## Shifting Structure Of Returns In MSCI World Index

May 01, 2014A normal distribution is a very common probability distribution. The term refers to the probability that a set of data or observations will fall between any two real numbers and for which a graphical representation (i.e. histogram) is symmetrical with a single central peak at the mean. A normal distribution is typically described as ‘bell-shaped’, with half of the distribution to the left of the mean and half of the distribution to the right of the mean. The spread of the data is dependent on the standard deviation; the smaller the standard deviation, the more concentrated the data.

In contrast to a normal distribution, the histogram for a non-normal set of observations is not symmetrical or bell-shaped. These kinds of configurations are described with variables such as skew and kurtosis and are frequently (though not always) related to data involving time measurements. Examples of non-normally distributed data include the waiting time at a doctor’s office, the price movement of an individual stock, or the cumulative earnings associated with a lifetime of work in a given profession.

The skewness of a distribution refers to a lack of symmetry. A histogram with negative skew has data that is skewed to the left while one with positive skew has more observations in the right tail. Generally, if skewness is less than -1 or greater than +1, the distribution is highly skewed; skew values between -1 and -½ or +½ and +1 are considered moderately skewed; and skewness between -½ and +½ indicates an approximately symmetric distribution.

Kurtosis is used to describe the extent to which a set of data is more or less “peaked” when compared to a normal distribution (with kurtosis of 0). Higher kurtosis (>0) is associated with the movement of observations toward the center and the tails of the distribution (i.e. out of the “shoulders”) while the opposite is true for a distribution with kurtosis <0.

We measure the average daily percent change for all stocks in the MSCI World Index and aggregate the results. In an upward trending market, we generally observe a negative skew and positive excess kurtosis. In downward trending market, we generally observe a positive skew and negative excess kurtosis. **In last 31 days, the distribution of daily returns within the MSCI World Index has shifted to a negative configuration of a positive skew and negative excess kurtosis. **This divergence suggests the underlying structure of the equity market is in flux. The nature of the drunk’s steps are changing.