Why Territory Factors Must Balance to Exposure
A territory factor changes the rate assigned to each exposure in that territory. To keep the plan's overall level consistent with a common base rate, the factors must be balanced over the exposures to which the plan applies. The relevant average is therefore determined by the plan's exposure distribution, not by how many territory labels there are.
This distinction matters whenever territories have different amounts of business. If one territory contains many more insured exposure units than another, its factor has more influence on the average rate across the plan. The exposure-weighted constraint makes that influence explicit. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
An exposure may be a policy-year, vehicle-year, or another suitable unit, depending on the product and rating plan. The choice must be consistent across territories. For example, a factor set intended for a distribution of earned vehicle-years should be balanced using that distribution, not a different distribution of policies or written premium.
The Weighted-Average Constraint
Let \(q_j\) denote a positive raw territory indication for territory \(j\), before balancing. The indications might be estimated relativities or comparable territory measures. To preserve their relative differences while making them exposure-balanced, multiply every indication by one common constant \(c\). Thus \(f_j=cq_j\). The balancing condition determines \(c\).
The derivation follows by multiplying both sides of the balancing condition by \(\sum_j e_j\): \(c\sum_j e_jq_j=\sum_j e_j\). Provided the exposures sum to a positive value and the weighted sum of raw indications is positive, division gives the expression for \(c\). The balanced factors are consequently
This normalization applies one common scale change, so any ratio of two raw indications is preserved: \(f_j/f_k=q_j/q_k\). It changes the overall level of the factors, not their relative pattern. It is the specific raw indications and the selected exposure distribution that determine the resulting scale.
Define what counts as one exposure and identify the distribution the factors are intended to represent.
Multiply each raw indication by its territory exposure count, add those products, and divide by total exposures.
Divide every raw indication by its exposure-weighted average, or equivalently multiply each by the reciprocal of that average.
Recalculate the exposure-weighted average of the final factors. It should equal one, apart from rounding.
Worked Applications
Worked Example: Balancing Raw Factors With Unequal Exposure Counts
A hypothetical property program has three territories. Its raw territory indications are 0.80, 1.10, and 1.60, and the target distribution contains 6,000, 3,000, and 1,000 exposures, respectively. Find exposure-balanced factors, preserving the raw relative differences.
Method. Calculate the exposure-weighted average of the raw indications, then divide each indication by that average. The common mistake to avoid is averaging the three indications equally, despite the unequal exposure counts.
Step 1: Total exposures. The total is \(6{,}000+3{,}000+1{,}000=10{,}000\).
Step 2: Weighted sum and average. The weighted sum is \(6{,}000(0.80)+3{,}000(1.10)+1{,}000(1.60)=4{,}800+3{,}300+1{,}600=9{,}700\). The raw weighted average is \(9{,}700/10{,}000=0.97\).
Step 3: Normalize. Divide each raw indication by \(0.97\):
Step 4: Check. Using the exact fractions, the weighted sum of the final factors is \(\{6{,}000(80)+3{,}000(110)+1{,}000(160)\}/97=970{,}000/97=10{,}000\). Dividing by 10,000 gives exactly 1. To six decimal places, the factors are 0.824742, 1.134021, and 1.649485. Their small displayed rounding differences do not change the exact balance.
Worked Example: Balancing Factors From Territory Loss-Cost Indications
A hypothetical liability program has comparable target loss-cost indications of $80, $100, and $140 per exposure in territories A, B, and C. The target distribution has 5,000, 3,000, and 2,000 exposures. Calculate territory factors relative to the exposure-weighted average indication.
Method. First find the weighted average loss-cost indication. Then express each territory's indication as a ratio to that average. The values here are illustrative comparable indications; the calculation balances these factors, not a complete premium rate with expense and profit provisions.
Step 1: Weighted average indication. The weighted total is \(5{,}000(80)+3{,}000(100)+2{,}000(140)=400{,}000+300{,}000+280{,}000=\$980{,}000\). Total exposures are \(5{,}000+3{,}000+2{,}000=10{,}000\), so the weighted average indication is \(\$980{,}000/10{,}000=\$98\) per exposure.
Step 2: Calculate factors. Divide each territory indication by $98:
Step 3: Check. The weighted sum is \(5{,}000(40/49)+3{,}000(50/49)+2{,}000(70/49)=(200{,}000+150{,}000+140{,}000)/49=10{,}000\). Thus the weighted-average factor is \(10{,}000/10{,}000=1\). The indication-weighted calculation and the ratio-to-average calculation give the same factors.
Worked Example: The Exposure Distribution Determines the Balance
A hypothetical rating plan has raw territory factors 0.70, 1.00, and 1.30. Under the target distribution, the territories have 2,000, 5,000, and 3,000 exposures. Find balanced factors. Then determine whether those same factors remain balanced if the distribution changes to 4,000, 3,000, and 3,000 exposures.
Method. Normalize the raw factors under the stated target distribution. Then test the resulting factors against the changed distribution. The important trap is assuming that a factor set balanced for one exposure mix must be balanced for every mix.
Step 1: Balance under the original mix. There are 10,000 exposures. The raw weighted sum is \(2{,}000(0.70)+5{,}000(1.00)+3{,}000(1.30)=1{,}400+5{,}000+3{,}900=10{,}300\). The raw weighted average is \(10{,}300/10{,}000=1.03\).
Step 2: Normalize. Divide each raw factor by 1.03:
Step 3: Test the changed mix. The raw-factor weighted sum under the new distribution is \(4{,}000(0.70)+3{,}000(1.00)+3{,}000(1.30)=2{,}800+3{,}000+3{,}900=9{,}700\). Its average is \(9{,}700/10{,}000=0.97\), not 1. The previously normalized factors therefore have a new weighted average of \(0.97/1.03=97/103=0.941748\), not 1.
Step 4: If balance is required for the changed mix, renormalize. Divide the raw factors by 0.97: \(0.70/0.97=70/97=0.721649\), \(1.00/0.97=100/97=1.030928\), and \(1.30/0.97=130/97=1.340206\). Their weighted sum is \(\{4{,}000(70)+3{,}000(100)+3{,}000(130)\}/97=970{,}000/97=10{,}000\), so their average under the changed mix is 1.
Interpretation and Common Traps
If \(B\) is a common base rate per exposure, the territory rate under a multiplicative plan is \(Bf_j\). The total indicated amount over the specified exposures is then \(B\sum_j e_jf_j\). When the factors are exposure-balanced, this is \(B\sum_j e_j\): the exposure-weighted average territory factor leaves the common base rate unchanged across that specified distribution.
That statement is conditional on the selected exposure distribution and on the plan's assumptions. The territories must cover the target exposures, the exposure measure must be consistent, and the factors must be applied as the intended multiplicative territory factors. If the target distribution changes, the weighted-average constraint may no longer hold, as the third example demonstrated.
A common incorrect calculation is to take the simple average of the factors in the first example: \((0.80+1.10+1.60)/3=1.166667\), and then divide each factor by 1.166667. That enforces an equal-territory average of one, not an exposure-weighted average of one. The correction is to use the actual exposure weights: the correct raw average is \(0.97\), so each raw indication is divided by \(0.97\). Equal weighting is appropriate only if the territories have equal exposures in the specified distribution.
| Check | What it means |
|---|---|
| Same exposure basis | Each \(e_j\) counts the same kind of exposure over the intended period. |
| Correct target mix | The weights represent the population for which the factors are intended to balance. |
| One common scale adjustment | Each raw indication is divided by the same weighted average, preserving relative factors. |
| Weighted-average verification | Compute \(\sum_j e_jf_j/\sum_j e_j\) after normalization; it should equal one. |
Check Your Understanding
For each hypothetical plan, identify the required exposure-weighted calculation and check whether the stated factor set is balanced.
- A two-territory plan has raw factors 0.90 and 1.20, with 4,000 and 6,000 exposures. Find the balanced factors to six decimal places. (Answer: weighted raw average \(=1.08\); factors \(0.833333\) and \(1.111111\).)
- Three territories have comparable indications of $75, $90, and $120 per exposure, with 2,000, 5,000, and 3,000 exposures. Find the relative factors to five decimal places. (Answer: weighted average indication \(=\$96\); factors \(0.78125\), \(0.93750\), and \(1.25000\).)
- Raw factors are 0.60, 1.00, and 1.40, with 1,000, 4,000, and 5,000 exposures. Calculate the raw weighted average and the balanced factors to five decimal places. (Answer: weighted average \(=1.16\); factors \(0.51724\), \(0.86207\), and \(1.20690\).)
- A plan has 3,000 exposures in territory A with factor 0.80, 2,000 in territory B with factor 1.10, and 5,000 in territory C with an unknown factor \(x\). What value of \(x\) makes the factors exposure-balanced? (Answer: \(x=1.08\).)
- A two-territory raw factor set is 0.75 and 1.25. It was balanced for 5,000 exposures in each territory, but the target mix changes to 8,000 and 2,000 exposures. Find the new weighted average of the old balanced factors and the factors rebalanced to the new mix. (Answer: old factors have new weighted average \(0.85\); rebalanced factors are \(0.882353\) and \(1.470588\).)