Japanese Yen (JPY) LIBOR rates: technical predictability overview

The original motivation for the technical, mostly correlation-based study of forex predictability, is “not supposed” to exist in liquid markets), may indeed take the front row sit at the LIBOR show. Perhaps, predictability of risk is a good phrase to discuss the subject matter at hand. Like the USD and EUR LIBOR reports, this document begins with historical LIBOR charts for the Japanese Yen, continues with volatility analysis, and culminates with correlations of logarithmic returns in JPY LIBOR.

Executive summary

JPY LIBORs show great variation of correlation patterns with loan term. Correlations of short-term LIBORs look dominated by singular events; time-dependency analysis is required to tell you more. Autocorrelations of short term LIBOR exhibit the now familiar “bipolar disorder” pattern with the period of one (for s/n-o/n) or a few days (for 1-week LIBOR). The smooth wave-like patterns of intermediate term USD and EUR LIBORs, about 70 days in period, are not found in JPY. As the term duration increases, the main correlation pattern becomes that of positive correlation between different maturity terms as well as inside individual time series (autocorrelation). Longer term maturities are seen to strongly influence the future of the shorter terms in a positively correlated fashion.

During the period under consideration, BOJ raised the Basic Discount rate only twice: on July 14, 2006, from 0.10 to 0.40, and on February 21, 2007, from 0.40 to 0.75. Thus the interesting stuff begins in 2006. The “invisible hand” of the market jumps the gun trying to anticipate the course of events almost regularly, to the extent this nervousness must represent a regular and significant speculative opportunity, if the market instruments tied to the LIBOR rates have the same features.

Table 1: Day-by-day volatilities (RMS) for the time series of logarithmic returns in JPY LIBOR in 2002-2008, various maturities

durationtime scalevolatility (RMS)
s/n-o/nday5.3×10-2
weekday4.4×10-2
monthday3.5×10-2
6 monthsday2.3×10-2
12 monthsday1.4×10-2

Unlike USD and EUR, volatility of JPY LIBOR goes down for larger maturities.

Distribution of logarithmic returns in s/n-o/n and 1-week JPY LIBOR rates

Fig.2: Distributions of logarithmic returns in JPY LIBOR rates. Volatility is a measure of the width of the return distribution.

The distributions of logarithmic returns on the day time scale look rather complex, possibly due to the fact that two histories with very different features are now mixed together: the pre-2006 and post-2006. The core distributions may be power-law (remember that with returns already containing logarithm and with the vertical axis explicitly logarithmic, we are looking at what is effectively a log-log plot, where any power law dependence would have looked linear, with different power law exponents resulting in different slopes), but the long tails certainly do not belong to the same, if any, power law as the core distribution.

LIBOR autocorrelations

As with the Euro and the US Dollar LIBORs and with some of the most volatile forex exchange rates, the most prominent feature of the s/n-o/n and 1-week JPY LIBOR autocorrelations is the “bipolar disorder” pattern seen from the bins with large negative signal surrounding the zero-time lag bin. (The expression “bipolar disorder” in relation to the market is credited to Benjamin Graham). Continuing with the psychiatric analogy, these indicate rapid (next day or two, depending on LIBOR term) changes in the mood of the credit market, a price action followed by an immediate correction.

JPY 3-month LIBOR autocorrelation, 1 day time scale JPY 6-month LIBOR autocorrelation, 1 day time scale

Fig.3:Autocorrelation of logarithmic returns in the historical JPY LIBOR is shown against the backdrop of statistical “noise”. The noise is obtained from martingale simulations based on the historical volatilities of LIBOR for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular time lag bin by analyzing 20 independent simulated pairs of uncorrelated time series. The RMS is a measure of accuracy in the determination of the correlation values, an irreducible uncertainty dependent on the amount of data and the time scale.

Fig.3 and subsequent figures ascertain the significance of the patterns by comparing with the statistical noise estimate, based on simulations devoid of correlations, but with volatility of the actual data. 1-week and longer term LIBOR autocorrelations are overall positive for the time lags of hunderds of days, with considerable evolution in shape. This is very different from forex exchange rates, and implies that in LIBOR, medium-range forecasting is straighforward for these duration terms: betting on the continuation of a trend is the winning strategy. In other words, trend following is possible with LIBOR — forex exchange rates, on the contrary, usually justify no such strategy, and you will not find wide positive correlation peaks in forex. Trend following is not a viable strategy with s/n-o/n LIBOR: here, betting on the next-day trend reversal or using longer range correlations, some of which just as sharp, seems to be the surest strategy.

Cross-correlations of LIBOR terms

Next, I am going to look at correlation between LIBOR rates of different maturities for various time lags. These help answer the question to what extent one LIBOR term can be predicted on the basis of any others. Here is the summary, followed by the data.

The covariance of different maturity terms (amplitude of the zero time-lag peak) is seen to go down as the difference in maturities grows; similar maturities are correlated tighter. The correlations become overall more positive between longer-term maturities. The cross-correlations of time series for different LIBOR terms show more positive correlation in the positive range of the time lag. In the figures, all correlated pairs are ordered so that shorter term is first, longer term second. The time lag being

t1-t2,

the conclusion is that often, longer term leads and shorter term follows.

Correlations between s/n-o/n and longer term LIBOR rates

Correlation between logarithmic returns in s/n-o/n and 1-week JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 1-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 3-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 6-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 12-month JPY LIBOR rates as a function of time lag, days

Fig.4: Correlation between logarithmic returns in s/n-o/n and, top to bottom: 1-week, 1-month, 3-month, 6-month and 12-month JPY LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlations between 1-week and longer term LIBOR rates

Correlation between logarithmic returns in 1-week and 1-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 3-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 6-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 12-month JPY LIBOR rates as a function of time lag, days

Fig.5: Correlation between logarithmic returns in 1-week and, top to bottom: 1-month, 3-month, 6-month and 12-month JPY LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlations between 1-month and longer term LIBOR rates

Correlation between logarithmic returns in 1-month and 3-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-month and 6-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-month and 12-month JPY LIBOR rates as a function of time lag, days

Fig.6: Correlation between logarithmic returns in 1-month and, top to bottom: 3-month, 6-month and 12-month JPY LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlations between 3-month and longer term LIBOR rates

Correlation between logarithmic returns in 3-month and 6-month JPY LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 3-month and 12-month JPY LIBOR rates as a function of time lag, days

Fig.7: Correlation between logarithmic returns in 3-month and, top to bottom: 6-month and 12-month JPY LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Euro LIBOR rates: technical predictability overview

The motivation for the technical, mostly correlation-based study of LIBORs was outlined in the USD LIBOR article. Forex correlation analysis made me believe that LIBORs are important for speculative forex forecasting. From a more academic standpoint, it is interesting to hone one’s analytic skills by expanding the range of application of the correlation techniques which yield intriguing results in forex to what’s behind the forex movements, namely to the interest rates. I’ve structured this document to begin with historical LIBOR charts for Euro, continue with volatility analysis, culminate with LIBOR autocorrelations which is my prime tool of predictability analysis and conclude with cross-correlations among Euro LIBORs of different maturity terms.

As far as the overnight LIBOR is concerned, the money markets jump the gun trying to anticipate the course of events almost regularly, to the extent this nervousness must represent a regular and significant arbitrage opportunity, if the market instruments tied to the LIBOR rates have the same features. The overnight LIBOR chart is full of arbupt jumps up and down; the upward trend which began in 2005 looks like a staircase with distinct steps; these sharp features are gone in 3-month data. Curiously, the 12-month chart has “extra” oscillations which do not correspond to the clear-cut treds in the shorter term LIBOR data. Apparently these represent (often mistaken) attempts of bankers to price in their anticipation of LIBOR evolution. For example, the anticipated interest rates drop in early 2008 never materialized for EUR. As a result of this and other similar episodes, 12-month LIBOR is more volatile than many shorter maturities; the edges of the maturity gamut are more volatile than the middle. This will be seen also from Table 1 in the Volatility section.

LIBOR volatility

I use RMS of the logarithmic return distribution as a quantitative measure of volatility in LIBOR interest rates.

Table 1: Day-by-day volatilities (RMS) for the time series of logarithmic returns in EUR LIBOR in 2002-2008, various maturities

durationtime scalevolatility (RMS)
s/n-o/nday3.8×10-2
weekday1.1×10-2
monthday0.56×10-2
3 monthsday0.37×10-2
12 monthsday0.88×10-2

Distribution of logarithmic returns in s/n-o/n and 1-week EUR LIBOR rates Distribution of logarithmic returns in 1-month, 3-month and 12-month EUR LIBOR rates

Fig.2: Distributions of logarithmic returns in EUR LIBOR rates, top: s/n-o/n and 1-week, bottom: 1-month, 3-month and 12-month maturity. Volatility is a measure of the width of the return distribution.

As with US Dollar LIBOR, the distributions of logarithmic returns on the day time scale are not lognormal. The s/n-o/n distribution looks broader than power-law (remember that with returns already containing logarithm and with the vertical axis explicitly logarithmic, we are looking at what is effectively a log-log plot, where any power law dependence would have looked linear, with different power law exponents resulting in different slopes).

LIBOR autocorrelations

Fig.4 shows the autocorrelations of logarithmic returns in the historical LIBOR time series for the Euro.

Autocorrelation in the logarithmic returns of s/n-o/n EUR LIBOR. Day time scale. Autocorrelation in the logarithmic returns of 1-week EUR LIBOR. Day time scale. Autocorrelation in the logarithmic returns of 1-month EUR LIBOR. Day time scale. Autocorrelation in the logarithmic returns of 3-month EUR LIBOR. Day time scale. Autocorrelation in the logarithmic returns of 6-month EUR LIBOR. Day time scale. Autocorrelation in the logarithmic returns of 12-month EUR LIBOR. Day time scale.

Fig.4: Autocorrelations in the historical EUR LIBOR is shown against the backdrop of statistical “noise”. The noise is obtained from martingale simulations based on the historical volatilities of LIBOR for the period under study. The noise is presented as mean plus-minus 1 RMS, where the RMS characterizes distribution of the correlation value obtained for each particular time lag bin by analyzing 20 independent simulated pairs of uncorrelated time series. The RMS is a measure of accuracy in the determination of the correlation values, an irreducible uncertainty dependent on the amount of data and the time scale. Top to bottom: s/n-o/n, 1-week, 1-month, 3-month, 6-month, and 12-month data.

As with the US Dollar LIBOR and with some of the most volatile forex exchange rates, the most prominent feature of these Euro autocorrelations is the “bipolar disorder” pattern seen from the bins with large negative signal surrounding the zero-time lag bin. (The expression “bipolar disorder” in relation to the market is credited to Benjamin Graham). Continuing with the psychiatric analogy, these indicate rapid (next day in case of the s/n-o/n LIBOR) changes in the mood of the patient (Mr Market), a price action followed by a correction. The correction comes next-day for s/n-o/n and on the fourth day for the 1-week term, and may last up to three days. Significance of this and other features is seen from comparison with random walk or martingale simulations respecting the volatility distributions of real LIBOR markets. 1-week LIBOR data show definite periodicity, the period being about 20 days. In 1-month and 3-month data, there is a change in the picture: the correlation is purely positive; there is still an oscillation, but with a longer period of perhaps 50 days. The pattern weakens in the 6-month term data and is practically gone in the 12-month.

Cross-correlations of LIBOR terms

Next, I am going to look at correlation between LIBOR rates of different maturities for various time lags.

Correlation between logarithmic returns in s/n-o/n and 1-week EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 1-month EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 3-month EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 12-month EUR LIBOR rates as a function of time lag, days

Fig.5: Correlation between logarithmic returns in s/n-o/n and, top to bottom: 1-week, 1-month, 3-month and 12-month EUR LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlation between logarithmic returns in 1-week and 1-month EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 3-month EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 12-month EUR LIBOR rates as a function of time lag, days

Fig.6: Correlation between logarithmic returns in 1-week and, top to bottom: 1-month, 3-month and 12-month EUR LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlation between logarithmic returns in 1-month and 3-month EUR LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-month and 12-month EUR LIBOR rates as a function of time lag, days

Fig.7: Correlation between logarithmic returns in 1-month and, top to bottom: 3-month and 12-month EUR LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Naturally, individual LIBOR maturities are generally positively correlated with one another. Terms of similar duration show more correlation.

US Dollar (USD) LIBOR Rates: Technical Predictability Overview

This article begins a series of analysis reports investigating a degree of predictability in the LIBOR rates, a popular capital cost indicator. The analysis is based on historical LIBOR interest rate data released by the British Bankers Association. I continue with the same technique proved useful in the predictability analysis of forex exchange rates, as our interest in the interest rates in general is in part provoked by the results of the latter analysis, namely:

  • sometimes, one forex exchage rate can “show the way” to a number of others, or in other words, foretell (in a probabilistic or statistical sense) their movement.
  • when that happens, it is usually the exchange rate with a large interest rate differential showing the way to the ones with lower interest rate differentials.

Autocorrelations in s/n-o/n USD LIBOR 2002-2008

Fig.1: LIBOR heartbeat: autocorrelation in logarithmic returns of historical USD LIBOR rates, s/n-o/n duration, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the recorded LIBOR volatility for the period under study (2002-2008). The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series. The LIBOR shows strong regular structure with a period of 10 business days (two weeks). Time lag is measured in days. The familiar jump-the-gun pattern (strong negative signal around zero time lag) seen sometimes in forex, is also visible here. This is the level of predictability one can only dream of in forex exchange rates, yet it is the interest rates that drive forex. Is LIBOR always that predictable?

Obviously, when exploring these “loopholes” or market inefficiencies for wealth generation, an algorithmic trader or a forex trading system (an automated decision making algorithm such as the one being built here on Forex Automaton™ site) must be mindful of the picture of LIBOR rates and its evolution, albeit in a somewhat different context than a long-term money manager. Being able to predict events, even in a weak statistical sense, is even better than merely following. Besides being useful via their implications for forex forecasting, LIBORs form an underlying indicator for derivatives of their own. LIBOR futures contracts and options on such contracts are traded on the CME. How does the predictability of LIBORs compare with that of currencies? Which one, LIBOR or forex, is more attractive to trade? Answering these questions, or providing a technical analysis framework to approach the answers, while leaving the fundamentals and event-driven trends aside, this series of articles about correlation features in LIBORs will serve as a useful compliment to our set of forex correlation analysis notes. I start this new series of articles with the all-important US Dollar LIBOR.

Executive Summary

The time series of US Dollar LIBORs is highly volatile; on the day scale, the distribution of “returns” (daily increments) is not only not lognormal but for some maturities not even power law. The main correlation pattern is positive correlation, between different terms as well as inside individual time series (autocorrelation). The heart-beat pattern of short-range LIBOR is replaced by longer-range waves and finally disappears in 12-month term data.

LIBOR charts

BBA tracks LIBORs for various loan durations (maturities). I focus on spot-next/overnight (s/n-o/n), 1 week, 1 month, 3 month, and 1 year maturities.

History of s/n-o/n USD LIBOR 2002-2008 History of 1 week USD LIBOR 2002-2008 History of 3-month USD LIBOR 2002-2008 History of 12-month USD LIBOR 2002-2008

Fig.2: Historical USD LIBOR rates charts, top to bottom: s/n-o/n, 1-week, 1-month, 3-month and 12-month. Time axis is labeled in MM-YY format.

Comparing the histories of different LIBOR maturities in different panels of Fig.2, you see their regular and sharp pattern becoming progressively less sharp as the maturity increases, with the shortest, s/n-o/n LIBOR, having the sharpest structure. The origin of the negative spikes surrounding the zero time-lag peak is quite clearly visible: the history of LIBOR is full of false alarm events when the rate suddenly changes its course (typically, jumps up) only to correct itself completely the next day. The money markets jump the gun trying to anticipate the course of events almost regularly, to the extent this nervousness must represent a regular and significant arbitrage opportunity, if the market instruments tied to the LIBOR rates have the same features.

LIBOR volatilities

Table 1: Day-by-day volatilities (RMS) for the time series of logarithmic returns in USD LIBOR in 2002-2008.

durationtime scalevolatility (RMS)
s/n-o/nday3.1×10-2
weekday1.4×10-2
monthday8.5×10-3
3 monthsday8.4×10-3
12 monthsday1.9×10-2

Distribution of logarithmic returns in s/n-o/n and 1-week USD LIBOR rates Distribution of logarithmic returns in 1-month, 3-month and 12-month USD LIBOR rates

Fig.3: Distributions of logarithmic returns in USD LIBOR rates, top: s/n-o/n and 1-week, bottom: 1-month, 3-month and 12-month maturity. Volatility is a measure of the width of the return distribution.

As you could have guessed from inspecting Fig.2, s/n-o/n LIBOR is more volatile than 1-week, whereas 12-month LIBOR is more volatile than 1-month and 3-month. Forex distributions on a shorter scale (one hour) look much less volatile than LIBOR, the “fat tails” in LIBOR distributions are truly amazing.


LIBOR autocorrelations

USD 1-week LIBOR autocorrelation, 1 day time scale USD 1-month LIBOR autocorrelation, 1 day time scale USD 3-month LIBOR autocorrelation, 1 day time scale USD 12-month LIBOR autocorrelation, 1 day time scale

Fig.4: Autocorrelation of logarithmic returns in the historical USD LIBOR is shown against the backdrop of statistical “noise”. The noise is obtained from martingale simulations based on the historical volatilities of LIBOR for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular time lag bin by analyzing 20 independent simulated pairs of uncorrelated time series. The RMS is a measure of accuracy in the determination of the correlation values, an irreducible uncertainty dependent on the amount of data and the time scale. Top to bottom: 1-week, 1-month, 3-month, and 12-month data.

As was seen in Fig.1, the pediodic “heartbeat” pattern in the overnight LIBOR autocorrelation shows no signs of abaiting for the range of lags as long as 200 days. In Fig.4, you see that this “heartbeat” is unique to the overnight LIBOR, while some “gun-jumping” (short range negativeness) is also seen in 1 week LIBOR loan term data, but is gone in longer terms. Fig.1 and Fig.4 ascertain the significance of the patterns by comparing with the statistical noise estimate, based on simulations devoid of correlations, but with volatility of the actual data. 1-week, 1-month and 3-month LIBOR autocorrelations are overall positive, with a strong short-range peak. This is very different from forex exchange rates, and implies that medium-range LIBOR forecasting is straighforward for these duration terms: betting on the continuation of a trend, no matter whether the trend is up or down, is the winning strategy. In other words, trend following is possible with LIBOR — forex exchange rates, on the contrary, usually justify no such strategy, and you will not find wide positive correlation peaks in forex. Neither does the 12-month LIBOR where the wide peak around zero time lag is gone.

Federal Open Markets Committee (FOMC) holds its regular meetings every 7 weeks, that is 35 business days. One might expect that regularity to result in a feature of some kind corresponding to this time scale. The wavelength of the 1-month and 3-month LIBOR oscillations, see Fig.4, which is about 70 days, may be linked to that, being a multiple of 35.

Cross-correlations of LIBOR terms

Next, I am going to look at correlation between LIBOR rates of different maturities for various time lags.

Correlation between logarithmic returns in s/n-o/n and 1-week USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 1-month USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 3-month USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in s/n-o/n and 12-month USD LIBOR rates as a function of time lag, days

Fig.5: Correlation between logarithmic returns in s/n-o/n and, top to bottom: 1-week, 1-month, 3-month and 12-month USD LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlation between logarithmic returns in 1-week and 1-month USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 3-month USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-week and 12-month USD LIBOR rates as a function of time lag, days

Fig.6: Correlation between logarithmic returns in 1-week and, top to bottom: 1-month, 3-month and 12-month USD LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Correlation between logarithmic returns in 1-month and 3-month USD LIBOR rates as a function of time lag, days Correlation between logarithmic returns in 1-month and 12-month USD LIBOR rates as a function of time lag, days

Fig.7: Correlation between logarithmic returns in 1-month and, top to bottom: 3-month and 12-month USD LIBOR rates as a function of time lag, days, shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical LIBOR volatilities for the period under study. The noise is presented as mean plus-minus 1 RMS, where RMS characterizes the distribution of the correlation value obtained for each particular bin by analyzing 20 independent simulated pairs of uncorrelated time series.

Naturally, individual LIBOR maturities are generally positively correlated with one another. Speaking of forecasting, more often than not we see more positive correlation in the positive range of time lag. All correlated pairs are ordered so that shorter term is first, longer term second. The time lag being

t1-t2,

the conclusion is that often, shorter term follows and longer term leads. 12-month term is seen to lead 1-week and 1-month; 3-month term is seen to lead s/n-o/n; so is 1-month.

Anticorrelation

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,xm) = k xmkx-k-1

for x greater or equal to xm. xm 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,xm) times x from xm 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 xm)

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× 1012 and xm=105. 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.

Table: Pearson correlation coefficient for the time series of logarithmic returns  in AUD/JPY and EUR/USD in various trading sessions in 2002-2008.

time scale Asia-Pacific session European session American session
hour0.140.130.11

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.

AUD/JPY and EUR/USD ntermarket correlation

Fig.1: Cross-correlation of AUD/JPY and EUR/USD, derived from the hour-by-hour logarithmic returns, for the three trading sessions.

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.

AUD/JPY and EUR/USD intermarket correlation European session

Fig.2: Cross-correlation of AUD/JPY and EUR/USD, derived from the hour-by-hour logarithmic returns, for the European (Eurasian) trading session shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical volatilities of AUD/JPY and EUR/USD in this trading session.

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

NZD/USD autocorrelation 1 hour time-lag bin

Fig.1: Autocorrelation of hourly logarithmic returns in NZD/USD. The time lag is in “business time” (periods without update ticks are excluded). The red band shows the level of noise as iferred from martingale simulations (see text).

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

NZD/USD bullish and bearish autocorrelation

Fig.3: NZD/USD bullish and bearish autocorrelations. Yellow: correlating only positive hourly returns. Blue: correlating only negative hourly returns.

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.

Table: Pearson correlation coefficient for the time series of logarithmic returns in EUR/USD and USD/CAD in various trading sessions in 2002-2008.

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.

EUR/USD and USD/CAD intermarket correlation

Fig.1: Cross-correlation of EUR/USD and USD/CAD, derived from the hour-by-hour logarithmic returns, for the three trading sessions.

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.

EUR/USD and USD/CAD intermarket correlation European session

Fig.2: Cross-correlation of EUR/USD and USD/CAD, derived from the hour-by-hour logarithmic returns, for the European (Eurasian) trading session shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical volatilities of EUR/USD and USD/CAD in this particular trading session.

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.

Table: Pearson correlation coefficient for the time series of logarithmic returns in EUR/USD and USD/JPY in various trading sessions in 2002-2008.

time scale Asia-Pacific session European session American session
hour0.150.160.12

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

EUR/USD and GBP/JPY intermarket correlation

Fig.1: Cross-correlation of EUR/USD and GBP/JPY, derived from the hour-by-hour logarithmic returns, for the three trading sessions.

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.

EUR/USD and GBP/JPY intermarket correlation European session

Fig.2: Cross-correlation of EUR/USD and GBP/JPY, derived from the hour-by-hour logarithmic returns, for the European (Eurasian) trading session shown against the backdrop of statistical noise (red). The noise is obtained from martingale simulations based on the historical volatilities of EUR/USD and GBP/JPY in this particular trading session.

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.

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

Euro/US Dollar and Pound Sterling/US Dollar are obviously correlated currency pairs. Due to the symmetry of the cross-correlation peak, a move in either pair can in principle be used to predict a move in the other: EUR/USD foretells GBP/USD and vice versa.

Table: Pearson correlation coefficient for the time series of logarithmic returns in EUR/USD and GBP/USD in various trading sessions in 2002-2008.

time scale Asia-Pacific session European session American session
hour0.660.730.76

The Asia-Pacific session shows the least correlation between the two currency pairs.

EUR/USD and GBP/USD intermarket correlation

Fig.1: Cross-correlation of EUR/USD and USD/JPY, derived from the hour-by-hour logarithmic returns, for the three trading sessions.

Fig.1 shows the intermarket correlation with one hour time scale and the range of lags of up to 12 hours, of interest to a day trader. The positive peak at the zero hour lag tells you that the currencies are correlated, or move in tandem. The height of the peak showing strength of the correlation varies session to session, we present the information textually in the table. The peak seems to be more than one bin wide, except for the Asia-Pacific session. The symmetry of the peak means that while it is true that a move in EUR/USD is followed by a move in the same direction in GBP/USD, it is equally true that an up or down move in GBP/USD may be followed by an up or a down move in EUR/USD. The market reaction is not instantaneous and it may take up to a couple of hours for the adjustment to finish (not true in the Asia-Pacific session). For trading EUR/USD and GBP/USD on the basis of the intermarket correlation strategy, the European and American sessions are the best time.