Normalizing Class Relativities After Credibility Weighting

Why Normalize Credibility-Weighted Relativities?

Credibility weighting produces a loss-cost indication for each class by balancing its observed experience with a collective complement. Turning those indications into relativities requires a further decision: the class relativities must be placed on a scale that is consistent with the overall loss-cost level used for the rating plan.

A set of raw relativities can preserve the relative differences between classes and still fail to reproduce the selected portfolio average when weighted by exposure. Normalization supplies one common scaling factor for all classes. It changes the overall level of the relativities, not their ratios to one another. The resulting indicated class loss costs then have the specified exposure-weighted average.

Definition: Let $C_j$ be the credibility-weighted loss cost for class $j$, $w_j$ its exposure weight, and $B>0$ a reference loss-cost level. The raw relativity is $r_j=C_j/B$. The exposure-weighted normalization factor is $$ N=\frac{1}{\bar r_w} =\frac{\sum_j w_j}{\sum_j w_j r_j}, \qquad \bar r_w=\frac{\sum_j w_j r_j}{\sum_j w_j}. $$ The normalized relativity is $R_j=Nr_j$. If $T$ is the specified target average loss cost, the indicated class loss cost is $T R_j$.

The weights must represent the exposure basis on which the target average is defined. Depending on the rating application, that might be earned car-years, policy-years, or another appropriate measure. The weights are not automatically claim counts, and they should not be switched between the calculation of the normalization factor and the portfolio target. Use costs and exposures for a consistent coverage, experience period, and loss-cost basis.

Deriving the Normalization Factor

The required condition is that the exposure-weighted average of the normalized relativities equal one. Write $W=\sum_j w_j$. For a common factor $N$,

\frac{\sum_j w_j R_j}{W} = \frac{\sum_j w_j(Nr_j)}{W} = N\frac{\sum_j w_jr_j}{W}.

Setting this weighted average equal to one gives $N=W/\sum_j w_jr_j=1/\bar r_w$. This is the unique common factor that satisfies the condition, provided the weights have a positive total and the weighted raw relativity is positive. The result is a scale adjustment: every raw relativity is multiplied by the same number, so for any two classes with positive relativities, $R_j/R_k=r_j/r_k$.

Since $r_j=C_j/B$, the reference level cancels from normalized relativities:

R_j = \frac{w_1+\cdots+w_J}{\sum_k w_k(C_k/B)}\frac{C_j}{B} = \frac{W C_j}{\sum_k w_k C_k}.

Consequently, the normalization can be calculated directly from credible loss costs without first choosing a reference level. Multiplying the normalized relativity by the target $T$ gives a class loss-cost indication. Its weighted average is

\frac{\sum_j w_j(T R_j)}{W} = T\frac{\sum_j w_j R_j}{W} = T.

This identity is the purpose of the normalization. It does not assert that the target is automatically the correct portfolio level; that target must be selected on its own actuarial basis. Normalization only ensures that the class indications reconcile to the target supplied.

Establish the credible class costs.
Use the credibility-weighted costs $C_j$ that are to underlie the class indications.

Fix the weights and target.
Choose the exposure basis $w_j$ and the specified portfolio average loss cost $T$, consistent with the application.

Calculate the normalization factor.
Compute $N=W/\sum_j w_jr_j$, or use the equivalent direct form for normalized relativities, $R_j=W C_j/\sum_j w_jC_j$.

Apply the target level and check.
Set the indicated class cost to $T R_j$, then verify that its exposure-weighted average is $T$.

Worked Applications

Worked Example: Three Classes and a New Target Level

A hypothetical insurer has credibility-weighted annual loss costs of $420, $560, and $700 for three classes. Their respective exposures are 1,000, 600, and 400 car-years. Raw relativities are formed using a $500 reference loss cost. Normalize the relativities and calculate indicated class costs that reproduce a target average loss cost of $540.

Method. Calculate the exposure-weighted raw relativity, divide each raw relativity by that average, and apply the target level. This avoids treating the unweighted mean of class relativities as the portfolio average.

The raw relativities are $420/500=0.84$, $560/500=1.12$, and $700/500=1.40$. Total exposure is $W=1{,}000+600+400=2{,}000$ car-years. The weighted raw relativity total is $1{,}000(0.84)+600(1.12)+400(1.40)=840+672+560=2{,}072$. Thus $\bar r_w=2{,}072/2{,}000=1.036$, and $N=1/1.036=250/259\approx0.965251$.

The normalized relativities are $0.84(250/259)=210/259\approx0.810811$, $1.12(250/259)=280/259\approx1.081081$, and $1.40(250/259)=350/259\approx1.351351$. Applying the $540 target gives costs of $540(210/259)\approx\$437.84$, $540(280/259)\approx\$583.78$, and $540(350/259)\approx\$729.73$, rounded to the nearest cent.

Check. The exposure-weighted cost total before rounding is $1{,}000(540)(210/259)+600(540)(280/259)+400(540)(350/259)$. The weighted normalized relativity total is $840+672+560$, scaled by $250/259$, or $2{,}000$. Therefore the weighted average cost is $540(2{,}000/2{,}000)=\$540$. The unweighted raw-relativity mean would be $(0.84+1.12+1.40)/3=1.12$; using it would not satisfy the exposure-weighted target condition.

Worked Example: Relativities Supplied Without a Common Level

A hypothetical specialty program has three classes with exposures of 200, 500, and 300 units. Its selected raw relativities are 0.80, 1.00, and 1.60. The target portfolio average loss cost is $360. Find the normalized relativities and class loss-cost indications.

Method. Normalize the supplied relativities directly with the stated exposure weights. Do not average the three values equally because the classes have unequal exposure.

The total exposure is $W=200+500+300=1{,}000$. The weighted raw relativity total is $200(0.80)+500(1.00)+300(1.60)=160+500+480=1{,}140$, so $\bar r_w=1{,}140/1{,}000=1.14$. The normalization factor is $N=1/1.14=50/57\approx0.877193$.

The normalized relativities are $0.80(50/57)=40/57\approx0.701754$, $1.00(50/57)=50/57\approx0.877193$, and $1.60(50/57)=80/57\approx1.403509$. Multiplying by $360 gives $(360)(40/57)\approx\$252.63$, $(360)(50/57)\approx\$315.79$, and $(360)(80/57)\approx\$505.26$, rounded to the nearest cent.

Check. The weighted normalized relativity total is $200(40/57)+500(50/57)+300(80/57)=(8{,}000+25{,}000+24{,}000)/57=1{,}000$. Its average is one, so the indicated costs average to $360(1)=\$360$. The unweighted average of the raw relativities is $1.1333\ldots$, not 1.14, and would produce a different, incorrect weighted reconciliation.

Worked Example: Normalize After Credibility Weighting

A hypothetical rating analysis uses a $600 collective complement for three classes. The experience loss costs are $510, $640, and $900, and the corresponding credibility weights are 0.25, 0.50, and 0.75. Class exposures are 400, 200, and 100 units. Use a $600 reference level to form raw relativities, normalize them, and set indicated costs to a target average of $650.

Method. First calculate each credibility-weighted cost, then normalize those costs as a set. Normalizing the experience costs before applying credibility would not normalize the class indications being used.

For each class, $C_j=Z_j\bar X_j+(1-Z_j)C_j^0$, where $C_j^0=\$600$. The first cost is $0.25(510)+0.75(600)=127.50+450=\$577.50$. The second is $0.50(640)+0.50(600)=320+300=\$620$. The third is $0.75(900)+0.25(600)=675+150=\$825$.

The raw relativities against $600 are $577.50/600=0.9625$, $620/600=1.033333\ldots$, and $825/600=1.375$. Total exposure is 700. The weighted raw relativity total is $400(0.9625)+200(1.033333\ldots)+100(1.375)=385+206.6667+137.5=729.1667$. Thus $\bar r_w=729.1667/700=25/24$, and $N=24/25=0.96$.

The normalized relativities are $0.9625(0.96)=0.924$, $1.033333\ldots(0.96)=0.992$, and $1.375(0.96)=1.32$. The indicated costs at $650 are $650(0.924)=\$600.60$, $650(0.992)=\$644.80$, and $650(1.32)=\$858.00$.

Check. The weighted normalized relativity total is $400(0.924)+200(0.992)+100(1.32)=369.6+198.4+132=700$. The weighted average relativity is one and the indicated average cost is therefore $650. All monetary amounts are exact to the cent in this calculation.

Interpretation and Conditions

Normalization is appropriate when the class pattern comes from credibility-weighted costs but the overall level is to be anchored to a specified average loss cost. The result separates two rating decisions: the normalized relativities describe the class pattern on an exposure-weighted average-one scale, while $T$ supplies the dollar level. If the target level changes but the class costs and exposure mix do not, the normalized relativities remain the same and the indicated class costs change in proportion to the new target.

The basic calculation assumes $w_j\geq0$, $W>0$, and a positive weighted sum of raw relativities. In the usual application, credible loss costs and the reference level are nonnegative, and at least one class with positive exposure has a positive indication. If the weighted indication is zero while the target is positive, no finite common multiplier can reconcile them; the input indications or the target basis must be reviewed. Zero-exposure classes do not affect the normalization, though a separate decision is needed if rates are required for them.

The exposure basis matters because normalization guarantees reconciliation only for the weights actually used. For example, a factor calculated with policy counts will not generally make the average equal the target when the target is defined on earned car-years. Similarly, a portfolio-mix change can alter the weighted raw average and therefore the factor, even when every class's raw indication is unchanged. Recalculate using the exposure mix associated with the target rather than carrying over a factor from an incompatible mix.

Common Exam Trap

A frequent error is to normalize by the simple arithmetic mean of the class relativities. That works only when all classes have equal exposure weights (or when the unweighted mean happens to equal the correctly weighted mean). Another error is to multiply by the weighted average instead of its reciprocal. If the raw weighted average is 1.14, multiplying each relativity by 1.14 makes the resulting weighted average $1.14^2$, not one; the factor must be $1/1.14$.

Key takeaway: Compute the exposure-weighted mean of the raw credibility-weighted relativities, divide each by that mean, and multiply the normalized relativity by the selected target average loss cost. The final exposure-weighted class loss cost must equal the target.

Check Your Understanding

For each problem, use the stated exposure basis and verify that the final weighted average matches the target.

Two classes have exposures 300 and 700 and credibility-weighted loss costs of $400 and $700. Find the common multiplier applied to the costs to reach a $620 target average. (Answer: $62/61\approx1.0164$; indicated costs are about $406.56 and $711.48.)
Three classes have raw relativities 0.70, 1.00, and 1.30 with exposures 250, 250, and 500. Find the normalization factor and normalized relativities. (Answer: $40/43\approx0.9302$; normalized relativities are $28/43\approx0.6512$, $40/43\approx0.9302$, and $52/43\approx1.2093$.)
Two classes have exposures 600 and 400 and credible costs of $520 and $650. Using a $500 reference, find the normalization factor for the raw relativities and the indicated costs at a $575 target. (Answer: $125/143\approx0.8741$; indicated costs are about $522.73 and $653.41.)
Two classes have exposures 100 and 300 and already credibility-weighted costs of $450 and $750. Find the common cost multiplier and show the resulting indicated costs at a $720 target. (Answer: $16/15\approx1.0667$; indicated costs are $480 and $800, with weighted average $720.)

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