Relativities, Base Rates, And Portfolio Revenue

From Fitted Factors to a Rate Table

A multiplicative rating model can assign different predicted rates to different risks while retaining a common scale. Turning those predictions into a usable rate table requires specifying that scale and deciding how the relativities will be centered. A base rate and normalized relativities make the table interpretable: each relativity expresses a risk’s rate relative to the portfolio average rate under a stated exposure mix.

The normalization convention matters. With exposure weights, the relativities average to one across the portfolio, so the base rate equals the average premium per exposure unit. Changing the exposure mix can change that average and the corresponding normalized factors, even if the underlying fitted factors have not changed.

Definition: Let rating cell $j$ have $e_j$ exposure units and a positive fitted factor $r_j$, for $j=1,\ldots,J$. The total exposure is $E=\sum_j e_j$, and the exposure-weighted mean fitted factor is $m=\sum_j e_j r_j/E$. The normalized relativity is $q_j=r_j/m$. Thus $\sum_j e_jq_j/E=1$. If the selected total portfolio premium is $T$, the normalized base rate is $B=T/E$, and the rate for cell $j$ is $Bq_j$.

An exposure unit is the unit to which the rate applies, such as one vehicle-year or one policy-year. It is not automatically the same as one policy: a policy in force for half a year might contribute 0.5 policy-year. Use weights consistent with the exposure basis of the rate.

In a multiplicative GLM with a log link, fitted rates can be written as $c r_j$, where $c$ is a scale and $r_j$ is the product of the applicable rating factors. The factors’ coding may determine which cell is assigned factor one, but that reference cell need not be the portfolio average. The normalization above changes the scale and recenters the factors without changing any cell’s fitted rate.

Why Exposure-Weighted Normalization Preserves Revenue

The calculation has two connected parts. First, normalize the factors using the specified exposure mix. Then choose the base rate as the target premium divided by total exposure. Together these steps guarantee that the resulting rate table reproduces the target total.

\sum_j e_j(Bq_j) =\frac{T}{E}\sum_j e_j\frac{r_j}{m} =\frac{T}{E}\frac{\sum_j e_jr_j}{m} =\frac{T}{E}\frac{Em}{m} =T.

The key equality is $\sum_j e_jr_j=Em$, which follows directly from the definition of the weighted mean $m$. This is an algebraic calibration to the selected portfolio premium; it does not establish that the selected target is adequate or that the factors are appropriate. Those are separate actuarial judgments.

The same result can be seen by starting with the unnormalized GLM scale. To match target premium $T$, set $c=T/\sum_j e_jr_j$. Since $\sum_j e_jr_j=Em$, this is $c=T/(Em)$. The normalized base rate satisfies $B=c m=T/E$, and $Bq_j=(cm)(r_j/m)=cr_j$. Normalization therefore preserves every fitted cell rate as well as the portfolio total.

Worked Example: Normalize Three Cells to a Target Premium

A hypothetical portfolio has 500 exposure units in cell A, 200 in cell B, and 300 in cell C. A fitted multiplicative model gives factors 0.60, 1.00, and 1.50, respectively. The selected total premium is $390,000. Find the exposure-weighted normalized relativities, base rate, cell rates, and cell premiums. State the method and check the total.

Method. Calculate the exposure-weighted mean of the raw factors, divide each raw factor by that mean, and set the normalized base rate equal to target premium divided by total exposure.

Step 1: Total exposure and weighted factor mean. Total exposure is $E=500+200+300=1{,}000$. The weighted factor total is $500(0.60)+200(1.00)+300(1.50)=300+200+450=950$. Therefore $m=950/1{,}000=0.95$.

Step 2: Normalize the factors. The relativities are $q_A=0.60/0.95=12/19\approx0.6315789$, $q_B=1.00/0.95=20/19\approx1.0526316$, and $q_C=1.50/0.95=30/19\approx1.5789474$. Their exposure-weighted average is $(500q_A+200q_B+300q_C)/1{,}000=(300+200+450)/(0.95\times1{,}000)=1$.

Step 3: Calculate the base rate and cell rates. The base rate is $B=390{,}000/1{,}000=\$390$ per exposure unit. The cell rates are $390(12/19)=\$246.315789\ldots$, $390(20/19)=\$410.526316\ldots$, and $390(30/19)=\$615.789474\ldots$.

Step 4: Calculate cell premiums and check. Using unrounded rates, the premiums are $500(390)(0.60/0.95)=\$123{,}157.8947\ldots$, $200(390)(1.00/0.95)=\$82{,}105.2632\ldots$, and $300(390)(1.50/0.95)=\$184{,}736.8421\ldots$. Their unrounded sum is exactly $390,000. Rounded to cents, the amounts are $123,157.89, $82,105.26, and $184,736.84; these displayed amounts sum to $389,999.99 because of independent rounding. The target is preserved before rounding.

Common mistake avoided: The first cell’s premium is not $123,000: $500\times390\times(0.60/0.95)=\$123{,}157.8947\ldots$. Retaining the full factor ratio until the final rounding step prevents this arithmetic error.

Revenue Allocation and Portfolio Mix

A rate table allocates the total premium across cells according to their exposure-weighted fitted factors. The share of total premium assigned to cell $j$ is $e_jr_j/\sum_k e_kr_k$. A cell with a higher factor receives a higher rate per exposure unit, but its share of total premium also depends on how much exposure is in that cell.

This is why normalization is tied to a portfolio mix. If the exposures change, the weighted mean $m$ may change. For a fixed target average rate, the normalized base rate remains that average, but the normalized relativities change because the portfolio average of the raw factors has changed. If the target is instead a fixed total premium, a change in total exposure also changes the average rate $T/E$.

Worked Example: Preserve a Target Across an Uneven Exposure Mix

A hypothetical book has 800 exposure units in cell A, 400 in cell B, and 200 in cell C. Its fitted factors are 0.80, 1.10, and 1.70. The selected premium target is $1,200,000. Find the weighted mean factor, normalized relativities, base rate, and segment premiums. Check the revenue total.

Method. Use exposure units—not an unweighted average of the three factors—to calculate the mean factor. Normalize the factors and then apply the target average rate to each normalized relativity.

Step 1: Weighted mean factor. Total exposure is $E=800+400+200=1{,}400$. The weighted factor total is $800(0.80)+400(1.10)+200(1.70)=640+440+340=1{,}420$. Thus $m=1{,}420/1{,}400=71/70\approx1.0142857$.

Step 2: Normalized relativities. The factors become $q_A=0.80/(71/70)=56/71\approx0.7887324$, $q_B=1.10/(71/70)=77/71\approx1.0845070$, and $q_C=1.70/(71/70)=119/71\approx1.6760563$. Their exposure-weighted mean is one because the weighted sum of the raw factors is $1{,}420$ and the normalization divides each by $1{,}420/1{,}400$.

Step 3: Base rate and cell rates. The base rate is $B=1{,}200{,}000/1{,}400=\$857.142857\ldots$ per exposure unit. The equivalent unnormalized scale is $c=1{,}200{,}000/1{,}420=\$845.070423\ldots$; $B=cm$. The cell rates $Bq_j=cr_j$ are approximately $676.056338, $929.577465, and $1,436.619718.

Step 4: Segment premiums. Cell A receives $800(676.056338\ldots)=\$540{,}845.0704\ldots$; cell B receives $400(929.577465\ldots)=\$371{,}830.9859\ldots$; and cell C receives $200(1{,}436.619718\ldots)=\$287{,}323.9437\ldots$. The unrounded premiums sum to $1,200,000. Rounded to cents they are $540,845.07, $371,830.99, and $287,323.94, which also sum to $1,200,000.00.

Check. The segment premium proportions are $640/1{,}420$, $440/1{,}420$, and $340/1{,}420$; they sum to one and match the premiums above. An unweighted mean of the factors would be $(0.80+1.10+1.70)/3=1.20$, which ignores the exposure distribution and would produce a different, incorrect set of normalized relativities for this book.

Use the Rate’s Exposure Basis

Normalization weights must match the unit used in the rate. If the rate is stated per vehicle-year, a policy covering two vehicles for half a year may contribute one vehicle-year, not merely one policy. Weighting by policy count in that situation can shift the relativities away from a weighted average of one over the actual exposures being priced.

The next example separates policy counts from exposure units. It also shows how using the wrong weights can cause a revenue shortfall or excess even when the normalization arithmetic itself is performed correctly.

Worked Example: Policy Counts Are Not Exposure Weights

A hypothetical insurer rates vehicle-years. Cell A has 200 policies averaging 0.5 vehicle-year each, cell B has 100 policies averaging 1 vehicle-year, and cell C has 50 policies averaging 2 vehicle-years. The factors are 0.50, 1.00, and 2.00. The target premium is $350,000. Find the proper normalized table and quantify the error from weighting by policy count.

Method. Convert each cell’s policy count to vehicle-year exposure first. Normalize using those exposures, because the rate applies per vehicle-year.

Step 1: Calculate exposures and factor totals. The exposures are $e_A=200(0.5)=100$, $e_B=100(1)=100$, and $e_C=50(2)=100$ vehicle-years. Thus $E=300$, and $\sum e_jr_j=100(0.50)+100(1.00)+100(2.00)=50+100+200=350$. The weighted mean factor is $m=350/300=7/6$.

Step 2: Normalize and set the base rate. The relativities are $q_A=(1/2)/(7/6)=3/7\approx0.4285714$, $q_B=1/(7/6)=6/7\approx0.8571429$, and $q_C=2/(7/6)=12/7\approx1.7142857$. The base rate is $B=350{,}000/300=\$1{,}166.666667$ per vehicle-year.

Step 3: Calculate premiums and check. The cell rates are $Bq_A=\$500$, $Bq_B=\$1{,}000$, and $Bq_C=\$2{,}000$ per vehicle-year. The premiums are $100(500)=\$50{,}000$, $100(1{,}000)=\$100{,}000$, and $100(2{,}000)=\$200{,}000$. They sum to the $350,000 target.

Wrong calculation and correction. If policy counts are used as weights, the count-weighted factor mean is $[200(0.50)+100(1.00)+50(2.00)]/350=300/350=6/7$, not the proper $7/6$. Dividing the factors by $6/7$ gives count-weighted relativities with average one, but their exposure-weighted sum is $350/(6/7)=408.3333$, not 300. Applying the $1,166.666667 base rate to those factors would produce about $408.3333(1{,}166.666667)=\$476{,}388.89$, not $350,000. The error is using policy counts when the premium rate is per vehicle-year.

Practical Checks and Common Traps

Before treating a normalized table as an implementation-ready rate plan, verify that the weights, target, and scale refer to the same basis. In particular, distinguish a fixed total premium from a fixed average rate. If exposure is $E$, a total premium target $T$ implies average rate $T/E$; it does not imply that the base rate is the unnormalized GLM scale $c$.

Do not assume the reference cell is average. A factor fixed at one by GLM coding is a reference level, not necessarily a portfolio-average risk.
Do not average factors equally unless the intended weights are equal. Equal weighting is appropriate only when the cells have equal exposure or when an equal-cell average is specifically intended.
Do not mix policy counts and exposure units. Use the units underlying the quoted rate and premium calculation.
Do not round factors too early. Keep unrounded weighted means and relativities through the premium reconciliation; round the final displayed rates or premiums.
Do not confuse calibration with adequacy. A table can exactly reproduce a selected premium target while that target or the underlying fitted factors still require actuarial review.

Key takeaway: For exposure weights $e_j$, normalize each fitted factor by $m=\sum_j e_jr_j/\sum_j e_j$, then set the base rate to $T/\sum_j e_j$. The resulting table has an exposure-weighted relativity of one and reproduces the selected total premium before rounding.

Check Your Understanding

For each problem, use the stated exposure basis and retain unrounded values until the final check.

A hypothetical portfolio has 400 exposure units with factor 0.75 and 600 with factor 1.25. The target premium is $420,000. Find the normalized relativities, base rate, and cell premiums. (Answer: Weighted mean factor 1.05; relativities 0.7143 and 1.1905; base rate $420 per exposure unit; premiums $120,000 and $300,000.)
A book has 100, 250, and 150 exposure units with factors 0.80, 1.20, and 1.60. Its target is $310,000. Find the weighted mean factor and normalized relativities. (Answer: Weighted mean factor 1.24; relativities 0.6452, 0.9677, and 1.2903.)
A rate is per equipment-year. Two cells have 80 policies averaging 0.5 equipment-year and 40 policies averaging 1 equipment-year. Their factors are 0.60 and 1.40, and the target premium is $44,000. Find the correct base rate and normalized relativities. (Answer: Exposures 40 and 40 equipment-years; weighted mean factor 1.00; relativities 0.6000 and 1.4000; base rate $550 per equipment-year.)
Two cells have 50 and 100 exposure units, with raw factors 2 and 4. The target premium is $180,000. Find the normalized base rate and rates for both cells. (Answer: Weighted mean factor $500/150=10/3$; normalized relativities 0.6000 and 1.2000; base rate $1,200; cell rates $720 and $1,440.)
A portfolio has 300 exposure units in cell A and 200 in cell B, with factors 0.90 and 1.40. A proposed table uses the equally weighted mean of the two factors for normalization and sets its base rate to the target average rate of $700. What total premium would that table produce for a target of $350,000, and does it preserve the target? (Answer: Equal-cell mean 1.15 gives relativities 0.7826 and 1.2174; their exposure-weighted sum is approximately 478.2609. The table produces approximately $334,782.61, so it does not preserve the target; the exposure-weighted mean factor is 1.10.)

What You'll Learn