19 Pages Posted: 25 Apr 2008
Date Written: December 20, 2007
Analysis of the NASDAQ Composite index in the 36 year period since its inception in 1971 reveals that the probability density function of 1-day returns is highly leptokurtotic; the maximum 1-day loss is more than 9 standard deviations from the mean. The excess kurtosis is, however, reduced by more than an order of magnitude when annual returns are considered. Monte Carlo simulations were performed to test whether this behavior could be attributed to random processes. Widely different 1-day cumulative distribution functions (CDFs) were sampled and it was found that the distribution of annual returns is independent of the shape of the sampling distribution and depends only on the mean and standard deviation of the 1-day returns. The kurtotic behavior of the Monte Carlo results is similar to that observed with the real data and the Monte Carlo simulations do a reasonably good job of bounding the annual return time series data. However, the simulated CDFs do not match the data, i.e., there is a considerably higher probability of loss than predicted by the simulations. This is highly suggestive of non-random behavior and is reinforced by the fact that the standard deviation of M-day returns from the data increased monotonically faster than square-root-M scaling of 1-day returns. To assess whether long-term dependence is present in the NASDAQ time series, rescaled range-statistic calculations were carried out to determine the Hurst exponent, H. It was found that H was 0.59 for 1-day returns and increased monotonically to a value of 0.87 for 250-day (annual) returns. However, most of this increase was also observed in simulated returns derived via a Gaussian random walk. Comparing the two resulted in a DH of 0.06 for 1-day returns and this difference diminished monotonically with increasing period, vanishing altogether after about 100 days.
Keywords: Randomness, Monte Carlo, Hurst Exponent
JEL Classification: C15, C52, C53
Suggested Citation: Suggested Citation