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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.

Research Paper www.risk.net/journal

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)

Research Paper www.risk.net/journal

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 jsnjsbut 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.

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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;

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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”

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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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