Actuarial Notes

Territory Ratemaking

Study guide: Exam 5

Territory Ratemaking is the derivation of geographic rate differentials. It is classification ratemaking with one distinctive property: geography is spatially correlated, so neighbouring territories inform one another and boundaries are themselves a decision.

\text{Territory Relativity}_i = \frac{\text{Pure Premium}_i}{\text{Pure Premium}_{\text{base}}}

\text{Rate}_i = \text{Base Rate} \times \text{Territory Relativity}_i

  • Geographic cost drivers: population and traffic density, crime, weather and catastrophe exposure, repair and medical costs, litigation propensity and local court behaviour, and — for property — building codes and fire protection class.
  • Territories are defined at some granularity (postal code, county, census tract) and then grouped into rating territories. Too fine and no cell is credible; too coarse and real cost differences are averaged away — the homogeneity/credibility trade-off in its sharpest form.
  • Spatial smoothing is what makes fine granularity workable: because loss cost varies continuously across space, a cell's estimate can borrow strength from its neighbours. Werner describes distance-based and adjacency-based smoothing, applied before territories are grouped.
  • Territory must be estimated alongside other rating variables, not before them. Urban territories carry different vehicle mixes, limit selections and driver populations, and a one-way territorial relativity absorbs all of it — see the univariate problem under Classification Ratemaking.
  • Catastrophe exposure must be handled separately. Coastal territories' relativities computed on experience including hurricane losses swing on whether a storm happened to make landfall during the period; the modelled catastrophe load belongs in the territory rate as its own component.
  • Territorial rating attracts the most social and regulatory scrutiny of any variable, because geography correlates with income and demographics. Some jurisdictions restrict or prohibit territory in certain lines.

Territory Relativities and Rates Base territory: 10{,}000 exposures, \$2{,}000{,}000 losses. Urban territory: 4{,}000 exposures, \$1{,}200{,}000 losses. The base rate is \$400.

Compute the urban relativity and rate.

Answer

\begin{align*}
\text{PP}_{\text{base}} &= \frac{\$2{,}000{,}000}{10{,}000} = \$200 \\[4pt]
\text{PP}_{\text{urban}} &= \frac{\$1{,}200{,}000}{4{,}000} = \$300 \\[6pt]
\text{Relativity} &= \frac{300}{200} = 1.50 \\[4pt]
\text{Urban rate} &= \$400 \times 1.50 = \$600
\end{align*}

With 4{,}000 exposures the urban indication is reasonably credible; a territory with 400 would need weighting toward the statewide average or a spatially smoothed neighbour estimate.

Smoothing and Grouping Territories A state is divided into 60 postal-code territories. Indicated relativities for five adjacent coastal codes, with their exposure volumes:

Code Exposures Indicated relativity
101 2{,}400 1.42
102 310 0.95
103 2{,}900 1.48
104 180 2.10
105 2{,}600 1.51

How should these be treated?

Answer The three large codes (101, 103, 105) tell a consistent story: coastal loss costs run about 1.45–1.50. The two small codes swing wildly in opposite directions — 0.95 and 2.10 — on 310 and 180 exposures respectively.

Those swings are noise, not geography. Loss cost does not fall by a third and then triple across three miles of the same coastline.

Two mechanisms, applied in order:

  1. Spatial smoothing. Estimate each cell partly from its neighbours. Codes 102 and 104 have little data of their own, so their smoothed estimates are pulled toward the \approx 1.47 level of the surrounding codes.
  2. Grouping. Combine all five into one rating territory at a credibility-weighted relativity near 1.47. The combined cell has 8{,}390 exposures — comfortably credible — and the grouping is defensible because the codes are contiguous and share the same cost drivers.

Taking the raw indications instead would charge one small coastal community half what its neighbours pay and another double, on the strength of a few hundred exposures — indefensible actuarially and unsustainable competitively.