Multifractal Analysis of Pegged and Floating Dollar-Real Exchange Rates

Ivana Stosic

Graduate Program in Economic Development, Department of Economics Vanderbilt University, VU Station B #351828, 2301 Vanderbilt Place

Nashville, Tennessee 37235-1828

José Rodrigo S. Silva and Tatijana Stosic

Departamento de Estatística e Informática, Universidade Federal Rural de Pernambuco Rua Dom Manoel de Medeiros s/n, Dois Irmãos, 52171-900, Recife-PE, Brasil.

E-mail: rodrigo.ufs@gmail.com, tastosic@gmail.com

Keyword: Exchange Rates, Multifractal Analysis, Dollar-Real.

Abstract: In this work we study dynamics properties of the Brazilin Real (BRL)/US Dollar

(USD) exchange rate before and after January 1999, when the Brazilian pegged exchange

rate system was substituted with a floating regime. Application of Multifractal Detrended

Fluctuation Analysis (MF-DFA) shows that BRL/USD exchange rate dynamics belong to a

class of multifractal processes. We examine how the transition from a pegged to floating

regime affects market dynamics, and find that after transition to floating regime, the

changes in multifractal spectrum indicate a more efficient market.

1. Introduction

The foreign exchange market (FX) is the world’s largest and most liquid financial

market. Its huge trading volume, the extreme liquidity, diversity of traders , geographical

dispersion, long trading hours, and variety of factors that affect exchange rates make the

foreign exchange market uniquely challenging for empirical analysis, forecasting and

model development. One crucial aspect of the foreign exchange market is the exchange

rate regime under which a country sets its exchange rate. Under a fixed exchange rate

regime, countries lose the ability to conduct monetary policy independently and are forced

to buy and sell foreign reserves as needed to maintain their exchange rates fixed. Under

a floating exchange rate regime, a currency’s value is determined by supply of and

demand for that currency in the foreign exchange market. Such a regime gives countries

freedom regarding how to conduct monetary policy. In Brazil the floating system was

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adopted in January 1999, after a severe currency crisis. Various economic, political, and

market factors affect FX, and it is still an open question how transitions from fixed to

floating exchange rate regimes affect market efficiency [2]. In this work we apply

Multifractal Detrended Fluctuation Analysis (MF-DFA) [1] to compare dynamics properties

of the BRL/USD exchange rate before and after January 1999, when Brazil switched its

foreign exchange rate regime from fixed to floating. To conduct this comparison, we

analyze returns and volatilities of daily closing exchange rates, and use the width of

multifractal spectrum as an indicator of degree of multifractality.

2. Data and methodology

2.1. Data

The data used in this work were obtained from the from http://finance.yahoo.com . We

analyze daily temporal series of the BRL/USD exchange rate for the period 02/01/1995-

24/01/2003.

2.2. Multifractal Detrended Fluctuation Analysis

The MF-DFA procedure is briefly described as follows. The original temporal series

Niix ,,1,)( …= is integrated to produce [ ] Nkxixkyk

i,,1,)()(

1…=−=∑ =

, where

∑ ==

N

iix

Nx

1)(

1is the average. Next, the integrated series )(ky is divided into nN non-

overlapping segments of length n , and in each segment the linear (or higher order

polynomial) least squares fit (representing local trend) is estimated. The integrated series

)(ky is then detrended by subtracting the local trend )(kyi (ordinates of straight line or

higher order polynomial segment) from the data in each segment and a q th order

fluctuation function is calculated as

[ ]

qN

i

qin

nik

i

n

q

n

kykynN

nF

/1

1

2/

1)1(

2)()(

11)(

−= ∑ ∑

= +−=

(1)

where, in general, q can take any real value except zero. Repeating this calculation for all

box sizes provides the relationship between fluctuation function )(nFq and box size n ,

where typically )(nFq increases with n according to a power law

. The generalized Hurst

exponent )(qh is obtained as the slope of the regression (least squares line fitting) of

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0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

Volatility

Before

After

f(α)

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

Returns

Before

After

f(α)

α

)(log nFq versus nlog . Another way to characterize multifractal process is the singularity

spectrum )()( qqf ταα −= where dqqdq /)()( τα =

and 1)()( −= qqhqτ . For a

monofractal signal, the singularity spectrum produces a single point in the f(α) plane,

whereas multifractal process yields a single humped function [1].

3. Results

We apply MF-DFA on daily returns )/ln( 1 ttt PPr+

= and daily volatilities tt rv = of

closing price tP of the BRL/USD exchange rate during the period 02/01/1995-14/01/1999

(pegged regime) and 15/01/1999-25/01/2003 (floating regime) with 1012 observations for

each series. The multifractal spectrum )(αf for the two time periods are shown in Fig.1.

After 01/15/1999, the incidence of large fluctuations of daily returns increases due to

transition to the floating regime, which results in wider and left skewed multifractal

spectrum indicating more complex dynamics. However, the value of the estimated Hurst

exponent, which can be roughly related to the position of maximum 0α [3], becomes

closer to 0,5, indicating an increase in market efficiency from the first to the second

period. Volatility time series display the opposite trend: while the position of maximum 0α

does not change, the degree of multifractality decreases after the transition to floating

regime indicating less clustering in volatility temporal series, and consequently more

efficient markets .

Figure 1: Multifractal spectrum of return and volatility time series before and after January

1999.

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4. Conclusion

In this work we apply Multifractal Detrended Fluctuation Analysis (MF-DFA) to compare

dynamics properties of the Brazilin Real (BRL)/US Dollar (USD) exchange rate before and

after January 1999, when Brazil switched its exchange rate regime from a pegged system

to a floating system. Our results show that BRL/USD exchange rate dynamics belong to a

class of multifractal processes, which is in agreement with results for other currency

exchange rates [1], supporting universality of the behavior of foreign exchange markets.

We also examine how the transition from pegged to floating regime affects market

dynamics and find opposite trends for returns and volatilities. After transition to a floating

regime, we observe in the multifractal spectrum that the position of maximum shifts

towards 0.5 for returns, while the width of the spectrum decreases for volatilities. These

changes in the multifractal spectrum indicate that moving from a fixed exchange rate

regime to a regime is followed by an increase in market efficiency.

5. Acknowledgments

This work was supported by Brazilian agencies CNPq and CAPES.

6. References

[1] J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H. E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series, Physica A 316 (2002) 87-114.

[2] F. G. Schmitt, L. Ma, T. Angounou, Multifractal analysis of the dollar-yuan and euro-yuan exchange rates before and after the reform of the peg, Quantitative Finance 11 (2011) 505-513.

[3] Y. Shimizu, S. Thurner, K. Ehrenberger, Multifractal spectra as a measure of complexity in human posture, Fractals 10 (2002) 103-116.

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