Chicken Little or Coefficient of Riskiness (COR)

Running around waving your arms and screaming “the Sky is Falling” is one way to communicate risk positions.  But as the story goes, it is not a particularly effective approach.  The classic story lays the blame on the lack of perspective on the part of Chicken Little.  But the way that the story is told suggests that in general people have almost zero tolerance for information about risk – they only want to hear from Chicken Little about certainties.

But insurers live in the world of risk.  Each insurer has their own complex stew of risks.  Their riskiness is a matter of extreme concern.  Many insurers use complex models to assess their riskiness.  But in some cases, there is a war for the hearts and minds of the decision makers in the insurer.  It is a war between the traditional qualitative gut view of riskiness and the new quantitative view of riskiness.  One tactic in that war used by the qualitative camp is to paint the quantitative camp as Chicken Little.

In a recent post, Riskviews told of a scale, a Coefficient of Riskiness.  The idea of the COR is to provide a simple basis for taking the argument about riskiness from the name calling stage to an actual discussion about Riskiness.

For each risk, we usually have some observations.  And from those observations, we can form the two basic statistical facts, the observed average and observed volatility (known as standard deviation to the quants).  But in the past 15 years, the discussion about risk has shifted away from the observable aspects of risk to an estimate of the amount of capital needed for each risk.

Now, if each risk held by an insurer could be subdivided into a large number of small risks that are similar in riskiness for each (including size of potential loss) and where the reasons for the losses for each individual risk were statistically separate (independent) then the maximum likely loss to be expected (99.9%tile) would be something like the average loss plus three times the volatility.  It does not matter what number is the average or what number is the standard deviation.

RISKVIEWS has suggested that this multiple of 3 would represent a standard amount of riskiness and become the index value for the Coefficient of Riskiness.

This could also be a starting point in looking at the amount of capital needed for any risks.  Three times the observed volatility plus the observed average loss.  (For the quants, this assumes that losses are positive values and gains negative.  If you want losses to be negative values, then take the observed average loss and subtract three times the volatility).

So in the debate about risk capital, that value is the starting point, the minimum to be expected.  So if a risk is viewed as made up of substantially similar but totally separate smaller risks (homogeneous and independent), then we start with a maximum likely loss of average plus three times volatility.  Many insurers choose (or have chosen for them) to hold capital for a loss at the 1 in 200 level.  That means holding capital for 83% of this Maximum Likely Loss.  This is the Viable capital level.  Some insurers who wish to be at the Robust level of capital will hold capital roughly 10% higher than the Maximum Likely Loss.  Insurers targeting the Secure capital level will hold capital at approximately 100% of the Maximum Likely Loss level.

But that is not the end of the discussion of capital.  Many of the portfolios of risks held by an insurer are not so well behaved.  Those portfolios are not similar and separate.  They are dissimilar in the likelihood of loss for individual exposures, they are dissimilar for the possible amount of loss.  One way of looking at those dissimilarities is that the variability of rate and of size result in a larger number of pooled risks acting statistically more like a smaller number of similar risks.

So if we can imagine that evaluation of riskiness can be transformed into a problem of translating a block of somewhat dissimilar, somewhat interdependent risks into a pool of similar, independent risks, this riskiness question comes clearly into focus.  Now we can use a binomial distribution to look at riskiness.  The plot below takes up one such analysis for a risk with an average incidence of 1 in 1000.  You see that for up to 1000 of these risks, the COR is 5 or higher.  The COR gets up to 6 for a pool of only 100 risks.  It gets close to 9 for a pool of only 50 risks.

 

cor

 

There is a different story for a risk with average incidence of 1 in 100.  COR is less than 6 for a pool as small as 25 exposures and the COR gets down to as low as 3.5.

Cor100

In producing these graphs, RISKVIEW notices that COR is largely a function of number of expected claims.  So The following graph shows COR plotted against number of expected claims for low expected number of claims.  (High expected claims produces COR that is very close to 3 so are not very interesting.)

COR4You see that the COR stays below 4.5 for expected claims 1 or greater.  And there does seem to be a gently sloping trend connecting the number of expected claims and the COR.

So for risks where losses are expected every year, the maximum COR seems to be under 4.5.  When we look at risks where the losses are expected less frequently, the COR can get much higher.  Values of COR above 5 start showing up with expected losses that are in the range of .2 and values above .1 are even higher.

cor5

What sorts of things fit with this frequency?  Major hurricanes in a particular zone, earthquakes, major credit losses all have expected frequencies of one every several years.

So what has this told us?  It has told us that fat tails can come from the small portfolio effect.  For a large portfolio of similar and separate risks, the tails are highly likely to be normal with a COR of 3.  For risks with a small number of exposures, the COR, and therefore the tail, might get as much as 50% fatter with a COR of up to 4.5. And the COR goes up as the number of expected losses goes down.

Risks with very fat tails are those with expected losses less frequent than one per year can have much fatter tails, up to three times as fat as normal.

So when faced with those infrequent risks, the Chicken Little approach is perhaps a reasonable approximation of the riskiness, if not a good indicator of the likelihood of an actual impending loss.

 

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