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, , as an exponential moving average of past prices (excluding p(t) itself) with a decay rate equal to n months. Long simulations can often only be performed on monthly data, so we use monthly closes. The signal vs_n(t) at the beginning of month t is constructed as

(20.1)

where the volatility σ_n is equal to the exponential moving average of the absolute monthly price changes, with a decay rate equal to n months. 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 ±σ_n⁻¹ of the underlying contract α⁶:

(20.2)

In the rest of the paper, we will focus on the choice n = 5 months, although the dependence on n will 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 (i.e., the fact that the average return is significantly different from zero) is therefore given by the Sharpe ratio times where N is the number of years over which the strategy is active. We will also define the drift μ 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.

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 important in practice, since diversification plays an important role in the performance of CTAs. However, since the purpose of this paper is to back-test 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 10-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

The Results

Our history of futures starts in 1960, mostly with commodities. As we can see from Figure 20.1 on the next page, the aggregated performance 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 20.2).

However, we might argue that this comes from the trivial fact that there is an overall drift μ 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 β with 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 +15% 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 Tables 20.2 and 20.3, 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 μ. Therefore, the performance shown in Figure 20.1 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 50 years or so, and the first 10 years of those 50 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 macroenvironments. This is the goal of the next section, which provides a convincing confirmation of the results based on futures.

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 U.S. 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.

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.

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 20.2 that the U.S. 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 20.4). 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 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.

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 Tables 20.5 and 20.6, 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 century), the time series are of reasonably good quality, i.e., prices are actually moving (no gaps) and there are no major outliers.

Validating the Proxies

We now want to check that the time series selected above, essentially based on spot data on 10-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 20.3 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 20.7 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 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 20.3.

Results of the Full Simulation

The performance of the trend-following strategy defined by Equation 2.2 over the entire time period (two centuries) is shown in Figure 20.4. 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, i.e., the t-statistic of “excess” returns. For comparison, the t-statistic of the drift μ of the same time series is 4.6. As documented in Table 20.8, 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 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 20.4, and we report the t-statistic for different periods in Table 20.9. The overall performance is in fact positive over every decade in the sample (see Figure 20.7). 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).

A Closer Look at the Signal

It is interesting to delve deeper into the predictability of the trend-following signal s_n(t), defined in Equation 2.1. Instead of computing the P&L given by Equation 2.2, we can instead look at the scatter plot of Δ(t) = p(t + 1) – p(t) as a function of s_n(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, i.e., Δ(t) = 
a + bs_n(t) + ξ(t), where a = 0:018 ± 0.003, b = 0:038 ± 0.002, and ξ is a noise term. The fact that a > 0 is equivalent to saying that the long-only strategy is, on average, profitable, whereas b > 0 indicates the presence of trends. However, it is not a priori obvious that we should expect a linear relation between Δ and sn. Trying a cubic regression gives a very small coefficient for the term and a clearly negative coefficient for the term, indicating that strong signals tend to flatten, as suggested by a running average of the signal shown in Figure 20.5. However, the strong mean reversion that such a negative cubic contribution would predict for large values of sn is suspicious. We have therefore instead tried to model a nonlinear saturation through a hyperbolic tangent (Figure 20.5):

(20.3)

which recovers the linear regime when |sn| ≪ s* but saturates for |sn| > 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 s* ≈ 0.89 and now b ≈ 0.075 (a linear fit is recovered in the limit s* → ∞). Interestingly, the values of a, b, and s* hardly change when n increases from 2.5 months to 10 months.

A Closer Look at the Recent Performance

The plateau in the performance of the trend over the last few years (see Figure 20.6) 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 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 20.7 the average P&L between time t – 10Y 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 20.7 also reveals that the 10-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 20.8). 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.

Interpretation

The above results show that long-term trends exist across all asset classes and are stable in time. As mentioned in the Introduction, trending behavior is also observed in the idiosyncratic component of individual stocks (Barroso and Santa-Clara 2013; Kent and Moskowitz 2013; Narasimhan and Titman 1993, 2011; Geczy and Samonov, 2016). 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 predict” 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).⁸ Studies 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 financial 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 reinforces 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 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 participants 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 20.5. 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).