Trends, Reversion, and Critical Phenomena in Financial Markets
TTrends, Reversion, and Critical Phenomenain Financial Markets
Christof SchmidhuberZurich University of Applied SciencesSchool of Engineering, Technikumstrasse 9CH-8401 Winterthur, [email protected] 3, 2020 a r X i v : . [ q -f i n . S T ] S e p bstract Financial markets across all asset classes are known to exhibit trends, which havebeen exploited by traders for decades. However, a closer look at the data reveals thatthose trends tend to revert when they become too strong. Here, we empirically measurethe interplay between trends and reversion in detail, based on 30 years of daily futuresprices for equity indices, interest rates, currencies and commodities.We find that trends tend to revert before they become statistically significant. Ourkey observation is that tomorrow’s expected return follows a cubic polynomial of to-day’s trend strength. The positive linear term of this polynomial represents trendpersistence, while its negative cubic term represents trend reversal. Their precise co-efficients determine the critical trend strength, beyond which trends tend to revert.These coefficients are small but statistically highly significant, if decades of datafor many different markets are combined. We confirm this by bootstrapping and out-of-sample testing. Moreover, we find that these coefficients are universal across assetclasses and have a universal scaling behavior, as the trends time horizon runs from afew days to several years. We also measure the rate, at which trends have become lesspersistent, as markets have become more efficient over the decades.Our empirical results point towards a potential deep analogy between financialmarkets and critical phenomena. In this analogy, the trend strength plays the role ofan order parameter, whose dynamics is described by a Langevin equation. The cubicpolynomial is the derivative of a quartic potential, which plays the role of the energy.This supports the conjecture that financial markets can be modeled as statistical me-chanical systems near criticality, whose microscopic constituents are Buy/Sell orders.
Keywords: trend following, mean reversion, futures markets, market efficiency, criticalphenomena, social networks
Introduction
It is well-known that financial markets across all asset classes exhibit trends. These trendshave been exploited very successfully by the tactical trading industry over the past decades,including the former ”turtle traders” [1] and today’s CTA industry.A close look at the available data reveals that those trends tend to revert as soon as theybecome too strong. In this paper, we demonstrate this based on 30 years of daily futuresreturns for equity indices, interest rates, currencies and commodities. We analyze trendswith 10 different time horizons, ranging from 2 days to 4 years, and empirically measurethe critical strength, beyond which trends tend to revert. Here, the ”strength” of a trend isdefined in terms of its statistical significance, namely as the t -statistics of the trend.In a first step, we measure the daily average return of a market as a function of thevalues of the 10 trend strengths on the previous day. In order to increase the statisticalsignificance of the results, we aggregate across different markets and time scales. Our keyobservation is that tomorrow’s average return can be quite accurately modeled by a polyno-mial of today’s trend strength. It consists of a positive linear term that is responsible for thepersistence of trends, and a negative cubic term that is responsible for the reversion of trends.Trends tend to revert beyond a critical trend strength, where the two terms balance eachother. The corresponding regression coefficients are small, but statistically highly significant.In a second step, we refine this quantitative analysis. Using multiple nonlinear regression,we empirically measure how the observed cubic function varies • with the time scale of the trends: we find that trends of medium strength persist atscales of several days to several years, while reversion dominates at shorter or longertime scales. We model this scale-dependence by polynomial regression as well. • with the asset class: we find that the available data do not allow us to fit differentmodel parameters to different asset classes. Within the limits of statistical significance,the model parameters are thus universal, i.e., independent of the asset.3 over time: we find that the patterns have gradually changed over the decades. Inparticular, trends have become less persistent, and there is little evidence that classicaltrend-following can perform as well in the future as it did in the past.Since financial market returns are only in a rough approximation independent, normally dis-tributed random variables, we cannot trust the standard significance analyses for regressionresults. Instead, we use bootstrapping and cross validation to confirm that our results arestatistically highly significant out-of-sample, and robust. Throughout this paper, we tryhard not to introduce a single parameter more than is absolutely necessary to capture theessence of the empirically observed patterns. We find that we may fit at most 6 parametersto our 30-year data set, and identify what we believe are the 4-6 most relevant parameters.While trends have been exploited by the systematic trading industry for decades, theyarrived relatively late in academia. Early observations on market trends appear, e.g., in[2, 3]. Early literature on the interplay of trends and reversion has focused on their cross-sectional counterparts (momentum and value) for single stocks [4]. With the advent ofalternative beta strategies [5, 6], trend-following has become an active academic researcharea [7, 8, 9, 10, 11, 12]. By now, there is an extensive literature on trend-following, includ-ing backtests of its performance more than a century into the past [13, 14], and efforts tooptimize trend-following strategies by machine learning methods [15]. For a recent review oftrend- and reversion strategies, see [16] and references therein.Much of the financial literature in this field tries to improve trading strategies, be it bynew trend signals, by new algorithms for mapping signals to position sizes, by identifyingmarket environments in which a given strategy works best or worst, or by reducing tradingcosts or risks. However, while the results reported in our article also have implications forinvestors (e.g., they signal when to exit trends), our key motivation for publishing themgoes much further: as discussed in section 5, the cubic polynomial, the scaling relations, andthe universality that we observe all point towards a potential deep analogy between financialmarkets and statistical mechanical systems near second-order phase transitions. This in turnsupports the idea that markets can be modeled in terms of ”social networks” of traders. Our4esults lay the empirical basis for systematically analyzing the nature of these networks.As a corollary, our observations also support a modified version of the efficient markethypothesis: they suggest that market inefficiencies do exist, but disappear before they be-come strongly statistically significant. In addition, our measurements quantify how marketshave become more efficient with respect to trends over the decades. Our analysis is based on historical daily log-returns for the set of 24 futures contracts shownin table 1. This set is diversified across four asset classes (equity indices, interest rates,currencies, commodities), three regions (Americas, Europe, Asia) and three commodity sec-tors (energy, metals, agriculture). We use futures returns, instead of the underlying marketreturns, because futures returns are guaranteed to be marked-to-market daily. Moreover,they are readily available for all asset classes and net of the risk-free rate, which also makesreturns in different currencies and interest rate regimes comparable with each other.
Table 1: Markets
America Europe AsiaEquities S&P 500 DAX 30 Nikkei 225TSE 60 FTSE 100 Hang SengInterest rates US 10-year Germany 10-year Japan 10-yearCanada 10-year UK 10-year Australia 3-yearCurrencies CAD/USD EUR/USD JPY/USDGBP/USD AUD/USDNZD/USDCommodities Crude Oil Gold SoybeansNatural Gas Copper Live CattleCom.-Sectors: Energy Metals Agriculture5or all contracts, we consider 30 years of daily price data, covering the period from Jan1, 1990, to Dec 31, 2019. The first two years 1990 and 1991 are merely used to compute thetrend strengths at the beginning of 1992 (see below), so the actual regression analysis coversonly 28 years. Daily prices P i ( t ) were taken from Bloomberg, where i labels the asset andfutures are rolled 5 days prior to first notice. We define normalized daily log-returns R i ( t ): R i ( t ) = r i ( t ) σ i , r i ( t ) = ln P i ( t ) P i ( t − , σ i = var( r i ) , µ i = mean( r i ) , (1)where the long-term daily risk premium µ i and the long-term daily standard deviation σ i of a market i are measured over the whole 30-year period. For some futures markets, thelog-returns r i ( t ) had to be backtracked or proxied as follows:1. TSE 60 futures: their history begins on Sep 9, 1999. Before, the TSE 60 futures returnsare proxied by the S&P 500 futures returns, which, in our analysis, thus have doubleweight during that period.2. Hang Seng index futures: their history begins on Apr 2, 1992. Before, their returnsare proxied by Nikkei 225 futures returns. As the regression analysis begins only onJan 1, 1992, this is a minor data correction.3. DAX futures: their history begins on Nov 26, 1990. Before, FTSE 100 futures returnsare used as a proxy. This data correction is also minor: it merely slightly affects theinitial trend strengths at the beginning of 1992, when the analysis begins.4. EUR futures: their history begins on May 21, 1998. Before, the Deutsche Mark is usedas a substitute for the Euro. We have reconstructed Deutsche Mark futures returnsfrom the spot exchange rate and German/U.S. Libor differentials.5. NZD futures: their history begins on May 9, 1997. Before, we have reconstructed thefutures returns from the spot exchange rate and the NZD/USD Libor differentials.6. German 10-year Bund futures: their history begins on Nov 27, 1990. Before, we havereconstructed futures returns from daily German 10-year and short-term interest rates,assuming a duration of 8. This data correction is also minor: it merely slightly affectsthe initial trend strengths at the beginning of 2002, when the analysis begins.6. Natural gas futures: their history begins on Apr 5, 1990. Before, 1.5-fold levered crudeoil futures returns are used as proxies for the natural gas futures returns, using the U.S.Libor rate as the cost of leverage. The 1.5-fold leverage reflects the higher volatility ofnatural gas compared with crude oil. Again, this data correction is minor, as it merelyaffects the initial trend strengths at the beginning of 1992, when the analysis starts. We will examine the interplay between trends and reversion at 10 different time scales: T k = 2 k business days with k ∈ { , , , ..., } This represents periods of approximately 2 days, 4 days, 8 days, 3 weeks, 6 weeks, 3 months,6 months, 1 year, 2 years, and 4 years. Thus, there are 10 different trend strengths at eachpoint in time. A given asset may well be, e.g., in a long-term up-trend at the 1-year timescale, and at the same time in a short-term down-trend at the 3-week time scale.
As reviewed in [17, 18], there are many different definitions of the strength of a trend, mostof which are highly correlated. For the purpose of this study, we need a definition that • has only a single free parameter, the horizon T (to avoid overfitting historical data) • can be computed recursively (which will later help to relate it to critical phenomena)Let us develop the most convenient such definition step by step. For a given time scale T , we define the trend strength φ i,T ( t ) of a market i on day t ∈ Z as a weighted average ofpast daily returns of that market - more precisely, of the normalized past daily log-returns(1) in excess of the long-term risk premium: φ i,T ( t ) = ∞ (cid:88) n =0 w T ( n ) · ˆ R i ( t − n ) with ˆ R i ( t − n ) = R i ( t − n ) − µ i σ i , (2)where w T ( n ) is a weight function for the time scale T . Removing the long-term risk premium µ i in (2) ensures that the long-term expectation value of the trend φ i,T is zero for each market7nd each time scale. We also normalize the weight function such that φ i,T has standarddeviation 1. Assuming that market returns on different days are independent from eachother (which is true to high accuracy), this implies: ∞ (cid:88) n =0 w T ( n ) = 1 . (3)With this normalization, φ i,T can be regarded as the statistical significance of the trend.E.g., φ i,T = 2 represents a highly significant up-trend, while φ i,T = − . step function (fig. 1, dotted line). In this case, thetrend strength φ i,T is just proportional to the average log-return over the past T days. Un-fortunately, this weight function leads to artificial jumps of the trend strength φ i,T on dayswhen nothing happens, except that an outlier return leaves the rolling time window.This can be avoided by an exponentially decaying weight function ˜ w T ( n ) (fig.1, dashedline). Moreovoer, the corresponding trend strength ψ can now be computed recursively:˜ w T ( n ) = M T e − n/T with normalization factor M T = (cid:112) − e − /T , (4) ψ i,T ( t ) = ∞ (cid:88) n =0 ˜ w T ( n ) · ˆ R i ( t − n ) = e − /T ψ i,T ( t −
1) + M T · ˆ R i ( t ) . (5)It can be verified that ˜ w T satisfies (3). However, ψ is quite volatile and jumps when anoutlier return enters the rolling time window. One way to solve this problem is to use thecommon definition of φ i,T in terms of a moving average crossover : one subtracts the averagelog-price of asset i over a longer time period L from the average log price of the same assetover a shorter time period S . As pointed out in [17], this corresponds to a wedge-like weightfunction (fig. 1, solid line). It makes the trend strength less volatile, as outlier returns affectit only gradually over the time period S . It also filters out short-term trends on time scalessmaller than S , which helps to seperate trends at different time scales from each other.8igure 1: Our trend strength is defined as a weighted sum of past log-returns. The grey areashows the weight function used in this paper, compared with three standard alternatives.All four weight functions shown here have the same average lookback period.Unfortunately, the moving price average has two parameters L, S (instead of just one parameter T ) that must be fitted to the data in any analysis, which tends to reduce thestatistical significance of the results. In this article, we will therefore use another similarweight function that involves only the single parameter T (fig. 1, grey area; for comparabilitywith the other weight functions, the figure shows w T/ instead of w T ): w T ( n ) = N T · ( n + 1) · exp( − nT ) with N T = (1 − e − /T ) √ − e − /T . (6)With this normalization factor N T , one can verify that (3) is indeed satisfied. Moreover,this definition, together with (5), allows for a recursive combined computation of the twovariants ψ, φ of the trend strength (which will be important in section 5): φ i,T ( t ) = ∞ (cid:88) n =0 w T ( n ) · ˆ R i ( t − n ) = e − /T φ i,T ( t −
1) + N T M T · ψ i,T ( t ) (7)The ”average lookback period” of this trend strength, i.e., the expectation value E [ n + 1] ofthe number of days we look back (where ”today”, i.e. n = 0, counts as a 1-day lookback), is E [ n + 1] = ∞ (cid:88) n =0 ( n + 1) · w T ( n ) · (cid:2) ∞ (cid:88) n =0 w T ( n ) (cid:3) − = T. (8)We have verified that, for a given horizon T , all definitions of the trend strength in factyield quite similar results in our regression analysis of section 4, as long as the weight func-tion rises gradually, decays gradually, and the average lookback period is the same. However,9e use (6,7) here, as it is the simplest mathematical function that satisfies these criteria, and has only the single free parameter T , and can be computed recursively. (6) was originallyintroduced by the author in 2008 at Syndex Capital Management, and has been used toreplicate Managed Futures indices as part of a UCITS fund from 2010-2014.To limit the impact of outlier values of φ i,T on our results, we will cut it off at ± . φ capi,T = min (2 . , max ( − . , φ i,T ) . Without this cap and floor, the regression analysis would yield higher R-squared’s, and theresults would be similar, but less robust. According to [11], the managed futures industrypractise for this cutoff is 2.0. We use the slightly higher value of 2.5, beacuse this allows usto study more precisely the regime where trends revert.
Table 2 displays a small extract of the resulting database for our analysis. Only two of the7305 business days and only three of the 24 markets are shown. The third column showsthe normalized daily log-returns (1), which have standard deviation 1. The 7305 businessdays cover only the 28-year period from Jan 1992 - Dec 2019, because the first two years1990-1992 were only used to compute the initial trend strengths at the beginning of 1992.The full table with 7305 ×
24 = 175 (cid:48)
320 lines is published along with this paper.
Table 2: Database
Trend strengths on previous day for 10 time scalesDay Market R i ( t ) 2d 4d 8d 3w 6w 3m 6m 1y 2y 4y1 S&P 500 -0.2 0.3 0.7 1.0 0.6 0.2 0.3 0.6 0.6 1.0 1.61 EUR/$ -0.1 0.2 0.2 0.0 -0.4 -0.6 -0.6 -0.8 -0.9 -0.7 -0.81 Gold -0.5 -0.3 -0.7 -0.7 0.1 1.1 1.5 1.1 0.4 0.1 -0.42 S&P 500 -0.3 -0.1 0.4 0.9 0.7 0.2 0.3 0.6 0.6 1.0 1.62 EUR/$ -1.0 -0.7 -0.2 -0.2 -0.4 -0.6 -0.6 -0.8 -0.9 -0.7 -0.82 Gold 0.8 0.3 -0.2 -0.6 0.1 1.1 1.5 1.1 0.4 0.1 -0.4 Qualitative Observations
This section begins with an exploratory analysis of our data. The analysis in this sectionis only qualitative, but it serves to motivate the specific quantitative, statistically rigorousregression analysis of the following section. We stress again that our aim is not to improvefutures trading strategies, which would have to include risk limits, trading cost minimization,and other features. Rather, we simply want to empirically measure and model the smallautocorrelations of market returns as accurately as possible as a basis for future work.
We use the data of table 2 to measure the expected daily return of a futures market as afunction of the trend strengths in that market on the previous day. To this end, we first con-struct 7305 · ·
10 = 1 (cid:48) (cid:48)
200 pairs of data. Each pair consists of the normalized log-return R i ( t ) in that market on day t , and one of the 10 trend strengths φ i,T ( t −
1) on the previousday. So each return appears in 10 data pairs, each time paired with a different trend strength.We then group those pairs into 15 bins of increasing trend strength from −∞ to -13/6,-13/6 to -11/6, ... ,-1/6 to+1/6, ... ,11/6 to 13/6, 13/6 to ∞ . The mean trend strengthwithin each bin is shown on the x-axis of fig. 2 (left). Within each bin, we average overthe normalized return on the day after the trend has been measured. To obtain statis-tically significant results, we aggregate over the 28 years of daily returns for each market,across all 24 markets, and across different time scales. Fig. 2 (left) shows the results for the 4monthly trend strengths (i.e., aggregated over T = 6 weeks, 3 months, 6 months, and 1 year).We observe that the average next-day return is close to zero at zero trend strength, andgrows linearly with the trend strength for small trend strengths. As the trend strengthincreases further, the average next-day return reaches a maximum. It then decreases againuntil it becomes zero somewhere below trend strength 2. For even stronger trends, theaverage return decreases dramatically. The same picture is mirrored on the left-hand sideof the graph for down-trends. We have verified that this pattern remains almost the same11f another day of delay is added, i.e., if the next-day return in our data pairs is replaced bythe return 2 days later.Figure 2: Left : The expectation value E ( r ) of the next days return of a futures market isa nonlinear function of the current trend strength φ . As verified by the extensive statisticalanalysis of section 4, it can be modeled by a cubic polynomial of φ , whose linear term bφ (with b >
0) represents trend-persistence, and whose cubic term cφ (with c <
0) representstrend-reversion.
Right : As confirmed by bootstrapping in section 4, the regression coefficients b and c corresponding to the linear and cubic terms are statistically highly significant.Thus, trends tend to revert when they become too strong. This makes sense intuitively:after strong trends, markets tend to be overbought or oversold, so one expects a reversion to”value”. Our analysis quantifies where exactly this happens: below a critical trend strengthof 2, before trends become strongly statistically significant. Note that this is not in line withclassical trend-following, which would follow the trend no matter how strong it becomes.The dashed line in fig. 2 (left) indicates the trading position that a classical trend-followerwould take as a function of the trend strength. In a next step, we analyze how the pattern observed in the previous sub-section depends onthe time scale. To this end, we refine the bins used above: we split each bin into 10 smaller12ins, one for each of the 10 time scales. The resulting 15 ×
10 refined bins are now too smalland the results too noisy. To reduce the noise, we average the next-day returns over blocksof 3x3 neigboring bins (resp. 3x2 or 2x3 bins at the borders, 2x2 bins at the corners of thematrix of 15 ×
10 bins), weighted by the number of returns in each bin. This yields the heatmap of fig. 3 (left). Fig. 2 (left) can be thought of as a horizontal cross-section through thisheat map along the dashed line.Figure 3:
Left : A heat map shows how the expectation value of tomorrows return dependsboth on todays trend strength φ and its time horizon. Fig. 2 (left) can be thought of as across-section of Fig. 3 (left) along the dashed line. Right : The polynomial regression analysisof section 4 models the pattern of Fig. 3 (left) by an elliptic regime within which trends arepersistent, and outside of which they revert. The values of the center and of the semi-axesof the ellipse are statistically highly significant, as confirmed by bootstrapping.We observe that the trend- and reversion pattern of fig. 2 (left) is strongest on time scalesfrom 1 month to 1 year. This is in line with the fact that typical trend-followers operate onthose time scales. As the time scale increases or decreases, the pattern becomes weaker. Theregion where markets trend seems to disappear both for time scales of the order of economiccycles (several years) and for intra-week time scales. The phenomenon of reversion, on theother hand, appears to remain strong at all time scales.13 .3 Counting Degrees of Freedom
As emphasized in [19], one must be very conservative in introducing new factors and pa-rameters in financial market models. Before modeling the observed patterns in detail, let ustherefore do a back-of-the-envelope calculation of how many parameters we can hope to fitin our model without over-fitting our daily return data, and what fraction of the variance ofthese returns we can hope to explain by trend factors.Our 7 (cid:48) ·
24 = 175 (cid:48)
320 daily log-returns are not independent, because the 24 marketsare correlated with each other. How many independent markets are there? The daily re-turns are normalized to have variance 1. For a portfolio that invests 1 /
24 in each market, wefind a variance of σ ∼ /
8, just as if it contained n m = 8 independent assets. A principalcomponent analysis confirms that the first 8 (resp. 12) principal components explain 65%(resp. 80%) of the variance of the returns of our 24 markets. In this sense, these returnseffectively live in a space of dimension n m ∼
8. Adding more markets to our 24 time seriesdoes not significantly increase n m .What is the highest annualized Sharpe ratio S that one can hope to achieve by systemat-ically trading a broadly diversified set of highly liquid futures markets based on trends andreversion? Experience with the Managed Futures (”CTA”) industry suggests that S can beat best 1. The small number of CTA’s that have achieved a higher Sharpe ratio for severalyears in a row presumably also pursue other strategies that are not purely based on trends,or they are not market-neutral (zero net exposure to each market over time).An annualized Sharpe ratio of S = 1 implies a daily Sharpe ratio ρ for each market of ρ = S √ · n m ∼ . . So the predicted next-day return of a market has a correlation of ρ = 0 .
02 with the actualnext-day return. E.g., if we only try to predict the sign of the next return, we can at besthope to be right on 51 and wrong on 49 out of 100 days. The adjusted R-squared (achievedout-of-sample in real trading) is then R adj ∼ ρ = 4 basis points (1 bp = 4 · − ). Clearly,14he variance of financial market returns is overwhelmingly due to random noise.If we fit k parameters to our data, and if our returns were independent and identicallydistributed (”iid”), then, for small R , the adjusted R-squared would be approximately R adj ∼ R − kN ∼ bp with N = 260 · n m · Y data points (9)for Y years of daily data. If we require that the correction for the in-sample bias does noterode more than 20% of our R , then we conclude that we cannot fit more than k ∼ N · bp ∼ In this section, we confirm and quantify the observations of the previous section by nonlinearregression based on ordinary least squares. To this end, we model the next-day return of amarket as a polynomial function of both the current trend strength in that market and itstime scale. This regression is performed directly on the underlying 1’753’200 pairs of data,not on the bins we have defined in the previous section. Thus, our results are independentof any choice of how to split the data into bins.
The graph in fig. 2 (left) suggests to model the next-day normalized log-return R ( t + 1) (1)as a polynomial of the current trend strength φ ( t ) (2) across all markets and time scales: R ( t + 1) = a + b · φ ( t ) + d · φ ( t ) + c · φ ( t ) + ... + (cid:15) ( t + 1) , (10)where (cid:15) represents random noise, and a measures the average risk premium µ i /σ i across allassets. Similar models with a polynomial random force have been postulated previously,15otably by econophysicists with a background in critical phenomena [20, 21, 22, 23]. Ourobservations of the previous section give clear empirical support for a polynomial ansatz.We will discuss the relationship with critical phenomena in more detail in section 5.We have performed a corresponding regression analysis on the 1’753’200 data pairs { r t +1 , φ t } . Using only the linear and cubic terms of (10) yields the results of table 3. Table 3
Regression 1 with linear and cubic termsCoefficient Value Error t-statistics a . ± .
41% 3 . b . ± .
44% 2 . c − . ± .
24% 2 . R .
31 bp 4 .
91 bp R adj .
07 bp 4 .
01 bpSince market returns cannot be assumed to be independent, identically distributed normalvariables, we cannot trust the usual estimates of the t-statistics, adjusted R-squared, andF-statistic. Instead, the test statistics shown in table 3 are measured empirically as follows: • The standard errors of the coefficients and their t-statistics are computed by boot-strapping: from the 7305 days, we randomly create 5000 new samples of 7305 dayseach, with replacement. I.e., some days occur several times in a new sample, whileother days do not occur at all. Regression on each new sample of days yields thedistribution of 5000 regression coefficients b and c shown in fig. 2 (right). The errorsof the coefficients in table 3 represent half the difference between the 84th and 16thpercentile, which equals the standard deviation in the case of a normal distribution. • The adjusted R-squared is computed by 15-fold cross validation: we split our 7305-daytime window into 15 sub-windows of 487 consecutive days each. For each sub-window,we predict the next-day returns based on the betas obtained by regression on the other164 sub-windows. The square of the correlation between the predicted and the actualreturns is the out-of-sample R -squared R adj reported in table 3. • Table 3 also reports R and R adj ”aggregated across time scales”. Those are based onusing the equally-weighted mean of the 10 trend strengths on each day to predict thenext-day return for each market. I.e., we combine the 10 different trend factors intoa single one, which naturally has a higher predictive power than each single factor byitself. This regression is thus performed on only 7305 ·
24 = 175 (cid:48)
320 pairs of data. • The F -statistics can be computed numerically to be F = 4 . p -Value of 0 . R adj and not F to compare the out-of-sampleexplanatory power of our models with each other.The regression results of table 3 confirm and quantify our conclusions from the previoussection. We see that the values of b and c - although very small - are statistically highlysignificant, despite the fact that market returns are neither normally distributed, nor inde-pendent, nor identically distributed. So is the average long-term risk premium a . The overallresult is significant at the 99% level. The aggregated out-of-sample R adj that combines thepredictions from all 10 time scales matches our initial expectation of 4 bp (9). Note that thecorrection for the aggregated in-sample bias, R − R adj = 0 . bp , is much bigger than whatwould have been expected if returns were ”iid”, namely 2 / (260 · ·
28) = 0 . bp .We have also tested the quadratic, quartic and quintic terms in φ T in (10). None of themturned out to be statistically significant at the 95% level. We therefore drop them from ouranalysis to avoid over-fitting the historical data (i.p., the t-statistics for d is below 1). Next, we try to refine our model by measuring the dependence of the coefficients b and con the time scale, the asset class, and the time period. We begin with the time scale T : we17odel the expected return as a function of both the trend strength and T , trying to replicatefig. 3 (left). We first repeat the linear and cubic regression (10) of the previous sub-sectionfor each of the 10 time scales seperately. The resulting coefficients b ( T ) and c ( T ) are plottedin fig. 4 (we neglect the overall risk premium a , which is not the focus of this paper).From the coefficient b of the linear term, which models trends, we observe that trend-following works best at time scales from 3 months to 1 year, where b peaks. This appears tobe in line with the time scales on which typical CTAs follow trends. Even at those scales,the critical trend strength φ c = (cid:112) − b/c ≤ . , beyond which trends tend to revert, is below 2. So trends never become strongly significant.For scales below a few days and above several years, b seems to go to zero, which means thattrends are not persistent there. This is consistent with the heat map in fig. 2 (right).Figure 4: Left:
The coefficient b ( T ) of the linear term (corresponding to the trending ofmarkets) peaks at time scales T of 3 months to 1 year. Its scale dependence is modeled bya parabola. Right:
The coefficient c of the cubic term (corresponding to the reversion ofmarkets) does not show a clear dependence on the time scale.On the other hand, the coefficient c of the cubic term, which ensures that trends revert,is quite stable, except that its magnitude appears to be somewhat lower for the 2- and 4-year18cales. The 2- and 4-year results must be taken with a grain of salt, though, as there are only14 independent 2-year trends and 7 independent 4-year trends in our 28-year time window.Indeed, a preliminary check based on 60 years of monthly returns resulted in c ∼ − .
6% atthe 8-year scale. The available data thus indicate that, unlike trend-following, mean rever-sion works at all time scales. This is also consistent with our earlier observation from theheat map in fig. 3 (left).To quantify these observations, we refine our regression ansatz (10). We continue tomodel the cubic coefficient c by a constant, but we model the dependence of the linearcoefficient b ( k ) on the logarithm k of the time scale T = 2 k by a parabola: b ( k ) = b − e · ( k − k ) ⇒ R ( t + 1) = b · (cid:110) − ( k − k ) (∆ k ) (cid:111) · φ ( t ) + c · φ ( t ) + (cid:15) ( t + 1) (11)with (∆ k ) = b/e . The critical trend strength φ c ( k ) = ( − b ( k ) /c ) / , at which the expectedreturn E ( R t +1 ) is zero (without the noise (cid:15) ), and beyond which trends revert, is then anellipse with semi-axes ∆ k and φ c ( k ). Altogether, we now fit 4 parameters to our data: • The ”persistence of trends” b , i.e. the value of b ( k ) at its peak • The ”strength of reversion” c • The range k ± ∆ k of the log of the time scales T = 2 k at which markets may trend. Table 4
Refined regression 2 with 4 parametersCoefficient Value Error t-Stat. b . ± .
51% 3 . c − . ± .
24% 2 . k . ± .
72 8.0∆ k . ± .
16 4 . R .
64 bp 7 .
51 bp R adj .
22 bp 5 .
81 bp19 nonlinear regression on the full underlying data set yields the results of table 4. Fig. 3(right) plots the elliptic region, which seperates the trend regime (inside) from the reversionregime (outside). For its second semi-axis, we find φ c ( k ) = 1 . ± .
32. This quantifies theempirical heat map in fig. 3 (left) and confirms that highly significant trends of strength(i.e., t -statistics) φ c ≥ b and c looks the same as in the univariate case (Fig. 2, right). Fig 5(left) plots the distribution of the values of the center k and the semi-axis ∆ k of the ellipsethat separates the trending regime from the reversion regime. The ”aggregate” R and R adj in table 4 now refer to a single factor that is a linear combination of the 10 trend strengthsfor the 10 time scales, weighted by a parabolic weight function proportional to b ( k ). Notethat the aggregated adjusted R-squared now exceeds our original expectation (9) of 4 bp .Figure 5: Left:
Distribution of the regression coefficients for the center k and width ∆ k ofthe elliptic region within which trends are persistent, as obtained by bootstrapping. Right :Distribution of the linear and cubic regression coefficients b, c for a rejected alternative model.We have tried to further refine ansatz (11). First, b ( k ) in fig. 4 (left) seems to be tilted tothe right, which could be accounted for by models such as b ( k ) ∼ b − e · ( k − k ) + f · ( k − k ) , or b ( k ) ∼ exp( f · k ) cos(( k − k ) / ∆ k ). We find that such models increase the adjusted R -squaredat best marginally. So we use the simplest model (11) in this paper, to avoid over-fitting the20istorical data. Second, we also tested for a polynomial dependence of c on k . The mostsignificant ansatz was that c ( k ) is also a parabola proportional to − b ( k ). In this case, thecritical trend strength is constant across all time scales, and the region within which marketstrend is rectangular instead of elliptic. The distribution of the parameters b ( k ) , c ( k ) thenturns out to have the shape of the stretched annulus shown in fig. 5 (right). However, thisscenario seems less likely, as it yields a much lower adjusted R-squared (0.77 bp). Can we refine our 4-parameter-model further by distingushing between asset classes, i.e., byfitting seperate regression parameters for equities, bonds, currencies and commodities?To test this, we have repeated the regression analysis of the previous section for these 4sub-sets of our data. Fig. 6 (left) shows the 16th, 50th and 84th percentile of the values of the4 regression parameters for each asset class, divided by the values of the regression parametersfor the overall sample. E.g., for equities, the quantiles for b are (1 . , . , . . , . , .
01) of the overall regression coefficient 2 .
00% (see table 3).Those multiples are what is shown in the first bar of fig. 6 (left).For each asset class, we observe that the values of all four parameters are within onestandard error of the overall parameter values. Thus, based on our data set, we cannotjustify fitting different parameters of our trend-reversion model to individual asset classes,let alone to individual assets. This is consistent with our back-of-the-envelope caculation ofsubsection 3.4, which suggests that we cannot fit as many as 4 × Lastly, we investigate how our patterns have been evolving in time. First, we split up the28-year (7305-day) time window into three non-overlapping sub-periods of 2435 days each:An early period from Jan 1992 to Apr 2001, a middle period from May 2001 to Aug 2010,and a late period from Sep 2010 to Dec 2019.21igure 6:
Left:
Ratios of the values of our 4 regression parameters for equities, interestrates, FX rates and commodities, divided by their overall values across all asset classes. Theratios do not differ significantly from 1.
Right:
The analoguous ratios for the early, middleand late third of the time period. At least b has decreased significantly over time.Fig. 6 (right) shows the 16th, 50th and 84th percentile of the values of the 4 regressionparameters for each of these time-windows, again divided by the values of the regression pa-rameters for the overall sample. The persistence of trends b has consistently and significantlydecreased over time. With less consistency, this can also be observed for strength of reversion c , while no clear trend is visible for the range k ± ∆ k , within which trends persist. Thedecrease of b is in line with the industry observation that trend-following no longer worksas well as it used to: markets seem to have become more efficient in this respect. Given thedecrease in trading costs, an increase in algorithmic trading, and an increase in assets undermanagement invested in trend-following, this is not surprising.To verify and quantify these observations, let us introduce time t , measured in years, withits origin t = 0 on Dec 31, 2005, the center of our 28-year time window. We now furtherrefine our model (11) by including linear trends in b and c while leaving k and ∆ k constant: b ( t ) = ¯ b · (1 − Q b · t ) , c ( t ) = ¯ c · (1 − Q c · t )The results of a regression analysis, including bootstrapping and cross-validation, are shown22n table 5. The decrease of c , which measures the strength of reversion, is only weaklysignificant. However, the decrease of b , which measures the persistence of trends, is significantat the 97% confidence level. In principle, we could compute the year Y , in which b ( t ) = 0: Y = 2005 + 1 Q b ∈ { , } with expectation value Y ∼ . Thus, if one were to take this linear down-trend of b literally, one would conclude that thephenomenon of persistent market trends may have already disappeared. However, there areother scenarios for the time decay of the persistence of trends that are consistent with ourdata. E.g., for an exponential decay scenario, in which trends never disappear, we find anonly slightly lower adjusted R-squared of 1 . bp instead of 1 . bp , with b ∼ ¯ b · e − Qt with decay rate Q ∼ (24 years) − . Table 5
Refined regression 4 with 6 parametersCoefficient Value Error t-Stat.¯ b . ± .
53% 3 . c − . ± .
22% 2 . k . ± .
55 10 . k . ± .
51 9 . Q b ± . Q c ± . R .
97 bp 8 .
90 bp R adj .
52 bp 6 .
97 bpWe have also tested scenarios where all 4 parameters or other subsets of them changeat different rates, but found that all of these scenarios significantly reduce the adjusted R-squared. It is left for future work to investigate the time evolution of the pattern of trendsand reversion in more detail. 23
Analogies with Critical Phenomena
In this section, we point out some striking analogies between the empirical observations ofsections 3 and 4 and critical phenomena in statistical mechanics. Analogies between financialmarkets and critical phenomena, such as scaling relations, have long been observed [24]. Ourresults go further: they seem to directly and specifically identify the trend strength with theorder parameter of a Landau-type mean field theory with a quartic potential.Analogies with critical phenomena are plausible, if financial markets are regarded asstatistical mechanical systems, whose microscopic constituents are the Buy/Sell orders ofindividual traders. It is conceivable that these orders can be modeled by degrees of freedomthat sit on the vertices of a hypothetical ”social network of traders”. These degrees of free-dom may interact with each other in analogy with spins on a lattice, thereby creating themacroscopic phenomena of trends (herding behavior) and reversion (contrarian behavior).To imitate these phenomena and their interplay, various spin- and agent models have beenproposed in the literature (see, e.g., [25, 26], and [27] for a recent review).Candidates for the ”social network of traders” include small-world networks [28], scale-free networks [29], or the Feynman diagrams of large-N field theory [30]. For a recent reviewof candidates for social networks, see [31]. To our knowledge, no convincing specific modelhas emerged as a consensus so far. Our results provide an empirical basis for accepting orrejecting such candidates: any statistical-mechanical model of financial markets, if accurate,must replicate the interplay of trends and reversion observed in this paper.To make this precise, let us first reap the benefits of our recursive definitions (5,7) of thetrend strength, which lead to simple differential equations in the ”continuum limit” T (cid:29) ddt + 2 T ) ψ ( t ) = 2 √ T · ˆ R i ( t ) , ( ddt + 2 T ) φ ( t ) = 2 √ T ψ ( t ) . (12)To be specific, let us focus on the 6-month time horizon, i.e., T = 2 = 128 trading days(the results for other horizons are similar). Combining (12) with the ansatz (10) implies the24ollowing second-order stochastic differential equation for the trend strength φ :( ddt + 164 ) φ ( t ) = − ∂∂φ V ( φ ) + 1256 (cid:15) ( t ) with V ( φ ) = − b · φ + | c | · φ , (13)with rescaled random noise (cid:15) . Its simpler cousin ψ in (5) obeys a first-order equation:( ddt + 164 ) ψ ( t ) = − ∂∂φ ˜ V ( ψ ) + 14 √ (cid:15) ( t ) with ˜ V ( ψ ) = − ˜ b · ψ + | ˜ c | · ψ , (14)with the following empirical parameter values, as measured by a regression analysis that isanalogous to that reported in section 4 for ψ : b = 2 . , c = − . , ˜ b = 1 . , ˜ c = − . . (14) is the purely dissipative Langevin equation, which is reminiscent of the earlier de-scription [20] of the dynamics of financial markets at intraday scales by another Langevinequation. In the theory of critical phenomena, the Langevin equation is well-known to de-scribe the dynamics of the order parameter of certain statistical mechanical systems nearsecond-order phase transitions [32, 33]. This is consistent with the conjecture that the trendstrength (defined as either φ or ψ ) plays the role of an order parameter, in analogy with themagnetization in spin models.To take the analogy further, statistical mechanical systems near second-order phase tran-sitions are characterized by universal critical exponents. E.g., a scalar field theory with a φ potential similar to the potentials V in (13,14) describes water and steam and other physi-cal systems in the same universality class (such CO or the Ising model) near their criticalpoints [33]. For all systems within this universality class, the parameters b and c show thesame scaling behavior as a function of the length scale L (e.g., b ∼ L κ for some exponent κ ).In critical dynamics, scaling with L also translates into a scaling with the time horizon T [32].In section 4, we have seen that - within the limits of statistical significance - the valuesof the coefficients b and c are the same for very different markets, such as equity indices,bonds, FX-rates, and commodities. The parameters k and ∆ k in (11), which characterize25ow b behaves under a rescaling of the time horizon T , are also the same. This could be anexpression of universality and scaling in financial markets. To confirm this, it will be key toexamine how the scaling behavior in (11) extends to intra-day and multi-year time horizons T = 2 k with k >
10 or k <
1. For example, it might reflect a complex critical exponent [34].Together with the stochastic differential equations (13,14), the empirically observed scalingbehavior may uniquely specify a particular social network that models financial markets.To conclude this section, let us compare with some previous work. In [21], a relatedmodel for the dynamics of asset prices was postulated. The role of the trend was played bythe deviation of the current asset price from its unknown ”value”. Terms of any order wereconsidered in the polynomial potential, and the corresponding classical solutions were dis-cussed. Compared with [21], our trends are measurable, and we focus on a quartic potential,empirically observe the values of its coefficients and their scale dependence, and provide asimple and intuitive map between the quadratic (quartic) terms and trends (reversion).In [22], another model with a polynomial random force similar to (10) was postulated.The trend strength was defined by a moving average crossover (which does not lead to exactdifferential operators such as (12) in the continuum limit). This model was applied in [23]to intraday returns for the USD/JPY and USD/EUR exchange rates during stress periods.Instead of our quartic potential with stable coefficients, only a cubic potential was measured.Morevoer, its coefficients, including their signs, were found to rapidly vary in time.However, these studies were based on very different data sets, namely tick data (instead ofdaily data) for single assets over time periods of several weeks (instead of decades). Thus, itis no surprise that the stable quartic potential (corresponding to the cubic trem in (10)) wasnot found in [23]: as we have seen, in order to detect it with strong statistical significance,one needs not only decades of data, but also aggregate them over a broadly diversified setof assets. Also, since the coefficient of the cubic potential reported in [23] varies rapidly intime, it can be expected to average out over long time scales. This is consistent with thefact that we do not observe a cubic potential in our empirical long-term analysis.26
Summary and Discussion
In this paper, we have empirically observed the interplay of trends and reversion in finan-cial markets, based on 30 years of daily futures returns across equity indices, interest rates,currencies and commodities. We have considered trends over ten different time horizons of T = 2 k days with k ∈ { , , ..., } , ranging from 2 days to approximately 4 years. Fora given market i on a given day t , we have defined the trend strength φ i,k ( t ) as the statis-tical significance ( t -statistics) of a smoothed version of its mean return over the past 2 k days.Our key results, as illustrated in figs 2 and 3, are the following: for a given market i andeach time horizon labeled by k , tomorrow’s normalized log-return R i ( t + 1) can accuratelybe modeled by a cubic polynomial of today’s trend strength in that market: R i ( t + 1) = α i + b · f k · φ i,k ( t ) + c · φ i,k ( t ) + (cid:15) i ( t + 1) . (15)Here, (cid:15) i represents random noise. α i is the normalized long-term risk premium of market i ,which has not been not the focus of this paper. Instead, we have concentrated on determiningthe coefficients b, c, and the function f k , which measure how the expected return of an assetvaries in time. As discussed, we interpret b as the persistence of trends, and c as the strengthof trend reversion. Within the limits of statistical significance, we find that they are universal,i.e., the same for all assets. Over the past 30 years, we find from table 4: b ∼ +2 . , c ∼ − .
6% (16)While the strength of reversion is approximately constant, we find that the persistence oftrends depends on the time horizon of the trend. Within the range of time scales consideredhere, it can be approximated by a parabolic function of the log of the time scale: f k ∼ − ( k − k ) ∆ k with k ∼ , ∆ k ∼ . (17)This implies that trends may only be stable if the log of the time horizon is within the range k ± ∆ k , corresponding to time scales from a few days to several years. The parameters k and ∆ k are also universal. By bootstrapping and cross-validation, we have found that all27our parameters in (16) and (17) are statistically highly significant out-of-sample.Let us now discuss these results. First, they imply that trends tend to revert above acritical trend strength, where the linear and cubic term in (15) balance each other. Thiscritical trend strength lies below 2 in all cases. In other words, by the time a trend has be-come statistically significant, such that it is obvious in a price chart, it is already over. Thissupports a variant of the efficient market hypothesis [35, 36, 37]: inefficiencies in financialmarkets are eliminated before they become strongly statistically significant.Despite being insignificant, small trends can add value for investors through tactical assetallocation strategies, if accompanied by appropriate risk management and broad diversifi-cation across assets. While this paper does not recommend investment strategies, we notethat the inclusion of the cubic term in (15) appears to be a major improvement over clas-sical trend-following, as it takes investors out of trends before they are likely to revert. Webelieve that publishing such strategies and subjecting them to an academic discussion andindependent review will ensure a high level of professionality in asset management.Trend-following has been very successful in the 80’s and 90’s, when it was the proprietarystrategy of a limited number of traders. By now, large amounts of capital have flown intothis strategy, so it can no longer be expected to provide a free lunch. Indeed, while we havenot observed a consistent weakening of the strength of reversion c , we have seen that thepersistence of market trends b has clearly decreased over the decades. This measures therate, at which markets are becoming more efficient with respect to trends.What will happen, when all investors try to exploit trends and reversion? Then bothphenomena should weaken, until they earn a moderate equilibrium return that just compen-sates for the systematic risk of these strategies and their implementation costs. In this sense,trend-following and mean reversion may just become ”alternative market factors” as partof the general market portfolio. In fact, the weakening of b that we have observed here in-dicates that this development is already well underway at least for traditional trend-following.28n a conceptual level, our precise measurement of trends and reversion reveals intriguinganalogies with critical phenomena in physics. They support the conjecture that financialmarkets can be modeled by statistical mechanical systems near second-order phase transi-tions. In such a model, Buy/Sell orders would represent microscopic degrees of freedom thatlive on a ”social network” of traders. The trend strength would play the role of an orderparameter, whose dynamics is described by the stochastic differential equations (13,14). To-gether with an extension of the scaling behavior (17) to shorter and longer horizons, theseequations provide an empirical starting point for developing such a model.If such a statistical mechanical theory of financial markets can be established, it willintroduce powerful concepts from field theory into finance, such as the renormalization group,critical exponents, and Feynman diagrams. This will lead to a new and deeper understandingof financial markets, and phenomena such as trends, reversion, and shocks will become moreaccessible to scientific analysis. Further research in this direction is underway. Acknowledgements
I would like to thank W. Breymann for discussions and encouragement to publish theseobservations, and A. Ruckstuhl for advice on statistical matters. I would also like to thank J.Behrens for many interesting conversations and cooperation at Syndex Capital Managementfrom 2008-2014, where some of these observations were initially made. This research issupported by grant no. CRSK-2 190659 from the Swiss National Science Foundation.