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11.11.2008
How abnormal was the stock market in October 2008?
October 2008 was certainly a spectacular month in the stock markets. Large daily changes occurred that surprised most investors. Yet, although many investors had not seen such wild gyrations of stock prices for a long time, there was a general sense that this had happened before.
Those of us who studied modern finance theory, however, were truly astonished by the sheer improbability of the events occurring in the stock markets during that fateful month. One of the basic assumptions used in almost all our finance models is that returns are normally distributed. These models are widely used to price derivatives and other complex financial products. What do these models tell us about the probabilities of the events that occurred in October?
The following table gives an answer. We selected the six largest daily percentage changes in the Dow Jones Industrial Average during October, and asked the question of how frequent these changes occur assuming that, as is commonly done in finance models, these events are normally distributed. The results are truly astonishing. There were two daily changes of more than 10% during the month. With a standard deviation of daily changes of 1.032% (computed over the period 1971-2008) movements of such a magnitude can occur only once every 73 to 603 trillion billion years. Since our universe, according to most physicists, exists a mere 20 billion years we, finance theorists, would have had to wait for another trillion universes before one such change could be observed. Yet it happened twice during the same month. A truly miraculous event. The other four changes during the same month of October have a somewhat higher frequency, but surely we did not expect these to happen in our lifetimes.
Table: Six Largest Movements of the Dow-Jones Industrial Average in October 2008
(2) The mean of the distribution is set to zero and the standard deviation computed over the whole sample (St. Dev. = 1.032%). (3) 10 21 = “Sextillion” in Western (Arabic) Numeral; “Trillion Billion” in US; modern British & Australian
So what happened in October? Is it really a miraculous month? Certainly not. Since 1928 the Dow-Jones has had daily returns above the threshold of five times one standard deviation (in absolute value) not less than 73 times! (see figure 1). Compare this to the normally distributed random shocks plotted in figure 2; it never crosses the five standard deviation boundaries. A five standard deviation event should only occur once every 7000 years in a world of normal distributions. It happened 73 times over an 80-year period.
Figure 1: Dow Jones Industrial Average 1928-2008
Figure 2: Random Normal Process
Our conclusion, therefore, should be not that these events are miraculous but that our finance models are wrong. By assuming that changes in stock prices are normally distributed, these models underestimate risk in a spectacular way. As a result, investors have been misled in a very big way, believing that the risks they were taking were small. The risks were very big. We now know why.
It is time to throw away the models and to go back to the drawing board. In the meantime banks should be forbidden to hold complex assets on their balance sheets that have been priced using one of these models.
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