Normalizing Relativities to an Overall Average

Why Normalize Class Relativities?

Class relativities express how class rates compare with a common rate level. A factor of 1.20 represents a rate 20% above that level; a factor of 0.80 represents a rate 20% below it. But a set of indicated factors does not automatically average to 1.00 across the portfolio. Its average depends on how much exposure is assigned to each class.

Normalization rescales all the indicated factors by the same multiplier so their average, using a specified set of exposure weights, is 1.00. This preserves the relative differences among classes while setting the overall level of the factors under that weighting basis. It does not establish that the factors are otherwise appropriate, nor does it guarantee a 1.00 average under a different portfolio mix.

Definition: Suppose there are $k$ classes, with indicated relativities $r_j$ and nonnegative exposure weights $w_j$, for $j=1,\ldots,k$. The weighted average of the indicated relativities is $$ \bar{r}_w=\frac{\sum_{j=1}^{k}w_jr_j}{\sum_{j=1}^{k}w_j}. $$ If the denominator and weighted sum are positive, the normalization multiplier is $$ c=\frac{1}{\bar{r}_w} =\frac{\sum_{j=1}^{k}w_j}{\sum_{j=1}^{k}w_jr_j}, $$ and the normalized relativity for class $j$ is $r_j^*=cr_j$. Thus $\sum_j w_jr_j^*/\sum_jw_j=1$.

The weight $w_j$ must represent the exposure basis chosen for the comparison. Depending on the analysis, it might be earned exposures, projected exposures, or another specified measure of class volume. Use the basis stated in the problem or selected for the analysis; do not silently substitute a different basis. If all weights are zero, the weighted average is undefined. A class with zero weight does not affect the normalization under that weighting basis.

This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. The central calculation is short, but interpreting its weights correctly is essential: the multiplier is determined by both the factors and the portfolio distribution used in the calculation.

How the Normalization Formula Works

The desired normalized factors $r_j^*$ must have weighted average 1.00. If every factor is multiplied by the same positive number $c$, the weighted average of the new factors is $c\bar{r}_w$. Choosing $c=1/\bar{r}_w$ therefore makes that average exactly 1.00. Because the same multiplier applies to every class, the ratio between any two positive factors is unchanged.

\frac{\sum_jw_jr_j^*}{\sum_jw_j} = \frac{\sum_jw_j(cr_j)}{\sum_jw_j} = c\bar{r}_w = \frac{1}{\bar{r}_w}\bar{r}_w =1.

This also gives a direct verification rule: after normalization, the sum of weighted normalized factors must equal the sum of the weights. Small differences may appear if the displayed factors are rounded, so perform the calculation with unrounded values and round only for presentation.

Confirm the inputs.
List each class factor and its weight. Check that the weights refer to the intended exposure basis and cover the classes being normalized.

Find the weighted factor total.
Multiply each factor by its weight and add the products, $\sum_jw_jr_j$. Also calculate the total weight, $\sum_jw_j$.

Calculate the multiplier.
Divide total weight by weighted factor total. Equivalently, divide 1 by the weighted average of the original factors.

Rescale and check.
Multiply every factor by the same multiplier. Verify that the weighted average of the unrounded normalized factors is 1.00.

Worked Calculations

Worked Example: Normalizing Three Classes

A hypothetical insurer is reviewing three classes. The indicated relativities are 0.80, 1.00, and 1.60. The selected exposure weights are 1,000, 2,000, and 500, respectively. Calculate the normalized factors and verify the weighted average.

Method. Calculate the weighted average using the stated exposure weights, then multiply every indicated factor by its reciprocal. This avoids treating classes with different exposure volumes as equally weighted.

Step 1: Calculate the total weight and weighted factor total.

\sum_jw_j=1{,}000+2{,}000+500=3{,}500, $$ $$ \sum_jw_jr_j =1{,}000(0.80)+2{,}000(1.00)+500(1.60) =800+2{,}000+800 =3{,}600.

Step 2: Find the weighted average and multiplier. The original weighted average is $3{,}600/3{,}500=36/35$. Therefore, the normalization multiplier is $c=35/36=0.972222\ldots$.

Step 3: Normalize each factor.

r_1^*=0.80\left(\frac{35}{36}\right)=0.7778,\qquad r_2^*=1.00\left(\frac{35}{36}\right)=0.9722, $$ $$ r_3^*=1.60\left(\frac{35}{36}\right)=1.5556.

Step 4: Verify before rounding. The weighted normalized total is $(35/36)(3{,}600)=3{,}500$, equal to the total weight. The weighted average is therefore $3{,}500/3{,}500=1.00$. The factors displayed above are rounded to four decimal places; the check uses unrounded values.

Common mistake avoided: taking the simple average of the three factors would give $(0.80+1.00+1.60)/3=1.1333\ldots$. That is not the specified exposure-weighted average because the class weights are unequal.

Worked Example: How the Weighting Basis Changes the Result

For a hypothetical product, three indicated factors are 0.70, 1.10, and 1.80. An analyst considers two possible exposure distributions: a current distribution with weights 400, 300, and 300, and a projected distribution with weights 700, 200, and 100. Normalize the factors separately under each distribution.

Method. Apply the normalization formula once for each stated weight set. The factors stay fixed, but their weighted averages—and therefore the multipliers—can differ.

Current distribution. The total weight is $400+300+300=1{,}000$. The weighted factor total is $400(0.70)+300(1.10)+300(1.80)=280+330+540=1{,}150$. The weighted average is $1.15$, so $c=1/1.15=20/23=0.869565\ldots$. The normalized factors are:

0.70\left(\frac{20}{23}\right)=0.6087,\qquad 1.10\left(\frac{20}{23}\right)=0.9565,\qquad 1.80\left(\frac{20}{23}\right)=1.5652.

The check is $(20/23)(1{,}150)=1{,}000$, equal to the current total weight.

Projected distribution. Its total weight is also $1{,}000$, but its weighted factor total is $700(0.70)+200(1.10)+100(1.80)=490+220+180=890$. The weighted average is $0.89$, so $c=1/0.89=100/89=1.123596\ldots$. The normalized factors are:

0.70\left(\frac{100}{89}\right)=0.7865,\qquad 1.10\left(\frac{100}{89}\right)=1.2360,\qquad 1.80\left(\frac{100}{89}\right)=2.0225.

The check is $(100/89)(890)=1{,}000$. Both sets average to 1.00 under their own weights, but the class factors differ because the distributions differ. Each displayed factor is rounded to four decimal places.

Common mistake avoided: reusing the current-distribution multiplier for the projected distribution. A normalization multiplier belongs to a particular set of weights; changing the weights requires recalculating it.

Worked Example: Checking the Aggregate Rate Effect

A hypothetical rating plan has three classes with exposure counts of 600, 300, and 100. Their indicated factors are 0.90, 1.20, and 1.50. The base rate is $500 per exposure. Normalize the factors and compare the aggregate indicated premium before and after normalization, holding the base rate and exposures fixed.

Method. First normalize using the given exposure counts. Then calculate aggregate premium as the sum, over classes, of exposure count times base rate times class factor.

Step 1: Find the multiplier. Total exposures are $600+300+100=1{,}000$. The weighted factor total is $600(0.90)+300(1.20)+100(1.50)=540+360+150=1{,}050$. Thus, the weighted average is $1.05$, and $c=1/1.05=20/21=0.952380\ldots$.

Step 2: Calculate normalized factors.

r_1^*=0.90\left(\frac{20}{21}\right)=0.8571,\qquad r_2^*=1.20\left(\frac{20}{21}\right)=1.1429,\qquad r_3^*=1.50\left(\frac{20}{21}\right)=1.4286.

Step 3: Compare aggregate premiums. Before normalization, the aggregate indicated premium is $600(\$500)(0.90)+300(\$500)(1.20)+100(\$500)(1.50)=\$270{,}000+\$180{,}000+\$75{,}000=\$525{,}000$. After normalization, the unrounded weighted factor total is $1{,}050(20/21)=1{,}000$, so aggregate premium is $1{,}000(\$500)=\$500{,}000$.

Check and interpretation. The normalized weighted average is 1.00, so the aggregate premium equals the base rate times total exposures. The factor values are displayed to four decimal places; aggregate premium is calculated using unrounded factors. This comparison holds the base rate and exposure counts fixed; it does not assert that a particular base rate is adequate for the insurer’s costs or objectives.

Common mistake avoided: summing the class factors or taking their simple average to determine premium impact. Aggregate premium depends on the product of exposure, base rate, and factor in each class.

Interpretation, Limits, and Exam Traps

Normalization is a scale adjustment, not a change in the pattern of the indicated factors. For any two classes with positive factors, \[ \frac{r_a^*}{r_b^*}=\frac{cr_a}{cr_b}=\frac{r_a}{r_b}. \] Thus, if one class’s indicated factor is twice another’s, it remains twice the other after both are normalized. Normalization changes the overall scale so that the factors average to 1.00 for the selected weights.

When a single base rate is multiplied by each class factor, and exposures and base rate are held fixed, the weighted-average condition has a useful aggregate interpretation. If $B$ is the base rate, then aggregate premium using normalized factors is \[ \sum_jw_jBr_j^*=B\sum_jw_j. \] That is, the exposure-weighted class factors produce the same aggregate premium as applying the base rate to all exposures. This is a mathematical consequence of the normalization condition, not a separate guarantee about the adequacy of the base rate.

The most important practical judgment is the weighting basis. If the goal is an average of 1.00 for projected exposure, historical weights may not achieve that goal when the class distribution has changed. Conversely, using projected weights when the intended basis is historical exposure answers a different question. State the weights, calculate the result under those weights, and interpret the normalized factors on that same basis.

Keep these checks in view:

Use the specified weights, not an unweighted average unless all weights are equal.
Use one common multiplier for every factor in the set being normalized.
Check that the weighted sum of normalized factors equals the total weight.
Do not expect normalized factors to average to 1.00 under every possible portfolio mix.
Round for presentation only after calculating the multiplier and completing the verification.

Key takeaway: Normalize by dividing each indicated factor by its weighted average. The result has a 1.00 weighted average for the chosen exposure basis, while the relative spacing among the factors is preserved.

Check Your Understanding

For each problem, use the stated weights, retain unrounded values for checks, and round displayed relativities to four decimal places.

A hypothetical program has factors 0.80, 1.00, and 1.40 with exposure weights 200, 300, and 500. Find the weighted average, normalization multiplier, and normalized factors. (Answer: weighted average $1.16$; multiplier $25/29=0.8621$; normalized factors $0.6897,\ 0.8621,\ 1.2069$.)
A hypothetical insurer indicates factors 0.90, 1.30, and 1.80 for classes with weights 600, 250, and 150. Calculate the normalized factors and verify the weighted-average condition. (Answer: weighted factor total $1{,}135$, total weight $1{,}000$, multiplier $200/227=0.8811$; normalized factors $0.7930,\ 1.1454,\ 1.5859$, with weighted average $1.00$.)
Two classes have indicated factors 0.75 and 1.50. Normalize first using weights 500 and 500, then using weights 200 and 800. Explain why the answers differ. (Answer: for weights $500,500$, multiplier $8/9$ and factors $0.6667,\ 1.3333$; for weights $200,800$, multiplier $20/27$ and factors $0.5556,\ 1.1111$. The weighting distributions differ.)
A hypothetical plan has weights 150, 250, and 100; factors 0.80, 1.10, and 1.70; and a base rate of $720. Normalize the factors and find aggregate premium after normalization. (Answer: weighted factor total $565$, total weight $500$, multiplier $100/113$; normalized factors $0.7080,\ 0.9735,\ 1.5044$; aggregate premium using unrounded factors is $500(\$720)=\$360{,}000$.)
An analyst uses the unweighted average of factors 0.60, 1.20, and 1.50, although the specified weights are 800, 150, and 50. Calculate the correct normalized factors and state the error in using the simple average. (Answer: weighted factor total $480+180+75=735$, total weight $1{,}000$, multiplier $200/147=1.3605$; normalized factors $0.8163,\ 1.6327,\ 2.0408$. The simple average is not the required exposure-weighted average.)

What You'll Learn