Rate Impacts of Territory Factor Changes

Why Measure the Premium Impact?

A territory factor revision changes the premium charged to risks assigned to the affected territories, assuming the other rating components remain unchanged. Before implementation, an actuary needs to quantify both the change for a representative risk and the aggregate effect for the portfolio. These are related but different calculations: a large percentage change in a small territory can have less effect on total premium than a modest change in a territory with substantial existing premium.

The calculation is a comparison of premiums under two rating plans for the same risks and exposures. It is not, by itself, a new indication of the appropriate territory factors. The factors are treated as given; the task is to measure their premium effect.

Definition: Let $B_j$ be the premium base for risk or portfolio segment $j$, excluding the territory factor but including the exposure and all other rating components held fixed. Let $f_{j,\mathrm{old}}$ and $f_{j,\mathrm{new}}$ be the old and revised territory factors. Then $P_{j,\mathrm{old}}=B_jf_{j,\mathrm{old}}$ and $P_{j,\mathrm{new}}=B_jf_{j,\mathrm{new}}$. When the old premium is known and the old factor is positive, $P_{j,\mathrm{new}}=P_{j,\mathrm{old}}(f_{j,\mathrm{new}}/f_{j,\mathrm{old}})$.

The premium base can be a policy’s premium before the territory factor, or the sum of such premiums for a group. For example, it may include limits, coverage, class, and other rating elements. It must exclude the territory factor being changed. If a territory factor is applied to only part of a policy’s premium, only that affected part belongs in the calculation; do not apply the territory change to unrelated premium.

Individual-Risk Impact

For one risk, the percentage change is determined by the ratio of the revised factor to the old factor. The risk’s base premium cancels when comparing the two premiums:

\frac{P_{\mathrm{new}}-P_{\mathrm{old}}}{P_{\mathrm{old}}} = \frac{Bf_{\mathrm{new}}-Bf_{\mathrm{old}}}{Bf_{\mathrm{old}}} = \frac{f_{\mathrm{new}}}{f_{\mathrm{old}}}-1.

This relationship assumes a positive old factor and an unchanged base premium. It means, for example, that moving a factor from 0.80 to 0.88 raises the premium by 10%, not by 8%. The factor’s absolute difference is not the percentage premium change; the factor ratio is.

Worked Example: Revised Premium for One Risk

A hypothetical small-equipment policy has a premium base of $1,250 before its territory factor. All other rating components remain fixed. The old territory factor is 0.80 and the revised factor is 0.92. Calculate the old and revised premiums and the percentage change.

Method. Apply the old and new territory factors to the same base. The common base makes the percentage change equal to the factor ratio minus one.

Old premium. $P_{\mathrm{old}}=\$1{,}250(0.80)=\$1{,}000$.

Revised premium. $P_{\mathrm{new}}=\$1{,}250(0.92)=\$1{,}150$.

Dollar and percentage changes. The dollar change is $\$1{,}150-\$1{,}000=\$150$. The percentage change is $(0.92/0.80)-1=1.15-1=0.15$, or 15.00%.

Check. Fifteen percent of the old $1,000 premium is $150, matching the calculated dollar increase. This method avoids treating the factor difference of 0.12 as a 12% premium increase.

Aggregate Impact for a Fixed Portfolio

For a portfolio divided into segments, calculate the old and revised premium in each segment, then sum. The segments may be territories, or groups of risks within territories if their bases or factor changes differ. The old premium is a natural weight for the factor ratio because it already reflects both the size of the segment’s premium base and its old territory factor.

P_{\mathrm{old}}=\sum_j P_{j,\mathrm{old}}, \qquad P_{\mathrm{new}}=\sum_j P_{j,\mathrm{old}} \left(\frac{f_{j,\mathrm{new}}}{f_{j,\mathrm{old}}}\right).

The portfolio percentage change is $(P_{\mathrm{new}}-P_{\mathrm{old}})/P_{\mathrm{old}}$. It is generally not the simple average of territory percentage changes. A simple average gives every territory the same weight regardless of its premium volume; the portfolio calculation weights each change by that territory’s share of old premium.

Derivation: For segment $j$, the fixed-base assumption gives $P_{j,\mathrm{old}}=B_jf_{j,\mathrm{old}}$. Solving for the base gives $B_j=P_{j,\mathrm{old}}/f_{j,\mathrm{old}}$. The revised premium is therefore $B_jf_{j,\mathrm{new}}=P_{j,\mathrm{old}}(f_{j,\mathrm{new}}/f_{j,\mathrm{old}})$. Summing this identity over the unchanged portfolio gives the aggregate formula above. Dividing the aggregate premium change by $P_{\mathrm{old}}$ shows that the portfolio percentage change is the old-premium-weighted average of the segment percentage changes.

Worked Example: Aggregate Change Using Existing Premiums

A hypothetical portfolio has $150,000 of old premium in Territory A and $250,000 in Territory B. Territory A’s factor changes from 1.00 to 1.04. Territory B’s factor changes from 1.25 to 1.20. Assume the policies, exposures, and all non-territory rating components stay fixed. Find the revised total premium and the aggregate percentage change.

Method. Since the amounts provided are old premiums, multiply each by its new-to-old factor ratio, not by the new factor alone.

Territory A. Its ratio is $1.04/1.00=1.04$. Its revised premium is $\$150{,}000(1.04)=\$156{,}000$, an increase of $6,000.

Territory B. Its ratio is $1.20/1.25=0.96$. Its revised premium is $\$250{,}000(0.96)=\$240{,}000$, a decrease of $10,000.

Aggregate. The old total is $\$150{,}000+\$250{,}000=\$400{,}000$. The revised total is $\$156{,}000+\$240{,}000=\$396{,}000$. Thus the decrease is $4,000, or $\$4{,}000/\$400{,}000=1.00\%$.

Check. The segment changes, +$6,000 and −$10,000, sum to −$4,000. The factor ratios avoid applying a new factor to a premium that already reflects the old factor.

When Premium Bases, Rather Than Old Premiums, Are Given

Sometimes the data are stated as premium bases excluding territory factors. In that case, apply each old factor to its base to obtain old premium, and each new factor to the same base to obtain revised premium. Do not use the old-premium formula until the old premium has been correctly calculated.

Worked Example: Aggregate Impact From Territory-Free Bases

A hypothetical portfolio has a territory-free premium base of $20,000 in Territory X and $27,000 in Territory Y. X’s factor changes from 0.90 to 1.00; Y’s changes from 1.20 to 1.10. Calculate old and revised aggregate premiums and the percentage change, rounding the percentage to two decimal places.

Method. Apply both factors to each territory’s unchanged base, then compare the portfolio totals.

Territory X. Old premium is $\$20{,}000(0.90)=\$18{,}000$. Revised premium is $\$20{,}000(1.00)=\$20{,}000$.

Territory Y. Old premium is $\$27{,}000(1.20)=\$32{,}400$. Revised premium is $\$27{,}000(1.10)=\$29{,}700$.

Totals and change. The old total is $\$18{,}000+\$32{,}400=\$50{,}400$. The revised total is $\$20{,}000+\$29{,}700=\$49{,}700$. The change is a decrease of $\$50{,}400-\$49{,}700=\$700$. As a percentage of old premium, the decrease is $\$700/\$50{,}400=0.013888\ldots$, or 1.39% to two decimal places.

Check. X increases by $2,000 and Y decreases by $2,700, for a net decrease of $700. This base-based calculation avoids using the wrong denominator or applying the new factor to the old premium as though the old factor were absent.

Common Mistakes and Practical Conditions

A frequent error is to multiply existing premium by the new factor rather than by the new-to-old factor ratio. In the preceding two-territory example, the incorrect calculation for Territory B would be $\$250{,}000(1.20)=\$300{,}000$. That is wrong because the $250,000 already includes the old factor of 1.25. The correct calculation is $\$250{,}000(1.20/1.25)=\$240{,}000$.

Another error is to take an unweighted mean of territory percentage changes. In the $400,000 portfolio, Territory A’s change is +4% and Territory B’s is −4%. Their simple average is 0%, but that is not the portfolio result: the old-premium weights are 37.5% for A and 62.5% for B, producing an aggregate decrease of 1%.

The fixed-portfolio calculation is appropriate when the question asks for a rate change’s mechanical premium effect on the same risks and exposures. It assumes unchanged exposure counts, territory assignments, policy characteristics, and other rating factors. If a stated policy has a premium component not subject to territory rating, isolate the affected component before applying the factor ratio. If the portfolio distribution or exposures are expected to change, the result is a scenario estimate under the stated distribution, not a prediction that those changes will not occur.

Key takeaway: For one risk, multiply its territory-free base by the new factor. For an existing-premium segment, multiply by the new-to-old factor ratio. For a portfolio, sum the segment premiums first and measure the dollar and percentage changes against the old total.

Check Your Understanding

For each hypothetical case, assume all rating components other than the stated territory factor remain unchanged.

A policy has a territory-free base of $900. Its factor changes from 0.75 to 0.84. What are its old and revised premiums, and what is the percentage change? (Answer: Old $675; revised $756; increase $81, or 12.00%.)
Territory A has $80,000 of old premium and a factor change from 1.10 to 1.21. Territory B has $120,000 of old premium and a change from 0.95 to 0.855. Find the revised portfolio premium and percentage change. (Answer: Revised $196,000; decrease $4,000, or 2.00%.)
Territory C has a base of $40,000 and factors changing from 0.80 to 0.90. Territory D has a base of $60,000 and factors changing from 1.20 to 1.10. Find old and revised totals and the percentage change. (Answer: Old $104,000; revised $102,000; decrease $2,000, or 1.92%.)
A portfolio has $200,000 old premium, of which $50,000 is in a territory whose factor increases by 8%. All remaining premium is unaffected. Find the revised total and percentage change. (Answer: Revised $204,000; increase $4,000, or 2.00%.)
Territory E has a $30,000 base and factors changing from 0.90 to 0.99. Territory F has a $70,000 base and factors changing from 1.20 to 1.14. Find the aggregate percentage change, rounded to two decimal places. (Answer: Old $111,000; revised $109,500; decrease $1,500, or 1.35%.)

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