Measuring & Managing Property Accumulation Chris Gross, The St Paul Companies November 2002 CAS Meeting

Terrorism Model vs. General Accumulation Management Any terrorism model is likely to be highly subjective, particularly with regard to frequency/probability of events. September 11 th demonstrates the principle that there will always be hazards that previously had not been contemplated. While terrorism and other hazard specific models should be pursued, prudence dictates that geographical concentration of exposure should be controlled regardless of perceived hazard.

A Basic Approach to Measurement of Concentration Map Exposures using latitude and longitude Sum across predefined geographical ranges to identify pockets of concentration

Exposures in Manhattan Size of bubble reflects insured value.

Problems with Basic Approach Binary summing (risks either fully in or fully out of an area) is highly sensitive to placement of areas. More sophisticated treatment is required for policies with multiple locations and and policy level financial characteristics (limits, attachments, etc.) Need to know not only location of “hot spots” (potentially numerous), but which policies/business units are contributing. Need to be able to succinctly answer the question “How is it changing?”

Addressing the Problem of Binary Summing In order to make measurement of concentration less sensitive to placement of investigated areas, we used a simple continuous damage function rather than a simple “in” or “out” approach.

Simple Damage Function Damage = Insured Value *.5^(distance/radius) “Radius” here is a parameter that indicates the general size of an event, defined as that distance at which 50% damage occurs. We have tended to concentrate on a radius of 200 meters, but do consider other radii.

General process For a given event location, repeat this process for all policies, sum to arrive at a total portfolio loss (L) for that location Repeat at various locations across the area being considered.

Converting the Graphic Image into Succinct Measures of Concentration The maximum is certainly an important measure, but insufficient. It is tempting to consider the volume beneath by the surface. –This simply returns a measure of the overall exposure, not how concentrated that exposure is. One could consider the volume beneath the surface, but above some threshold (excess aggregate loss) –What threshold should you use? We chose instead to look at L squared, and consider the volume under that surface. –This gives a measure that is subject not only to the amount of exposure, but how concentrated that exposure is.

Succinct Measure of Concentration- continued Helpful if measure has a value that is can be related intuitively. Consider a single, solitary risk viewed in terms of this model, summing the square of the modeled damage across the area that affects that risk. The total value will be approximately equal to: Where “resolution” = (“radius”)/(event spacing). Ex. (radius = 200m, event grid spaced 100m x 100m  resolution=2)

Single Risk Equivalent Exposure Dividing the previous sum by the first term in the equation (referred to hence as “c”), and taking the square root, simply returns the exposure for the single, solitary risk. So to make our accumulation measure more intuitive, we define “Single Risk Equivalent Exposure” (SREE) as: This measure is analogous to a standard deviation, in that if –insured locations are far apart (independent), –insured locations are at exactly the same spot (fully interdependent),

Contribution of an Individual Policy to Property Accumulation Simply allocate L^2 for a specific event to each policy in proportion to the loss contributed, i.e. We then can define: A way to describe SREE at the policy level, is that it reflects not only the exposure of the policy, but how close the policy is geographically to other risks. For example, a $100 MM location with nothing else around it will have a SREE of $100 MM. If it is close to a lot of other exposures, it very well may have a SREE of $300 MM.

Contribution to Property Accumulation by Business Unit SREE can be easily aggregated to any level desired by simply squaring, summing, and taking the square root. Doing this across policies within a business unit can give us a measure of SREE by business unit (whether within a city, or across numerous cities)

Managing Property Accumulations Monitor SREE over time by city, business unit, and in total, particularly relative to premium volume One way to more actively manage accumulation risk is to calculate a minimum premium by policy equal to f * SREE^2 where the factor f (which will be very small) is adjusted up or down by management in order to have the desired effect If property accumulation risk is growing beyond the corporate comfort level, increase f. If property exposure is dropping more than intended, reduce f.

Managing Property Accumulations -- continued This minimum premium method helps create consistency between how property accumulation risk is managed across cities (using the same factor nationwide) Even if this method is not used explicitly within underwriting guidelines, sorting policies by shortfall of actual premium from minimum premium can be a very helpful tool for management to identify policies where the accumulation risk being added is possibly out of line with the profitability of the account.