Rescaling Relativities Without Changing Rates

Why Rescaling Is Possible

A multiplicative rating plan calculates an indicated rate by starting with a base rate and multiplying by the relativities for the risk’s factor levels. This makes the base rate and the relativities work together: changing one component can be offset by changing another. Consequently, two different-looking rating tables can produce exactly the same indicated rate for every risk.

This equivalence matters when interpreting and checking a rating plan. A change to a table’s numbers does not necessarily mean the rates have changed. The key is to check whether every relativity for one factor has been multiplied by the same positive constant and whether the base rate has been adjusted in the opposite direction.

Definition: For a rating plan with base rate $B$ and $K$ rating factors, the indicated rate for a risk with factor levels $j_1,\ldots,j_K$ is $$ R(j_1,\ldots,j_K)=B\prod_{k=1}^{K} r_{k,j_k}, $$ where $r_{k,j_k}$ is the relativity for the risk’s level $j_k$ of factor $k$. In the rescaling considered here, one factor’s relativities are all multiplied by the same constant $c>0$, and the base rate is divided by $c$.

This tutorial concerns deterministic rating calculations. The base rate and relativities are assumed to be positive, as is $c$. No interest rate, mortality assumption, or probability distribution is needed: the result is an algebraic property of the rating formula, not a claim about a loss distribution.

The Rescaling Result

Suppose factor $q$ has relativity $r_{q,j}$ at level $j$. Define a new table by multiplying every relativity for that factor by the same positive constant $c$:

r'_{q,j}=c\,r_{q,j}\quad\text{for every level }j, \qquad B'=\frac{B}{c}.

All relativities for the other factors are left unchanged. For a risk at level $j_q$ of factor $q$, the new indicated rate is

R'=B'\,r'_{q,j_q}\prod_{k\ne q}r_{k,j_k} =\frac{B}{c}\,(c\,r_{q,j_q})\prod_{k\ne q}r_{k,j_k} =B\,r_{q,j_q}\prod_{k\ne q}r_{k,j_k} =R.

The cancellation works for each possible level $j_q$, not just for one selected risk. That is why the operation preserves the entire schedule of indicated rates. It also explains the condition that the same $c$ must be applied to every level of the selected factor: a single base-rate adjustment can cancel a common multiplier, but generally cannot cancel different multipliers at different levels.

Theorem: In a multiplicative rating plan, multiplying every relativity for one factor by a common constant $c>0$ and dividing the base rate by $c$ leaves every indicated rate unchanged, provided all other factor relativities remain unchanged.

Worked Applications

Worked Example: Rescaling One Factor in a Two-Factor Plan

A hypothetical annual rating plan has base rate $B=\$600$. A vehicle-use factor has relativities 0.90 for occasional use and 1.25 for regular use. A territory factor has relativity 0.80 for a particular territory. The indicated rate for a regular-use risk in that territory is to be preserved while every vehicle-use relativity is multiplied by $c=1.20$. Find the new base rate, both transformed vehicle-use relativities, and the indicated rate before and after rescaling.

Method and assumptions. This is a deterministic multiplicative rating calculation. No interest, mortality, or probability distribution applies. Divide the base rate by $c$, multiply each vehicle-use relativity by $c$, and leave the territory relativity unchanged.

Step 1: Calculate the original rate. The original rate is $$ R=\$600(1.25)(0.80)=\$600(1.00)=\$600.00. $$

Step 2: Calculate the transformed values. The new base rate is $$ B'=\frac{\$600}{1.20}=\$500. $$ The vehicle-use relativities become $$ 0.90(1.20)=1.08,\qquad 1.25(1.20)=1.50. $$ The territory relativity remains 0.80.

Step 3: Recalculate the indicated rate. $$ R'=\$500(1.50)(0.80)=\$500(1.20)=\$600.00. $$ The result matches the original $\$600.00$. The common mistake this calculation avoids is adjusting the selected relativity but forgetting to adjust the base rate by the reciprocal factor.

Worked Example: Checking a Full Relativity Table

A hypothetical plan has base rate $B=\$900$. Factor A has levels with relativities 0.80, 1.00, and 1.30; factor B has relativities 0.75 and 1.20. An analyst proposes multiplying all factor A relativities by 1.25 and changing the base rate to $\$720$. Check whether the proposal preserves rates, and calculate the rate for a risk at factor A level 1.30 and factor B level 1.20 under both tables.

Method and assumptions. The rating formula is deterministic and multiplicative; no interest rate, mortality assumption, or probability distribution enters the calculation. The proposed base-rate change is consistent with the rescaling rule if $\$900/1.25=\$720$.

Step 1: Transform the entire factor A table. $$ 0.80(1.25)=1.00,\qquad 1.00(1.25)=1.25,\qquad 1.30(1.25)=1.625. $$ Factor B is unchanged at 0.75 and 1.20.

Step 2: Calculate the original indicated rate. $$ R=\$900(1.30)(1.20)=\$900(1.56)=\$1{,}404.00. $$

Step 3: Calculate the proposed indicated rate. $$ R'=\$720(1.625)(1.20)=\$720(1.95)=\$1{,}404.00. $$ Thus the proposal preserves this rate. More generally, it preserves every rate because the same multiplier 1.25 was applied to every factor A level and the base rate was divided by 1.25. The transformed factor A level that was 1.00 is now 1.25; retaining that level’s old value would not implement the proposed rescaling.

Worked Example: Inferring the Scale Change From Two Tables

A hypothetical rating plan’s original base rate is $\$480$, and its factor X relativities are 0.90, 1.10, and 1.60. A revised table lists 1.08, 1.32, and 1.92 for the same levels, with a base rate of $\$400$. Factor Y is unchanged; the risk considered has factor Y relativity 0.85. Determine whether the two tables are equivalent and compare the rate for the risk at the factor X level originally assigned 1.60.

Method and assumptions. The entries are deterministic multiplicative rating components, with no interest, mortality, or probability distribution assumption. Compare the new-to-old relativity ratio at each level, then check the base-rate adjustment and compute the selected rate.

Step 1: Find the ratio at each factor X level. $$ \frac{1.08}{0.90}=1.20,\qquad \frac{1.32}{1.10}=1.20,\qquad \frac{1.92}{1.60}=1.20. $$ Every level has the common multiplier $c=1.20$.

Step 2: Check the base rate. $$ \frac{\$480}{1.20}=\$400, $$ which is the revised base rate. Therefore, the tables satisfy the rescaling rule.

Step 3: Compare the selected rates. Under the original table, $$ R=\$480(1.60)(0.85)=\$480(1.36)=\$652.80. $$ Under the revised table, $$ R'=\$400(1.92)(0.85)=\$400(1.632)=\$652.80. $$ The rates agree. The common mistake avoided here is checking only one relativity ratio: equivalence requires the same ratio at every level of the rescaled factor.

Anchors and What the Table Represents

A table can use a reference level, or anchor, whose relativity is set to 1. That convention helps identify how the other levels compare with the reference level. It is a feature of the table’s representation; the rate-invariance result itself does not require a reference relativity to equal 1.

If a reference level has relativity 1, rescaling every level by $c$ changes its relativity to $c$. Thus the rescaled table no longer satisfies the convention that the reference relativity equals 1, unless $c=1$. This does not contradict rate invariance: the base rate has also changed to offset the rescaling. If a rating plan must retain the original anchor convention, an arbitrary non-unit rescaling is not a valid way to represent that same convention.

The result also clarifies how to audit a proposed change. The base-rate ratio and the factor-relativity ratios must be reciprocals: if every selected-factor relativity is multiplied by $c$, then the new base rate must be the old base rate divided by $c$. A change to just one level, or a base-rate adjustment in the same direction rather than the reciprocal direction, will generally change indicated rates.

Identify the selected factor.
List its old and new relativities at every level; do not infer equivalence from a single level.

Check for a common multiplier.
For each level, divide the new relativity by the old relativity. All ratios must equal the same positive $c$.

Check the base rate.
The required new base rate is $B/c$, not $cB$.

Verify a complete risk profile.
Recalculate a rate using the old and new tables, retaining the other factor relativities. The results should agree.

Common Exam Trap

Suppose one factor’s relativities are all multiplied by 1.50. A tempting but incorrect calculation is to multiply the base rate by 1.50 as well. For a risk whose selected-factor relativity is 1.20 and whose other factor product is 0.90, an old base rate of $\$800$ gives an original rate of $$ \$800(1.20)(0.90)=\$864. $$ If the relativity becomes $1.20(1.50)=1.80$ and the base rate is incorrectly changed to $\$800(1.50)=\$1{,}200$, the new rate is $$ \$1{,}200(1.80)(0.90)=\$1{,}944, $$ not $\$864$. Both components have been scaled in the same direction, so their effects compound instead of canceling. The correction is to divide the base rate by 1.50: $B'=\$800/1.50$. Then $$ \left(\frac{\$800}{1.50}\right)(1.80)(0.90)=\$864. $$

Forgetting the condition that the multiplier is common across levels is another important error. If one level is multiplied by 1.20 and another by 1.30, there is no single base-rate adjustment that cancels both changes for all risks. In that case, rates at the affected levels generally change.

Key takeaway: Rescaling is a change in how the rating plan’s base rate and one factor’s relativities are expressed, not necessarily a change in indicated rates. Multiply every relativity for that factor by $c>0$, divide the base rate by $c$, and verify that the same $c$ applies at every level.

Check Your Understanding

For each problem, apply the rescaling rule and check the stated rates.

A hypothetical plan has base rate $\$720$, a selected-factor relativity of 1.25, and another factor relativity of 0.90. All selected-factor relativities are multiplied by 0.80. Find the new base rate, the transformed selected relativity, and the original and new indicated rates for this risk. (Answer: New base rate $\$900$; transformed relativity 1.00; both rates $\$810.00$.)
A plan has base rate $\$500$ and factor A relativities 0.70, 1.10, and 1.50. All are multiplied by 1.20, and factor B stays at 0.80 for a risk whose original factor A relativity is 1.50. Find the new base rate and compare indicated rates. (Answer: New base rate $\$416.666\ldots$; both rates $\$600.00$.)
A plan has base rate $\$640$ and factor A relativities 0.75, 1.25, and 1.50. The base rate changes to $\$800$, and the third factor A relativity changes from 1.50 to 1.20; all other levels are claimed to be rescaled by the same constant. What is the multiplier, what are the transformed factor A relativities, and what happens to the rate for a risk using the middle level and an unchanged other-factor relativity of 1.10? (Answer: $c=0.80$; new factor A relativities 0.60, 1.00, and 1.20; both rates $\$880.00$.)
An old table has factor C relativities 0.60, 1.00, and 1.40, and base rate $\$1{,}200$. A proposed table has relativities 0.90, 1.50, and 2.10 and base rate $\$800$. With another unchanged factor relativity of 0.95, compare the rate using the third factor C level. (Answer: The common multiplier is 1.50, so the tables are equivalent; both rates are $\$1{,}596.00$.)
A base rate of $\$1{,}000$ and two selected-factor relativities, 1.20 and 0.90, are rescaled by 1.25 and 0.80, respectively. A risk also has an unchanged third-factor relativity of 1.10. Find the new base rate and compare its indicated rate with the original. (Answer: The multiplier product is 1.00, so the new base rate is $\$1{,}000$; both rates are $\$1{,}188.00$.)

What You'll Learn