Separating Territory Effects from Class Effects

Why Territory and Class Effects Must Be Separated

A territory’s observed loss cost depends both on the risk associated with that territory and on the rating classes represented there. If one territory has more high-cost classes, its aggregate loss cost may be higher even when risks in corresponding classes have identical experience across territories. A raw comparison can therefore confuse territory effect with class mix.

The objective is a like-for-like comparison: compare territories within the same class, or recalculate each territory’s experience using the same class weights. This is a focused application of exposure-mix standardization. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For territory $t$ and class $c$, the observed class-specific loss cost is $L_{tc}=U_{tc}/E_{tc}$, where $U_{tc}$ is the adjusted loss amount and $E_{tc}$ is the corresponding exposure. A within-class territorial comparison compares $L_{tc}$ across territories for a fixed class. A standardized territory loss cost is $\sum_c w_cL_{tc}$, using the same class weights $w_c$ for every territory, with $w_c\geq 0$ and $\sum_cw_c=1$.

In this tutorial, loss amounts and exposures are assumed to have already been made comparable for the purpose of the indication, including any needed development, trend, or premium-basis adjustments. No discounting or mortality model is used, and no probability distribution is assumed: the calculations compare adjusted observed experience. A loss cost is loss per exposure, not necessarily premium per exposure.

Within-Class Comparisons

For a fixed class, compare the territorial loss costs directly. If territory $A$ has class-$c$ loss cost $L_{Ac}$ and territory $B$ has $L_{Bc}$, the ratio $L_{Ac}/L_{Bc}$ indicates the observed difference for that class. Comparing like with like removes the difference in class proportions from that particular comparison.

A single territory factor is most readily supported when the within-class ratios are reasonably consistent across classes. In particular, a simple multiplicative model is

L_{tc}=b\,T_tC_c,

where $b$ is a common base, $T_t$ is the territory factor, and $C_c$ is the class factor. Under this model, the ratio between two territories does not depend on class:

\frac{L_{Ac}}{L_{Bc}}=\frac{T_A}{T_B}.

This is a modeling assumption, not a mathematical consequence of the data. If the observed ratios differ substantially by class, do not present their differences as one uniform territorial effect without qualification. A common-weight standardized result can still describe the comparison for a chosen reference mix, but it is a mix-specific summary.

Worked Example: Comparing Territories Within Each Class

A hypothetical insurer compares adjusted loss experience in two territories. Class 1 has 100 exposures and $120,000 of losses in Territory A, and 200 exposures and $216,000 in Territory B. Class 2 has 100 exposures and $180,000 of losses in Territory A, and 50 exposures and $81,000 in Territory B. Calculate the class-specific loss costs and compare the territories. Assume the losses are already adjusted to a common basis; there is no discounting or mortality assumption, and no distribution is imposed.

Method: Calculate loss per exposure separately in each class, then form the Territory A-to-B ratio. This avoids treating a difference in class composition as a territorial difference.

For Class 1:

L_{A1}=\frac{\$120{,}000}{100}=\$1{,}200,\qquad L_{B1}=\frac{\$216{,}000}{200}=\$1{,}080.

The Class 1 ratio is $\$1{,}200/\$1{,}080=1.1111$, or about 11.11% higher in Territory A.

For Class 2:

L_{A2}=\frac{\$180{,}000}{100}=\$1{,}800,\qquad L_{B2}=\frac{\$81{,}000}{50}=\$1{,}620.

The Class 2 ratio is $\$1{,}800/\$1{,}620=1.1111$, also about 11.11% higher in Territory A. The matching ratios are consistent with a common relative territory difference across these two classes. They do not, on their own, establish that the observed indications are fully credible or that other adjustments are unnecessary.

Check: The aggregate loss costs are $300,000 divided by 200 exposures, or $1,500, in Territory A; and $297,000 divided by 250 exposures, or $1,188, in Territory B. Their ratio is $1{,}500/1{,}188=1.2626$, not 1.1111. The discrepancy arises because the aggregate mixes differ: Class 2 is half of Territory A’s exposures, but only one-fifth of Territory B’s. The within-class comparison avoids this mix distortion. The common mistake is to interpret the 1.2626 aggregate ratio as the territory effect.

Standardizing to a Common Class Mix

Within-class ratios provide separate comparisons by class. Standardization gives a single summary by applying the same reference class weights to each territory. The weights could represent a selected portfolio mix or another clearly defined reference distribution. They must be common across territories for the comparison to control for class mix.

Standardization formula: Given common class weights $w_c$ summing to 1, the standardized loss cost for territory $t$ is $L_t^{\mathrm{std}}=\sum_c w_cL_{tc}$. For territories $A$ and $B$, their standardized ratio is $L_A^{\mathrm{std}}/L_B^{\mathrm{std}}$, provided the denominator is positive.

The reason this works is that the formula holds the mix fixed. An observed aggregate loss cost instead uses each territory’s own exposure weights. To see the distinction, define $q_{tc}=E_{tc}/\sum_jE_{tj}$, the actual fraction of territory $t$’s exposures in class $c$. Then:

\frac{\sum_c U_{tc}}{\sum_c E_{tc}} =\frac{\sum_c E_{tc}L_{tc}}{\sum_c E_{tc}} =\sum_c q_{tc}L_{tc}.

If the $q_{tc}$ differ by territory, the aggregate comparisons use different weights. Replacing $q_{tc}$ with the same reference weights $w_c$ for every territory removes that particular source of non-comparability. It does not remove differences in experience within classes, nor does it prove that one common territory factor applies at every class level.

Choose the comparison basis.
Confirm that losses and exposures are comparable across territories, and identify the classes included.

Calculate class loss costs.
For every territory and class, divide adjusted losses by matching exposures.

Compare like with like.
Calculate within-class ratios, or apply one common set of class weights to each territory’s class loss costs.

Interpret the result in context.
Report the reference mix for a standardized indication and check whether class-specific comparisons support a common territory effect.

Worked Example: Standardizing When Class Results Differ

A hypothetical portfolio has two classes. In Territory North, Class A has 90 exposures and $81,000 of adjusted losses; Class B has 30 exposures and $63,000. In Territory South, Class A has 30 exposures and $29,700; Class B has 90 exposures and $162,000. Standardize both territories to weights 60% for Class A and 40% for Class B. Assume adjusted losses are comparable, with no discounting, mortality model, or assumed probability distribution.

Method: Calculate the four class loss costs, then apply the same 0.60 and 0.40 weights in both territories. This answers the comparison for the specified reference mix.

In North, Class A costs $\$81{,}000/90=\$900$ per exposure, and Class B costs $\$63{,}000/30=\$2{,}100$. Its standardized cost is:

L_{\mathrm{North}}^{\mathrm{std}} =0.60(\$900)+0.40(\$2{,}100) =\$540+\$840 =\$1{,}380.

In South, Class A costs $\$29{,}700/30=\$990$, and Class B costs $\$162{,}000/90=\$1{,}800$. Its standardized cost is:

L_{\mathrm{South}}^{\mathrm{std}} =0.60(\$990)+0.40(\$1{,}800) =\$594+\$720 =\$1{,}314.

The standardized South-to-North ratio is $\$1{,}314/\$1{,}380=0.9522$, to four decimal places. For the specified 60%/40% reference mix, South’s standardized loss cost is about 4.78% lower.

The within-class ratios are $990/900=1.10$ for Class A and $1{,}800/2{,}100=0.8571$ for Class B. They do not agree, so the standardized ratio is a weighted summary, not evidence of a uniform territory factor across classes. The common mistake is to report 0.9522 as an unqualified effect that applies equally to both classes. The calculation instead states the result for the particular weights chosen.

Derivation and Interpretation

The standardization formula follows directly from constructing the same hypothetical class mix in each territory. Imagine assigning one unit of reference exposure weight $w_c$ to class $c$. That class contributes $w_cL_{tc}$ to the average loss cost in territory $t$. Adding the contributions across classes gives:

L_t^{\mathrm{std}}=\sum_c w_cL_{tc}.

Because the weights are identical for all territories, differences in this sum cannot be caused simply by one territory having a larger observed share of a high-cost class. The result still depends on the chosen weights. If the territorial differences vary by class, changing the reference mix can change the standardized ratio. State the weights or their interpretation whenever reporting the result.

Worked Example: Three Classes and Two Different Aggregate Results

A hypothetical insurer compares Central and East using three classes. Central’s class exposures and adjusted losses are: Class 1, 50 and $40,000; Class 2, 25 and $30,000; Class 3, 10 and $20,000. East’s are: Class 1, 40 and $35,200; Class 2, 30 and $39,600; Class 3, 20 and $44,000. Standardize to weights 50%, 30%, and 20%, respectively. Assume losses are already adjusted to the same basis; there is no discounting, mortality model, or assumed probability distribution.

Method: Calculate the class loss costs, apply the reference weights, and compare the result with the unstandardized aggregate ratio. This checks whether the actual class mixes affect the raw comparison.

Central’s class costs are $\$40{,}000/50=\$800$, $\$30{,}000/25=\$1{,}200$, and $\$20{,}000/10=\$2{,}000$. Therefore:

L_{\mathrm{Central}}^{\mathrm{std}} =0.50(\$800)+0.30(\$1{,}200)+0.20(\$2{,}000) =\$400+\$360+\$400 =\$1{,}160.

East’s class costs are $\$35{,}200/40=\$880$, $\$39{,}600/30=\$1{,}320$, and $\$44{,}000/20=\$2{,}200$. Its standardized cost is:

L_{\mathrm{East}}^{\mathrm{std}} =0.50(\$880)+0.30(\$1{,}320)+0.20(\$2{,}200) =\$440+\$396+\$440 =\$1{,}276.

The standardized East-to-Central ratio is $\$1{,}276/\$1{,}160=1.1000$. Each class-specific ratio is also 1.10: $880/800=1.10$, $1{,}320/1{,}200=1.10$, and $2{,}200/2{,}000=1.10$. Thus the class comparisons are consistent with a 10% higher East indication in this example.

For comparison, Central’s aggregate cost is $\$90{,}000/85=\$1{,}058.82$, while East’s is $\$118{,}800/90=\$1{,}320$. Their aggregate ratio is $1{,}320/1{,}058.82=1.2467$, to four decimal places. The raw aggregate suggests a much larger difference because the territories have different class proportions. Standardization and the within-class checks both recover 1.10. The common mistake is to divide total losses by total exposures and assume that the resulting ratio isolates territory.

Assumptions and Common Traps

Separation is only as sound as the comparison basis. Before interpreting a territory indication, verify that class definitions are applied consistently and that the losses and exposures have been adjusted on a comparable basis. A difference in adjustment, exposure definition, or class assignment can remain in the comparison even after standardizing the class mix.

The simple multiplicative model $L_{tc}=bT_tC_c$ implies a common territory ratio across classes. Check that implication with within-class results. If ratios differ, a standardized value can still be useful, but it summarizes the territories for the stated weights rather than proving a single class-independent effect. The later subject of territory interactions treats more explicitly how territorial differences may vary by class; no such model is needed to perform the comparisons here.

Approach	What it compares	Interpretation to preserve
Within-class ratio	Two territories for one fixed class	Specific to that class
Common-weight standardization	Territories under the same selected class mix	Specific to the stated reference weights
Raw aggregate ratio	Observed territory totals using each territory’s own mix	May combine territory differences with class-mix differences

Do not compare different mixes as if they were identical. Aggregate loss costs can shift because class proportions differ.
Do not change the standard weights by territory. Different weights reintroduce mix differences into the standardized comparison.
Do not hide inconsistent class results. A weighted summary does not make class-specific territory ratios equal.
Do not confuse loss cost with premium. The comparison here is loss per exposure; a premium-based measure requires its own clearly defined basis.

Key takeaway: To separate territory from class mix, compare territories within each class or standardize each territory to identical class weights. Interpret a single territorial factor only when the class-level evidence and the modeling assumptions support that summary.

Check Your Understanding

For each problem, use adjusted losses and exposures as stated; assume no discounting, mortality model, or probability distribution unless specified.

In a single class, Territory West has 80 exposures and $96,000 of adjusted losses; Territory East has 120 exposures and $126,000. Calculate each loss cost and the West-to-East ratio. (Answer: West $1,200; East $1,050; ratio 1.1429.)
Territory A has class loss costs of $700 and $1,700; Territory B has $770 and $1,530. Using common weights 40% and 60%, calculate both standardized costs and the A-to-B ratio. (Answer: A $1,300; B $1,226; ratio 1.0604.)
In a class, Territory Pine has 40 exposures and $52,000 of adjusted losses; Territory Lake has 100 exposures and $115,000. What is the Pine-to-Lake within-class ratio, and what is its interpretation? (Answer: Pine $1,300; Lake $1,150; ratio 1.1304, so Pine’s observed loss cost is about 13.04% higher in this class.)
Class 1 and Class 2 loss costs are $900 and $2,100 in Territory North, and $990 and $1,890 in Territory South. Standardize both using weights 70% and 30%, then calculate the South-to-North ratio. (Answer: North $1,260; South $1,260; ratio 1.0000.)
Two classes have common-weight values 25% and 75%. Territory Red has class costs $800 and $1,600; Territory Blue has $880 and $1,440. Calculate each standardized cost and their Red-to-Blue ratio, and state why it is not necessarily a common class-level effect. (Answer: Red $1,400; Blue $1,300; ratio 1.0769; the class-specific ratios differ, so the standardized result is specific to the chosen weights.)

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