Fixed-Mix On-Leveling Across Groups

Why Standardize the Exposure Mix?

A portfolio’s average premium per exposure can change for two distinct reasons: premium rates within groups may change, or the portfolio may contain a different proportion of exposures in each group. For example, if a higher-premium territory grows as a share of a book, the overall average premium can rise even when no territory’s rate changes. Comparing aggregate averages alone can therefore confuse a mix change with a rate-level change.

Fixed-mix on-leveling addresses this comparison problem. It calculates each period’s group-level premium rates, then averages those rates using the same selected group weights in every period. The standardization does not change the underlying group rates; it holds the comparison mix constant.

Definition: For group $g$ in period $t$, let $P_{g,t}$ be earned premium and $E_{g,t}$ be the corresponding exposure. The group-level average premium rate is $r_{g,t}=P_{g,t}/E_{g,t}$, provided $E_{g,t}>0$. A fixed-mix average for period $t$ is $R_t^*=\sum_g w_g^*r_{g,t}$, where the chosen standard weights satisfy $w_g^*\geq 0$ and $\sum_g w_g^*=1$. The weights are held constant across the periods being compared.

Weights may represent a selected base-period mix, a target portfolio mix, or another explicitly chosen standard. If the weights are based on exposures, $w_g^*=E_g^*/\sum_h E_h^*$, where $E_g^*$ is the standard number of exposures assigned to group $g$. The choice should suit the comparison; there is not one universally correct mix.

This method is applied after the group-level premium measures have been put on a comparable basis. If an earlier period’s premiums need to be adjusted to a common rate level, apply the appropriate on-level factors to that period’s group premiums (or group rates) before calculating the fixed-mix average. A mix adjustment and a rate-level adjustment do different jobs: one standardizes group weights, while the other adjusts premiums for rate changes.

How the Fixed-Mix Comparison Works

The observed portfolio average in period $t$ uses that period’s actual exposure weights, $w_{g,t}=E_{g,t}/\sum_h E_{h,t}$. It is therefore $R_t=\sum_g w_{g,t}r_{g,t}$. By contrast, $R_t^*$ uses the same $w_g^*$ in every period. A comparison such as $R_2^*/R_1^*$ describes the relative change in the standardized average, while $R_2^*-R_1^*$ gives its dollar-per-exposure change.

R_t=\frac{\sum_g P_{g,t}}{\sum_g E_{g,t}} =\sum_g w_{g,t}r_{g,t}, \qquad R_t^*=\sum_g w_g^*r_{g,t}, \qquad \text{relative change}=\frac{R_2^*}{R_1^*}-1.

The distinction follows directly by dividing total premium by total exposure. Since $P_{g,t}=E_{g,t}r_{g,t}$, total premium is $\sum_g E_{g,t}r_{g,t}$. Dividing by $\sum_g E_{g,t}$ gives $\sum_g (E_{g,t}/\sum_h E_{h,t})r_{g,t}$, which weights each group rate by its actual share of exposure. Substituting fixed weights instead answers a different question: what would the average be if every period had the same chosen mix?

This standardization removes the arithmetic effect of changing group proportions, but it does not by itself prove that a rate change caused every difference in group rates. Changes in coverage, exposure definitions, underwriting, or other factors can also affect observed premiums per exposure. Interpretation depends on the groups and measures being sufficiently comparable across periods.

Worked Example: A Changing Mix With Stable Group Rates

A hypothetical insurer compares two periods for two territories. In Period 1, Territory A has 80 exposures and $48,000 earned premium; Territory B has 20 exposures and $18,000 earned premium. In Period 2, A has 20 exposures and $12,000 earned premium; B has 80 exposures and $72,000 earned premium. Calculate each actual portfolio average and each fixed-mix average using equal weights. What does the comparison show?

Method. Calculate each territory’s premium per exposure, then average those group rates using actual weights for the portfolio results and 50%/50% weights for both fixed-mix results. The common mistake to avoid is attributing the aggregate increase to higher group rates without checking the group mix.

Step 1: Group rates. In Period 1, A’s rate is $$48{,}000/80=$600$, and B’s rate is $$18{,}000/20=$900$. In Period 2, A’s rate is $$12{,}000/20=$600$, and B’s rate is $$72{,}000/80=$900$. Neither group rate changed.

Step 2: Actual portfolio averages. Period 1 has 100 exposures and $66,000 premium, so its average is $$66{,}000/100=$660$. Period 2 has 100 exposures and $84,000 premium, so its average is $$84{,}000/100=$840$. The unadjusted average increased by $$840/$660-1=0.272727\ldots$, or 27.27% to two decimal places.

Step 3: Fixed-mix averages. With equal weights, Period 1 is $0.5($600)+0.5($900)=$750$. Period 2 is $0.5($600)+0.5($900)=$750$. The fixed-mix change is $$750/$750-1=0\%$.

Check. Territory B’s share increased from 20% to 80%. Since B has the higher rate, the actual portfolio average rose even though both group rates stayed constant. The fixed-mix result correctly shows no change in the standardized group-rate average.

Applying Group-Specific On-Level Factors

Sometimes the group rates being compared are not yet at a common rate level. In that case, apply the specified on-level factor to the applicable group premium or rate, and then use the fixed weights. The factor and its direction must match the intended comparison date; do not apply a portfolio-wide factor automatically if the group-level factors differ.

Worked Example: On-Leveling Before Standardizing the Mix

A hypothetical two-territory book has the following Period 1 data: Territory A earned $30,000 on 60 exposures, and Territory B earned $32,000 on 40 exposures. To place Period 1 at the comparison rate level, apply factors of 1.10 to A and 1.10 to B. In Period 2, A earned $16,500 on 30 exposures and B earned $61,600 on 70 exposures. Compare the periods using Period 1’s 60%/40% mix as the standard. Also calculate the unadjusted actual average in Period 2.

Method. On-level Period 1 by group, calculate the group rates for both periods, and weight each period with 60%/40%. The trap is to use Period 2’s actual 30%/70% mix in the standardized comparison.

Step 1: Period 1 group rates and on-leveling. A’s observed rate is $$30{,}000/60=$500$; after the factor, it is $$500(1.10)=$550$. B’s observed rate is $$32{,}000/40=$800$; after the factor, it is $$800(1.10)=$880$. Equivalently, the on-level premiums are $$30{,}000(1.10)=$33{,}000$ and $$32{,}000(1.10)=$35{,}200$.

Step 2: Period 2 group rates. A’s rate is $$16{,}500/30=$550$. B’s rate is $$61{,}600/70=$880$.

Step 3: Fixed-mix comparison. Period 1’s standardized average is $0.60($550)+0.40($880)=$330+$352=$682$. Period 2’s standardized average is also $0.60($550)+0.40($880)=$682$. Thus the fixed-mix change is $0\%$.

Step 4: Actual Period 2 average. Period 2 has $78,100 total premium and 100 exposures, so its actual average is $781 per exposure. Period 1’s actual, not on-leveled, average was $$62{,}000/100=$620$. Those two actual averages are not a like-for-like rate-level comparison: Period 1 needs its on-level adjustment, and the actual mix also differs. The comparison after on-leveling and standardizing shows equal group rates at the common level.

Check. The on-leveled Period 1 group rates equal the Period 2 group rates, so any fixed mix gives the same standardized averages. The Period 2 actual average of $781 is higher because its mix is 30% A and 70% B, rather than the 60% A and 40% B used for standardization.

Group-Specific Adjustments and Interpretation

A fixed-mix result can also combine group-specific adjustments with different group rates. The calculation remains a weighted average, but each input must represent the intended comparison basis. The following example illustrates why a substantial change in actual portfolio average can coexist with a small change in fixed-mix average.

Worked Example: Different On-Level Factors by Group

A hypothetical insurer has two groups. In Period 1, Group A’s premium rate is $400 and Group B’s is $1,000. Factors of 1.08 for A and 0.95 for B convert these rates to the comparison level. In Period 2, the rates at that comparison level are $450 and $900, respectively. Period 1’s actual mix is 80% A and 20% B; Period 2’s is 20% A and 80% B. Compare the periods at a standard mix of 70% A and 30% B, then compare with the actual averages.

Method. Adjust each Period 1 group rate separately, then apply the same 70%/30% weights to both periods. The common error is to multiply the overall Period 1 average by one factor even though the given factors differ by group.

Step 1: On-level Period 1 rates. Group A becomes $$400(1.08)=$432$. Group B becomes $$1{,}000(0.95)=$950$.

Step 2: Fixed-mix averages. Period 1’s standardized average is $0.70($432)+0.30($950)=$302.40+$285=$587.40$. Period 2’s is $0.70($450)+0.30($900)=$315+$270=$585.00$. The relative change is $$585/$587.40-1=-0.004086\ldots$, or a decrease of 0.41% to two decimal places.

Step 3: Actual averages. Period 1’s on-leveled actual average is $0.80($432)+0.20($950)=$345.60+$190=$535.60$. Period 2’s actual average is $0.20($450)+0.80($900)=$90+$720=$810$. Comparing these actual averages gives $$810/$535.60-1=0.512323\ldots$, or an increase of 51.23% to two decimal places.

Check. The two comparisons differ because the actual shares of the higher-rate group B changed substantially, from 20% to 80%. At the fixed 70%/30% mix, the standardized average decreased slightly; the actual portfolio average increased. These are different measures answering different questions.

Assumptions, Choices, and Common Traps

Fixed-mix standardization is an arithmetic comparison, not a causal model. Its interpretation is strongest when group definitions are consistent, exposures are measured on a comparable basis, and group premium rates are calculated for comparable coverage and periods. A group with no exposure in a period has no observed rate for that period; the calculation then needs a defensible estimate or a different comparison design, rather than silently treating the missing rate as zero.

The chosen standard weights affect the dollar average. Two analysts can obtain different standardized averages by selecting different mixes, while both calculations are correct for their stated standards. If the purpose is to quantify change, use the same weights in both periods and clearly state them. Do not compare a fixed-mix result in one period with an actual-mix result in another.

Key takeaway: First make group-level premium rates comparable, including any needed on-level adjustment. Then apply one common set of exposure weights to every period. The resulting comparison isolates changes in the weighted group rates from changes in the portfolio’s exposure mix.

Check Your Understanding

For each problem, use the stated standard mix and distinguish the standardized result from any actual-mix average requested.

A book has two groups with rates of $300 and $700 per exposure. The actual mix is 90% in the first group, while the standard mix is 25% in the first group and 75% in the second. Find the fixed-mix average and the actual average. (Answer: fixed-mix average $600; actual average $340 per exposure.)
Period 1 group rates are $500 and $900, with a standard mix of 60% and 40%. Period 2 rates are $550 and $990. Find both fixed-mix averages and the relative change. (Answer: $660 and $726 per exposure; increase of 10.00%.)
Period 1 group rates are $250 and $800. Apply on-level factors of 1.20 and 0.90, respectively, and then use standard weights of 40% and 60%. Find the on-leveled fixed-mix average. (Answer: $552 per exposure.)
Three group rates are $400, $600, and $1,100. The standard weights are 20%, 50%, and 30%; the actual weights are 50%, 20%, and 30%. Find both averages and their difference. (Answer: fixed-mix average $710; actual average $650; fixed-mix average is $60 higher.)
Two periods have rates of $450 and $750 in Period 1, and $495 and $825 in Period 2. Use a standard mix of 40% in the first group and 60% in the second. If Period 2’s actual mix is 80%/20%, find the fixed-mix change and Period 2’s actual average. (Answer: fixed-mix averages $630 and $693, a 10.00% increase; Period 2 actual average $561 per exposure.)

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