Eur. Phys. J. B 20, 537–541 (2001) T HE EUROPEAN P HYSICAL JOURNAL B c  EDP Sciences Societ` a Italiana di Fisica Springer-Verlag 2001 False Euro (FEUR) exchange rate correlated behaviors and investment strategy M. Ausloos 1, a and K. Ivanova 2 1 GRASP and SUPRAS, B5, Sart Tilman, 4000 Li` ege, Belgium 2 Department of Meteorology, Pennsylvania State University, University Park, PA 16802, USA Received 31 August 2000 Abstract. We have searched for correlations and anticorrelations with respect to currencies as CHF, DKK, JPY, and USD in order to understand the EUR behavior. In order to do so we have invented a false euro (FEUR) dating back to 1993 and have derived simulated exchange rates of the FEUR. Within the Detrended Fluctuation Analysis (DFA) statistical method we have obtained the power law behavior describing the rms. deviation of the fluctuations as a function of time. We have compared the time-dependent exponent for these four exchange rates, and observe the role of the DEM, and the other currencies forming the EUR. A simple investment strategy based on the local DFA technique shows one can obtain appreciable gains, even taking into account some modest transaction fee. We compare the time dependent α exponent of the DFA for various exchange rates as in a correlation matrix for estimating respective influences. PACS. 05.45.Tp Time series analysis – 05.45.Gg Control of chaos, applications of chaos – 74.40.+k Fluctuations (noise, chaos, nonequilibrium superconductivity, localization, etc.) 1 Introduction The Euro is a new currency which will replace 11 Eu- ropean currencies in 2002, i.e. Austrian Schilling (ATS), Belgian Franc (BEF), Finnish Markka (FIM), German Mark (DEM), French Franc (FRF), Irish Pound (IEP), Italian Lira (ITL), Luxembourg Franc (LUF), Dutch Guilder (NLG), Portuguese Escudo (PTE), Spanish Pe- seta (ESP). It is already used for psychological purposes during the present ‘transition period’, and its exchange rate is already quoted [1]. The 11 currencies (BEF=LUF) and their exchange rates with respect to the EUR are shown in Table 1 of reference [2]. A data series can be artificially constructed for the EUR exchange rate toward other currencies, e.g. the Swiss Franc (EUR/CHF) following the artificial rule: 1EUR/CHF = 11  i=1 γ i 11 C i /CHF (1) where the γ i ’s are the conversion rates and the C i ’s denote the respective currencies. The same may hold true for EUR/DKK, EUR/JPY, and EUR/USD which are the only ones considered here below. The normalized evolution of ten (since LUF = BEF) currency (forming EUR) exchange rates vs. a e-mail: marcel.ausloos@ulg.ac.be USD, – assuming an exchange rate = 1 on January 01, 1993 has been shown in Figure 1 in reference [2]. The other EUR/DKK, EUR/CHF, and EUR/JPY cases are similarly obtained. Such data is not shown here for lack of space. The number of data points has been equalized, as done in reference [2] and is equal to N = 1902, spanning the interval time from January 1, 1993 till June 30, 2000. The fluctuations can be as large as 30% for ESP, FIM, and PTE and 20% for the others. The DFA technique [3] has often been described and is not recalled here. It leads to investigating whether the function 〈F 2 (τ )〉 has a scaling behavior, i.e. 〈 1 τ (k+1)τ  n=kτ +1 [y(n) − z (n)] 2 〉∼ τ 2α (2) where y(n) is the investigated time series and z (n) is hereby a linear function fitting at best the data in the τ interval which is considered. Let it be recalled that in reference [2] two different scaling ranges were found for the EUR/DKK; one, from four to 25 days (5 weeks) with α =0.37 ± 0.01, and another, after that for up to 300 days (61 weeks) with more Brownian-like correlations, i.e. α =0.48 ± 0.03. The time scale invariance for EUR/CHF, EUR/USD, and EUR/JPY holds from 5 days (one week) to about 300 days (one year) showing Brownian types of correlations.