Swiss Franc (CHF) LIBOR: technical predictability overview

I’ve outlined the original motivation to study historical LIBOR data from predictability point of view in the USD LIBOR article. I continue with the logarithmic returns technique that proved useful in forex. Like the previous reports, this document begins with historical LIBOR charts for the Swiss Franc, continues with volatility analysis, and culminates with autocorrelations and correlations. You will see that predictable patterns in CHF LIBORs vary with duration term. Autocorrelations of short-term LIBORs show fast (about 4-day period) oscillation. For 3-month and 6-month terms, the main correlation pattern does not develop 70-day period waves on top of positive background, in contrast to USD and EUR LIBORs, but keeps oscillating between positive and negative autocorrelation values, with the oscillation period longer than that of the shorter terms. The autocorrelation of 12-month LIBOR remains similar to 6-month instead of becoming more uniformly positive as it does for JPY or more jittery as it does for USD, EUR and GBP.

LIBOR charts

History of s/n-o/n CHF LIBOR 2002-2008 History of 1 week CHF LIBOR 2002-2008 History of 1-month CHF LIBOR 2002-2008 History of 3-month CHF LIBOR 2002-2008 History of 6-month CHF LIBOR 2002-2008 History of 12-month CHF LIBOR 2002-2008

Fig.1: Historical CHF LIBOR rates charts, top to bottom: s/n-o/n, 1-week, 1-month, 3-month, 6-month and 12-month. Time axis is labeled in MM-YY format.

The evolution of visual features with maturity here resembles that of other currencies, except for the fact that for CHF, longer maturities show considerably less “initiative” in developing their own trends. Their history looks more like a dumb version of the shorter ranges. For the short maturities, the markets jump 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. This will be seen qunatitatively in the correlation plots.

LIBOR volatility

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

durationtime scalevolatility (RMS)
s/n-o/nday7.3×10-2
weekday4.8×10-2
monthday2.6×10-2
3 monthsday2.2×10-2
6 monthsday2.2×10-2
12 monthsday2.4×10-2

Volatility of CHF LIBOR seems to go down with duration in a more reliable fashion than for other currencies, in particular, USD and EUR. Like JPY, logarithmic returns in CHF look very volatile — this is because the market think about interest rate variations in the “absolute”, not relative sense, and because JPY and CHF interest rates are low, therefore interest rate moves worthy of market’s attention are relatively large for these markets.

Distribution of logarithmic returns in s/n-o/n and 1-week CHF LIBOR rates Distribution of logarithmic returns in 3-month, 6-month and 12-month CHF LIBOR rates

Fig.2: Distributions of logarithmic returns in CHF LIBOR rates, top: s/n-o/n and 1-week, bottom: 1-month, 6-month and 12-month maturity. Volatility is a measure of the width of the return distribution.

The distribution of logarithmic returns look 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

As with some of the most volatile forex exchange rates and all LIBORs looked at so far, the most prominent feature of the s/n-o/n and 1-week CHF 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.

CHF s/n-o/n LIBOR autocorrelation, 1 day time scale CHF 1-week LIBOR autocorrelation, 1 day time scale CHF 1-month LIBOR autocorrelation, 1 day time scale CHF 3-month LIBOR autocorrelation, 1 day time scale CHF 6-month LIBOR autocorrelation, 1 day time scale CHF 12-month LIBOR autocorrelation, 1 day time scale

Fig.3:Autocorrelation of logarithmic returns in the historical CHF 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 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.

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-month and longer term LIBOR autocorrelations have broad peaks around zero. This is very different from forex exchange rates, and implies that in LIBOR, medium-range (several days) forecasting is straighforward for these maturities: betting on the continuation of a trend is the winning strategy. In other words, trend following is possible with medium-range LIBOR.

The predictive positive zero-lag peak of 1-month and longer maturities has to be contrasted with the opposite feature seen in shorter maturities, namely the “bipolar disorder”, a tendency to form patterns where the strategy of betting on the trend reveral is more likely to succeed. This tendency shows up in the negative correlation magnitude at the lag that corresponds to the time it takes for the trend reversal. In CHF LIBOR, s/n-o/n and 1-week data, the time is no more than 2-3 days. Trend following is not a viable strategy with s/n-o/n and 1-week 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. The zero-lag correlation magnitudes between different LIBOR terms are presented as Pearson correlation coefficents in the table. The figures focus on the correlation shapes at the time lags surrounding the zero-lag peak.

Table 1: Pearson correlation coefficients between CHF LIBOR in 2002-2008, various maturities

durations/n-o/n1-week1-month3-month6-month12-month
s/n-o/n10.730.510.320.250.17
1-week10.620.460.350.28
1-month10.770.660.51
3-month10.850.66
6-month10.85
12-month1

The correlation of different maturity terms (which is roughly the square root of the zero time-lag peak amplitude) is seen to go down as the difference in maturities grows; similar maturities are correlated tighter. Overall, the evolution with LIBOR maturity term is the same as described for autocorrelations.

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

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

Correlations between 1-week and longer term LIBOR rates

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

Correlations between 1-month and longer term LIBOR rates

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

Correlations between 3-month and longer term LIBOR rates

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

Pound Sterling (GBP) LIBOR Rates: Technical Predictability Overview

The original motivation for the technical, mostly correlation-based study of LIBORs was outlined in the USD LIBOR article. Like the USD, EUR and JPY LIBOR reports, this document begins with historical LIBOR charts for the Pound Sterling, continues with volatility analysis, and culminates with correlations of logarithmic returns in GBP LIBOR. You will see that predictable patterns in GBP LIBORs show great variation with loan term. Autocorrelations of short-term LIBORs look jittery on the next-day time scale. Autocorrelations of short term (s/n-o/n and 1-week) LIBOR exhibit the now familiar “bipolar disorder” pattern with the characteristic time period of no more than 2-3 days. The smooth wave-like patterns of intermediate term USD and EUR LIBORs, about 70 days in period, are also found in GBP. As the term duration increases, the main correlation pattern becomes that of predictive (non-zero time lag) positive correlation between different maturity terms as well as inside individual time series (autocorrelation).

LIBOR charts

History of s/n-o/n GBP LIBOR 2002-2008 History of 1 week GBP LIBOR 2002-2008 History of 1-month GBP LIBOR 2002-2008 History of 3-month GBP LIBOR 2002-2008 History of 6-month GBP LIBOR 2002-2008 History of 12-month GBP LIBOR 2002-2008

Fig.1: Historical GBP LIBOR rates charts, top to bottom: s/n-o/n, 1-week, 1-month, 3-month, 6-month and 12-month. Time axis is labeled in MM-YY format.

The most striking feature is the high volatility of s/n-o/n and 1-week LIBOR rates in 2002 which gradually goes down. For the short maturities, the markets jump 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. Longer maturities develop patterns of their own while the shorter ones are dominated by the basic step-like pattern modulated by the short-range neurosis. This will be seen qunatitatively in the correlation plots.

 

LIBOR volatility

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

durationtime scalevolatility (RMS)
s/n-o/nday7.8×10-2
weekday2.6×10-2
monthday5.8×10-2
3 monthsday3.8×10-2
6 monthsday6.6×10-2
12 monthsday7.1×10-2

Volatility of GBP LIBOR seems to have no easy pattern in its dependence on duration term.

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

Fig.2: Distributions of logarithmic returns in GBP LIBOR rates, top: s/n-o/n and 1-week, bottom: 1-month, 6-month and 12-month maturity. Volatility is a measure of the width of the return distribution.

The distribution of logarithmic returns on the day time scale looks rather complex, reflecting the evolution of the LIBOR pattern with time — the jittery picture of 2002 will certainly result in a different logarithmic return distribution than the 2006 and 2007. 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 GBP 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.

GBP s/n-o/n LIBOR autocorrelation, 1 day time scale GBP 1-week LIBOR autocorrelation, 1 day time scale GBP 1-month LIBOR autocorrelation, 1 day time scale GBP 3-month LIBOR autocorrelation, 1 day time scale GBP 6-month LIBOR autocorrelation, 1 day time scale GBP 12-month LIBOR autocorrelation, 1 day time scale

Fig.3: Autocorrelation of logarithmic returns in the historical GBP 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: s/n-o/n, 1-week, 1-month, 3-month, 6-month, and 12-month data.

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 (but not 12-month) 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 (several days) forecasting is straighforward for these maturities: 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, generally justify no such strategy, and you will not find wide positive peaks in the forex return correlations.

The predictive positive zero-lag peak of 1-month and longer maturities has to be contrasted with the opposite feature seen in shorter maturities, namely the “bipolar disorder”, a tendency to form patterns where the strategy of betting on the trend reveral is more likely to succeed. This tendency shows up in the negative correlation magnitude at the lag that corresponds to the time it takes for the trend reversal. In GBP LIBOR, s/n-o/n and 1-week data, the time is no more than 2-3 days. Trend following is not a viable strategy with s/n-o/n and 1-week 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. 1-month, 3-month and 6-month figures show oscillations with what looks like 70 to 75 day period (counting business days only). Not sure what this has to do with periodicity of BOE meetings — but 70 days is twice the FOMC’s regular period. In fact, similar periodicity has been seen in the USD and EUR LIBOR autocorrelations for comparable maturities.

 

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.

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

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

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.