Most decision makers are familiar with the statistical average and standard deviation measures. But risk management typically focuses on unlikely “tail” events. The financial crisis helped popularize the term “fat tails” to represent the idea that these extreme events are more likely than we might have believed. To move beyond “thin tailed” models, we need a way to describe the fatness of the tail.

## Extrapolating the Tail of the Risk Model

The statistical approach to building a model of risk involves collecting observations and then using the data—along with a general understanding of the underlying phenomena—to choose a probability distribution function (PDF).

This process is often explained in terms of “fitting” one of several common PDFs to the data. But an alternate view of the process would be to think of it as an extrapolation, because most observed values are near the mean. Under the so-called Normal PDF, we expect observations to fall within one standard deviation of the mean about two-thirds of the time, and within two standard deviations almost 98% of the time. When modeling annual phenomena, it is unlikely that we will have even one observation to guide the fit at the 99^{th} percentile[i].

So, in most cases, we really are using the shape of the PDF to extrapolate into the tail. But we often gloss over that fact. Model documentation sometimes states the PDF used for extrapolation, but rarely discusses why that PDF was chosen and almost never mentions the importance of the modeler’s judgment in selecting the parameters that determine extreme values via extrapolation.

## A New Measure: Coefficient of Riskiness

During the financial crisis David Viniar, CFO of Goldman Sachs, famously observed “*we are seeing things that were 25 standard deviation moves, several days in a row.”*

That might have suggested he was using the wrong model. But our own work with insurance risk models shows that “tail” events can be many multiples of standard deviation away from the mean. This is the idea of the “coefficient of riskiness”: it’s a new way to describe fatness of tails.

We define the coefficient of riskiness (CoR) as the number of standard deviations that the 99.9^{th} percentile value is from the mean[ii].

CoR = (V_{.999} –* µ)/σ*

We use mean and standard deviation in defining the CoR not because they are the mathematically optimal way to measure extreme value tendencies, but because they are the two risk modeling terms most widely known to business leaders.

These three metrics—mean, standard deviation, and CoR—let us describe a PDF’s average value, typical level of fluctuation, and potential for producing extreme results.

## Examples From Insurance Risk Models

Examining a large number of models Willis Re constructed for our insurance company clients (covering all perils and lines of business), we see quite a wide range of CoR values.

## Communicating Riskiness with CoR

The CoR measure offers a way to explain fatness of tails to business leaders without getting into complicated mathematics. If adopted widely, CoR could come to be used like the Richter Scale for earthquakes or the Saffir-Simpson Hurricane Scale. If you were presenting a model of hurricanes or earthquakes and mentioned that you had modeled a “4” as the most severe event, property underwriters would have some sense of what that meant, even if they don’t know anything about the details of catastrophe modeling. They can form an opinion about whether 4 is reasonable for the most severe event produced by the model, and participate in a discussion on that basis.

Similarly, CoR could facilitate discussion of model severity. If you believe that Viniar’s comment about 25 standard deviations was based on sound data (rather than an exaggeration to make a point), you would doubtless reject the validity of the Normal PDF, which has CoR = 3. Were non-technical users of risk models to gain an appreciation of which risks have CoR = 3 and which have CoR = 12, that could be a large advance in understanding a very important risk characteristic.

Few people understand the science or math behind the Richter Scale, but everyone living in an earthquake zone can experience a shake and come pretty close to nailing the Richter Score of that event without any fancy equipment – and they know how to prepare for a quake of magnitude 4, 5, or 6. By bringing the Coefficient of Riskiness into our business conversations, we can help business leaders develop intuition about what the risk models imply about preparing for extreme events of all kinds.

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*The above is a summary of a new paper, titled **Into the Tails of Risk: An intervention into the process of risk evaluation**, which was awarded the “Best Practical Paper” prize at the 2015 ERM Symposium. *

[i] Some one-year loss calculations are performed by calculating a value for a much shorter period and extending that calculation to the full year by making an heroic assumption about the relationship between that short period and the full year. That substitutes a problem from that time period assumption for the lack of actual data about full year risk. And whether practitioners realize it or not, that process is an extrapolation into the unknown.

[ii] The 99.9 percentile is chosen to be beyond the values most often used from the model. All of the ideas presented here about CoR would apply with a different chosen reference point.

Dave,

Thanks for this very interesting reading.

It reminds me of the article “Just how much capital is at risk?” published in the Standard & Poor’s reinsurance highlights 2012. The authors proposed to use the ratio between the 1:250yr loss and the 1:50yr loss as a measure of the tail heaviness of a reinsurer’s loss experience. A reinsurer with a lighter tail would experience higher losses as proportion of the 1:250yr loss therefore underestimating its catastrophe risk exposure.

I would be interest to know you thoughts on this.

Regards,

Giovanni

Willis Re Analytics