Anticorrelation, or anti-correlation, means the same as negative correlation. To say that A and B are anticorrelated means that A tends to move up when B moves down and vice versa. This can be said when discussing both autocorrelation or cross-correlation. The term is not as widely used in financial contexts as it is in physics.

## How many people trade forex every day? A Pareto estimate.

To estimate how many people trade forex every day, we use Pareto distribution. This is a power-law, highly asymmetric distribution, encountered in many contexts, including distribution of wealth. Pareto’s probability density distribution is

f(x|k,x_{m}) = k x_{m}^{k}x^{-k-1}

for x greater or equal to x_{m}. x_{m} is the minimum value of x, in our case — the minimum value contributed by a trader into the total turnaround. This number defines who we call a trader for the purpose of this estimate, a threshold so to speak.

According to the definition of a mathematical expectation E[x], it is an integral of f(x|k,x_{m}) times x from x_{m} to infinity and

L E[x] = T

where T is the total turnaround and L is the number of trades that create it. The integral is easy to deal with for k > 1 and we obtain

L = T(k-1)/(k x_{m})

Now to the less definitive stuff. To the best of my knowledge as of now the total daily forex turnaround is about $3 trillions. A reasonable definition of a trader is someone who trades at least one standard size lot a day, that is, $100,000. Thus T=3× 10^{12} and x_{m}=10^{5}. There is a fair amount of uncertainty as to what the Pareto index k is for this market. There is a famous 80-20 rule applicable in many contexts. In this case it would mean that 20% of traders contribute 80% of the turnaround. The 80-20 rule is just a particular instance of a Pareto distribution corresponding to Pareto index of 1.161. With this input, we estimate to have about **four million forex transactions a day.** That’s what we’ve called L. The number of traders creating L transactions a day depends on the structure of the relationships between them, that is, who trades with whom. Imagine a trader as a point on a plane, the L transactions being the links connecting the points, and you get the picture — there are various ways of connecting the points. Our problem is that we’ve estimated (with Pareto’s help) the number of links but we do not know the number of points they connect. One situation, extreme in a sense and only applicable for the sake of academic argument, would be to assume that each trader has a unique partner (kind of a monogamous marriage). Then the number of traders n is simply L times 2. Of course such a system is not capable of moving money. But as you will see, it requires the highest number of traders (**eight million traders?!**) to create a given turnaround, and is interesting as a limiting case. Another situation is the egalitarian one where every trader is equally likely to trade with every other trader. This is also not realizable in practice, but it corresponds to the equation:

L = n(n-1)/2

Because n is much larger than one, n is simply

n = (2L)^{1/2}

If such topology of trading links were the case, only about 3 thousand traders (with Pareto wealth distribution) could create our present trading volume. Such topology may be considered an idealized limit, the Holy Grail of the online trading business. The reality is probably in between these two extremes, with traders being not completely isolated, but forming relatively isolated clusters around brokers, banks, hedge funds and similar institutions. The relationships between those higher level entities are then much closer to the egalitarian model — a tightly knit community where every member knows every other member.

If the estimates of the number of traders, given the trading volume, differ so much depending on the ogranization of the market, then the really interesting conclusion is the converse: the real reason for the spectacular growth in the forex trading volume seen in the past few years probably has at least as much to do with changes in the organization of the market as it does with purely economic reasons.

## AUD/JPY and EUR/USD 2002-2008: Intermarket Correlations (Leader-Follower)

Australian Dollar/Japanese Yen and Euro/US Dollar are weekly correlated. A positive correlation tail with time lags up to 3 hours is seen indicating that EUR/USD tends to lag behind AUD/JPY.

time scale | Asia-Pacific session | European session | American session |
---|---|---|---|

hour | 0.14 | 0.13 | 0.11 |

AUD/JPY and EUR/USD are weakly correlated on average for the period. The correlation is the least pronounced in the American session, most pronounced in the Asia-Pacific session.

The fact that most of the correlation is concentrated at the 0 lag means that the correlation (reported in the table) works out *mostly *on the time scale of up to 1 hour. The tail of positive correlation to the left of the 0 lag indicates that there is a “tail” of predictable action in EUR/USD lagging behind AUD/JPY. It is the strongest in the European and American sessions. Even though the Asia-Pacific session has the strongest correlation between the two currency pairs within the 0-lag time bin (see the table), it has the weakest correlation away from 0 and thus must be the worst for forecasting on the basis of this correlation feature.

To judge how reliable the correlation signal at the non-zero lags is, one has to compare the signal with the noise level obtained from the martingale simulations.

Fig.2 demonstrates the non-flat (although quite predictable) behaviour of the noise level with time lag. This can not be ignored otherwise one risks over-interpreting the picture. The area around zero is fairly safe since the noise is at the minimum when the lag is at an integer number of days. Based on the level of the noise, the tail in the first couple of bins to the left of the 0 peak (which means EUR/USD is trailing AUD/JPY) looks like a real effect. We are probably looking at the “risk aversion”/”risk appetite” mood swings where the AUD/JPY having a very strong interest rate differential can indeed lead the show.

## NZD/USD 2005-2008: Predictability Overview

The correlation patterns we see in one of the world’s most volatile exchange rates, the New Zealand Dollar/US Dollar exchange rate, are very similar to those seen in AUD/JPY.

The interest rate differential has been in favor of the New Zealand Dollar.

### The basic autocorrelation

As before we employ autocorrelation as a straightforward, inter-disciplinary, non-proprietary technique to test market efficiency in the NZD/USD market. In Fig.1 we look for features on the time scale of up to two days such as to suite the time scale of day trading or swing trading. The hatched red band shows the range of statistical noise (namely its expectation plus minus its RMS deviation). Statistical noise was obtained by simulating 20 independent time series of the length corresponding to that of the NZD/USD series, each one constructed to reproduce the measured distribution of returns for the time period under study (including the fat tails!), but completely devoid of correlations (martingale time series). From these, the expectation and RMS of the autocorrelation amplitude in each time lag bin were calculated. The one-hour time lag “contrarian” feature (a significant anticorrelation) we saw in this type of plot for other currency pairs involving USD ( AUD/USD ) is quite strong in the NZD/USD autocorrelation. It is noteworthy that the negative feature around 0 is more than one bin wide, it involves the -3 hour bin as well. The autocorrelation being an average of a product of hourly returns taken with a lag, this negativity means that we are way too frequently (more frequently than in the corresponding martingale time series) taking a product of opposite sign returns — or that the product of the opposite sign returns by far outweighs that of the same sign returns. Because trend reversals on the time scale of one to three hours happen either too often or are too lucrative, NZD/USD, like GBP/JPY, AUD/USD and AUD/JPY analyzed before, may well be the market where winning strategy requires being a contrarian on a short time scale.

A group of time lag bins 12-24 hours away from show a significantly positive correlaiton. In other words, the currency pair has a tendency to repeat its moves 12-24 hours after they happened — a feature worth a closer look as a forecasting mechanism.

### Bull/bear asymmetry in NZD/USD

In Fig.3 we construct autocorrelations of the subsamples of the full time series (the “bullish” and “bearish” ones) selected by taking only positive and negative returns respectively. The 24 hour cycle of bullish and bearish action, clearly seen in most other currency pairs, is not well pronounced here for some reason. In this regard, NZD/USD is similar to AUD/JPY.

Typically, the “bearish” correlation has a higher amplitude whenever the base currency has a higher interest rate. This has been seen with AUD/USD , AUD/JPY , USD/JPY , GBP/JPY , USD/CAD , (although the interest rate differential has not been that high, it is in favor of USD), CHF/JPY , EUR/JPY, EUR/CHF. Conversely, the “bullish” correlation has a higher amplitude whenever the quote currency has a higher interest rate, as seen with EUR/AUD and EUR/GBP. While in the case of classic carry-trade currency pairs such as AUD/JPY I associated this feature with the unwinding of the carry-trade, the underlying mechanism is likely to be similar for other currency pairs. It seems, you can “jump on the bandwagon” of selling a high yield currency with more confidence than doing the opposite, as the higher amplitude and a bump in the NZD-bearish plot demonstrate.

The fact that one can read the sign of interest rate differential off the public forex quotes via basic correlation analysis indeed goes against the efficient market dogma and indicates that despite large liquidity such interest rate differentials are not completely discounted by the markets and there remain profit opportunities for algorithmic trading .

### Summary

The NZD/USD currency pair has been showing a “contrarian” trend reversal tendency which is likely to be part of a stable wave-like pattern. Therefore, NZD/USD is not completely “efficient” from the point of view of basic two-point correlation analysis. Long term prospects of NZD/USD are the subject of fundamental analysis and are outside the scope of this article. Cross-correlations with other markets are to be discussed in the up-coming articles. In this report we use data for the period from 00:00 2005-08-16 to 00:00 2008-02-01 (New York time).

## EUR/USD and USD/CAD 2002-2008: Intermarket Correlations (Symmetric Predictive)

Euro / US Dollar and US Dollar/ Canadian Dollar present another example of symmetrically cross-anticorrelated currency pairs.

time scale | Asia-Pacific session | European session | American session |
---|---|---|---|

hour | -0.38 | -0.42 | -0.43 |

EUR/USD and USD/CAD are anticorrelated on average for the period. The anticorrelation is the least pronounced in the Asia-Pacific session.

The fact that most of the anticorrelation is concentrated at the 0 lag bin means that the anticorrelation (reported in the table) works out *mostly *on the time scale of up to 1 hour. The peak seems to be more than one bin wide, except for perhaps the Asia-Pacific session. In Fig.2, we show statistical significance of the signal.

As Fig.2 demonstrates, the main challenge while working with trading session-specific correlations is the non-flat (although quite predictable) behaviour of the noise level with time lag. The symmetry of the peak means that while it is true that a move in EUR/USD foretells an opposite direction move in USD/CAD, it is equally true that an upward or downward move in USD/CAD foretells a downward or upward move in EUR/USD, respectively. (As always on this site, “foretells” should be understood in the statistical sense). The market reaction is not instantaneous. But the width of the peak lets one estimate how much time the markets take to play out their recation: it may take up to a couple of hours for the adjustment to fully finish (not true in the Asia-Pacific session) — significant signals with two-hour lags are confidently visible in Fig.2.

Data from 2002-08-20 through 2002-02-01 were used in this report.

## EUR/USD and GBP/JPY 2002-2008: Intermarket Correlations (Leader-Follower)

Euro/US Dollar and British Pound/Yen do not seem to share any investment themes. Nevertheless these are correlated currency pairs, with a hint of a leader-follower relationship.

time scale | Asia-Pacific session | European session | American session |
---|---|---|---|

hour | 0.15 | 0.16 | 0.12 |

EUR/USD and USD/JPY are weakly correlated on average for the period. The correlation is the least pronounced in the American session.

The fact that most of the correlation is concentrated at the 0 lag means that the correlation (reported in the table) works out *mostly *on the time scale of up to 1 hour. The tail of positive correlation to the right of the 0 lag indicates that there is a “tail” of predictable action in EUR/USD lagging behind GBP/JPY. It is seen in the European and American sessions. To judge how reliable it is, one has to compare the signal with the noise level obtained from the martingale simulations.

As Fig.2 demonstrates, the main challenge while working with trading session-specific correlations is the non-flat (although quite predictable) behaviour of the noise level with time lag. This can not be ignored otherwise one risks over-interpreting the picture. The area around zero is fairly safe since the noise is at the minimum when the lag is at an integer number of days. Based on the level of the noise, betting on EUR/USD following the lead of GBP/JPY seems to be a risky strategy. But if you decide to do that, the European or American session would be the best time.