Monday, March 23, 2009

Extremely Improbable Events Happen All the Time

Risk managers assert in their defense that the current economic crisis was an unforeseeable, low probability event. From Andrew Haldane’s paper, Why Banks Failed the Stress Test:

Risk managers are of course known for their pessimistic streak. Back in August 2007, the Chief Financial Officer of Goldman Sachs, David Viniar, commented to the Financial Times:

“We are seeing things that were 25-standard deviation moves, several days in a row”

To provide some context, assuming a normal distribution, a 7.26-sigma daily loss would be expected to occur once every 13.7 billion or so years. That is roughly the estimated age of the universe. A 25-sigma event would be expected to occur once every 6 x 10124 lives of the universe. That is quite a lot of human histories.

How is it possible that extremely low probability events occur? The answer is that, while many events are highly probable over a short period of time, over longer periods events are extremely improbable. It is highly probable that when you go to bed tonight, you will get up in the morning from the same bed. But, think back to where you went to bed twenty years ago, and the events in your life that brought you to where you go to sleep now. How likely was it that you ended up where you are? That you have the job you have? That you have the spouse and kids you do?

From Carl Bialik’s The Numbers Guy blog:

We tend to fixate on those events that are memorable, after they happen. Peter H. Westfall, a statistician at Texas Tech University, notes that any given order of a shuffled 52-card deck has about a one in 10 to the 68th power probability of happening, including the sequence in which all 52 cards appear in order. “Everything we see has about a zero probability,” Westfall said. “Calculating these probabilities after the fact is kind of meaningless.”

The present we’re living has impossibly low odds of occurring.

Bank risk managers acted as though every tomorrow would be similar to the short term past, and didn’t account for less probable but still very possible outcomes (like house prices declining) which could rapidly create a much different environment in just a year or two (like the one we’re living in now).