The data used in this paper refer to the period between April/1/2003 and November/7/2005, corresponding to a total of 618 observations per day for each asset and its respective ADR. The assets used were from Petrobras (PETR4), Telemar (TNLP4), Vale do Rio Doce (VALE5) and Banco Itaú (ITAU4).
The data used can be divided into three variables:
• Daily closing price of the ADRs of Brazilian enterprises traded in the North-American market in dollars (effective price);
• Daily closing price of Brazilian stocks, corresponding to the ADRs, traded in Real in the Brazilian market;
• Daily exchange rate, Real (R$) per dollar (US$).
The data related with the ADRs were obtained from the Economática, whereas the data on Brazilian stocks were obtained from the webpage of EasyInvest, a broker, and, finally, those data regarding the daily exchange rate were obtained from the webpage of IPEADATA.
These companies were chosen due to the level of importance of these assets in the BOVESPA index. From all the companies in the sample the dates that did not represent negotiations in both markets were excluded because there was no business with either the stock or the ADR. This fact is due to the difference of the dates of Brazilian and American holidays.
A new variable, called “ADR theoretical price”, was calculated on the basis of the closing prices of the company’s stocks, the dollar rate and the number of corresponding stocks to each ADR. That is to say:
t it
it C
*N PT = PA
where:
PTit = ADRi theoretical price of (US$) at instant t.
PAit = Stocki Price (R$) at instant t.
N = Number of corresponding stocks for each ADR.
Ct = Daily exchange rate (R$/US$) at instant t.
This calculation allows for a comparison between the “ADR effective price” (PE) and the ADR theoretical price”
(PT), since the currencies and amounts were equaled. By this, a comparison test of unconditional means (Test T-Student for two pair samples for the means) can be performed. By disregarding the transition cost, the unconditional
means of these two variables are expected to converge to the same point. This result is expected, since should the means be differing, there is room for continuous arbitrage.
The software used in this study was SPSS – Statistical Package for the Social Sciences, version 10.0. In the first moment, the level of linear dependence (Pearson Correlation) between variables PE and PT for each of the companies in the sample was studied. As expected, the calculated correlation coefficients show high linear dependence between the series; all companies had correlation above 99%. (Table 1).
Table 1: Correlations of Test T for two pair samples for the means
Despite the high correlation, it cannot be said that there are no arbitrage possibilities between these markets. It can be concluded that if such opportunity does exist, it is not so common for a chance to appear.
After this verification, an option was made on working with a single variable for each company, which is expressed through the ratio between PE and PT. That is:
it it
it PT
R = PE
where:
PEit = ADRi Effective Price at instant t PTit = ADRi theoretical price at instant t
With this variable, it is possible to verify the per cent difference between the theoretical price and the effective of the ADR of the i-th company.
In case of an efficient market (disregarding the transaction costs), this variable should equal to one, which is not verified. Therefore, when this variable is greater than one, there is an opportunity window, which can be used by acquiring stocks at the local market and selling the ADRs at the American market. When this ratio is lower than one, arbitrage is given as of the purchase of the ADRs at the US market and the sales corresponding to this ADR at the Brazilian market. In order to statistically test whether the unconditional mean of this ratio is equal to one, the T-Student test was applied for one sample (Table 2). This test points to frequent arbitrage possibilities for companies Telemar and Vale do Rio Doce, since the T test rejected (5% level) the hypothesis that the mean for such ratio is equal to one. As for companies Petrobras and Itaú, it cannot be said the same, once the hypothesis tested can be accepted (5% level).
Table 2: T-Test for a sample (One-Sample) Company Observations Correlation Sig.
Petrobras 618 99,967% 0,000
Telemar 618 99,733% 0,000
Vale 618 99,973% 0,000
Itaú 618 99,971% 0,000
-1,693 617 ,091 -4,888E-04 -1,06E-03 7,818E-05
-2,790 617 ,005 -8,421E-04 -1,43E-03 -2,49E-04
-3,702 617 ,000 -1,162E-03 -1,78E-03 -5,46E-04
-,295 617 ,768 -8,536E-05 -6,54E-04 4,829E-04
PETR4 TNLP4 VALE5 ITAU4
t df Sig. (2-tailed)
Mean
Difference Lower Upper
95% Confidence Interval of the
Difference Test Value = 1
As this test is used to test the unconditional mean of the data series, it only manages to indicate those assets with frequent arbitrage opportunities. However, the arbitrage possibilities may appear in some windows, which are not constant. In order to verify that, Table 3 shows the descriptive statistics for the series related with the ratio variable for all companies used.
It can be observed (Table 3) that the series of this variable for Petrobras has a minimum of 0.96, pointing to a 4%
distance between the theoretical and effective prices of the ADR; in spite of having an unconditional mean statistically equal to one, this instant can present arbitrage possibilities, that is, there may be arbitrage windows. The same can be said of company Itaú, but the difference between maximum prices reached 3%.
Table 3: Descriptive Statistics
618 ,96 1,03 ,9995 7,177E-03
618 ,93 1,03 ,9992 7,504E-03
618 ,93 1,02 ,9988 7,802E-03
618 ,98 1,03 ,9999 7,194E-03
618 PETR4
TNLP4 VALE5 ITAU4
Valid N (listwise)
N Minimum Maximum Mean Std. Deviation
In an attempt to forecast the opening of arbitrage windows, the autocorrelation (ACF) and the partial autocorrelation (PACF) of the series were studied in order to verify whether the conditional means for the series are constant.
Moreover, the Ljung-Box Test was applied with an aim at verifying whether the autocorrelation is statistically significant.
The variable studied regarding company Petrobras has significant ACF and PACF (Figure 1 and Figure 2, respectively) for the first lag. For companies Telemar and Itaú there is no statistically significant linear dependence for any lag (Figure 3, Figure 4, Figure 7 and Figure 8). The series for the variable related with company Vale do Rio Doce also has significant ACF and PACF (Figure 5 and Figure 6), which indicates a certain predictability by using the ARIMA models. The Ljung-Box test indicated (by using a 5% level of significance) that the series of Petrobras and Vale do Rio Doce have a significant linear dependence; by this, these data can be modeled by ARIMA models.
Figure 1: Autocorrelation PETR4
Figure 5: Autocorrelation VALE5
The ARIMA model that is best adjusted for the Petrobras series is AR (1), the estimated coefficient of which is 0.2858 and the constant is 0.9988. As for the Vale do Rio Doce series it is the ARMA (1,1) with self-regressive coefficient (AR) of 0.9510, movable mean coefficient (MA) of 0.7555 and constant of 0.9980.
These models allow for predicting the result of the ratio between the effective price and the theoretical price of the ADRs, and by this, it helps make the decision as to when to perform the operation and when to render resources available in order to put it into effect. Once the opportunity windows are modeled, the investor will be able to foresee when the market will adjust itself by closing the arbitrage window.