The Fallacy of Rare Events and Fat Tails

Rare events such as the 1987 stock market crash or the Google stock drop of last Friday are unpredictable. Those who try to convince investors and traders to constantly plan for such events essentially attempt to neutralize them and turn them into spectators of the markets. Realizing returns above risk-free return becomes very difficult for those who constantly anticipate rare events and try to minimize their financial impact.

I always enjoy reading the Attain Capital Management blog. But that does not mean I agree with everything. In their latest post, Black Monday 1987 and the 100 foot tall man, they state:

“…we can’t use normally distributed curves for analyzing the stock market or any other market for that matter. Or for analyzing bands popularity, book sales, or wealth.”

Image source: Wikipedia

I wonder if analyzing past market returns using the correct distribution is equivalent to trading a market successfully. Or for that purpose, if analyzing wealth properly for the fact that Bill Gates’ fortune is a +1000 sigma event can help anyone to be a successful entrepreneur, i.e. run her business better. Even economic policy cannot be affected by a proper analysis of wealth using a power law distribution instead of a normal distribution. For example, you just found out that the distribution of income is not normal and that the distribution of stock market returns follows a power law. What are you going to do about that? In the wealth case, are you going to start confiscating wealth to change the distribution to normal? In the latter case, are you going to limit your exposure to the point that a rare event cannot affect you seriously?

In the economic policy case we all know the fate of an attempt in the past century in certain unfortunate parts of the world to force a “normalization” of the wealth distribution, which is known as the Pareto distribution. The political system that tried to force that collapsed creating more pain and chaos than originally it planned to fight, actually causing a rare event itself, the rarest of all kinds, which was the collapse of itself.

We can also guess the projected returns of traders and fund managers who always anticipate rare events. Their returns will be dismal because their exposure to the market must be very low in relation to available capital. This is an example:

Suppose a trader with $100K gets a long signal for SPY, the ETF that tracks the S&P 500 index. The trader can tolerate a drop of 2% in his bankroll for this particular trade. The entry price is at $145 and he will exit with a loss if the price gets to $140 or with a profit if the price gets to $150. It is easy to calculate that the position size to accomplish his risk and reward objectives is 400 shares, although he could actually purchase a total of 690 shares with his capital based on entry price.

Now, let us see what happens if the trader gets scared and he wants to factor in the possibility that while his position is open, SPY can drop 20% in one day, just like the S&P 500 index did back in 1987. In this case, it it easy to calculate that he should buy only 69 shares, one order of magnitude less than what he could buy if fully invested. Of course, this diminishes any potential for profit from a good signal. In the previous case, the purchase of 400 shares allowed a potential 2% gain (400 shares x $5). In this case, the potential for profit is only 69 shares x $5 = $345.

I hope you get the message from the above simple case of trading. It is always important to know that rare events can happen and they will happen. It is also good to understand why we should not plan our life based on such rare events except in cases that its existence is threatened. Actually, one  legitimate area of application of rare event theory in my opinion is in life threatening situations but economics, investing and trading get neutralized if such events are constantly accounted for. Returns are minimized and progress is hindered because nobody wants a loan and no bank will loan to anyone because of the threat of such events, and more chaos results than it would in the case of such events. The conclusion is that planning for rare events can in turn cause a chain of rare events worse than those initially considered, just in the case of the collapse of certain economic systems enforced through political ideology.

A better suggestion is to never risk money that you cannot afford to lose.  This takes care of any rare event that can occur. Even better is this one: Risk only half of what you can afford to lose and keep the other half in case of rare events. Actually, rare events present great opportunities for those who have kept cash aside.

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