Balancing Across Multiple Rating Variables

Why Multiple Rating Variables Matter

A territory factor is applied alongside other rating variables, such as class or vehicle use. The premium associated with a territory therefore depends not only on its number of exposures, but also on the base premiums of the classes and other groups within it. To balance territory relativities for the whole rating plan, the weights must reflect those premiums.

This tutorial focuses on applying territory indications across a cross-classified rating plan while holding the other factors fixed. It does not estimate separate territory and class effects. Its central question is narrower: given a set of territory indications and the premium weights of all the cells, what common adjustment makes the new territory factors neutral to aggregate premium?

Definition: A cross-classified cell is a group of exposures defined by a combination of rating variables, such as territory and class. For cell $c$ in territory $t$, let $B_{tc}$ be its base premium before applying the territory factor. It includes the exposure count and all applicable premium components and non-territory factors. The territory's base-premium weight is $W_t=\sum_c B_{tc}$, and its total base premium is $B=\sum_t W_t$.

If the raw indication for territory $t$ is $r_t$, applying that indication to each cell in the territory produces an indicated premium of $r_t B_{tc}$. The raw indication is a multiplier, not a dollar amount. A single balancing multiplier can adjust all the raw territory indications without changing their relative proportions.

The Premium-Neutral Balancing Formula

Let $f_t$ denote the balanced territory factor. To preserve the aggregate premium at the base level, the premium after applying territory factors must equal the total base premium. Thus the balancing target is $\sum_t W_t f_t=B$. If each factor is the corresponding raw indication times a common multiplier $\lambda$, then:

f_t=\lambda r_t \qquad\text{and}\qquad \lambda=\frac{\sum_t W_t}{\sum_t W_t r_t}.

The numerator is the premium before territory factors; the denominator is the aggregate premium indicated by the raw territory factors. The formula assumes that the weights are the base premiums for the same exposures being rated, that the non-territory factors remain fixed, and that the raw factors are applied uniformly to all cells within each territory. The denominator must be positive. If the rating plan has a different stated premium target, use that target in place of $B$.

Why the formula works: After balancing, the indicated premium is $\sum_t W_t f_t=\sum_t W_t(\lambda r_t)=\lambda\sum_t W_t r_t$. Setting this equal to the target $B=\sum_t W_t$ and dividing by the positive indicated premium gives $\lambda=B/\sum_t W_t r_t$. Therefore the new factors preserve aggregate premium by construction.

This is a portfolio-level constraint. It does not require each class, or each territory-by-class cell, to retain its previous premium. The cross-classified cells determine the territory weights, but the balancing target is the stated aggregate total.

Worked Example: Balancing Two Territories Across Two Classes

Method: Sum cell base premiums within each territory, then apply one premium-neutral multiplier to the raw territory indications. All amounts are hypothetical.

A rating plan has two classes. Before territory factors, Class A has a premium of $100 per exposure and Class B has a premium of $200 per exposure. Territory 1 has 600 Class A exposures and 200 Class B exposures. Territory 2 has 1,200 Class A exposures and 400 Class B exposures. The raw territory indications are 0.80 and 1.20, respectively. Find balanced factors that preserve the aggregate base premium.

Step 1: Calculate the cell base premiums. Territory 1 has $600\times \$100=\$60{,}000$ in Class A and $200\times \$200=\$40{,}000$ in Class B. Its weight is $W_1=\$60{,}000+\$40{,}000=\$100{,}000$. Territory 2 has $1{,}200\times \$100=\$120{,}000$ in Class A and $400\times \$200=\$80{,}000$ in Class B. Its weight is $W_2=\$120{,}000+\$80{,}000=\$200{,}000$.

Step 2: Find the base and raw indicated totals. The base total is $\$100{,}000+\$200{,}000=\$300{,}000$. The raw indicated total is $\$100{,}000(0.80)+\$200{,}000(1.20)=\$80{,}000+\$240{,}000=\$320{,}000$.

Step 3: Normalize the indications. The multiplier is $\lambda=\$300{,}000/\$320{,}000=0.9375$. Thus $f_1=0.80(0.9375)=0.7500$, and $f_2=1.20(0.9375)=1.1250$.

Step 4: Check the result. The adjusted total is $\$100{,}000(0.7500)+\$200{,}000(1.1250)=\$75{,}000+\$225{,}000=\$300{,}000$, which equals the base total. The same result follows cell by cell: $60{,}000(0.75)+40{,}000(0.75)+120{,}000(1.125)+80{,}000(1.125)=45{,}000+30{,}000+135{,}000+90{,}000=\$300{,}000$.

Common mistake avoided: Applying the raw factors without normalization would produce $320,000, not $300,000. The balanced factors are 0.7500 and 1.1250.

Choosing Weights in the Rating Plan

The appropriate weight depends on the balancing target. When the target is premium neutrality under the existing non-territory rating structure, use each territory's base premium, not simply its exposure count. If exposures have different base premiums because their classes or other rating variables differ, equal exposure weights do not represent equal shares of the premium being balanced.

Quantity	What it measures	Use in premium-neutral balancing?
Exposure count	Number of insured units in a territory	Not by itself, unless all exposures have the same base premium
Base premium by cell	Exposure count multiplied by the applicable premium before the territory factor	Yes; sum across cells to obtain $W_t$
Raw indicated premium	Base-premium weight multiplied by the raw territory indication	Use in the denominator of the balancing multiplier

Equivalently, $W_t$ can be computed as the territory's exposure count times its average base premium per exposure, provided that average correctly reflects the mix of cells. Summing cell-level base premiums is often safer because it makes the contribution of each class or other rating group explicit.

Worked Example: Why Exposure Weights Can Fail

Method: Compare exposure weighting with base-premium weighting for a plan with different average premiums by territory. The target is unchanged aggregate premium.

A hypothetical portfolio has two territories. Territory 1 has 2,000 exposures with an average base premium of $50, so its base-premium weight is $2{,}000\times \$50=\$100{,}000$. Territory 2 has 1,000 exposures with an average base premium of $300, so its weight is $1{,}000\times \$300=\$300{,}000$. The raw indications are 0.90 and 1.10.

Step 1: Calculate the correct premium weights. The base total is $\$100{,}000+\$300{,}000=\$400{,}000$. The raw indicated total is $\$100{,}000(0.90)+\$300{,}000(1.10)=\$90{,}000+\$330{,}000=\$420{,}000$.

Step 2: Find balanced factors using premium weights. The multiplier is $\lambda=\$400{,}000/\$420{,}000=20/21$. The factors are $0.90(20/21)=18/21=6/7$, or approximately 0.8571, and $1.10(20/21)=22/21$, or approximately 1.0476.

Step 3: Verify the target. The adjusted premium is $\$100{,}000(6/7)+\$300{,}000(22/21)=\$85{,}714.29+\$314{,}285.71=\$400{,}000.00$.

Common mistake avoided: Equal exposure weighting gives a raw average of $(2{,}000(0.90)+1{,}000(1.10))/3{,}000=2{,}900/3{,}000=0.9667$. Normalizing by this average gives factors $0.90/(29/30)=27/29$ and $1.10/(29/30)=33/29$. But their premium-weighted total is $\$100{,}000(27/29)+\$300{,}000(33/29)=\$12{,}600{,}000/29$, or approximately $434,482.76. That is not the $400,000 target. Exposure weighting would be appropriate only for an exposure-weighted target, or when base premiums are identical across exposures.

Applying One Territory Factor Across the Cells

Once a territory factor is balanced, apply it to every cell in that territory while leaving the other factors unchanged. The total premium is found by adding the adjusted cell premiums. This cell-level check is useful even when the factor formula is correct: it can reveal an omitted class, an incorrect weight, or inconsistent treatment of a factor.

Worked Example: Balancing Three Territories Across the Same Classes

Method: Aggregate the base premiums across classes within each territory, normalize all three raw indications with one multiplier, and verify the result from the cell totals. All amounts are hypothetical.

A two-class rating plan has these base-premium cells, in thousands of dollars: Territory 1 has $70,000 in Class A and $50,000 in Class B; Territory 2 has $60,000 in Class A and $120,000 in Class B; Territory 3 has $40,000 in Class A and $60,000 in Class B. The raw territory indications are 1.10, 0.90, and 1.30. Find the premium-neutral factors.

Step 1: Sum each territory's cells. The weights are $W_1=\$70{,}000+\$50{,}000=\$120{,}000$, $W_2=\$60{,}000+\$120{,}000=\$180{,}000$, and $W_3=\$40{,}000+\$60{,}000=\$100{,}000$. The base total is $\$120{,}000+\$180{,}000+\$100{,}000=\$400{,}000$.

Step 2: Calculate the raw indicated total. It is $\$120{,}000(1.10)+\$180{,}000(0.90)+\$100{,}000(1.30)=\$132{,}000+\$162{,}000+\$130{,}000=\$424{,}000$.

Step 3: Calculate the multiplier and factors. The multiplier is $\lambda=\$400{,}000/\$424{,}000=50/53$, approximately 0.9434. The balanced factors are $1.10(50/53)=55/53\approx1.0377$, $0.90(50/53)=45/53\approx0.8491$, and $1.30(50/53)=65/53\approx1.2264$. Factors are shown to four decimal places.

Step 4: Verify using the territory weights. The adjusted total is $\$120{,}000(55/53)+\$180{,}000(45/53)+\$100{,}000(65/53)$. In thousands of dollars, the numerator is $120(55)+180(45)+100(65)=6{,}600+8{,}100+6{,}500=21{,}200$; $21{,}200/53=400$. The adjusted total is therefore $400,000.

The same check can be made by summing the adjusted cells: the Class A total is $[70(55)+60(45)+40(65)]/53=9{,}150/53$ thousand dollars, and the Class B total is $[50(55)+120(45)+60(65)]/53=12{,}050/53$ thousand dollars. Their sum is $21{,}200/53=400$ thousand dollars.

Common mistake avoided: Normalizing each territory independently within each class would use multiple balancing multipliers and would no longer apply one balanced territory schedule to the plan. For the stated aggregate premium target, use one multiplier across the territory factors and check the combined cell total.

Interpretation and Exam Traps

A balanced set of territory factors preserves the specified total premium; it does not make each factor equal to 1. Nor does it imply that each territory's premium remains unchanged. Factors above or below 1 redistribute premium across territories, while the common multiplier controls the aggregate.

Using the wrong denominator: The denominator is the aggregate premium indicated by the raw factors, $\sum_t W_t r_t$, not the exposure count or the sum of the factors.
Mixing weight definitions: If the target is premium neutrality, use base-premium weights calculated before the territory factor. Do not substitute exposure weights just because the exposure counts are readily available.
Omitting a rating variable: If a territory contains multiple classes, include the base premium from every applicable class when calculating its weight.
Changing the weights mid-calculation: Use the same base-premium definition for the base total and the indicated total. A factor already included in the weights should not be applied a second time.
Rounding too early: Keep the multiplier and factors at full precision while checking the total; round only the displayed final factors.

Key takeaway: For a premium-neutral territory revision across multiple rating variables, sum the pre-territory base premiums across the plan, calculate one multiplier as base total divided by raw indicated total, and apply that multiplier to every raw territory indication. The result is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical plan, find the balanced territory factors for the stated aggregate premium target. Treat all stated cell premiums as base premiums before territory factors.

Two territories have base-premium weights of $90,000 and $210,000, with raw indications of 1.20 and 0.90. Find the balanced factors to four decimal places. (Answer: Multiplier $100/99$; factors 1.2121 and 0.9091.)
Territory A has two cell base premiums of $30,000 and $70,000; Territory B has cell base premiums of $80,000 and $120,000. Their raw indications are 0.85 and 1.15. Find the common multiplier and balanced factors. (Answer: Multiplier $20/21$; factors 0.8095 and 1.0952.)
A portfolio has 1,000 exposures in Territory 1 at an average base premium of $60 and 2,000 exposures in Territory 2 at an average base premium of $90. The raw indications are 1.05 and 0.95. Find the premium-neutral factors to four decimal places. (Answer: Multiplier $40/39$; factors 1.0769 and 0.9744.)
Three territories have base-premium weights of $50,000, $100,000, and $150,000, with raw indications 0.80, 1.10, and 1.20. Find the balancing multiplier and the three factors. (Answer: Multiplier $10/11$; factors 0.7273, 1.0000, and 1.0909.)
A plan has two classes. Their base premiums by territory are: Territory X, $40,000 and $60,000; Territory Y, $120,000 and $80,000. Raw indications are 1.25 for X and 0.90 for Y. Find premium-neutral territory factors and verify the aggregate target. (Answer: Base weights $100,000 and $200,000; multiplier $8/9$; factors 1.1111 and 0.8000; adjusted total $300,000.)

What You'll Learn