Since the late 1960s, scholars in financial economics and practitioners in financial services have operated under the general assumption that incremental changes in exchange traded financial assets follow a normal distribution, as defined by the German mathematician Friedrich Gauss (1777 – 1855). In finance, what a statistician would call standard deviation around the mean is referred to as volatility. Under that guise it has become the most widely used metric for investment risk. Eugene Fama, a titan of modern financial theory, highlighted decades ago that this alleged normal distribution is characterised by ‚fat tails’ (extreme outliers). The notion of a normal distribution of incremental price changes is in essence based on the central limit theorem. According to this theorem, under certain conditions, frequency distributions approach normal with increased sample size. There is no general agreement whether the conditions for the application of the central limit theorem to financial markets are given or not. The popularity of volatility as a metric is clearly based on the assumption that these conditions are met, even if that assumption may be incorrect. By implication, when distortions from normal are encountered, they will be explained as being related to sample size, and dismissed. Measures of distortion, such as skew and kurtosis, are virtually unheard of in the everyday context of volatility. It is already rare that volatility is shown together with the mean incremental value, without which it has no meaningful context.

### Considerations

Obviously, days are more frequent than weeks, which are more frequent than months, quarters, and years. So the choice of time interval has a corresponding impact on sample size. Good statistical practice (if not the central limit theorem) demands adequate sample size. All else being equal, a large sample size is better than a small one. Thus, regardless of elected history, daily increments will allow the greatest sample size. But daily rates are not always the relevant increment. Traders may focus on daily increments or even intra-day changes, investors are probably more concerned with months and quarters than with days and weeks. A long history is not always available, and a history that is too long can become just as irrelevant as one that is too short. Mean incremental change multiplied by the number of increments equals the start-to-end change. For that very reason, the mean is used more often than any other measure of central tendency. Thus, the shorter the interval, the smaller the absolute value of the mean. But standard deviation does not follow that law. In contrast to mean values, interval specific variance is not equivalent to a simple change of denomination. Standard deviation is not another form of showing the same information, it is information in its own right. It sheds light on the sample that is not evident from the sample’s mean value. A market might appear to have low volatility when measured with one interval, and could seem highly volatile when measured with different intervals. By the same token, one frequency distribution could be heavily distorted, while another one possibly emulates a perfectly symmetrical and mean-centred ‚bell curve’. Does this matter? It certainly does. The more pronounced a distribution’s distortion, the less valid are subsequent calculations, or conclusions, that require distribution to be normal. Conceivably, and depending on interval choice, diametrically opposing conclusions are statistically true with regards to volatility for one and the same investment. When volatility is used as proxy for the comparative analysis of risk without an understanding of the nature of distortions in that distribution, or their magnitude, the lid is removed from Pandora’s box.

### Description Of Data

As part of my ongoing research related to the development and refinement of alternative metrics applicable to investing, I have examined frequency distributions of numerous equity indices from December 1999 to June 2023. Although my investigation was not focussed on variance, I have collected and examined a large body of data on that subject as a by-product of my analyses. Multiple calculations were performed on each market, using intervals of days, weeks, months and quarters. Different bank holidays apply in each country, resulting in a range of sample sizes for daily intervals, somewhat below 6’000. For longer intervals sample size is identical across markets, ranging from n= 94 (quarters) to n= 1’226 (weeks). All calculations in all markets refer to identical 23 ½ years of history. A list of man-made disasters in this era could include: two burst market bubbles (dot.com, collapse of several giant financial institutions), failed political gambles (Brexit), a primate as Head Of State (choose one), attempted and ongoing insurrection against democratic rule of a nuclear superpower (USA), military invasion of a sovereign state, … the list seems endless. Except for the unexpected death of James Bond, this piece of history seems a fairly ordinary episode in humankind’s fast-paced evolution toward enlightenment. Presumably, it is representative of general market behaviour.

I will illustrate just how pronounced distribution distortions can be, even with large samples sizes. While this will be no surprise to scholars in finance, more casual users of volatility may find this perplexing. A number of exhibits will help illustrate the data and my arguments: Firstly, I will present detailed frequency plots for daily and quarterly rates of change in one sample market (Exhibits 1 & 2). The choice fell on Switzerland because this markets is somewhat less volatile than many of its global peers. Plots display three standard deviations left and right, divided into twelve distribution bins. Such histograms incorporate the bell-curve (probability density function) representing normal distribution, a few auxiliary graphs, and numerical information. Further, I will demonstrate that for identical market history, the shape of frequency distributions shifts markedly when measured in different intervals. Exhibit 3 incorporates, still only for one sample market, distribution data of all four time intervals. Expanding the illustration geographically, tables with data for ten national equity indices show values for skew (Exhibit 4) and excess kurtosis (Exhibit 5).

### The Evidence

Let’s review review the frequency distribution of daily rates of change for the Swiss market index, as shown in Exhibit 1 below. At first, this distribution could appear relatively normal. Closer inspection reveals that it is quite morphed. An excessive number of values are found in the centre, at +0.5 and -0.5. Besides, these mean hugging bins are very unequal. In fact, the entire right hemisphere is more populated than the left. When expressed in numbers, this profile manifest as skew of roughly -0.1 (left sloping) and an excess kurtosis of almost 7.7 (indicating extreme outliers). 98.4% of observations are found inside the ‚visible’ range, somewhat short of 99.7%. While skew may look mild, kurtosis is genuinely high. 76 observations (equal to 1.3% of sample) are more extreme than expected. Kurtosis is sensitive to the number and position of outliers. A glance at the full range of increments (worst and best) reveals how far the extremes are removed from the mean, ,and on which side. We already know (from skew and other indicators of symmetry) that the distribution is left sloping, but skew is not sufficient to locate outliers by hemisphere. Returning to the heavily populated area of the distribution, we find bins -1.0 and 1.0 underpopulated, partially compensating for the rich excess only a little further in. Even so, that leaves 588 observations too many (i.e.: 10% excess over the normal 68%) inside one standard deviation.

Whatever expectation of future incremental change(s) a knowledge of daily volatility may or may not permit, better have a look at the plot for quarterly rates of change coming up next, in Exhibit 2. If there has been some evidence of symmetry among daily rates of change, then this is clearly not so in the distribution of quarterly changes. 63% of observations are above the mean. Avoid any short-lived consolation from milder excess kurtosis of 0.36, even in the complete absence of any outlier. Bin 3.0 is totally vacant, here that is explained by the sample size, which permits only a coarse distribution. Compared to daily rates of change, skew is obviously much more pronounced. In this distribution, skew is not even the most prominent aspect. That accolade belongs to the distribution being bi-modal: it has two frequency peaks, quite apart from one another at 0.5 and -2.0 respectively. The difference in content from 0.5 to 1.0 is excessive, whereas -1.0 and -0.5 are equal. More than half the data are found between mean and upper inflection point, but less than a quarter between mean and lower inflection. There is nothing normal at all about this distribution and none of that is evident from standard deviation and neither skew nor kurtosis could have revealed multiple distribution peaks, let alone locate them.

Having provided two detailed analyses of distribution characteristics to caution against over-reliance on distribution parameters, I will now present a summary of all four distributions for the same market, in Exhibit 3. In the table, columns D (in grey) and Q (in green) correspond to the detailed analyses shown before. Columns W (weekly, in red) and M (monthly, in blue) complete the illustration of distribution for Switzerland as example. Take a moment to peruse the data in either graphical and/or numerical form.

When comparing values for skew and kurtosis, it is important to ensure they were generated using the same method, as there are several methods available. I am using the modified Pearson formula for skew, and Cochran’s formula for kurtosis. Exhibit 4 shows that most markets, regardless of interval, are left-sloping which manifests in negative values. The median is higher than the mean. Incremental changes below the mean are either more numerous and/or more pronounced than those found above it. While this is not always so, is very often the case, for good reasons. Investors act on fear with greater urgency than on greed. Accordingly, positive skew in that table is quite rare. Another general feature is for skew to diminish in size with shorter intervals and larger sample size. With larger samples, the mean diminishes and the median approaches the mean. Hence, with larger sample size, skew must indeed become less pronounced. For the era shown here, all markets slope left in their distribution of daily increments. All but China do so for weeks. For daily rates of change, a single decimal would be insufficient to distinguish skew from one market to another. China and Japan are the most symmetric in their daily distribution, Australia and Canada the least. For months, China and India slope right. In the distribution of quarters, they are further joined by South Korea, while the other seven markets slope left. See through the aforementioned general patters and consider the order of magnitude within each column, and across markets. Note how that changes: Switzerland is ‚middle of the range’ with regards to symmetry of daily distribution but the least symmetric market when looking at quarters.

Excess Kurtosis (Exhibit 5) shows similar changes in distribution conformity as a function of interval choice. Kurtosis tends to be more pronounced in the distribution of short intervals compared to that of long intervals. This too is a general phenomenon. Indirectly, it also relates to sample size: But the inherent sensitivity of this statistic lies in the width of range of values, less so in the sample size. For daily rates of change, Canada and USA stand out has having the fattest tails. Change the lens (interval), and the order of markets by this statistic is automatically reshuffled. Move from days to weeks, and suddenly the Swiss market has the fattest tail. The market in Japan has severe outliers in the daily distribution, yet abnormally few in the distribution of quarters, the only negative excess kurtosis seen in that column and the entire table.

### Conclusions

At this point two facts should become evident:

- Regardless of the interval, frequency distribution is rarely normal enough.
- Changing the interval results in frequency distributions that are quite different from one another, even if they all reflect the same market history.

Frequency distribution of incremental rates of change in financial assets is highly complex. Distribution is never adequately described by standard deviation. The inclusion of skew and kurtosis as measures of distortion may yield valuable additional information, but these metrics have their own vulnerabilities and can not be relied upon. In academia, long term analyses are feasible, but in any commercial context, shorter time horizons prevail. There, fixed-length frames of observational relevance are deployed, trailing along with the passing of time. While this is not shown here, consider that distribution parameters are anything but stable. They fluctuate substantially across any market phase. A single new data entry may change everything. The lack of dependability of many distribution metrics is mostly a function of financial market behaviour. But beyond that, every metric also has its own strength and weakness. Personally, I consider standard deviation particularly prone to misrepresentation and poorly suited as an observational tool for a multitude of reasons. Even if other approaches for measuring variance are more stable and subjectively better, no metric can overcome the fact that financial markets do not easily fit into any chosen box. Only by examining the corresponding histogram can all critical distribution aspects of a specific sample be revealed. If two markets appear similar, or even alike, then probably not for long.

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