren-
cies, but also cross-sectionally in stock markets. The so-called momentum anomaly
consists in buying the past winners and selling the past losers in a market-neutral way,
again with a high statistical significance across many decades and different geographi-
cal zones (see Barroso and Santa-Clara 2013; Kent and Moskowitz 2013; Narasimhan
and Titman 1993; see also Narasimhan and Titman (2011) and Geczy and Samonov
(2013) for recent reviews). Although interesting in its own right (and vindicating the
hypothesis that trend following is universal (Asness et al 2013)), we will not study
this particular aspect of trend following in the present paper.
The outline of the paper is as follows. In Section 2, we define the trend-following
indicator used for this study and test its statistical significance on available futures
data. We start with futures since they are the preferred instruments of trend followers
in finance. Also, their prices are unambiguously defined by transparent market trades,
and not the result of a proprietary computation. In Section 3, we carefully examine,
for each asset class, how the available time series can be extended as far in the past as
possible. In Section 4 we then present our results over two centuries, and show how
exceptionally stable long trends have been. We examine more deeply the linearity of
the signal, and find that the trend predictability saturates for large values of the signal,
which is needed for the long-term stability of markets. And finally, in Section 5, we
discuss the significance of the recent performance of the trend in light of this long-term
simulation.
Research Paper www.risk.net/journal
44 Y. Lempérière et al
2 TREND FOLLOWING ON FUTURES SINCE 1960
2.1 Measuring trends
We choose to define our trend indicator in a way similar to simulating a constant risk
trading strategy (without costs). More precisely, we first define the reference price
level at time t,hpin;t, as an exponential moving average of past prices (excluding
p.t/ itself) with a decay rate equal to nmonths. Long simulations can often only
be performed on monthly data, so we use monthly closes. The signal sn.t/ at the
beginning of month tis constructed as
sn.t/ Dp.t 1/ hpin;t1
n.t 1/ ;(2.1)
where the volatility nis equal to the exponential moving average of the absolute
monthly price changes, with a decay rate equal to nmonths. The average strength of
the trend is then measured as the statistical significance of fictitious profits and losses
(P&Ls) of a risk managed strategy that buys or sells (depending on the sign of sn)a
quantity ˙1
nof the underlying contract ˛:6
Q˛
n.t/ DX
t0<t
sgnŒsn.t0/ p.t0C1/ p.t0/
n.t01/ :(2.2)
In the rest of the paper, we will focus on the choice nD5months, although the
dependence on nwill be discussed. Of course, different implementations can be
proposed. However, the general conclusions are extremely robust against changes
to the statistical test or to the implemented strategy (see, for example, Bartas and
Kosowski 2012; Clare et al 2012; Szakmary et al 2010).
In the following, we will define the Sharpe ratio of the P&L as its average return
divided by its volatility, both annualized. Since the P&L does not include interest
earned on the capital, and futures are self-financed instruments, we do not need to
subtract the risk-free rate to compute the Sharpe ratio. The t-statistic of the P&L (ie,
the fact that the average return is significantly different from zero) is therefore given
by the Sharpe ratio times pN, where Nis the number of years over which the strategy
is active. We will also define the drift of a time series as the average daily return of
the corresponding instrument, which would be the P&L of the long-only strategy if
financing costs were to be neglected.
2.2 The pool of assets
Since we wish to prove that trend following is a universal effect not restricted to any
one asset, we would like to test this signal on as large a pool as possible. This is also
6We call this a fictitious P&L since no attempt is made to model any realistic implementation costs
of the strategy.
Journal of Investment Strategies 3(3)
Two centuries of trend following 45
important in practice, since diversification plays an important role in the performance
of CTAs. However, since the purpose of this paper is to backtest the trend on a very
large history, we voluntarily limit ourselves to the contracts for which a long data set
is available. This naturally makes the inclusion of emerging markets more difficult.
Therefore, for indexes, bonds and currencies, we only consider the following seven
countries: Australia, Canada, Germany, Japan, Switzerland, the United Kingdom and
the United States. We believe the results of this section would only be improved by
the choice of a wider pool.
We also need to select a pool of commodities. In order to have a well-balanced
pool, we chose the following seven representative contracts: crude oil, Henry Hub
natural gas, corn, wheat, sugar, live cattle and copper.
In summary, we have a pool made up of seven commodity contracts, seven ten-year
bond contracts, seven stock index contracts and six currency contracts. All the data
used in the current paper comes from Global Financial Data (GFD).7
2.3 The results
Our history of futures starts in 1960, mostly with commodities. As we can see from Fig-
ure 1 on the next page, the aggregated performance P˛Q˛
n.t/ looks well-distributed
in time, with an overall t-statistic of 5.9, which is highly significant. The Sharpe ratio
and t-statistic are only weakly dependent on n(see Table 2 on page 47).
However, we might argue that this comes from the trivial fact that there is an overall
drift in most of these time series (for example, the stock market tends to go up over
time). It is therefore desirable to remove this “long” bias, by focusing on the residual
of the trend-following P&L when the ˇwith the long-only strategy has been factored
in. In fact, the correction is found to be rather small, since the trend-following P&L
and the long-only strategy are only C15% correlated. Still, this correction slightly
decreases the overall t-statistic of the trend-following performance, to 5.0.
In order to assess the significance of the above result, we break it down into different
sectors and decades. As shown in Table 2 on page 47 and Table 3 on page 47, the
t-statistic of the trend-following strategy is above 2.1 for all sectors and all decades,
and above 1.6 when debiased from the drift . Therefore, the performance shown in
Figure 1 on the next page is well-distributed across all sectors and periods, which
strongly supports the claim that the existence of trends in financial markets is indeed
universal. One issue, though, is that our history of futures only goes back fifty years
or so, and the first ten years of those fifty is only made up of commodities. In order to
test the stability and universality of the effect, it is desirable to extend the time series
to go back further in the past, in order to span many economic cycles and different
7See www.globalfinancialdata.com.
Research Paper www.risk.net/journal
46 Y. Lempérière et al
FIGURE 1 Fictitious P&L, as given by (2.2), of a five-month trend-following strategy on a
diversified pool of futures.
1970 1980 1990 2000 2010
0
20
40
60
80
t-statistic D5.9 (corresponding to a Sharpe ratio D0.8). Debiased t-statistic D5.0.
TABLE 1 Sharpe ratio and t-statistic of the trend .T / and t-statistic of the debiased trend
.T /for different time horizons n(in months), since 1960.
Time-scale SR t-statistic t-statistic
n(months) (T)(T)(T)
2 0.8 5.9 5.5
3 0.83 6.1 5.5
5 0.78 5.7 5.0
7 0.8 5.9 5.0
10 0.76 5.6 5.1
15 0.65 4.8 4.5
20 0.57 4.2 3.3
macroenvironments. This is the goal of the next section, which provides a convincing
confirmation of the results based on futures.
3 EXTENDING THE TIME SERIES: A CASE-BY-CASE APPROACH
We now try to find proxies for the futures time series that are reasonably correlated
with the actual futures prices on the recent period but allow us to go back in the
Journal of Investment Strategies 3(3)
Two centuries of trend following 47
TABLE 2 Sharpe ratio and t-statistic of the trend .T / for nD5, of the debiased trend
.T /and of the drift component of the different sectors, and the starting date for each
sector.
SR t-statistic t-statistic SR t-statistic Start
Sector (T)(T)(T)()() date
Currencies 0.57 3.6 3.4 0.05 0.32 May 1973
Commodities 0.8 5.9 5.0 0.33 2.45 Jan 1960
Bonds 0.49 2.8 1.6 0.58 3.3 May 1982
Indexes 0.41 2.3 2.1 0.4 2.3 Jan 1982
TABLE 3 Sharpe ratio and t-statistic of the trend .T / for nD5, of the debiased trend
.T /and of the drift component for each decade.
SR t-statistic t-statistic SR t-statistic
Period (T)(T)(T)()()
1960–1970 0.66 2.1 1.8 0.17 0.5
1970–1980 1.15 3.64 2.5 0.78 2.5
1980–1990 1.05 3.3 2.85 0.03 0.1
1990–2000 1.12 3.5 3.03 0.79 2.5
>2000 0.75 2.8 1.9 0.68 2.15
past a lot further. Natural candidates are spot prices on currencies, stock indexes and
commodities, and government rates for bonds. We shall examine each sector indepen-
dently. Before doing so, however, we should mention other important restrictions on
the use of the historical data. First, we expect trends to develop only on freely traded
instruments, where price evolution is not distorted by state interventions. Also, we
require a certain amount of liquidity, in order to have meaningful prices. These two
conditions, free-floating and liquid assets, will actually limit us when we look back
in the distant past.
3.1 Currencies
The futures time series goes back to 1973. In the previous period (1944–71), the
monetary system operated under the rules set out in the Bretton Woods agreements.
According to these international treaties, the exchange rates were pegged to the US
dollar (within a 1% margin), which remained the only currency that was convertible
into gold at a fixed rate. Therefore, no trend can be expected on these time series,
where prices are limited to a small band around a reference value.
Research Paper www.risk.net/journal
48 Y. Lempérière et al
Prior to this, the dominant system was the Gold Standard. In this regime, the
international value of a currency was determined by a fixed relationship with gold.
Gold in turn was used to settle international accounts. In this regime we also cannot
expect trends to develop, since the value of the currency is essentially fixed by its
conversion rate with gold. In the 1930s, many countries dropped out of this system,
massively devaluing in a desperate attempt to manage the consequences of the Great
Depression (the “beggar thy neighbor” policy). This also led to massively managed
currencies, with little hope of finding any genuine trending behavior.
All in all, therefore, it seems unlikely that we can find a free-floating substitute for
our futures time series on foreign exchange prior to 1973.
3.2 Government rates
Government debt (and default!) has been around for centuries (Reinhart and Rogoff
2009), but in order to observe a trend on interest rates we need a liquid secondary
market, on which the debt can be exchanged at all times. This is a highly nontrivial
feature for this market. Indeed, throughout most of the available history, government
debt has been used mostly as a way to finance extraordinary liabilities, such as wars. In
other periods of history, debt levels gradually reduced, as the principals were repaid,
or washed away by growth (as debt levels are quoted relative to GDP).
As a typical example, we can see in Figure 2 on the facing page that the US debt,
inherited from the War of Independence, fell to practically zero in 1835–6, during the
Jackson presidency. There is another spike in 1860–65, during the American Civil
War, which then gets gradually washed away by growth. We have to wait until World
War I to see a significant increase in debt, which then persists until today. Apart
from Australia, whose debt has grown at a roughly constant rate, and Japan, whose
turning point is around 1905, during the Russo-Japanese War, the situation in all other
countries is similar to that of the United States. From this point onward, the debt has
never been repaid in its totality in any of the countries we consider in this study, and
has mostly been rolled over from one bond issuance to the next.
Another more subtle point can explain the emergence of a stable debt market: at the
beginning of the twentieth century, the monetary policy (in its most straightforward
sense: the power to print money) was separated from the executive instances and
attributed to central banks, supposedly independent of the political power (see Figure 4
on the facing page). This move increased the confidence in the national debt of these
countries, and helped boost subsequent debt levels.
All of this leads us to the conclusion that the bond market before 1918 was not
developed enough to be considered as “freely traded and liquid”. Therefore, we start
our interest rate time series in 1918. We should note as well that we exclude from the
Journal of Investment Strategies 3(3)
Two centuries of trend following 49
FIGURE 2 Global debt of the US government (a) in billions of US dollars and (b) as a
fraction of GDP.
1800 1850 1900 1950 2000
0.0001
0.01
1
100
10 000
1800 1850 1900 1950 2000
0
0.2
0.4
0.6
0.8
1.0
1.2
(b)(a)
TABLE 4 Starting date of the central bank’s monopoly on the issuance of notes.
Country Start
United States 1913
Australia 1911
Canada 1935
Germany 1914
Switzerland 1907
Japan 1904
United Kingdom 1844
The Bank of England does not have this monopoly in Scotland and Ireland, but regulates the commercial banks that
share this privilege.
time series World War II and the immediate post-war period in Japan and Germany,
where the economy was heavily managed, therefore leading to price distortions.
3.3 Indexes and commodities
For these sectors, the situation is more straightforward. Stocks and commodities were
actively priced throughout the nineteenth century, so it is relatively easy to get clean,
well-defined prices. As we can see from Table 5 on the next page and Table 6 on
the next page, we can characterize trend-following strategies for over two centuries
on some of these time series. Apart from some episodes that we excluded, such as
World War II in Germany and Japan, where the stock market was closed, or the period
through which the price of crude oil was fixed (in the second half of the twentieth
Research Paper www.risk.net/journal
50 Y. Lempérière et al
TABLE 5 Starting date of the spot index monthly time series for each country.
Country Start
United States 1791
Australia 1875
Canada 1914
Germany 1870
Switzerland 1914
Japan 1914
United Kingdom 1693
TABLE 6 Starting date of the spot price for each commodity.
Commodity Start
Crude oil 1859
Natural gas 1986
Corn 1858
Wheat 1841
Sugar 1784
Live cattle 1858
Copper 1800
century), the time series are of reasonably good quality, ie, prices are actually moving
(no gaps) and there are no major outliers.
3.4 Validating the proxies
We now want to check that the time series selected above, essentially based on spot
data on ten-year government bonds, indexes and commodities, yield results that
are very similar to those we obtained with futures. This will validate our proxies
and allow us to extend, in the following section, our simulations to the pre-1960
period.
In Figure 3 on the facing page we show a comparison of the trend applied to
futures prices and to spot prices in the period of overlapping coverage between the two
data sets. From 1982 onward we have futures in all four sectors and the correlation
is measured to be 91%, which we consider to be acceptably high. We show the
correlations per sector calculated since 1960 in Table 7 on the facing page and observe
that the correlation remains high for indexes and bonds but is lower for commodities,
with a correlation of 65%. We know that the difference between the spot and futures
prices is the so called “cost of carry”, which is absent for the spot data, this additional
Journal of Investment Strategies 3(3)
Two centuries of trend following 51
FIGURE 3 Trend on spot and on futures prices.
1960 1970 1980 1990 2000 2010
0
20
40
60
80
100 Futures data
Spot data
The overall agreement since the late 1960s (when the number of traded futures contracts becomes significant) is
very good, although the average slope on spots is slightly smaller, as expected.
TABLE 7 Correlation between spot and futures trend-following strategies.
Spot–future
Sector correlation
Commodities 0.65
Bonds 0.91
Indexes 0.92
Even though the “cost of carry” plays an important role for commodities, the trends are still highly correlated.
term being especially significant and volatile for commodities. We find, however, that
the level of correlation is sufficiently high to render the results meaningful. In any
case the addition of the cost of carry can only improve the performance of the trend
on futures and any conclusion regarding trends on spot data will be further confirmed
by the use of futures data.
We therefore feel justified in using the spot data to build statistics over a long
history. We believe that the performance will be close to (and in any case, no worse
than) that on real futures, in particular because average financing costs are small, as
illustrated by Figure 3.
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52 Y. Lempérière et al
FIGURE 4 Aggregate performance of the trend on all sectors.
1800 1850 1900 1950 2000
0
100
200
300
t-statistic D10.5. Debiased t-statistic D9.8. Sharpe ratio D0.72.
TABLE 8 Sharpe ratio and t-statistic of the trend .T /, of the debiased trend .T /and of
the drift component of the different sectors, with the starting date for each sector.
SR t-statistic t-statistic SR t-statistic Start
Sector (T)(T)(T)()() date
Currencies 0.47 2.9 2.9 0.1 0.63 1973
Commodities 0.28 4.1 3.1 0.3 4.5 1800
Bonds 0.4 3.9 2.7 0.1 1 1918
Indexes 0.7 10.2 6.3 0.4 5.7 1800
4 TREND OVER TWO CENTURIES
4.1 Results of the full simulation
The performance of the trend-following strategy defined by (2.2) over the entire time
period (two centuries) is shown in Figure 4. It is visually clear that the performance is
highly significant. This is confirmed by the value of the t-statistic, which is found to be
above 10, and 9.8 when debiased from the long-only contribution, ie, the t-statistic of
“excess” returns. For comparison, the t-statistic of the drift of the same time series
is 4.6. As documented in Table 8, the performance is furthermore significant on each
individual sector, with a t-statistic of 2.9 or higher, and 2.7 or higher when the long
Journal of Investment Strategies 3(3)
Two centuries of trend following 53
TABLE 9 Sharpe ratio and t-statistic of the trend and of the drift over periods of fifty
years.
SR t-statistic SR t-statistic
Period (T)(T)()()
1800–1850 0.6 4.2 0.06 0.4
1850–1900 0.57 3.7 0.43 3.0
1900–1950 0.81 5.7 0.34 2.4
After 1950 0.99 7.9 0.41 2.9
bias is removed. Note that the debiased t-statistic of the trend is in fact higher than
the t-statistic of the long-only strategy, with the exception of commodities, where it
is slightly worse (3.1 versus 4.5).
The performance is also remarkably constant over two centuries: this is obvious
from Figure 4 on the facing page, and we report the t-statistic for different periods in
Table 9. The overall performance is in fact positive over every decade in the sample (see
Figure 7 on page 55). The increase in performance in the second half of the simulation
probably comes from the fact that we have more and more products as time goes on
(indeed, government yields and currencies both start well into the twentieth century).
4.2 A closer look at the signal
It is interesting to delve deeper into the predictability of the trend-following signal
sn.t/, defined in (2.1). Instead of computing the P&L given by (2.2), we can instead
look at the scatter plot of .t / Dp.t C1/ p.t/ as a function of sn.t/. This
gives a noisy blob of points with, to the naked eye, very little structure. However, a
regression line through the points leads to a statistically significant slope, ie, .t / D
aCbsn.t/ C.t/, where aD0:018 ˙0:003,bD0:038 ˙0:002 and is a noise
term. The fact that a>0is equivalent to saying that the long-only strategy is, on
average, profitable, whereas b>0indicates the presence of trends. However, it is not
a priori obvious that we should expect a linear relation between and sn. Trying a
cubic regression gives a very small coefficient for the s2
nterm and a clearly negative
coefficient for the s3
nterm, indicating that strong signals tend to flatten, as suggested
by a running average of the signal shown in Figure 5 on the next page. However,
the strong mean reversion that such a negative cubic contribution would predict for
large values of snis suspicious. We have therefore instead tried to model a nonlinear
saturation through a hyperbolic tangent (Figure 5 on the next page):
.t / DaCbstanh sn.t /
sC.t/; (4.1)
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54 Y. Lempérière et al
FIGURE 5 Fit of the scatter plot of .t / Dp.t C1/ p.t/ as a function of sn.t/,fornD5
months, and for futures data only.
–3 –2 –1 0 2
s
–0.10
–0.05
0
0.05
0.10
0.15 Hyperbolic tangent
Data (running avg)
Linear
1 3
We do not show the 240 000 points on which the fits are performed, but rather show a running average over 5000
consecutive points along the x-axis.We also show the results of a linear and hyperbolic tangent fit. Note the positive
intercept a0.02, which indicates the overall positive long-only bias. The best fit to the data is provided by the
hyperbolic tangent, which suggests a saturation of the signal for large values.
which recovers the linear regime when jsnjsbut saturates for jsnj>s
. This
nonlinear fit is found to be better than the cubic fit as well as the linear fit, as it prefers
a finite value s0:89 and now b0:075 (a linear fit is recovered in the limit
s!1). Interestingly, the values of a,band shardly change when nincreases
from 2.5 months to 10 months.
4.3 A closer look at the recent performance
The plateau in the performance of the trend over the last few years (see Figure 6 on
the facing page) has received a lot of attention from CTA managers and investors.
Among other explanations, the overcrowdedness of the strategy has frequently been
evoked to explain this relatively poor performance. We now want to reconsider these
conclusions in the context of the long-term simulation.
First, it should not come as a surprise that a strategy with a historical Sharpe
ratio below 0.8 shows relatively long drawdowns. In fact, the typical duration of a
drawdown is given by 1=S2(in years) for a strategy of Sharpe ratio S. This means
that, for a Sharpe of 0.7, typical drawdowns last two years, while drawdowns of four
Journal of Investment Strategies 3(3)
Two centuries of trend following 55
FIGURE 6 Recent performance of the trend.
2000 2005 2010 2015
490
500
510
Since 2011, the strategy is virtually flat.
FIGURE 7 Ten-year cumulated performance of the trend (arbitrary units).
1850 1900 1950 2000
10
20
30
The horizontal line is the historical average.
Research Paper www.risk.net/journal
56 Y. Lempérière et al
FIGURE 8 Performance of a three-day trend on futures contracts since 1970.
1960 1970 1980 1990 2000 2010
0
200
400
600
The effect seems to have completely disappeared since 2003 (or has maybe even inverted).
years are not exceptional (see Bouchaud and Potters (2003) and Seager et al (2014)
for more on this topic).
To see how significant the recent performance is, we have plotted in Figure 7 on
the preceding page the average P&L between time t10Y and time t. We find that,
though we are currently slightly below the historical average, this is by no means an
exceptional situation. A much worse performance was in fact observed in the 1940s.
Figure 7 on the preceding page also reveals that the ten-year performance of trend
following has, as noted above, never been negative in two centuries, which is again a
strong indication that trend following is ingrained in the evolution of prices.
The above conclusion is however only valid for long-term trends, with a horizon of
several months. Much shorter trends (say, over three days) have significantly decayed
since 1990 (see Figure 8). This is perfectly in line with a recent study by the Winton
group (Duke et al 2013). We will now propose a tentative interpretation of these
observations.
4.4 Interpretation
The above results show that long-term trends exist across all asset classes and are
stable in time. As mentioned in Section 1, trending behavior is also observed in
the idiosyncratic component of individual stocks (Barroso and Santa-Clara 2013;
Journal of Investment Strategies 3(3)
Two centuries of trend following 57
Geczy and Samonov 2013; Kent and Moskowitz 2013; Narasimhan and Titman 1993,
2011). What can explain such universal, persistent behavior of prices? We can find
two (possibly complementary) broad families of interpretation in the literature. One
explanation assumes that agents underreact to news and only progressively include the
available information in prices (Hong and Stein 1999; Kent et al 1998). An example
of this could be an announced sequence of rate increases by a central bank over
several months, which is not immediately reflected in bond prices because market
participants tend to only believe in what they see and are slow to change their previous
expectations (“conservatism bias”). In general, changes of policy (for governments,
central banks or indeed companies) are slow and progressive. If correctly anticipated,
prices should immediately reflect the end point of the policy change. Otherwise, prices
will progressively follow the announced changes and this inertia leads to trends.
Another distinct mechanism is that market participants’ expectations are directly
influenced by past trends: positive returns make them optimistic about future prices
and vice versa. These “extrapolative expectations” are supported by “learning to pre-
dict” experiments in artificial markets (Hommes et al 2008; Smith et al 1988), which
show that linear extrapolation is a strongly anchoring strategy. In a complex world
where information is difficult to decipher, trend following – together with herding –
is one of the “fast and frugal” heuristics (Gigerenzer and Goldstein 1996) that most
people are tempted to use (Bouchaud 2013). Survey data also points strongly in this
direction (Greenwood and Shleifer 2014; Menkhoff 2011; Shiller 2000).8Studies
of agent-based models in fact show that the imbalance between trend following and
fundamental pricing is crucial in accounting for some of the stylized facts of finan-
cial markets, such as fat tails and volatility clustering (see, for example, Barberis et
al 2013; Giardina and Bouchaud 2003; Hommes 2006; Lux and Marchesi 2000).9
Clearly, the perception of trends can lead to positive-feedback trading, which rein-
forces the existence of trends rather than making them disappear (Bouchaud and Cont
1998; DeLong et al 1990; Wyart and Bouchaud 2007).
On this last point, we note that the existence of trends far predates the explosion
of assets managed by CTAs. The data shown above suggests that CTAs have neither
substantially increased nor substantially reduced the strength of long-term trends in
major financial markets. While the degradation in recent performance, although not
8Anecdotally, based on a long history of Capital Fund Management (CFM) inflows and outflows,
our experience suggests that professional investors have a strong tendency to “chase performance”,
ie, to invest in CFM’s funds after a positive rally and redeem after negative performance.
9Within their model, Giardina and Bouchaud (2003) show that, without an element of trend follow-
ing, markets quickly reach an “efficient” stationary state where nothing much happens.
Research Paper www.risk.net/journal
58 Y. Lempérière et al
statistically significant, might be attributed to overcrowding of trending strategies,
it is not entirely clear how this would happen in the “extrapolative expectations”
scenario, which tends to be self-reinforcing (see, for example, Wyart and Bouchaud
(2007) for an explicit model). If, on the other hand, underreaction is the main driver
of trends in financial markets, we may indeed see trends disappear as market partic-
ipants better anticipate long-term policy changes (or indeed policy makers become
more easily predictable). Still, the empirical evidence supporting a behavioral trend-
following propensity seems to us strong enough to advocate extrapolative expectations
over underreactions. It would be interesting to build a detailed behavioral model that
explains why the trending signal saturates at high values, as evidenced in Figure 5 on
page 54. One plausible interpretation is that, when prices become more obviously out
of line, fundamentalist traders start stepping in, and this mitigates the impact of trend
followers, who are still lured in by the strong trend (see Bouchaud and Cont (1998),
Lux and Marchesi (2000) and Greenwood and Shleifer (2014) for similar stories).
5 CONCLUSIONS
In this study, we established the existence of anomalous excess returns based on
trend-following strategies across all asset classes and over very long time scales. We
first studied futures, as is customary, then spot data that allows us to go far back in
history. We carefully justified our procedure, in particular by comparing the results
on spot data in the recent period, which shows a strong correlation with futures, with
very similar drifts. The only sector where we found no way to extend the history is
for foreign exchange, since the idea of a free-floating currency is a rather recent one.
We found that the trend has been a very persistent feature of all the financial
markets we looked at. The overall t-statistic of the excess returns has been around 10
since 1800, after accounting for the long-only bias. Furthermore, the excess returns
associated to trends cannot be associated to any sort of risk premium (Lempérière et
al 2014; Narasimhan and Titman 2011). The effect is very stable, across both time and
asset classes. It makes the existence of trends one of the most statistically significant
anomalies in financial markets. When analyzing the trend-following signal further,
we found a clear saturation effect for large signals, suggesting that fundamentalist
traders do not attempt to resist “weak trends”, but might step in when their own signal
becomes strong enough.
We investigated the statistical significance of the recent mediocre performance of
the trend, and found that this was actually in line with a long historical backtest.
Therefore, the suggestion that long-term trend following has become overcrowded
is not borne out by our analysis and is compatible with our estimate that CTAs only
contribute a few percent of market volumes. Still, the understanding of the behavioral
causes of trends, and in particular the relative role of “extrapolative expectations”
Journal of Investment Strategies 3(3)
Two centuries of trend following 59
versus “underreaction” or “conservative biases”, would allow us to form an educated
opinion on the long-term viability of trend-following strategies. It is actually not obvi-
ous how crowdedness would deteriorate trend-following strategies, since more trend
following should speed up trends as managers attempt to “front-run” the competi-
tion. Figure 8 on page 56, however, adds to the conundrum by showing that faster
trends have actually progressively disappeared in recent years, without ever showing
an intermediate period where they strengthened. Coming up with a plausible mecha-
nism that explains how these fast trends have disappeared would be highly valuable
in understanding the fate of trends in financial markets.
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