indicators, rather than purely examining the returns, that it is possible to detect a significant impact of the news. Recently, Zumbach et al. (2000) have developed a scale of market shocks by integrating different volatility measures and relating a shock to its probability of occurrence. This measure is able to clearly identify turbulences on the market as well as to quantify the effect of news (Zumbach et al., 2000).

5.4 DISTRIBUTIONAL PROPERTIES OF RETURNS

We mentioned in the introduction to this chapter the variety of opinions about the distributions of FX returns and the corresponding data-generating process. In this section, we do not want to propose a new model for the probability distribution function, but rather examine empirically what type of behavior is observed when returns are measured at different frequencies. We shall first present general results on the entire distribution and note that they are fat-tailed. Then, instead of looking at the center of the distribution, we shall present an alternative way to characterize the distribution by looking at the behavior of the tails.

There are many possible models of distribution functions, but this variety is greatly reduced when considering the tails of the distributions. The tail of a distribution can be described by using only one parameter, the tail index a. The empirical estimation of the tail index is difficult and requires large numbers of observations. The availability of high-frequency data makes this possible in practice. The methods, the empirical results, and their interpretation are presented in Section 5.4.2.

5.4.1 Finite Variance, Symmetry and Decreasing Fat-Tailedness

In this subsection, we analyze the probability distribution of returns of financial assets. The probability distribution associates each movement size with a certain probability of occurrence. In the case of empirical data, the domain of possible return values is divided into boxes, and one counts the frequency of occurrence in each box. One important issue in the case of tick-by-tick data is that this data are irregularly spaced in time, tj. We have already discussed in Chapters 2 and 3 the different ways of constructing a homogeneous time series. Here we chose to take linearly interpolated prices. This is the appropriate method for interpolating in a series with independent random increments for most types of analyses. An alternative method discussed earlier, taking the last valid price before the gap as representative for the gap interval, must be avoided in a study of distributions as it would lead to a spurious large return from the last valid price within the gap to the first real price after the gap.

In Tables 5.1 and 5.2, we present the empirically computed moments of the distributions for the major FX rates against the USD and the major FX rates against the DEM.3 The means are close to zero, as compared to the standard deviations,

3 At least three of these cross rates have disappeared with the introduction of the Euro. Nevertheless, we think it is still interesting to report the results for them because they show the convergence of those

TABLE 5.1 Moments of return distributions for USD FX rates This table gives an empirical estimation of the first 4 moments of the unconditional return distribution at different time intervals for the major currencies against the USD for the period from January I, 1987, to December 31, 1993. The term kurtosis refers to the excess kurtosis, so a normal distribution has a kurtosis value of zero.

and the absolute values of the skewness are, except in very few cases, significantly smaller than 1. We can conclude from these facts that the empirical distribution is almost symmetric. The mean values are slightly negative (except for GBP-USD where the currencies are inverted) because during this period (from January 1, 1987, to December 31,1993) we have experienced an overall decline of the USD. For all time horizons, the empirically determined (excess) kurtosis exceeds the value 0, which is the theoretical value for a Gaussian distribution. For the shortest

currencies to the Euro by exhibiting lower variances than the others. They present a good example of the influence of external factors on the statistical behavior of financial asset prices.

TABLE 5.2 Moments of return distributions for DEM FX rates

This table gives an empirical estimation of the first 4 moments of the unconditional return distribution at different time intervals for the major currencies against the DEM for the period from January I, 1987, to December 31, 1993. The term kurtosis refers to the excess kurtosis, so a normal distribution has a kurtosis value of zero.

time intervals, the kurtosis values are extremely high. Another interesting feature is that all of the rates show the same general behavior, a decreasing kurtosis with increasing time intervals. At intervals of around 1 week, the kurtosis is rather close to the Gaussian value.

Tables 5.1 and 5.2 suggest that the variance and the third moment are finite in the large-sample limit and that the fourth moment may not be finite. Some solid evidence in favor of these hypotheses is added by the tail index studies that follow, mainly the results of Table 5.3. Indeed, the larger the number of observations, the larger the empirically computed kurtosis. At frequencies higher than 10 min,