How Territory Factors Fit Into a Rating Plan
A territory factor adjusts a rate to reflect the territory assigned to a risk. It is one component of a rating plan, not a dollar amount and not, by itself, the premium charged. To calculate a premium, the factor must be combined with a base rate, an exposure amount, and any other applicable rating factors.
The focus here is how to apply an already specified territory factor within a rating plan. The factor is treated as a given input; the calculation does not establish how territories are defined or how factors are estimated from experience. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
For example, a territory factor of 1.12 means the territory-adjusted rate is 1.12 times the corresponding rate before the territory adjustment, with other inputs unchanged. It does not mean that the policy premium is $1.12, or that every policy in that territory has a premium 12% higher than every policy in another territory. A policy’s class, exposure, and other rating characteristics also affect its premium.
From Relativity to Rate and Premium
A base rate has units such as dollars per vehicle-year or dollars per insured-value unit. Rating factors are dimensionless multipliers. Exposure converts a rate per unit into a premium amount; the other factors adjust the rate for the characteristics represented in the plan. For a simple multiplicative plan, the order of multiplication does not change the result.
The product formula assumes that the stated factors apply multiplicatively to the same base rate and exposure, and that no separate minimum premium, fixed fee, cap, or rounding rule alters the result. If a plan includes such provisions, apply them as specified rather than assuming the product formula is the final amount billed.
Worked Example: Calculating Premiums With Different Territory, Class, and Exposure Inputs
A hypothetical commercial plan has a base rate of $400 per exposure unit. Policy A has 1.5 units, territory factor 1.05, and class factor 0.90. Policy B has 1.25 units, territory factor 1.20, and class factor 0.80. There are no other factors, fees, or minimum premiums. Calculate both premiums and compare the premium ratio with the territory-factor ratio. These are direct rating calculations; no interest, mortality, or probability model is involved.
Method. Apply the multiplicative rating formula separately to each policy. This avoids treating the territory factor as if it were the whole premium or attributing differences in exposure and class to territory.
Policy A. The base amount for its exposure is \( \$400 \times 1.5 = \$600 \). Apply the territory factor: \( \$600 \times 1.05 = \$630 \). Then apply the class factor: \( \$630 \times 0.90 = \$567 \). Thus, \(P_A=\$567.00\).
Policy B. The base amount for its exposure is \( \$400 \times 1.25 = \$500 \). Apply the territory factor: \( \$500 \times 1.20 = \$600 \). Then apply the class factor: \( \$600 \times 0.80 = \$480 \). Thus, \(P_B=\$480.00\).
The B-to-A premium ratio is \( \$480/\$567 = 160/189 \approx 0.8466 \). The B-to-A territory-factor ratio is \( 1.20/1.05 = 8/7 \approx 1.1429 \). The premium ratio is not the territory-factor ratio because the policies have different exposure amounts and class factors. Check: applying each set of inputs directly gives $567.00 and $480.00, matching the calculated premiums.
When a Premium Ratio Equals a Territory-Factor Ratio
A territory-factor comparison isolates the territory effect only when the other inputs are held fixed. Suppose two otherwise identical risks have the same base rate, exposure, class factor, and all other applicable factors, but different territory factors. The common quantities cancel when the premiums are divided.
This cancellation is the reason a factor ratio can describe the relative premium effect of territory under controlled conditions. If an exposure, class, or other factor differs, it does not cancel, and the full premium ratio includes that difference as well.
Worked Example: Isolating the Territory Effect
A hypothetical policy has a base rate of $500 per unit, 2 exposure units, a class factor of 1.10, and no other adjustment. Its territory factor changes from 0.95 to 1.14, while every other input remains unchanged. Calculate both premiums, the premium change, and the relative increase. No interest, mortality, or probability model is involved.
Method. Calculate the common premium amount before territory, then apply each territory factor. Because the other inputs are fixed, the premium ratio must equal the territory-factor ratio.
The common amount before territory is \( \$500 \times 2 \times 1.10 = \$1{,}100 \). At factor 0.95, the premium is \( \$1{,}100 \times 0.95 = \$1{,}045.00 \). At factor 1.14, the premium is \( \$1{,}100 \times 1.14 = \$1{,}254.00 \). The dollar increase is \( \$1{,}254-\$1{,}045=\$209.00 \).
The premium ratio is \( \$1{,}254/\$1{,}045=1.20 \). The territory-factor ratio is \( 1.14/0.95=1.20 \). Therefore, the premium increases by \(1.20-1=0.20\), or 20%. Check: 20% of $1,045.00 is $209.00, exactly the calculated dollar increase. The equality of the ratios depends on the unchanged base rate, exposure, class factor, and other inputs.
Why the Multiplicative Formula Works
The formula follows from applying each rating adjustment to the amount produced by the preceding parts of the plan. Begin with a base rate \(B\) per exposure unit. For \(E\) units, the unadjusted premium is \(BE\). Applying territory multiplies that amount by \(T_j\), giving \(BET_j\). Applying a class factor \(C\) gives \(BET_jC\); applying further factors extends the product in the same way.
This derivation also explains why adding factor deviations is generally wrong. A factor of 1.10 multiplies the rate by 1.10; a second factor of 1.10 multiplies the resulting amount again. Together they produce \(1.10 \times 1.10 = 1.21\), not \(1+0.10+0.10=1.20\). The multiplicative result includes the effect of the second adjustment on the already adjusted amount.
Worked Example: Separating the Territory Rate From the Policy Premium
A hypothetical insurer’s rating plan uses a base rate of $850 per exposure unit. A policy has 3 units, territory factor 0.92, class factor 1.15, and an additional plan factor of 1.05. There are no fees, minimums, or caps. Find (1) the territory-adjusted base rate per unit before the other factors and (2) the full indicated premium. No interest, mortality, or probability model is involved.
Method. First calculate the rate after territory alone. Then apply exposure and the remaining factors to obtain the policy premium. This keeps the dimensionless territory relativity distinct from both dollar quantities.
The territory-adjusted base rate per unit is \( \$850 \times 0.92 = \$782.00 \) per unit. The full premium is \( \$782.00 \times 3 \times 1.15 \times 1.05 \). Step by step, \( \$782.00 \times 3 = \$2{,}346.00 \); \( \$2{,}346.00 \times 1.15 = \$2{,}697.90 \); and \( \$2{,}697.90 \times 1.05 = \$2{,}832.795 \). Rounded to the nearest cent only at the final step, the indicated premium is $2,832.80.
The factor 0.92 is not a premium of $0.92. The territory-adjusted rate is $782.00 per unit, and the full policy premium is $2,832.80 after exposure, class, and the additional factor are included. Check: multiplying the original inputs gives \(850 \times 3 \times 0.92 \times 1.15 \times 1.05=2{,}832.795\), which rounds to the same amount.
Territory Relativity and Territory-Specific Indicated Premium
A territory relativity describes one component of a rating plan. A territory-specific indicated premium is the dollar result for a particular risk after applying the territory factor together with the base rate, exposure, and all other relevant rating variables. In the simple plan above, it is \(P_j\). It cannot be determined from \(T_j\) alone.
The word “indicated” does not change the units: a factor remains dimensionless, while a premium is an amount of money. A territory-adjusted rate per exposure unit is also distinct from a full policy premium. For example, \(BT_j\) is a rate per unit before other factors; \(BET_jCD\) is a premium when the listed factors apply.
This distinction matters when reviewing rating outputs. A larger territory factor implies a larger premium only for risks that are otherwise alike under the same plan. It does not imply that every risk in that territory has a higher premium than every risk in another territory. Different exposures and rating characteristics can produce a different ordering of total premiums.
Common Mistakes and Exam Traps
- Reporting the factor as a dollar premium. A relativity such as 1.08 is a multiplier. Convert it to a dollar rate or premium by applying it to the appropriate base and other inputs.
- Comparing premium ratios when the risks differ. The ratio \(T_a/T_b\) equals the premium ratio only if the other multiplicative inputs are the same. If they differ, calculate the full ratio or compare the territory factors separately.
- Adding percentage adjustments instead of multiplying factors. Two successive 10% increases produce a factor of \(1.10 \times 1.10=1.21\), a 21% increase, not 20%.
- Ignoring plan provisions outside the product. A fixed fee or minimum premium can make the final billed amount fail to change in direct proportion to the territory factor. Follow the plan’s stated order and rules.
- Rounding intermediate values. Keep full precision through the calculation and round the final monetary result as required. Intermediate rounding can create avoidable differences.
Check Your Understanding
For each hypothetical rating-plan problem, calculate using the stated inputs and distinguish the factor from the resulting rate or premium.
- A plan has a base rate of $300 per unit, territory factor 1.08, class factor 0.95, and 2 units of exposure. With no other adjustments, what is the premium? (Answer: $615.60)
- Two otherwise identical risks have territory factors 0.90 and 1.08. What is the ratio of the second risk’s premium to the first, and what is the percentage increase? (Answer: 1.20; 20%)
- A $720 per-unit base rate applies to a risk with 1.5 units, territory factor 0.85, and class factor 1.20. Calculate the full premium, then state the territory-adjusted base rate per unit before class adjustment. (Answer: $1,101.60 premium; $612.00 per unit)
- Risk A has a $400 base rate, 1 unit, territory factor 1.10, and class factor 1.00. Risk B has the same base rate, 1.25 units, territory factor 1.00, and class factor 0.90. Calculate both premiums. Does their premium ratio equal their territory-factor ratio? (Answer: A is $440.00; B is $450.00; no, the premium ratio is \(450/440\approx1.0227\), while the territory-factor ratio is \(1.00/1.10\approx0.9091\))
- A policy’s pre-territory premium is $1,250. Its territory factor changes from 0.96 to 1.08, with all other inputs fixed. Find the old and new premiums and the dollar increase. (Answer: $1,200.00; $1,350.00; increase of $150.00)