Why Normalize Relativities to a Portfolio Mix?
A multiplicative rating plan expresses a class rate as a base rate multiplied by a class relativity. The relativity communicates how a class compares with the reference level, but the reference level need not represent the average risk in the actual portfolio. Normalizing the relativities to a specified mix makes their exposure-weighted average equal to 1.00. This gives the base rate a direct interpretation: it is the average indicated rate per exposure unit for that mix.
Normalization is a re-expression of the same class rates, not a rate change. The base rate must change along with the relativities. If the relativities are divided by a common factor but the base rate is left alone, every indicated class rate changes. The calculations below use hypothetical rating plans in which each exposure receives exactly one class relativity.
The weights must correspond to the portfolio mix being used. If the units are earned exposures, use earned exposures; if the rating basis is another defined exposure measure, use that measure consistently. A class with more exposure should generally contribute more to the weighted average than a class with less exposure. An unweighted average of class relativities is appropriate only if the classes have equal exposure in the portfolio being normalized.
How the Base Rate and Relativities Move Together
The normalization factor \(c\) is the average relativity under the selected mix. Dividing every relativity by \(c\) shifts the weighted average to 1.00. To keep the rate for each class unchanged, multiply the base rate by that same \(c\). The new base rate is therefore the weighted average of the old class rates per exposure unit.
The first equality is the normalization condition. The second is the rate-preservation condition. The third follows by substituting the definition of \(c\): the adjusted base is the exposure-weighted average of the original indicated class rates. These results require a positive \(c\), as is the case for positive relativities and a portfolio with positive exposure. If all relativities are zero, there is no positive scale factor that can be used for this normalization.
List each class’s exposure on the chosen basis and check that the exposures sum to a positive total.
Multiply each class relativity by its exposure, add the products, and divide by total exposure.
Divide every relativity by the weighted average and multiply the original base rate by that average.
Confirm that the weighted average of the normalized relativities is 1.00 and that each class’s rate is unchanged.
Worked Example: Normalize Three Classes and Preserve Their Rates
A hypothetical insurer has a one-factor rating plan with a base rate of $600 per exposure unit. The class relativities are 0.75, 1.00, and 1.50. The target portfolio contains 400, 300, and 300 exposure units in those classes, respectively. Find the normalized relativities, the adjusted base rate, and verify the class rates. Use six decimals when rounding normalized relativities.
Method. Calculate the exposure-weighted average relativity, divide each relativity by it, and multiply the base rate by it. This avoids treating the classes as equally represented when their exposures differ.
The total exposure is \(400+300+300=1{,}000\). The exposure-weighted relativity total is \(400(0.75)+300(1.00)+300(1.50)=300+300+450=1{,}050\). Therefore, \(c=1{,}050/1{,}000=1.05\).
Divide each original relativity by 1.05: \(0.75/1.05=5/7=0.714286\), \(1.00/1.05=20/21=0.952381\), and \(1.50/1.05=10/7=1.428571\), rounded to six decimals.
Adjust the base rate: \(B'=600(1.05)=\$630\) per exposure unit. The original class rates are \(600(0.75)=\$450\), \(600(1.00)=\$600\), and \(600(1.50)=\$900\). Using the exact normalized values, the new rates are \(630(5/7)=\$450\), \(630(20/21)=\$600\), and \(630(10/7)=\$900\).
Check. The normalized weighted average is \([400(5/7)+300(20/21)+300(10/7)]/1{,}000=(2{,}000/7+2{,}000/7+3{,}000/7)/1{,}000=1\). The weighted average class rate is \([400(450)+300(600)+300(900)]/1{,}000=\$630\), equal to the adjusted base rate.
Common mistake avoided: leaving the base at $600 after changing the relativities would reduce every class rate by the factor \(1/1.05\). The base must be multiplied by the same normalization factor used to divide the relativities.
Weights Must Reflect the Portfolio Mix
Policy counts, claim counts, and equal class weights are not substitutes for exposure weights unless they are the specified basis for the target mix. For example, a class with many short-term policies might contribute less earned exposure than a class with fewer full-year policies. Using the wrong weights produces a different normalization factor and therefore a different adjusted base rate.
The relativities may still be useful comparisons when the portfolio mix changes, but the average relativity and corresponding normalized base rate depend on the mix. If the target mix changes, recalculate \(c\) using the new weights. Do not interpret the resulting change in average rate as a change in every class’s underlying indicated rate: the change can result from the portfolio having shifted toward higher- or lower-rated classes.
Worked Example: Exposure Weights Versus Policy Counts
A hypothetical commercial package has three classes with relativities 0.80, 1.50, and 2.00 and a base rate of $1,000 per exposure unit. The classes have 700, 200, and 100 exposure units, but their policy counts are 500, 300, and 200. Normalize to the stated exposure mix and find the adjusted base rate and class rates. Round normalized relativities to six decimals.
Method. Use exposure units, not policy counts, because the requested mix is an exposure mix. The count data are a deliberate trap: they would assign different weights to the classes.
Total exposure is \(700+200+100=1{,}000\). The weighted relativity total is \(700(0.80)+200(1.50)+100(2.00)=560+300+200=1{,}060\). Thus \(c=1{,}060/1{,}000=1.06\).
The normalized relativities are \(0.80/1.06=40/53=0.754717\), \(1.50/1.06=75/53=1.415094\), and \(2.00/1.06=100/53=1.886792\), rounded to six decimals. The adjusted base rate is \(B'=1{,}000(1.06)=\$1{,}060\).
The original class rates are \(1{,}000(0.80)=\$800\), \(1{,}000(1.50)=\$1{,}500\), and \(1{,}000(2.00)=\$2{,}000\). The exact normalized factors give \(1{,}060(40/53)=\$800\), \(1{,}060(75/53)=\$1{,}500\), and \(1{,}060(100/53)=\$2{,}000\). Their exposure-weighted average is \([700(800)+200(1{,}500)+100(2{,}000)]/1{,}000 =(560{,}000+300{,}000+200{,}000)/1{,}000=\$1{,}060\).
Check and common mistake avoided: the weighted average normalized relativity is \([700(40/53)+200(75/53)+100(100/53)]/1{,}000 =(28{,}000+15{,}000+10{,}000)/(53{,}000)=1\). Using policy counts instead would give \(c=[500(0.80)+300(1.50)+200(2.00)]/1{,}000=1.25\), which is not the requested exposure-weighted normalization.
Normalization Can Change When the Mix Changes
For a fixed set of class relativities, the class-to-class comparisons do not change when the portfolio mix changes. The normalization factor does change, because it is an average under a particular mix. Consequently, the normalized relativities and adjusted base rate also change together. Each class’s indicated rate remains the same if the original base rate and original relativities have not otherwise changed.
Worked Example: Re-normalize After a Portfolio Shift
A hypothetical rating plan has a base rate of $700 and class relativities 0.80, 1.10, and 1.60. The original mix has 500, 300, and 200 exposure units. A proposed target mix has 200, 300, and 500 exposure units. Calculate the normalization factor and adjusted base for each mix, and verify that the class rates do not change. Round normalized relativities to six decimals.
Method. Calculate each mix’s weighted average relativity separately. Reusing the old normalization factor for the proposed mix would normalize to the wrong portfolio.
For the original mix, total exposure is \(500+300+200=1{,}000\). The weighted relativity total is \(500(0.80)+300(1.10)+200(1.60)=400+330+320=1{,}050\). Thus \(c_{\text{old}}=1.05\), and the adjusted base is \(700(1.05)=\$735\). The normalized relativities are \(0.80/1.05=0.761905\), \(1.10/1.05=1.047619\), and \(1.60/1.05=1.523810\), rounded to six decimals.
For the proposed mix, the weighted relativity total is \(200(0.80)+300(1.10)+500(1.60)=160+330+800=1{,}290\), again over 1,000 exposure units. Thus \(c_{\text{new}}=1.29\), and the adjusted base is \(700(1.29)=\$903\). The new normalized relativities are \(0.80/1.29=80/129=0.620155\), \(1.10/1.29=110/129=0.852713\), and \(1.60/1.29=160/129=1.240310\), rounded to six decimals.
The original class rates are $560, $770, and $1,120. With the proposed-mix normalization, the exact products are \(903(80/129)=7(80)=\$560\), \(903(110/129)=7(110)=\$770\), and \(903(160/129)=7(160)=\$1{,}120\). The new portfolio’s weighted average rate is \([200(560)+300(770)+500(1{,}120)]/1{,}000 =(112{,}000+231{,}000+560{,}000)/1{,}000=\$903\).
Check and common mistake avoided: using the old factor 1.05 for the proposed mix would make its weighted average normalized relativity \(1.29/1.05\), not 1.00. Recalculate from the target weights whenever the mix changes. The portfolio average rate changes from $735 to $903 because the hypothetical mix contains more exposure in the higher-relativity class; the individual class rates remain unchanged.
Interpretation and Practical Checks
An average relativity of 1.00 is meaningful only relative to the stated weights. It does not mean that each class has a relativity of 1.00, nor that the base rate is an unconditional average for every possible portfolio. It is the average rate for the specified mix and rating basis under the one-factor structure described here.
A useful audit is to compare both sides of the rate-preservation identity for every class:
Use unrounded values for the calculation and round only for presentation. If rounded normalized factors are used in production, their displayed products may differ slightly from the unrounded indicated class rates. A second check is to calculate the exposure-weighted average of the normalized relativities; it should be 1.00, apart from any small difference caused by displayed rounding.
- Use the target mix: a normalization to one portfolio does not automatically apply to another.
- Use the specified weight basis: do not substitute policy counts for exposure units without justification.
- Adjust both components: divide the relativities by \(c\) and multiply the base rate by \(c\).
- Check class rates: normalization should not alter the rate \(Br_j\) for any class.
This scaling is useful when a rating table needs a portfolio-centered reference while preserving the underlying indicated rates. Its main limitation is that the weights must match the rating structure and target portfolio being summarized. If multiple rating factors are involved, a portfolio average rate depends on the full rating cells and their exposures; averaging one factor’s relativities alone should not be treated as the average of all rates without accounting for the other factors.
Check Your Understanding
For each problem, use the stated exposure mix and keep full precision until the final answer.
- A hypothetical plan has relativities 0.80, 1.10, and 1.60 and respective exposures 500, 300, and 200. Find the weighted average relativity and normalized relativities to six decimals. (Answer: \(c=1.05\); normalized relativities \(0.761905,\ 1.047619,\ 1.523810\).)
- A base rate is $800. Relativities are 0.70, 1.30, and 1.80, with 60%, 25%, and 15% of exposure in the classes. Find the weighted average relativity and adjusted base rate. (Answer: \(c=1.015\); adjusted base rate \(=\$812.00\).)
- A hypothetical plan has a $450 base rate, relativities 0.90, 1.00, and 1.50, and exposures 400, 400, and 200. Find the normalization factor, adjusted base, and the original and normalized class rates. (Answer: \(c=1.06\); adjusted base \(=\$477.00\); class rates remain \(\$405.00,\ \$450.00,\ \$675.00\).)
- A plan has relativities 0.80, 1.40, and 2.00. Policy counts are 500, 300, and 200, but exposure units are 600, 300, and 100. With a $900 base rate, normalize to exposure units. (Answer: \(c=1.10\); adjusted base \(=\$990.00\); normalized relativities \(0.727273,\ 1.272727,\ 1.818182\).)
- A $720 base rate applies to relativities 0.60, 1.20, and 1.50. The target mix is 100, 500, and 400 exposure units. Find the weighted average relativity and adjusted base, then verify the portfolio average rate using the original class rates. (Answer: \(c=1.26\); adjusted base \(=\$907.20\); original class rates are \(\$432.00,\ \$864.00,\ \$1{,}080.00\), whose weighted average is \(\$907.20\).)