End-to-End Territory Factor Selection Workshop

From Territory Data to Selected Factors

Territory factor selection is a chain of decisions, not a single calculation. A sound indication can be distorted by incomplete data, different mixes of risks, an unsuitable complement, or factors that are not balanced to the intended portfolio basis. The objective of an end-to-end review is to make each step explicit and to confirm that the final factors produce a reasonable result when applied to the policies they will rate.

This workshop uses hypothetical data. For the calculations, assume that the loss costs supplied for each territory and class are already adjusted to a consistent coverage and time basis, including any development, trend, and rate-level adjustments required for the analysis. The focus here is the selection workflow, not the calculation of those prior adjustments.

Definition: A territory indication is a measure of the relative cost or rate level suggested by experience assigned to a territory. A territory factor is the multiplier applied to the applicable base rate for a risk in that territory. An indication is evidence for selection; it is not automatically the final factor.

Start With Data and Basis Checks

Before calculating an indication, confirm that the data describe the intended book and that territory assignments are usable. At a minimum, reconcile exposure counts and loss totals to their source summaries; check for missing or duplicated records, invalid territory codes, and unexpected changes in territory definitions. Confirm that exposures and losses use compatible periods and coverage definitions. If the input is already adjusted, document the adjustment basis and verify that it is the same across territories.

Next, inspect the class detail within each territory. Territory averages can differ because the underlying class mix differs, even when the territory has no genuine cost difference within any class. A mix adjustment addresses this comparison problem by evaluating each territory using the same standard class weights.

Method: Let $LC_{tc}$ be the adjusted loss cost for territory $t$ and class $c$, and let $w_c^{\mathrm{std}}$ be the selected standard proportion for class $c$, with $\sum_c w_c^{\mathrm{std}}=1$. The territory's standard-mix loss cost is $$ LC_t^{\mathrm{std}}=\sum_c w_c^{\mathrm{std}}LC_{tc}. $$ This standardization is appropriate when the class-level costs are comparable across territories and the same class definitions and standard weights are meaningful for all territories. It does not correct for unmeasured differences within a class.

Worked Example: Checking and Standardizing Two Territories

A hypothetical insurer reviews two territories, East and West. Its adjusted class-level data are shown below. The standard mix is 50% Class A and 50% Class B. Calculate each territory's class loss costs, standard-mix loss cost, and relativity to a $1,600 standard-mix benchmark.

Territory	Class	Exposures	Adjusted losses
East	A	120	$144,000
East	B	80	$144,000
West	A	60	$84,000
West	B	140	$280,000

Step 1: Check totals and calculate class loss costs. East has $120+80=200$ exposures and adjusted losses of $$144{,}000+$144{,}000=$288{,}000$. Its Class A loss cost is $$144{,}000/120=$1{,}200$, and its Class B loss cost is $$144{,}000/80=$1{,}800$. West has $60+140=200$ exposures and losses of $$84{,}000+$280{,}000=$364{,}000$. Its Class A loss cost is $$84{,}000/60=$1{,}400$, and its Class B loss cost is $$280{,}000/140=$2{,}000$.

Step 2: Apply the common standard mix. East's standardized loss cost is $0.50($1{,}200)+0.50($1{,}800)=$600+$900=$1{,}500$. West's is $0.50($1{,}400)+0.50($2{,}000)=$700+$1{,}000=$1{,}700$.

Step 3: Divide by the benchmark. East's relativity is $$1{,}500/$1{,}600=0.9375$. West's is $$1{,}700/$1{,}600=1.0625$. These are preliminary indications, not yet credibility-weighted or balanced factors.

Check and trap avoided. The unstandardized territory averages are $$288{,}000/200=$1{,}440$ for East and $$364{,}000/200=$1{,}820$ for West. Comparing those directly would mix territory differences with the territories' different class distributions. The standard-mix comparison instead uses identical 50/50 weights in both territories.

Apply Credibility, Then Balance

A territory's observed experience may be less stable than a broader complement, such as a related portfolio indication. Credibility weighting combines the two only when they are stated on the same basis. Let $I_t$ be the territory indication, $C_t$ its complement, and $Z_t$ the selected credibility, where $0\leq Z_t\leq1$. The credibility-weighted indication is $$ S_t=Z_t I_t+(1-Z_t)C_t. $$ The credibility value should come from the applicable credibility method and data conditions; it should not be chosen merely to make a desired factor appear. If the indication and complement are expressed as loss costs, keep both in loss-cost units for the blend. If they are factors, blend factors on a consistent reference basis.

After credibility weighting, convert the selected indications to preliminary factors using a stated reference level. If the factors are intended to preserve an overall portfolio level, balance them using the relevant weights. For example, where $W_t$ is the current premium at a unit territory factor, a neutral balance satisfies $$ \frac{\sum_t W_tF_t}{\sum_t W_t}=1. $$ This condition assumes the goal is to preserve aggregate premium at the current overall level. If the selected overall rate level is intended to change, balance to that target level instead. The balancing weights must represent the portfolio to which the factors will apply; equal territory weights are not generally appropriate.

The normalization follows directly. If preliminary factors are $R_t$, their weighted average is $\bar R_W=\sum_t W_tR_t/\sum_t W_t$. Set $F_t=R_t/\bar R_W$. Then $$ \frac{\sum_t W_tF_t}{\sum_t W_t} =\frac{\sum_t W_t(R_t/\bar R_W)}{\sum_t W_t} =\frac{\bar R_W\sum_t W_t}{\sum_t W_t\bar R_W}=1. $$ Thus normalization changes the overall scale while preserving the relative pattern among territories.

Worked Example: Credibility Weighting and Factor Balancing

Continue with the hypothetical East and West standard-mix loss costs of $1,500 and $1,700. Suppose the complement is $1,600 for both territories; selected credibility values are 0.40 for East and 0.60 for West. Use $1,600 as the reference cost, then balance the preliminary factors to a neutral average using equal portfolio weights.

Step 1: Blend each territory with the complement. East's selected cost is $0.40($1{,}500)+0.60($1{,}600)=$600+$960=$1{,}560$. West's selected cost is $0.60($1{,}700)+0.40($1{,}600)=$1{,}020+$640=$1{,}660$.

Step 2: Convert to preliminary factors. East is $$1{,}560/$1{,}600=0.9750$. West is $$1{,}660/$1{,}600=1.0375$. With equal weights, their weighted average is $(0.9750+1.0375)/2=1.00625$.

Step 3: Normalize and select practical factors. The balanced East factor is $0.9750/1.00625=156/161\approx0.968944$. The balanced West factor is $1.0375/1.00625=166/161\approx1.031056$. Rounded to three decimal places, select 0.969 for East and 1.031 for West.

Check and trap avoided. The rounded factors average to $(0.969+1.031)/2=1.000$, so they remain balanced under the stated equal weights. Using unbalanced indications as final factors would produce an average factor of 1.00625 and would raise the overall level by 0.625% under these weights. The example's equal weights are an explicit assumption, not a general rule.

Selection and Portfolio Impact Review

A selected factor is a practical decision informed by the indication, credibility, portfolio basis, and business constraints. Review the direction and size of each indicated change; investigate unusual factors, sharp differences between adjacent territories, and factors that conflict with credible experience. Smoothing, boundary stability, and experience with little data may inform judgment, but any departure from the indicated result should be documented rather than hidden in the arithmetic.

Rounding can disturb balance. After selecting rounded factors, recompute the weighted average using the intended portfolio weights. If a material difference arises, consider a suitably precise factor schedule or make a documented adjustment consistent with the selection objectives. Do not assume that factors remain balanced simply because the unrounded values were balanced.

Finally, translate the factor changes into premium impact. A factor percentage change is not the same as the change in total premium: the portfolio's policy counts, base premiums, and current factors determine the aggregate result. The review should report both the total dollar change and the percentage change from current premium, and should identify the portfolio and assumptions used.

Worked Example: Quantifying the Premium Impact

A hypothetical portfolio has 400 policies in Territory X and 600 in Territory Y. Each policy has a $900 base premium before the territory factor. Current factors are 1.000 for X and 0.950 for Y; selected factors are 1.050 and 0.970, respectively. Assume no other rating variables or premium changes. Calculate current premium, selected premium, and the percentage change.

Step 1: Calculate current premium. Territory X contributes $400($900)(1.000)=$360{,}000$. Territory Y contributes $600($900)(0.950)=$513{,}000$. Total current premium is $$360{,}000+$513{,}000=$873{,}000$.

Step 2: Calculate selected premium. X contributes $400($900)(1.050)=$378{,}000$. Y contributes $600($900)(0.970)=$523{,}800$. Total selected premium is $$378{,}000+$523{,}800=$901{,}800$.

Step 3: Calculate the change. The dollar increase is $$901{,}800-$873{,}000=$28{,}800$. The percentage increase is $$28{,}800/$873{,}000=0.0329897$, or 3.30% rounded to two decimal places.

Check and trap avoided. The selected Y premium is $600\times900\times0.970=$523{,}800$, not $522,600. Replacing it with the incorrect amount would understate the selected total and the premium increase. The portfolio impact is 3.30%, not the simple average of the two territory factor changes, because the territories have different policy counts and current factors.

Diagnostics and Common Exam Traps

A disciplined review checks each link between data and premium. The following questions help identify where an apparent result may be misleading:

Are the inputs comparable? Loss costs on different coverage, time, or adjustment bases cannot be directly compared as territory differences.
Was the same standard mix used? Territory-specific weights defeat the purpose of mix standardization.
Is the complement aligned? Credibility blending is meaningful only when the complement and territorial indication are on compatible bases.
Do balancing weights represent the target book? Use weights consistent with the rate structure and portfolio objective, not arbitrary equal weights.
Was balance rechecked after rounding? The actual selected schedule, rather than only its unrounded version, determines the implemented average.
Does the premium-impact denominator match the question? A percentage change in aggregate premium uses current aggregate premium as its denominator.

The central justification for the workflow is that each operation answers a different question: data checks ask whether the evidence is usable; mix adjustment asks how territories compare under a common composition; credibility addresses the stability of each indication; balancing sets the intended overall level; and premium-impact review tests what the selected factors do to the actual portfolio. Skipping a stage can make a numerically precise result inappropriate.

Key takeaway: Treat territory factor selection as a documented sequence. Verify the data basis, standardize the mix, apply a justified credibility blend, balance to explicit weights, check the rounded schedule, and calculate the portfolio premium impact from the actual rating inputs.

Check Your Understanding

Work each problem through the stated basis and check the arithmetic before comparing your result.

A territory has adjusted Class A and B loss costs of $1,000 and $1,500. The standard mix is 40% A and 60% B. A second territory has corresponding costs of $1,200 and $1,800. Find both standard-mix costs and the second territory's relativity to the first. (Answer: $1,300 and $1,560; relativity 1.2000.)
A territory indication is $240 per exposure, its complement is $200, and its credibility is 0.25. Calculate the credibility-weighted cost. (Answer: $210 per exposure.)
Preliminary factors are 0.900 for Territory A and 1.200 for Territory B. Their balancing weights are 300 and 700. Normalize the factors to a weighted average of 1 and report each to four decimal places. (Answer: A = 0.8108; B = 1.0811.)
A portfolio has 200 policies with a $750 base premium and 300 policies with the same base premium. Current factors are 1.000 and 0.900; selected factors are 1.050 and 1.000. Find current and selected aggregate premium and the percentage change, rounded to two decimal places. (Answer: $352,500 current; $382,500 selected; increase $30,000, or 8.51%.)
A territory's standard-mix indicated cost is $1,320. Its complement is $1,250, and credibility is 0.40. Using a $1,300 reference cost, calculate the credibility-weighted cost and its preliminary relativity. (Answer: $1,278; relativity $1{,}278/1{,}300=0.9831$, rounded to four decimal places.)

What You'll Learn