aa r X i v : . [ q -f i n . S T ] A ug Christmas Jump in LIBOR
Vikenty Mikheev and Serge E. MiheevAugust 2019
Abstract
A short-term pattern in LIBOR dynamics was discovered. Namely, 2-monthLIBOR experiences a jump after Xmas. The sign and size of the jump dependson data trend on 21 days before Xmas.
Key Words:
LIBOR; short term approximation; pattern; swap market; Christ-mas jump
In 1986, a new benchmark interest rate was introduced and named London InterbankOffered Rate (LIBOR). At LIBOR major banks of the world lend to one another inthe international interbank market for short-term loans. From mathematical point ofview, LIBOR is a sequence of daily changing real values.LIBOR plays crucial role in Swap Market, where people exchange their loan in-terests and can win or lose money depending on their right or wrong predictions ofLIBOR dynamics. For example, a person P got a one-million-dollar loan with 5%interest and a person E borrowed the same amount but with the interest 2%+LIBOR.After some time they decide to exchange their interest rates because P thinks thatLIBOR will go lower than 3% but E hopes that it will go higher than 3%. Boththeir opinions are based on some prediction methods, even if it is just an intuition.We intend to bring another prediction tool into the game. A curious reader mayfind more complex models and measures on LIBOR for different problems [Jamsh],[Schoen], [Moreni] and [Hinch].Thus, here we are not interested to LIBOR nature per se but to its volatility only.More precisely, we study the behavior of LIBOR after Christmas from December 26to December 31.So, how does Christmas affect dynamics of LIBOR until the next holiday?1
Definition and Models
The research we conducted indicates convincingly that a jump is present. But whatis a jump in a discrete sequence of numbers? The following seems to be the mostacceptable
Definition 1
We introduce two approximations for discrete data. One for a certainperiod before considered date (in this case Dec 25) and the other one for a certainperiod after the date. Let them accordingly have forms: B + A ( x ) and B + A ( x ) ,where x is a date and A , A are continuous functions. Moreover, A ( Dec
25) = A ( Dec
25) = 0 . The difference B − B is the jump. It is easy to see that the so-defined jump depends on the type of approximation,i.e. on how the functions A , A are selected; and on the amount of the input data.We still have to decide the amount of input data to calculate the approximating pair B , A . Notice that the amount of the data for A and B is three pairs (date, LIBORof this date) only because there are exactly three working banking days between Xmasand NYE. Data source is available at [IBORate] or multiple other sources.Variability of the data due to random factors leads to the choice of the simplestapproximation. We use linear approximating functions, which coefficients may befound by linear regression. We restrict ourselves to LIBOR data for the last 22 years,because it is natural to expect the evolution of LIBOR behavior over the years.So, for some year j in some set J taken sequentially with no gaps from { } data are taken for 15 banking days x ′− , ..., x ′− (corresponding to 21 calendardays) preceding Xmas of year j . As all the days are in December, for simplicity offollowing constructions we may decrease them by 25 without problem of passing daysto another month. So, x i := x ′ i − , i = − , ..., −
1. Each x i corresponds to y i ,which is the annual interest rate of LIBOR for 2 months on day x i . Using them webuild a linear regression ˆ y ( x ) = a j x + b j , (1)or y i = ˆ y ( x ) + error i , where x is a December day minus 25.That is, in terms of Definition 1, B := b j , A ( x ) := a j x . Since the current trendof LIBOR (meaning the rate of growth or decrease) does not change a lot over a shorttime interval, it is almost the same before and after Xmas. Therefore, we seek anapproximation after Xmas in the following form: ˆ y ( x ) = a ′ j x + b ′ j , where a ′ j = a j and x is a December day decreased by 25. In terms of Definition 1, B := b ′ j , A ( x ) := a ′ j x ≡ A ( x ). Hence, there is only one unknown parameter B . It can also be found2y linear regression procedure or it can be calculated simply as the average of values y i − a j x i , where i runs 1 , , x i ∈ { − , ..., − } = { , ..., } (There areexactly 3 bank days between Xmas and New Year.)Thus, for each selected year j there is a relationship ( b j , a j ) → ˆ b j . According toDefinition 1 the difference ∆ j := ˆ b j − b j is the jump we have been looking for. Havingsuch connections over 22 years, one can try to find a pattern. To do that, we turnto linear-quadratic regression in shorten variant deprived pure square.This time weapproximate on two-dimensional nodes, in other words the approximating function isof the form: F ( a, b ) := β + β a + β b + β ab (2)with an approximation table F J ( a j , b j ) ≈ ∆ j , j ∈ J ⊂ { , . . . , } . Subindex J at F points at which subset of years over the past 22 has been chosen to constructthe regression. The remaining years will be used to verify the statistical reliability ofthe result. We conducted the process above for several different numbers of years for F regression(from 5 to 20 years), different LIBOR data (Overnight, 1 month, 2 months etc). Themost convincing results were have obtained with the following set ups: 21 calendarday regression for each year from 15-year interval; 2-month-loan values of LIBOR.Observe the results in the Table 1.Table 1: Model ∆ ≈ β + β a + β b + β ab and its predictions with p-values p for2019 year.Data 2000-14 2001-15 2002-16 2003-17 2004 - 2018Pred. on 2015 2016 2017 2018 2019 c β -2.96E-3 4.7E-4 4.1E-4 0.00327 0.00473 ( p =0.138) c β -9.286 -9.337 -9.321 -9.278 -9.265 ( p =1.43E-13) c β p =0.088) c β p =5.05E-09)Adj R L mean L mean b ∆ = 0 . − . a − . b + 2 . a b (3)It may be activated at Dec 24 2019 as following:At this day extract data from [IBORate] for bank days since Dec 21 till Dec24 (in 2019, of course). Build 15 pairs ( x i , y i ) , i = − , ..., −
1. Put them intoany program to find linear regression, for example our code in R can be foundin https://github.com/keshmish/Chistmas-Jump-in-LIBOR.git
The result of itswork is two numbers: corresponding to free term is b , the other one is a .Substitution them to (3) yields the jump.And what is about last Christmas [LiborGone] of LIBOR existence, 2020? Atpresent the data for 2019 year do not exist. Therefore we can only propose the samecoefficients as in (3). It can worse the prediction but quite a little. At 2020 Xmaswill happened in Tuesday, so Dec 24 also is bank day. So, this day one should do thesame that we suggested above for 2019. Namely, take 15 pairs of data ( x i , y i ) , i = − , . . . , − a , b . Then put them in the right hand side of the formula (3) on the place of a , b , respectively.In fact, a reader can observe that predictors of (2) don’t change too dramaticallywith the shift of 15 regression years by one. Thus, the prediction for 2020 with2004-2018 model may appear to be quite good.The prediction of the jump can be used to predict the mean LIBOR after Xmasbefore NYE ( L meanj ). Let us show some formulas.According to Definition 1 the jump with approximations above is ∆ j = ˆ b j − b j ,where ˆ b j = arg min b { P i =1 ( a j x i + b − y i ) } , which is equivalent to ˆ b j = P i =1 ( y i − a j x i ).Hence ∆ j = 13 X i =1 ( y i − a j x i ) − b j = 13 X y i − X i =1 ( a j x i + b j ) . (4)Notice that the last term of (4) is nothing but predicted after Christmas meanvalue of LIBOR ( ˆ L meanj ) according to the regression for j -th year. Thus, L meanj = ˆ L meanj + ∆ j = ˆ L meanj + ˆ∆ j + (∆ j − ˆ∆ j ) . If as estimate of L meanj we take ˆ L meanj + ˆ∆ j , then its absolute error equals to ˆ∆ j − ∆ j .The latter difference according to our calculations for years 2015, 2016, 2017, 2018was always less by absolute value than | ˆ∆ j | .4 Conclusion
We have found a short-term pattern in LIBOR dynamics. Namely, 2-month LIBORexperiences a jump after Xmas. The sign and size of the jump depends on data trendon 21 days before Xmas. The results are obtained in the form of the jump per se andas mean predicted value of LIBOR between Xmas and NYE. A swap market playermay try to use this information to predict behaviour LIBOR to do a better game onhis part. For Xmas of 2019 one on a date of Dec 24 can compute a and b accordingto (1) on 21 calendar days and use the formula (3) to predict the jump after Xmas. Institute for Mathematics and its Applications ( https://ima.umn.edu/node ), U.of Minnesota, Fadil Santosa, Daniel Spirn, Davood Damircheli, Anthony Nguyen,Samantha Pinella.
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Chapman and Hall/CRC , New York, 2005.[Moreni] N. Moreni and A. Pallavicini. Parsimonious HJM modelling for multipleyield curve dynamics.
Quantitative Finance , 2(14):199-210, Routledge, 2014[Hinch] M. Hinch, M. McCord and S. McGreal. LIBOR and interest rate spread: sen-sitivities of the Australian housing market.
Pacific Rim Property ResearchJournal , 1(25):73-99, Routledge, 2019.[LiborGone] John Heltman. Libor is going dark in 2021, andsome banks aren’t ready.
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