Why Territory Revisions Require Rebalancing
A territory factor is meaningful only in relation to the territories, exposures, and premium base used to select it. If a boundary moves or risks are reassigned, the mix of business under each factor changes. Even when the selected relativities themselves remain unchanged, the weighted statewide average can change. The factors may therefore need to be recalculated to preserve the intended statewide level.
The central task is to identify the quantity the factors must balance. An exposure-weighted normalization gives the factors an average of one across the revised exposures. A premium-weighted balance preserves aggregate premium under the current premium base. These are different objectives, and they generally use different weights.
The \(r_j\) values are the relativities or indications before the final common scaling. The \(w_j\) values must reflect the revised territory structure and the balancing objective. For exposure-weighted normalization, \(w_j\) is the revised exposure count or exposure measure. For premium-weighted balancing, \(w_j\) is the current premium before applying the territory factor. A weight should not be carried forward from the old territory map merely because it was used in a prior calculation.
What Changes When Boundaries or Membership Change?
A boundary revision can affect both the weights and the indications. First, exposures and current premium must be reassigned to the new territories. Second, if experience is being used to estimate territory differences, losses and exposures must be assigned to the revised boundaries as well. A factor calculated for an old territory is not automatically an appropriate indication for a new territory made from parts of several old ones.
When historical data can be restated under the new map, calculate each revised territory’s measure from the restated data. For example, if territory indications come from loss costs, calculate losses divided by exposures for each revised territory, then compare those loss costs with the statewide loss cost. This yields indications on the new geographic basis. If the data cannot be restated, any use of old indications to approximate a new territory requires an explicit method and assumptions; simple averaging of old factors is not generally justified.
Specify which risks, exposure units, and historical records belong to each territory under the new boundaries.
Aggregate revised exposures for exposure normalization, or current premium before territory factors for premium balance.
Use experience restated to the revised map if the indications are experience-based. Do not treat a changed boundary as if the underlying indication necessarily stayed the same.
Scale all indications by the same \(c\), using weights consistent with the stated target.
Substitute the final factors into the weighted total and confirm that it equals the target.
Deriving the Balancing Multiplier
Suppose the desired weighted average factor is one. The balanced factors have the form \(f_j=c r_j\), where the same positive multiplier \(c\) applies to every territory. Requiring the weighted total to equal the total weight gives
Because \(c\) is common to all territories, it can be factored out of the left-hand side:
If the weights are nonnegative and at least one \(w_j r_j\) is positive, the denominator is positive, so solving for \(c\) gives
This derivation also shows why a single multiplier preserves the relative pattern of indications: for any two territories with positive indications, \(f_j/f_m=r_j/r_m\). The multiplier changes the overall level, not the indicated relationship between territories. This normalization is not appropriate if the selected factors are meant to retain a different target average, or if separate constraints are imposed on particular territories.
Exposure-Weighted Normalization After a Revision
Worked Example: New Exposure Mix, Same Initial Indications
A hypothetical insurer has three revised territories with exposures of 1,000, 600, and 400 units. Preliminary indications are 1.20, 0.90, and 0.70. Normalize the factors so their revised exposure-weighted average is one. Before the boundary revision, the exposure counts were 800, 800, and 400. Show why using the old weights would produce a different result.
Method. Apply exposure-weighted normalization using the revised exposure counts. The revised total exposure is \(1{,}000+600+400=2{,}000\).
The revised weighted total of the unnormalized indications is
Thus \(c=2{,}000/2{,}020=100/101\), or approximately 0.990099. The revised factors are
Factors are displayed to four decimal places; the balance check uses the unrounded values. The weighted total is \((100/101)(2{,}020)=2{,}000\), so the weighted average is \(2{,}000/2{,}000=1\).
Using the obsolete exposures instead would give an indicated total of \(800(1.20)+800(0.90)+400(0.70)=1{,}880\), with total exposure \(2{,}000\). That would produce a different multiplier, \(2{,}000/1{,}880\), which does not balance the revised exposure mix. The common mistake is to retain the old weights after the map changes; the correction is to calculate the multiplier from the revised exposures.
Recompute Indications When Membership Changes
Worked Example: Combining Experience Under the Revised Map
A hypothetical insurer revises its map. Two former areas are combined into a new West territory, while a third becomes the new East territory. After assigning historical records to the new boundaries, West has 500 exposures and 50,000 dollars of adjusted losses from one former area, plus 300 exposures and 42,000 dollars of adjusted losses from the other. East has 400 exposures and 60,000 dollars of adjusted losses. Use loss cost relativities and normalize them to a statewide exposure-weighted average of one.
Method. Recalculate loss costs on the revised boundaries, then express each as a ratio to the statewide loss cost. The new West totals are \(500+300=800\) exposures and \(50{,}000+42{,}000=92{,}000\) dollars of losses. Its loss cost is \(92{,}000/800=\$115\) per exposure. East’s loss cost is \(60{,}000/400=\$150\) per exposure.
The statewide totals are \(800+400=1{,}200\) exposures and \(92{,}000+60{,}000=\$152{,}000\) of losses. The statewide loss cost is \(152{,}000/1{,}200=\$126.6667\) per exposure. The unnormalized relativities are therefore
The statewide loss cost is an exposure-weighted average of the territory loss costs. Consequently, the unrounded relativities already have exposure-weighted average one:
The normalized factors are approximately 0.9079 for West and 1.1842 for East. The check confirms that the factors balance on the revised exposure base. The common mistake is to average the former territories’ factors without their exposure and loss information; the correction is to aggregate the restated losses and exposures first, then calculate the revised indications.
Premium-Weighted Balance Under Current Rates
An exposure-weighted average of one does not necessarily preserve total current premium. If territories have different base premiums per exposure, exposure weights and premium weights differ. For a premium-preservation objective, let \(P_j\) be the current premium in revised territory \(j\) before applying its territory factor. To preserve the total current premium, require \(\sum_j P_j f_j=\sum_j P_j\), giving \(c=\sum_j P_j/\sum_j P_jr_j\).
Worked Example: Preserving Premium After Reassigning Business
A hypothetical rating plan has revised current premium weights of \$600,000, \$250,000, and \$150,000. Preliminary territory indications are 0.90, 1.20, and 0.80. These weights differ from the former premium distribution because policies were reassigned after a boundary change. Calculate premium-balanced factors that preserve the \$1,000,000 aggregate premium.
Method. Use premium weights, not exposure counts, because the target is preservation of current aggregate premium. The weighted indicated premium is
The total current premium is \$1,000,000, so \(c=1{,}000{,}000/960{,}000=25/24\). The factors are
Displayed to four decimal places, the factors are 0.9375, 1.2500, and 0.8333. Using unrounded values, the indicated total premium after applying the factors is \((25/24)(\$960{,}000)=\$1{,}000{,}000\), exactly the current total. Thus the factors preserve aggregate premium while retaining the relative pattern of the three indications.
A common mistake is to calculate \(c\) from exposure counts when the requirement is to preserve premium. That would balance a different weighted average. The correction is to use the premium immediately before application of the territory factor, and to rebuild that premium weight after territory membership changes.
Practical Checks and Common Traps
A balancing calculation is only as sound as its inputs. Before finalizing the factors, confirm that each weight corresponds to the revised territories and that the indication and weighting bases are consistent with the intended use. For experience-based indications, check that losses and exposures use the same revised boundaries and that the loss measures are on a consistent basis. For premium balance, use the specified current premium base, not premium already multiplied by the proposed factors.
| Objective | Weight to use | Balance check |
|---|---|---|
| Exposure-weighted average factor of one | Revised exposures | \(\sum_j E_j f_j=\sum_j E_j\) |
| Preserve current aggregate premium | Revised current premium before territory factors | \(\sum_j P_j f_j=\sum_j P_j\) |
| Revised loss-cost indications | Restated losses and exposures used to calculate loss costs | Compare each revised territory loss cost with the statewide loss cost |
- Old weights after a revision. Reassignment changes the portfolio mix. Recalculate the weights rather than carrying forward the old territory distribution.
- Unweighted averaging. The arithmetic mean of the factors is generally not the exposure- or premium-weighted average required by the plan.
- Mixing the objective and the weights. Exposure normalization and premium preservation answer different questions. State the target first, then select its weights.
- Rounding before balancing. Use full precision to calculate \(c\) and the final check. Round factors only for presentation or implementation, and recheck the effect of implementation rounding.
Check Your Understanding
For each problem, identify the relevant balancing basis and verify the weighted result.
- A hypothetical plan has revised exposures of 500 and 300, with unnormalized indications 1.10 and 0.80. Find the exposure-normalized factors, to four decimal places. (Answer: \(c=80/79\); factors 1.1139 and 0.8101.)
- Current premium before territory factors is \$400,000 in one revised territory and \$600,000 in another. Their indications are 0.95 and 1.10. Find the premium-balanced factors, to four decimal places. (Answer: \(c=25/26\); factors 0.9135 and 1.0577.)
- After a boundary revision, territory A has 200 exposures and \$30,000 in adjusted losses; B has 300 exposures and \$30,000 in adjusted losses; C has 500 exposures and \$75,000 in adjusted losses. Calculate the revised loss-cost relativities to the statewide average, to four decimal places. (Answer: statewide loss cost \$135; factors \(8/9=0.8889\), \(8/9=0.8889\), and \(10/9=1.1111\).)
- A revised three-territory map has exposures of 200, 300, and 500, with indications 0.80, 1.00, and 1.40. Find the common exposure-normalizing multiplier and the factors, to four decimal places. (Answer: \(c=25/29\); factors 0.6897, 0.8621, and 1.2069.)
- Following reassignment, current premium weights are \$500,000, \$300,000, and \$200,000. Indications are 0.90, 1.20, and 0.80. Find factors that preserve aggregate premium, to four decimal places. (Answer: \(c=100/97\); factors 0.9278, 1.2371, and 0.8247.)