Exponential Distributions, Part 1 - Expect More Bad Luck Than Good

In a recent blog post, I discussed how JPMorgan got burned by assuming derivatives behaved according to a normal distribution when in fact they followed an exponential distribution.  I want to delve a bit further into some of the counter-intuitive aspects of exponential distributions because I have found most people are not familiar with them.

Randomness

First it is important to understand that all of these distributions I am referring to deal with randomness.  When the next event cannot be predicted precisely, such as tomorrow's stock market close or the roll of the dice, it is an example of randomness.  However, if a population of related random events is studied as a group, the randomness will take on a particular form.  The histogram for rolling a single die will be flat, with each value from 1 through 6 having an equal 1/6th chance of occurring.  Sometimes those histograms take the shape of a bell curve (normal distribution) or an exponential distribution or something completely different.  It just happens that normal and exponential curves appear a lot in the real world.

Erlang Distributions, shown to the right, are just one family out of many exponential curves.  The mathematics of Erlang functions can be viewed here, but it's not necessary to know the math in order to understand their behaviour.  Most queues in the real world follow an Erlang curve (think the line-up at the bank or how long you will be kept on hold by your company's help desk).  Exponential distributions come in many shapes and sizes, but they share some common attributes that differ significantly from the Normal Distribution.  These mathematical differences create real-world consequences that must be understood in order to manage them appropriately.

Exponential Attribute #1:  Assemetry
Or Expect More Bad Luck Than Good

Normal distributions are symmetrical about their mean, which implies random variation will occur equally above and below the average value.  If you equate randomness with luck, and one side of the mean as good and the other as bad, then you will get approximately equal amounts of good luck (above the mean) and bad luck (below the mean).

Exponential distributions however are lop-sided.  This is because there is a minimum value but no practical maximum, and because the mean is relatively close to the minimum.  For example, inpatient length of stay in a hospital cannot be less than 1 day but it can extend out to hundreds of days on rare occasions.  The duration of a telephone call cannot be less than zero seconds but it can extend into hours or even days if it is a computer modem that's making the call.

How is this important?  

It means unusual random events will almost ALWAYS be on the long side of the distribution.  You will not get enough randomly small events to balance out the randomly large events as you would expect in a normal distribution.  For instance, if shoe size in a population happens to follow a normal distribution, we would expect to find an unusually large set of feet (i.e. more than 3 sigma above the mean) on a rare occasion (about 1 out of every 370 people).  Similarly we would also expect to find an unusually small set of feet (i.e. less than 3 sigma below the mean) on an equally rare occasion.  If our sample size of the population is sufficient, those large and small outliers would balance each other out and our mean value would be unaffected.  Not so with exponential outliers.  When you get an inpatient who stays in a hospital ICU bed for 2 years (such as a spinal injury from a diving accident) you will never get an inpatient who stays -2 years to balance things out.    That one outlier event will significantly impact the average length of stay for that ICU and probably for the entire hospital.

Therefore, exponential distributions bring unbalanced randomness.  Warren Buffet found this to be true of the insurance industry, as he states in his 2005 Letter to Shareholders:
"One thing, though, we have learned – the hard way – after many years in the business: Surprises in insurance are far from symmetrical. You are lucky if you get one that is pleasant for every ten that go the other way."

JP Morgan learned this as well.  Days where derivative contracts lost big money were not balanced out by days where those same contracts earned big money.  The random behaviour of that system was severely unbalanced.  The bad luck exceeded the good luck.

Except it wasn't luck.  It was predictable randomness.

If your system's randomness follows an exponential curve, then you must plan for more bad luck than good.


In Part 2 we will look at another implication of this asymmetry, namely the large tail of an exponential distribution.