What a Relativity Entry Says
A rating table is easier to read once you identify its reference level. The entry for another level tells you how its indicated rate compares with the rate for that reference level, with the other rating characteristics held fixed. It is a ratio, not a dollar amount and not, by itself, a percentage change.
For example, an entry of 1.20 means the level’s indicated rate is 1.20 times the reference-level rate: it is 20% higher. An entry of 0.85 means the rate is 0.85 times the reference-level rate: it is 15% lower. An entry of 1.00 means the rate is the same as the reference-level rate for that factor.
The reference level gives the comparison point. In the usual presentation, its entry is 1.00 by construction. That does not mean the reference level has zero risk, zero premium, or a rate of one dollar. It means only that its rate is being used as the denominator for comparisons within that factor.
| Factor-table entry | Interpretation relative to reference | Percentage change from reference |
|---|---|---|
| Greater than 1.00 | Indicated rate is higher | \((r-1)\times 100\%\) higher |
| Equal to 1.00 | Indicated rate is equal | 0% change |
| Between 0 and 1.00 | Indicated rate is lower | \((1-r)\times 100\%\) lower |
These interpretations assume the rating plan applies the factor multiplicatively and that the comparison holds other characteristics fixed. A factor below 1.00 is still positive; it indicates a reduction relative to the reference level, not a negative rate. An entry exactly equal to 1.00 indicates no adjustment from this factor’s reference level.
Why the Entry Is a Ratio
The interpretation follows directly from the multiplicative rating structure. Let \(B\) denote the base rate, let \(r_j\) be the entry for level \(j\) of a factor, and let \(C\) denote the product of any other factors that are unchanged in the comparison. The indicated rate for level \(j\) is \(B C r_j\). For the reference level, whose entry is 1.00, it is \(B C\). Thus:
The base rate and all unchanged factors cancel. That is why the table entry itself is the ratio to the reference, provided the rating plan is multiplicative and the comparison changes only this factor’s level. If a plan includes interactions or other adjustments that change between the two risks, the simple one-factor comparison need not apply.
Worked Interpretations
A useful reading sequence is to locate the reference level, read the factor entry, and compare that entry with 1.00. If a dollar comparison is requested, multiply the reference-level rate by the entry. When comparing two non-reference levels, divide their entries in the order required by the question.
Worked Example: Converting Entries to Changes From the Reference
A hypothetical insurer’s rating table for vehicle use has a reference level of Personal use with an entry of 1.00. The entries are 0.84 for Pleasure-only use and 1.32 for Local delivery use. For otherwise identical risks, the indicated rate for Personal use is $600. Interpret both other entries and calculate their indicated rates.
Method. Treat each entry as a multiplier of the reference-level rate. The percentage change from the reference is the factor entry minus 1.00, expressed as a percentage.
Pleasure-only use. The entry is 0.84, so the rate is 0.84 times the Personal-use rate:
The percentage change is \(0.84-1.00=-0.16\), or a 16% decrease. Thus the indicated rate is $504.00, which is $96.00 below $600.00; \(96/600=0.16\), confirming the 16% decrease.
Local delivery use. The entry is 1.32, so:
The percentage change is \(1.32-1.00=0.32\), or a 32% increase. The dollar difference is \(\$792-\$600=\$192\), and \(\$192/\$600=0.32\), confirming the increase.
Common mistake avoided: The entry 0.84 does not mean an 84% decrease. It means 84% of the reference rate remains, so the decrease is \(1.00-0.84=0.16\), or 16%. The entry 1.32 means 132% of the reference rate, which is 32% higher.
Check. Dividing each calculated rate by $600 gives \(504/600=0.84\) and \(792/600=1.32\), the original table entries.
Worked Example: Comparing Two Non-Reference Levels
A hypothetical property rating table uses Zone A as its reference level. The entries are 1.00 for Zone A, 1.12 for Zone B, and 0.96 for Zone C. For an otherwise identical property, the Zone A indicated rate is $875. Find the Zone B and Zone C indicated rates, then express Zone B’s rate as a percentage higher than Zone C’s.
Method. First apply each entry to the reference rate. For the final comparison, divide the Zone B rate by the Zone C rate, because the question asks how much higher B is relative to C. The denominator must therefore be C’s rate.
Zone B.
Zone B is 12% above the reference because \(1.12-1.00=0.12\). The dollar increase is \(\$980-\$875=\$105\), and \(\$105/\$875=0.12\).
Zone C.
Zone C is 4% below the reference because \(1.00-0.96=0.04\). The dollar decrease is \(\$875-\$840=\$35\), and \(\$35/\$875=0.04\).
Compare B with C. The ratio is \(980/840=1.166666\ldots\). Therefore Zone B is \(1.166666\ldots-1=0.166666\ldots\), or approximately 16.67%, higher than Zone C.
Common mistake avoided: The difference between the table entries is \(1.12-0.96=0.16\), but that is not a 16% increase relative to Zone C. The relative increase uses Zone C as the denominator: \((1.12-0.96)/0.96=0.166666\ldots\), or 16.67%.
Check. The dollar difference is \(\$980-\$840=\$140\), and \(\$140/\$840=0.166666\ldots\), agreeing with the ratio calculation.
Worked Example: Reading Entries From Both Sides of 1.00
A hypothetical insurer’s annual-mileage factor table has these entries: 1.00 for the reference mileage band, 0.90 for a lower-mileage band, and 1.25 for a higher-mileage band. The reference-level indicated rate is $720. State each band’s percentage change from the reference, calculate its rate, and compare the higher-mileage rate with the lower-mileage rate.
Method. Use 1.00 as the reference multiplier. For the comparison between the two non-reference bands, divide the higher-mileage entry by the lower-mileage entry.
Lower-mileage band. Its entry is 0.90, which is \(1.00-0.90=0.10\), or 10%, below the reference. Its rate is:
Higher-mileage band. Its entry is 1.25, which is \(1.25-1.00=0.25\), or 25%, above the reference. Its rate is:
Compare the two bands. The ratio is \(1.25/0.90=1.388888\ldots\). So the higher-mileage rate is approximately 38.89% above the lower-mileage rate. As a dollar check, \(\$900-\$648=\$252\), and \(\$252/\$648=0.388888\ldots\).
Common mistake avoided: The difference \(1.25-0.90=0.35\) is a difference in factor values, not the percentage increase from the lower-mileage level. To calculate that increase, use the lower level as the denominator: \(0.35/0.90=0.388888\ldots\), or 38.89%.
Check. The two calculated rates divided by $720 return their table entries: \(648/720=0.90\) and \(900/720=1.25\).
Reading Tables Without Misstating the Result
An entry above 1.00 implies an increase only relative to the reference level for that factor. It does not say that every risk at that level has the same final premium, because other characteristics can also affect the indicated rate. Likewise, an entry below 1.00 indicates a decrease relative to the reference, not necessarily a decrease relative to every other level in the table.
When a question asks for a percentage change, state the comparison being made. For a level compared with the reference, use 1.00 as the denominator through the difference from 1.00. For two non-reference levels, use the level named after “relative to” or “compared with” as the denominator. The direction matters: a percentage increase from a smaller value and a percentage decrease from a larger value are not generally equal.
- Entry above 1.00: higher than the reference by \((r-1)\times100\%\).
- Entry below 1.00: lower than the reference by \((1-r)\times100\%\).
- Entry equal to 1.00: the same indicated rate as the reference for this factor.
- Two non-reference entries: divide the first level’s entry by the comparison level’s entry to obtain their rate ratio.
Keep the scope of the interpretation precise: factor entries describe modeled rate relationships in the rating plan. They are not statements that one individual risk has a known, certain loss outcome. This distinction helps separate the table’s indicated relationship from the uncertainty inherent in insurance losses.
Check Your Understanding
For each item, identify the reference comparison and show how the entry translates into a rate relationship.
- A hypothetical rating table has a reference entry of 1.00 and another level’s entry is 0.93. What percentage change does the other level indicate relative to the reference? (Answer: 7% lower.)
- A reference-level indicated rate is $640 and a level has entry 1.18. Calculate the level’s indicated rate and percentage change from the reference. (Answer: $755.20, or 18% higher.)
- Two non-reference levels have entries 0.82 and 1.05. How much lower is the 0.82 level than the 1.05 level, as a percentage of the 1.05 level? (Answer: \((1.05-0.82)/1.05=0.219047\ldots\), or approximately 21.90% lower.)
- A table shows 1.00 for its reference level and 1.00 for a second level. What does the second entry imply about the rate comparison for this factor, and what does it not imply about the overall premium? (Answer: The factor makes no rate difference between these levels; it does not imply that the overall premium is $1 or that other factors have no effect.)
- Two levels have entries 1.15 and 0.92. How much higher is the 1.15 level relative to the 0.92 level? (Answer: \(1.15/0.92=1.25\), so it is 25% higher.)