# All Stock Market Indices Are Fata Morganas

May 4th, 2014
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Have you ever wondered what information a stock market index actually gives? An index point is not a fixed unit in time and does not have any historical significance.

## Two measurements during a marathon

Imagine the following two measurements:

• In the first measurement, the average time is measured which 30 runners need to run a marathon.
• In the second measurement the same runners start a marathon. However, after 10 kilometres the 10 slowest runners are replaced by 10 fresh. After 20 kilometres again the 10 slowest runners are replaced by 10 fresh and after 30 kilometres this happens again. Again the average time is measured.

In which measurement will the average time over 30 runners be the smallest? And what will happen to the average time if the stopwatch is running more slowly during the second measurement? In the beginning of the second measurement, there are 60 seconds in a minute and at the end this increased to 380 seconds in a minute. Does it still make sense to compare the two measurements?

If your answer to this question is no, you will probably have no problem qualifying stock market indexes such as the AEX, DAX, S&P 500 and the Dow Jones Industrial Average as fata morganas.

## All Stock Market Indices are Fata Morganas

The parallel with stock market indices is easy to draw. An index is calculated on the basis of a set of shares. Every index has its own formula and the formula results in the number of points of the index. However, this set of shares changes regularly. For a new period the value is based on a different set of shares. It is very strange that these different sets of shares are represented as the same unit. In less than ten years twelve of the thirty companies (i.e. 40%) in the Dow Jones were replaced. Over a period of sixteen years, twenty companies were replaced, a figure of 67%. This meant that over a very short period we were left comparing a basket of today's apples with a basket of yesterday's pears.

Even more disturbing is the fact that with every change in the set of shares used to calculate the number of points, the formula also changes. This is done because the index, which is the result of two different sets of shares at the moment the set is changed, must be the same for both sets at that point in time. The index graphs must be continuous lines. For example, the Dow Jones is calculated by adding the shares and dividing the result by a number. Because of changes in the set of shares and the splitting of shares the divider changes continuously. At the moment the divider is 0.15571590501117 but in 1985 this number was higher than 1. An index point in two periods of time is therefore calculated in different ways:

••••••••••••••••••••••••••••
Dow 1985 = (x1 + x2 +..+x30) / 1
Dow 2014 = (x1 + x2 +.. + x30) / 0.15571590501117
••••••••••••••••••••••••••••

In the 1990s many shares were split. To make sure the result of the calculation remained the same both the number of shares and the divider changed. An increase in share value of 1 dollar of the set of shares in 2014 results is 6.4 times more points than in 1985. The fact that in the 1990s many shares were split is probably the cause of the exponential growth of the Dow Jones index. At the moment the Dow is at 16,437 points. If we used the 1985 formula it would be at 2,559 points.

The most remarkable characteristic is of course the constantly changing set of shares. Generally speaking, the poorly performing companies are removed from the set and companies in the growth phase, which perform well, are added. This greatly increases the chance that the index will rise rather than go down, you don't need a probability calculation

## Should the European Union reindex the stock markets?

It makes no sense to compare the number of points with the number of points of a stock market index from 30 years ago. The time is ripe for the European Union, after the introduction of the euro, to reindex European stockmarkets to 100 points. A fata morgana is an optical phenomenon, and so is the graph of a stock market index.

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