From Rating Factors to a Cell Rate
A rating plan may classify a risk using several characteristics, such as territory and vehicle use. A rating cell is the combination of one selected level from each factor. To calculate the rate for that cell, use the base rate and the relativities assigned to those selected levels. The calculation is multiplicative: each selected relativity scales the rate produced by the other selections.
This approach is useful because each factor can be shown in a compact table while the resulting rate reflects all the selected characteristics together. The calculation is straightforward, but it is important to select exactly one applicable relativity from each factor and multiply them rather than add their percentage changes.
The base rate and all cell rates must be on the same coverage and exposure-unit basis. For example, if the base rate is an annual rate per vehicle-year, the calculated cell rate is also an annual rate per vehicle-year. The cell rate is not automatically the total premium for a policy with multiple exposure units.
Why the Relativities Multiply
A factor relativity represents a proportional adjustment. If the rate before applying factor \(A\) is \(B\), applying its relativity gives \(B r_A\). If factor \(C\) then applies to that resulting rate, its proportional adjustment gives \((B r_A)r_C\). By associativity of multiplication, this is \(B r_A r_C\).
This calculation assumes that the rating plan specifies separate multiplicative adjustments and that each selected factor level applies to the same rate basis. In particular, the formula has no additional interaction adjustment: the combined effect is fully represented by multiplying the two selected relativities. If the plan specifies a separate factor for a particular combination of levels, that term would also be needed; it is not part of the two-factor formula here.
The order of multiplication does not change the result. That does not mean the adjustments can be added. For example, a factor of 1.20 followed by a factor of 0.80 produces an overall multiplier of \(1.20 \times 0.80=0.96\), not a net change of 20% minus 20%, or zero.
Calculating a Two-Factor Table
A small table makes the selection process visible. Each cell in the body of the table corresponds to one combination of factor levels. To calculate a cell rate, identify its row and column, take the corresponding relativities, and multiply both by the base rate.
Worked Example: Completing a Territory and Use Table
A hypothetical insurer uses an annual base rate of $500 per vehicle-year. Its territory relativities are 1.20 for Territory A and 0.90 for Territory B. Its use relativities are 0.80 for personal use and 1.25 for business use. Calculate the annual rate for each of the four cells.
Method. Multiply the $500 base rate by one territory relativity and one use relativity for each cell. The common mistake to avoid is adding percentage changes instead of applying the two multipliers.
Territory A, personal use. Select 1.20 and 0.80: $$ 500 \times 1.20=600,\qquad 600 \times 0.80=\$480.00. $$
Territory A, business use. Select 1.20 and 1.25: $$ 500 \times 1.20=600,\qquad 600 \times 1.25=\$750.00. $$
Territory B, personal use. Select 0.90 and 0.80: $$ 500 \times 0.90=450,\qquad 450 \times 0.80=\$360.00. $$
Territory B, business use. Select 0.90 and 1.25: $$ 500 \times 0.90=450,\qquad 450 \times 1.25=\$562.50. $$
The completed table is:
| Territory | Personal use, relativity 0.80 | Business use, relativity 1.25 |
|---|---|---|
| A, relativity 1.20 | $480.00 | $750.00 |
| B, relativity 0.90 | $360.00 | $562.50 |
Check. Each displayed rate equals $500 times its row relativity times its column relativity. For example, the Territory B business rate is \(500 \times 0.90 \times 1.25=\$562.50\), matching the table.
Reading the Selections Carefully
For each cell, the selected levels—not all levels appearing in the table—determine the rate. A risk classified in one territory does not receive the relativities for other territories. Likewise, a risk with personal use does not also receive the business-use relativity. The cell identifies a single applicable level from each factor.
A useful calculation layout is to write the base rate and selected factors before doing arithmetic:
State the selected level of each factor in the risk description.
Use the entries corresponding to the selected levels only.
Keep the rate's stated coverage and exposure-unit basis.
Verify that every factor is included once and that the result has rate units, not a dimensionless multiplier.
Worked Example: Applying the Selected Levels to a New Base Rate
A hypothetical small-commercial program has a base rate of $640 per insured unit-year. The territory relativities are 1.15 for urban and 0.85 for rural. The operating-pattern relativities are 0.90 for occasional operation and 1.10 for regular operation. Calculate the rates for an urban risk with regular operation and a rural risk with occasional operation.
Method. For each risk, select its territory factor and operating-pattern factor, then multiply both by the base rate. This avoids the mistake of using an unselected factor merely because it appears in the same table.
Urban, regular operation. The selected relativities are 1.15 and 1.10: $$ 640 \times 1.15=736, $$ $$ 736 \times 1.10=\$809.60. $$ The cell rate is $809.60 per insured unit-year.
Rural, occasional operation. The selected relativities are 0.85 and 0.90: $$ 640 \times 0.85=544, $$ $$ 544 \times 0.90=\$489.60. $$ The cell rate is $489.60 per insured unit-year.
Check. The first cell's combined multiplier is \(1.15 \times 1.10=1.265\), so \(640 \times 1.265=\$809.60\). The second cell's combined multiplier is \(0.85 \times 0.90=0.765\), so \(640 \times 0.765=\$489.60\). Both checks agree with the stepwise calculations.
Cell Rate Versus Policy Premium
The product of the base rate and the relativities gives a rate on the base rate's exposure-unit basis. If the policy has a specified number of exposure units and the pricing instruction is to multiply the rate by those units, that calculation follows the cell-rate calculation. Keep the two quantities distinct: a rate is stated per exposure unit, while the resulting premium is an amount for the stated exposure.
Worked Example: Converting a Cell Rate to a Premium
A hypothetical equipment policy has a base rate of $875 per equipment-year. The region relativity is 0.92 for inland risks and 1.08 for coastal risks. The use relativity is 1.30 for delivery use and 0.75 for recreational use. An inland delivery risk has 1.5 equipment-years of exposure. Calculate its cell rate and the premium obtained by multiplying that rate by the stated exposure.
Method. First calculate the cell rate using the inland and delivery relativities. Then multiply that rate by 1.5 equipment-years to obtain the premium. The common mistake to avoid is calling the cell rate itself the premium, or multiplying exposure into the relativities before identifying the cell.
The inland delivery cell rate is: $$ 875 \times 0.92=805, $$ $$ 805 \times 1.30=\$1{,}046.50\text{ per equipment-year}. $$ Multiplying by the exposure gives: $$ \$1{,}046.50 \times 1.5=\$1{,}569.75. $$ Thus the indicated amount under the stated calculation is a $1,046.50 rate per equipment-year and a $1,569.75 premium for 1.5 equipment-years.
Check. The combined relativity is \(0.92 \times 1.30=1.196\). Then \(875 \times 1.196=\$1{,}046.50\), and \(1{,}046.50 \times 1.5=\$1{,}569.75\). The units are consistent: dollars per equipment-year times equipment-years gives dollars.
Common Exam Trap: Adding Percentage Changes
Suppose two selected factors have relativities 1.20 and 0.80. It is tempting to describe the first as a 20% increase and the second as a 20% decrease, then conclude that they cancel. That calculation is wrong because the decrease applies proportionally to the rate after the increase.
For a $500 base rate, the incorrect additive calculation gives $500. The correct sequential calculation is \(500 \times 1.20=600\), followed by \(600 \times 0.80=\$480\). The second factor reduces $600 by 20%, which is $120, not the original $500 multiplied by 20%.
Check Your Understanding
For each problem, calculate the requested cell rate from the given base rate and selected relativities.
- A hypothetical service plan has a base rate of $400 per unit-year. The terrain relativity is 1.10 and the operating relativity is 0.90 for the selected cell. What is its rate? (Answer: $396.00 per unit-year)
- A hypothetical fleet has a base rate of $600 per vehicle-year. North and south territory relativities are 1.20 and 0.85; local and long-haul use relativities are 0.90 and 1.15. Calculate the north-local and south-long-haul rates. (Answer: $648.00 and $586.50 per vehicle-year, respectively)
- A hypothetical property program has a base rate of $750 per insured location-year. The selected region relativity is 1.08 and the selected construction relativity is 0.80. Calculate the cell rate. (Answer: $648.00 per location-year)
- A hypothetical equipment policy has a base rate of $920 per equipment-year. For one cell, the selected location relativity is 1.25 and the selected use relativity is 0.88. Calculate the cell rate. (Answer: $1,012.00 per equipment-year)
- A hypothetical service contract has a base rate of $540 per contract-year. Area A and B relativities are 1.10 and 0.90; standard and premium service relativities are 0.80 and 1.25. Calculate the Area A premium-service and Area B standard-service rates. (Answer: $742.50 and $388.80 per contract-year, respectively)