Stats in Risk   

Probabilities were invented in 1600s.  Kahneman showed that people round off their probabilities to: won’t happen, will happen, or maybe.  We are not good a judging how much insurance we need.  Robert Shiller points out that with stats we are like the cultures that count one, two, and, many.  They are able to remember hundreds of different plants but they do not count. 

In emergent theory outcomes are dependent on millions of little things, independent events, that accumulate.   Think of humans from DNA.

Unexpected events are often caused by a failure of the independence of these events.   For example changes in the stock market indices should be random since market changes are based on news, which is random and the indices cancel out individual stock risk.  Are stock returns independent variables?  They are until group think sets in. 

Fat tail events complicate this model.   In Finance, low probability shocks that shouldn’t happen occur.This is why geometric returns are more useful.  Outliers exist .  Random shocks to the economy are normally distributed. In nature the normal distribution is not the only distribution that occurs and some

other distributions have fatter tails.   -  Kurtosis

The Central limit theorem is probably the most important theory of statistics.  if you have independent identically distributed random variables and they have a finite variance  then the distribution of an average of these variables converges to a normal distribution as the number is increased.  The normal curve is so common because so many things we observe are averages of separate events.   Note that he normal distribution does not have fat tails.   This is why things work most of the time.

One reason that the normal distribution is so common because most events we observe are an average of independent events.   In a normal distribution the tails drop off.   It assumes the underlying variables have a finite variance. 

Other stat concepts include

Variance – is the sum of weighted probabilities of deviation from the mean

And standard deviation = square root of the variance

Covariance - how two different random variables move together?

 Can be positive, negative, or zero if unrelated

Correlation -  is scaled  -1  to +1

P = cov(x,y) /(s1,s2)

 

Low covariance is important in reducing risk.  We may not get expected return if the observations are not independent.

In the Law of large numbers, which is another stat concept– although there are a lot of independent shocks if they are independent, there is not much risk.   Variance goes to zero as n goes to infinity.   Insurance relies on this.

Value at Risk  VaR  was invented in 1987 to measure corporate risk.  Companies would calculate that there is a 5% probability   of loosing a million dollars.  It was found not to work well in 2008.

CoVar  was invented by  Brunnermier at Princeton.  It is  Value at risk of financial institutions conditional on other institutions being under distress.  This is supposed to be a newer more accurate model

Majority rule can be a good way to decide the better of two options.  Intrinsic justifications do not concern themselves with the quality of the decisions.  Voting outcomes can track truth if three conditions hold.  The voters are better choosers than random, are independent, and they vote sincerely.    One problem is that people  have preexisting cultural mindsets which determine how they look at information or even what new information they consider.   Only ten percent of people can explain what nano-technology is but 80% have an opinion on how safe it is.  Page argues that good decisions depend on sufficient cognitive diversity of the group or having people use different models to arrive at a solution.   A  smart diverse group of people is needed to get to the right answer.      

I am not sure how this ties to religion or even political parties, which encourage uniformity of thought.   Most people do not choose their political party or religion.  

 Shiller Finance Lecture

 wisdom of crowds

Scott Page wisdom of crowds