A new book on the likelihood of rare events may make some of the discussion on tail risks more real. The Improbability Principle - Why coincidences, miracles, and rare events happen every day by David Hand provides a short description of extreme events - why they are more likely and why we misunderstand them. If you are looking for depth and details about some of the key statistics behind on these interesting topics, you will be disappointed. However, if you want a free ranging discussion on many interesting topics concerning rare events, this is is a nice place to start. You will be fascinated by the stories of the author and you will want to learn more after this simple taste on the world of extremes.
The Improbability Principle in statistics is simple. Extremely improbable events are commonplace, or certainly more likely than what many think. As the sample size or set of opportunities gets bigger and bigger, the mean becomes more well behaved but any extreme event is more likely to occur. But the author also bring in a set of associated principles that affect how we view the world. For example, there is Borel's Law that says, events with sufficiently small probability never occur or at least we must act in all circumstances as if they were impossible. He gives us the laws of the extreme and unusual jungle.
Chance is probability and luck is just chance with a notion of good or bad. We do have to make judgements about good or bad when we say that events will happen. This is just inevitable. In fact, there is also a law of inevitability, something must happen and over enough time it will happen. This is associated with the law of truly large numbers states that with a large enough number of opportunities and outrageous event will occur. Yes, there is the law of large numbers which states that averages will fluctuate less as the sample gets bigger, but the extremes will also more likely occur. This is associated with the law of combinations where highly improbable becomes almost inevitable. We have to appreciate the law of the probability lever, a small change in the shape of the probability distribution will have a significant change in the chance of an event occurring. The stock market crash was a rare event but with a small change in volatility or kurtosis in the price distribution, there could be a multiple increase in the likelihood of an event. A fat tailed distribution makes a stock market crash not just more likely but almost inevitable.
Of course, we fool ourselves if we are given enough data. There is also a law of selection describing our biases, if we look at enough events, we will find what we are looking for. This is just another way of saying that we have a hindsight bias. Odd events and patterns will occur and if we use selectivity we will find these odd events. News report are full of oddities because we have selected these unusual events. We also find patterns and similarity because of the law of near enough. Events which are sufficiently similar are regarded as identical. We are always looking for patterns and we find then whether they are there or not. However, patterns are still likely to occur.
We should expect extremes and tail events. So the job of the risk manager is not to predict these events but always structure the portfolio so that if these inevitable rare events do occur, the portfolio's performance will not be devastated.