Why Geography Requires Careful Ratemaking
Location can be highly relevant to expected insurance costs. Hypothetical drivers in different areas may face different theft frequency, traffic density, repair costs, or weather exposure. Property risks may differ in distance to emergency services or in exposure to a defined peril. At the same time, residence patterns can differ across protected groups. A location factor can therefore reflect expected losses and also act as a proxy associated with a protected trait.
Neither fact, by itself, settles whether a rating approach is appropriate. A geographic difference in observed losses does not automatically establish that every geographic definition is justified, and a difference in average premiums across groups does not by itself prove that the location factor is impermissible. A sound analysis identifies the loss measure being rated, explains why the geography is relevant, tests how the result changes under reasonable alternatives, and evaluates group effects in the applicable regulatory context.
Measure the Geographic Indication First
For a simple experience indication, let \(Z\) denote a geographic category, \(L_z\) the aggregate losses assigned to that category, and \(E_z\) its earned exposures. If losses have been developed to a common valuation and coverage basis, a basic indicated pure premium is \(r_z=L_z/E_z\). The measure must use a consistent exposure unit and loss definition across territories. It is not a complete final rate: expenses, other rating variables, trend, credibility, and applicable constraints may also matter.
For a set of geographic categories, the portfolio average pure premium is the exposure-weighted average, not the unweighted average of territory rates. A commonly used relative factor is each territory’s indicated pure premium divided by that portfolio average. This normalization makes the exposure-weighted mean factor equal to one.
The last equality follows by substitution: \(\sum_z E_z f_z=\sum_z E_zr_z/\bar r=(\sum_zL_z)/\bar r=\sum_zE_z\). Thus the factors are centered on the portfolio’s exposure mix. A different normalization base can be chosen for a rating plan, but the selected base and its implications should be explicit.
Worked Example: Territory Rates and Group Composition
A hypothetical insurer reviews two territories using losses and exposures on the same basis. Territory A has 1,000 exposures and $120,000 of losses; Territory B has 500 exposures and $90,000 of losses. In a separate fairness review, Group 1 has 700 exposures in A and 300 in B; Group 0 has 300 in A and 200 in B. Calculate the pure premiums, normalized factors, and average indicated premium by group.
Method. Divide losses by exposures for each territory, then calculate the portfolio average using total losses over total exposures. For group averages, weight each territory’s pure premium by that group’s exposure distribution.
Territory A. Its pure premium is \(r_A=\$120{,}000/1{,}000=\$120\) per exposure.
Territory B. Its pure premium is \(r_B=\$90{,}000/500=\$180\) per exposure.
Portfolio average. Total losses are \(\$120{,}000+\$90{,}000=\$210{,}000\), and total exposures are \(1{,}000+500=1{,}500\). Therefore \(\bar r=\$210{,}000/1{,}500=\$140\).
Normalized factors. \(f_A=\$120/\$140=0.8571\) and \(f_B=\$180/\$140=1.2857\), rounded to four decimal places. The exposure-weighted check is \((1{,}000(0.8571429)+500(1.2857143))/1{,}500=1.0000\), allowing for rounding.
Group averages. Group 1 has 1,000 exposures: \(700/1{,}000=0.70\) in A and \(300/1{,}000=0.30\) in B. Its indicated average is \(0.70(\$120)+0.30(\$180)=\$138\). Group 0 has 500 exposures: \(300/500=0.60\) in A and \(200/500=0.40\) in B. Its indicated average is \(0.60(\$120)+0.40(\$180)=\$144\).
Check and interpretation. Weighting the group averages by their exposures gives \((1{,}000(\$138)+500(\$144))/1{,}500=\$140\), matching the portfolio average. The group difference of \(\$6\) arises here from different geographic distributions; the calculation does not show that Group 1 and Group 0 have different indicated rates within either territory. The common mistake is to compare group average premiums without checking the underlying territory mix.
How Geography Produces a Proxy Association
Let \(G\) denote a protected group used for an appropriate audit, and let \(r_z\) be the geographic indication. If the rate within a territory does not vary by group, the average indicated premium for group \(g\) is
This expression follows by partitioning the group’s exposures by territory: each territory contributes its share \(P(Z=z\mid G=g)\) multiplied by the indicated premium assigned there. Consequently, two groups can have different average indicated premiums even when the same territory factor applies to everyone in each territory. The difference is explained arithmetically by the combination of territory rates and group-specific location distributions. That decomposition describes a rate pattern; it does not decide whether the pattern is lawful or whether the territories are sufficiently justified.
A useful review distinguishes at least three questions. First, does the geography capture a credible expected-cost difference on the relevant coverage and time basis? Second, is the chosen boundary or location measure a defensible representation of the underlying risk? Third, what rate and access consequences follow for protected groups under that definition? The answers require evidence and jurisdiction-specific review; a statistical association alone cannot answer all three.
Compare Geographic Definitions
A coarse definition can combine locations with materially different expected costs. A fine definition can better distinguish risk, but may make individual boundaries more consequential and can create greater variation in rates across nearby locations. These are not merely technical choices: the map, unit of measurement, and treatment of boundary cases affect both the indicated premiums and which groups are exposed to them.
| Definition approach | Potential actuarial value | Proxy and implementation questions |
|---|---|---|
| Broad territories | Stable, interpretable averages across larger areas | Can mask local risk variation; ask whether unlike locations are being pooled |
| Postal or administrative areas | Convenient assignment using familiar boundaries | Boundaries may not match hazard patterns; small boundary changes can reassign risks |
| Smaller geographic cells | Can represent local variation more directly | Can make local differences more rate-sensitive; examine group distributions and operational precision |
| Continuous location measures | May represent distance or hazard gradients without relying only on named areas | Check that the measure has a defensible relationship to expected costs and is consistently calculated |
A comparison should keep other elements as consistent as practicable: the loss definition, experience period, exposure basis, trend, coverage, and treatment of other rating variables. Otherwise an apparent change attributed to the map may actually come from a different data or modeling choice. Where permitted, an alternative geography should be assessed not just by its fit to historical losses but also by its stability, explainability, operational feasibility, and group-level effects.
Worked Example: Coarse Territory Versus Local Areas
A hypothetical insurer considers whether to split a broad territory C into two local areas. C1 has 100 exposures and $8,000 in losses; C2 has 100 exposures and $24,000 in losses. Group 1 has 80 exposures in C1 and 20 in C2. Group 0 has 20 in C1 and 80 in C2. Compare the coarse and local indications, including the groups’ average indicated premiums.
Coarse method. Combined losses are \(\$8{,}000+\$24{,}000=\$32{,}000\), and combined exposure is \(100+100=200\). The broad-territory indication is \(\$32{,}000/200=\$160\) per exposure. Under this definition, both groups receive the same indicated amount, \(\$160\), because every exposure is in C.
Local method. C1’s indication is \(\$8{,}000/100=\$80\). C2’s indication is \(\$24{,}000/100=\$240\). Relative to the overall \(\$160\) average, the local factors are \(80/160=0.5000\) and \(240/160=1.5000\). Their exposure-weighted mean is \((100(0.5000)+100(1.5000))/200=1.0000\).
Group averages under the local method. Group 1’s distribution is 80% C1 and 20% C2, so its average is \(0.80(\$80)+0.20(\$240)=\$64+\$48=\$112\). Group 0’s distribution is 20% C1 and 80% C2, so its average is \(0.20(\$80)+0.80(\$240)=\$16+\$192=\$208\). Their exposure-weighted combined average is \((100(\$112)+100(\$208))/200=\$160\).
Check and interpretation. The local definition distinguishes the two observed loss levels but also produces a \(\$96\) difference in group averages from their different location distributions. The coarse definition removes that geographic premium difference in this example but does not distinguish the observed local loss levels. Neither arithmetic result alone selects the appropriate map: the decision requires review of the expected-cost evidence, reasonable alternatives, group effects, and applicable requirements. A common mistake is to call the coarse method fair merely because these two group averages match, or to call the fine method justified merely because the local loss averages differ.
Separate Geographic Effects From Other Risk Mix
A raw territory comparison can also reflect differences in other rating variables. If one area contains more risks from a class with higher expected losses, its unadjusted average can be higher even when location adds no difference within that class. An adjusted comparison asks what the geographic indications would be under a common distribution of the other variables. This is a diagnostic, not a substitute for checking data quality, model specification, or whether the adjustment itself is appropriate.
Worked Example: Adjusting for a Different Class Mix
A hypothetical book has two areas, East and West, and two risk classes, Low and High. East has 80 Low exposures with $8,000 in losses and 20 High exposures with $4,000 in losses. West has 20 Low exposures with $2,000 in losses and 80 High exposures with $16,000 in losses. Calculate raw area averages and compare them with averages standardized to a 50% Low, 50% High mix.
Raw East average. East has \(80+20=100\) exposures and losses of \(\$8{,}000+\$4{,}000=\$12{,}000\), giving \(\$12{,}000/100=\$120\).
Raw West average. West has \(20+80=100\) exposures and losses of \(\$2{,}000+\$16{,}000=\$18{,}000\), giving \(\$18{,}000/100=\$180\).
Within-class rates. Low has \(\$8{,}000/80=\$100\) in East and \(\$2{,}000/20=\$100\) in West. High has \(\$4{,}000/20=\$200\) in East and \(\$16{,}000/80=\$200\) in West.
Standardized comparison. Applying the same 50%/50% class mix to each area gives \(0.50(\$100)+0.50(\$200)=\$150\) for East and the same \(0.50(\$100)+0.50(\$200)=\$150\) for West.
Check and interpretation. The raw \(\$60\) difference is entirely associated with the different class mixes in these data; there is no within-class area difference. The standardized averages agree. This does not establish that geography never matters; it shows why a raw average alone is not enough to attribute a difference to location. A common mistake is to take \(\$180/\$120=1.5\) as a geographic factor without checking the class composition.
Practical Review and Exam Traps
A disciplined review proceeds from the risk measure to the geographic definition and then to its group implications. Keep the analysis tied to the intended use of the factor: an observed association with historical losses supports prediction only to the extent that the data and model are suitable for the proposed rate application. Do not infer that residence itself causes the losses merely because a location category is predictive.
State the loss basis, exposure unit, experience period, and geographic assignment rule. Confirm that territories use comparable data and development.
Identify the plausible cost mechanism represented by location, such as hazard or repair cost, and distinguish that rationale from mere correlation with a group.
Recalculate indications under reasonable alternative boundaries or location measures while holding other assumptions consistent where possible.
Compare geographic distributions and indicated premiums across relevant groups, then document what the evidence does and does not establish under applicable rules.
A frequent exam error is to treat group-average premiums as if they were within-territory comparisons. The two quantities answer different questions: the group average combines territory rates with that group’s location mix, whereas a within-territory comparison holds location fixed. Another error is to average territory rates without exposure weights; the portfolio indication is based on total losses divided by total exposures. State which comparison is being made and show the weights.
Check Your Understanding
Work each problem before checking the answer in parentheses. Dollar amounts are hypothetical.
- A territory has 400 exposures and $52,000 in losses. A second territory has 600 exposures and $108,000 in losses. Find both pure premiums and the portfolio average. (Answer: $130 and $180; portfolio average $160 per exposure.)
- Two territories have indicated premiums of $90 and $150. Group A has 60% of its exposures in the first territory; Group B has 30% there. Find each group’s average indicated premium and their difference. (Answer: Group A $114; Group B $132; Group B is $18 higher.)
- A broad territory contains North with 50 exposures and $3,000 in losses, and South with 150 exposures and $21,000 in losses. Find the broad rate, the two local rates, and local factors normalized to the broad rate. (Answer: Broad rate $120; North $60 with factor 0.5000; South $140 with factor 1.1667.)
- In Area 1, Low risks have a rate of $80 and High risks a rate of $160. In Area 2, the corresponding rates are $80 and $160. Each area is standardized to 40% Low and 60% High. Find both standardized averages and state whether the rates show a within-class area difference. (Answer: $128 in each area; no within-class area difference.)
- A portfolio has 200 exposures in Area A at $100 each and 300 exposures in Area B at $160 each. Group X has 120 exposures in A and 80 in B. Calculate the portfolio average and Group X’s average indicated premium. (Answer: Portfolio average $136; Group X average $124.)