The Limits of Separate Multiplicative Factors

Why Separate Factors Have a Limit

A multiplicative rating plan is useful because it builds a class rate from a base rate and a set of factor relativities. But when each factor acts separately, the structure makes a specific commitment: the effect of changing one rating characteristic is the same at every level of the other characteristics. If the indicated rates show a distinct joint effect, a table of separate factors alone cannot represent it exactly.

This is a structural limitation, not necessarily evidence that the rating plan is poorly designed. A separate-factor table is a deliberate simplification. The practical question is whether its fixed-product structure is adequate for the class rates being represented.

Definition: For two binary rating characteristics $A$ and $B$, let $r_{00}$, $r_{10}$, $r_{01}$, and $r_{11}$ denote the rates for the four combinations of reference or changed levels. A separate multiplicative-factor structure has rates $r_{ab}=r_{00} f_A^a f_B^b$, where $f_A$ and $f_B$ are the relativities for the changed levels. It therefore requires $r_{00}r_{11}=r_{10}r_{01}$. A departure from this condition is a joint effect relative to the separate-factor structure.

This condition applies to rates on a comparable basis: the same coverage, time period, exposure unit, and rate definition. The factors are assumed to be positive and to apply consistently across the other characteristic's levels. The condition concerns the structure of the indicated rates; it does not say that a single observed class rate is known without uncertainty.

Deriving the Fixed-Product Condition

Take the rate for the reference combination as $r_{00}=B$, where $B$ is the base rate. Let $f_A$ be the relativity for changing $A$ from its reference level, and $f_B$ the relativity for changing $B$. Separate factors imply:

r_{10}=Bf_A,\qquad r_{01}=Bf_B,\qquad r_{11}=Bf_Af_B.

Multiplying the reference rate by the joint rate gives $B(Bf_Af_B)=B^2f_Af_B$. Multiplying the two single-change rates gives $(Bf_A)(Bf_B)=B^2f_Af_B$. Thus the products must agree:

r_{00}r_{11}=r_{10}r_{01}.

Equivalently, the effect of changing $A$ must be the same whether $B$ is at its reference or changed level:

\frac{r_{10}}{r_{00}}=\frac{r_{11}}{r_{01}}=f_A.

The same requirement applies to $B$: $r_{01}/r_{00}=r_{11}/r_{10}=f_B$. If the two ratios for either characteristic differ, no single relativity for that characteristic can reproduce all four rates while retaining the other separate factor.

Worked Examples

Worked Example: Checking a Two-Factor Rate Table

A hypothetical insurer is reviewing annual rates per vehicle for a particular coverage. Characteristic $A$ is a high-use class, and characteristic $B$ is a specialized vehicle class. The indicated rates are $r_{00}=\$240$, $r_{10}=\$336$, $r_{01}=\$300$, and $r_{11}=\$462$. Find the single-factor relativities from the reference classes, predict the joint rate under separate factors, and measure the departure of the indicated joint rate from that prediction.

Method. Use the reference class to obtain each single-factor relativity, then apply both to the base rate. This tests whether one factor for each characteristic is sufficient.

The base rate is $B=\$240$. The indicated relativity for $A$, comparing the $A$-only class with the reference class, is $336/240=1.40$. The indicated relativity for $B$ is $300/240=1.25$.

The separate-factor prediction for the joint class is:

\$240(1.40)(1.25)=\$240(1.75)=\$420.

The indicated joint rate is $\$462$, which is $\$462-\$420=\$42$ above the separate-factor prediction. As a ratio, the indicated rate is $462/420=1.10$, or 10% higher.

A second check compares the effect of $A$ across the two levels of $B$. At the reference level of $B$, it is $336/240=1.40$. At the changed level of $B$, it is $462/300=1.54$. These differ, so a single $A$ relativity does not match all four indicated rates. The product check confirms this: $240(462)=110{,}880$, while $336(300)=100{,}800$.

Check. Applying the derived separate factors gives $\$420$, while the stated joint indication is $\$462$. The ratio $462/420=1.10$ agrees with the calculated 10% departure. The common mistake this avoids is treating the factor-derived $\$420$ as though it were the observed or indicated joint rate.

Worked Example: Finding a Missing Rate Under the Separate-Factor Assumption

A hypothetical rating table has two binary characteristics. Three annual rates per policy are $r_{00}=\$180$, $r_{10}=\$225$, and $r_{01}=\$252$. First find the joint rate required by separate factors. Then compare it with a subsequently reviewed joint-class indication of $\$330$.

Method. Rearrange the fixed-product condition to solve for the missing rate:

r_{11}=\frac{r_{10}r_{01}}{r_{00}}.

Substitution gives $r_{11}=(225)(252)/180$. The numerator is $56{,}700$, and $56{,}700/180=315$. Thus the separate-factor structure requires a joint rate of $\$315$.

The reviewed joint indication is $\$330$, so it is $\$330-\$315=\$15$ higher. Its ratio to the separate-factor prediction is $330/315=22/21$, approximately $1.0476$; the joint indication is approximately 4.76% higher.

The single-factor relativities inferred from the two reference comparisons are $225/180=1.25$ and $252/180=1.40$. They predict $180(1.25)(1.40)=180(1.75)=315$, which agrees with the fixed-product calculation. Using $\$330$ as the missing rate instead would violate the restriction: $180(330)=59{,}400$, whereas $225(252)=56{,}700$.

Check. The required rate is $\$315$, and the reviewed indication exceeds it by $\$15$. The common mistake this avoids is multiplying the two single-change rates directly and forgetting to divide by the reference rate.

Worked Example: Translating a Joint Effect Into Portfolio Premium

A hypothetical insurer is reviewing annual rates per exposure unit for a product with two binary characteristics: territory and usage. The reference rate is $\$500$. The territory-only rate is $\$650$, and the usage-only rate is $\$600$. The indicated rate for the class with both changed characteristics is $\$858$. The portfolio has 60 reference exposures, 20 territory-only exposures, 30 usage-only exposures, and 10 exposures with both characteristics. Calculate the joint rate predicted by separate factors and the portfolio premium difference if the indicated joint rate is used instead.

Method. Derive each relativity from its single-change class, calculate the separate-factor joint rate, and then compare total premiums using the stated exposure counts.

The territory relativity is $650/500=1.30$, and the usage relativity is $600/500=1.20$. The separate-factor joint rate is:

\$500(1.30)(1.20)=\$500(1.56)=\$780.

The indicated joint rate, $\$858$, is $\$858-\$780=\$78$ higher per exposure unit. Its ratio to the separate-factor prediction is $858/780=1.10$.

At the separate-factor rates, the four class premiums are $60(\$500)=\$30{,}000$, $20(\$650)=\$13{,}000$, $30(\$600)=\$18{,}000$, and $10(\$780)=\$7{,}800$. The total is $\$30{,}000+\$13{,}000+\$18{,}000+\$7{,}800=\$68{,}800$.

Using the indicated joint rate instead changes only the last class premium: $10(\$858)=\$8{,}580$. The resulting total is $\$30{,}000+\$13{,}000+\$18{,}000+\$8{,}580=\$69{,}580$. The difference is $\$69{,}580-\$68{,}800=\$780$, also equal to $10(\$858-\$780)=10(\$78)$.

Check. The per-exposure difference is $\$78$; with 10 joint-class exposures, the total difference is $\$780$. The common mistake this avoids is applying the joint-class difference to every portfolio exposure rather than only to exposures in that class.

Interpreting a Departure

A mismatch between an indicated rate and a separate-factor prediction means the table cannot reproduce both with the same set of separate relativities. It does not, on its own, establish why the mismatch exists. The underlying experience may support a distinct joint effect, or the class indication may be affected by limited data, differences in mix, or other uncertainty. An actuary should not interpret one departure as proof of a stable relationship without considering the basis and reliability of the indications.

If the joint effect is supported and important for the rating objective, one possible structural response is a factor for the joint combination. In the first worked example, that factor relative to the separate-factor prediction is $462/420=1.10$. Applied to that joint class, it raises the predicted rate from $\$420$ to $\$462$. This is a way to describe the missing relationship, not a conclusion that every rating plan should include such a factor.

A small class may also be handled differently from a well-supported class, depending on the ratemaking process and applicable constraints. The central calculation remains the same: keep the separate-factor prediction distinct from the class indication, then quantify the difference on the relevant rate or premium basis.

Common Exam Trap

The most important trap is to assume that a multiplicative table can match any four class rates simply because the rates are positive. With two binary characteristics, separate factors have only a base rate and two relativities. Those three quantities determine the fourth rate; it is not free to vary independently.

For example, a wrong approach to the first table would be to use $1.40$ for $A$, $1.25$ for $B$, and then claim that the joint indication is $\$462$. The product calculation is actually $\$240(1.40)(1.25)=\$420$, not $\$462$. The correction is to report the separate-factor prediction as $\$420$, compare it with the $\$462$ indication, and state that the table does not reproduce that joint rate without an additional joint adjustment or a different structure.

Key takeaway: Separate multiplicative factors require the joint class rate to equal the base rate times each single-factor relativity. For two binary characteristics, test this with $r_{00}r_{11}=r_{10}r_{01}$. If it fails, quantify the departure and keep the factor-based prediction separate from the joint-class indication.

Check Your Understanding

Work through each problem using rates on a consistent basis. Answers are given for checking.

A hypothetical table has a reference rate of $\$240$, an $A$-only rate of $\$300$, and a $B$-only rate of $\$264$. Under separate factors, what joint rate is predicted? (Answer: $\$330$)
A table has $r_{00}=\$200$, $r_{10}=\$260$, and $r_{01}=\$230$. Find the joint rate required by separate factors. (Answer: $\$299$)
For the table in question 2, the indicated joint rate is $\$345$. By what factor does it exceed the separate-factor prediction? (Answer: $345/299$, approximately $1.154$)
A hypothetical portfolio has separate-factor rates of $\$400$, $\$480$, $\$500$, and a predicted joint rate of $\$600$. The indicated joint rate is $\$630$, with 12 exposures in that joint class. What is the total premium difference for that class? (Answer: $\$360$)
A two-factor table has $r_{00}=\$150$, $r_{10}=\$180$, $r_{01}=\$210$, and $r_{11}=\$252$. Does it satisfy the separate-factor condition? Show the two cross-products. (Answer: Yes; $150(252)=37{,}800$ and $180(210)=37{,}800$)

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