Why Separate Rate Changes from Mix Changes?
Total premium can change for several reasons at once. The average premium charged within rating groups may change, the distribution of exposures among those groups may shift, or the total number of exposures may rise or fall. Looking only at total premium does not reveal which of these occurred.
For example, if a hypothetical insurer writes a larger share of exposures in a group with higher premiums, its overall average premium can increase even when no group’s rate changes. Conversely, average premium can decrease despite rate increases if exposures shift toward lower-premium groups. A fixed-mix comparison separates these effects by asking what the later period’s group premiums would imply if the group distribution had stayed at its initial-period composition.
The groups must be defined consistently in both periods, and the exposure unit must be comparable: for example, one vehicle-year in each period, rather than vehicle-years in one period and policies in another. The group premiums should also be measured on a consistent basis, such as the same coverage and premium definition. The calculations below are accounting comparisons; they do not by themselves adjust for trend, changes in coverage, or other differences between periods.
The Fixed-Mix Decomposition
Use period 0 as the reference mix. First compare the period 1 group premiums using period 0 weights. This gives a fixed-mix rate index. Then compare the actual period 1 average premium, which uses period 1 weights, with that fixed-mix average. This second ratio is the mix effect under this specified order of comparison.
The exposure-volume ratio is \(E_1/E_0\). Together, these factors give an exact decomposition of the total-premium ratio:
The fixed-mix rate index is an aggregate comparison, not necessarily the rate change for every group. If group rate changes differ, the index summarizes them using the reference-period exposure shares. The mix effect is not a rate change: it reflects how the later period’s exposure shares redistribute exposure among groups with different period 1 premiums.
Why the Decomposition Holds
The justification follows by multiplying the factors and cancelling the intermediate average premiums. Since \(P_t=E_t\bar p_t\), the total-premium ratio is the exposure-volume ratio times the ratio of average premiums. Insert the fixed-mix average \(\sum_g w_{0g}p_{1g}\) into that average-premium ratio by multiplying and dividing by the same positive quantity.
The first ratio on the right compares premiums at a common mix, so it isolates the group-premium change under the chosen reference weights. The second compares two mixes at the same period 1 group premiums. This is why the product reproduces the total-premium change exactly, even when group rate changes and exposure shares both change.
The allocation between rate and mix depends on the order of comparison. The method above first changes group premiums at period 0 weights, then changes weights at period 1 premiums. Reversing that order can give different individual rate and mix factors because it assigns the interaction between rate and mix differently. The product of the factors still reconciles to the same overall average-premium change.
Worked Example: Equal Initial Shares and a Changed Mix
A hypothetical insurer has two rating groups. In period 0, each group has 100 exposure units. Premiums per exposure are $150 for Group A and $450 for Group B. In period 1, Group A has 60 exposures and Group B has 90; their premiums are $165 and $495, respectively. Calculate the exposure-volume ratio, fixed-mix rate index, mix effect, and total-premium ratio.
Method. Use the period 0 exposure shares for the rate comparison, then compare the actual period 1 average with the fixed-mix period 1 average.
The period 0 total exposure is \(100+100=200\), and the period 1 total is \(60+90=150\). Thus the exposure-volume ratio is \(150/200=0.7500\). The period 0 shares are \(w_{0A}=100/200=0.5\) and \(w_{0B}=0.5\). The period 1 shares are \(w_{1A}=60/150=0.4\) and \(w_{1B}=90/150=0.6\).
The period 0 average premium is \((0.5)(150)+(0.5)(450)=300\). At the period 1 premiums but with period 0 shares, the fixed-mix average is \((0.5)(165)+(0.5)(495)=330\). Therefore, the fixed-mix rate index is \(330/300=1.1000\).
The actual period 1 average premium is \((0.4)(165)+(0.6)(495)=66+297=363\). The mix effect is \(363/330=1.1000\). Total premium in period 0 is \(100(150)+100(450)=\$60{,}000\). In period 1 it is \(60(165)+90(495)=\$9{,}900+\$44{,}550=\$54{,}450\). The total-premium ratio is \(54{,}450/60{,}000=0.9075\).
Check. The factors multiply to \(0.7500(1.1000)(1.1000)=0.9075\), matching the direct total-premium ratio. The average premium rises because both group premiums rise and the share in the higher-premium group increases, while total exposures decline. A common mistake is to set the mix effect to 1.0000 merely because period 0 had equal group shares; the later shares are not equal, and the calculated mix effect is 1.1000.
Worked Example: A Shift Toward a Lower-Premium Group
A hypothetical program has 80 exposures in Group A and 120 in Group B in period 0. Their premiums are $200 and $500. In period 1 there are 150 exposures in Group A and 50 in Group B, with premiums of $220 and $450. Find the fixed-mix rate index, mix effect, and total-premium ratio.
Method. Compare group premiums first at the period 0 mix, then measure the effect of the period 1 shares at period 1 premiums.
Total exposure is 200 in each period. Period 0 shares are \(0.4\) and \(0.6\), so the period 0 average premium is \((0.4)(200)+(0.6)(500)=80+300=380\). Applying period 1 premiums to those same shares gives \((0.4)(220)+(0.6)(450)=88+270=358\). The fixed-mix rate index is \(358/380=0.9421\) to four decimal places.
Period 1 shares are \(150/200=0.75\) and \(50/200=0.25\). The actual period 1 average premium is \((0.75)(220)+(0.25)(450)=165+112.50=277.50\). The mix effect is \(277.50/358=0.7751\) to four decimal places.
Period 0 total premium is \(80(200)+120(500)=\$16{,}000+\$60{,}000=\$76{,}000\). Period 1 total premium is \(150(220)+50(450)=\$33{,}000+\$22{,}500=\$55{,}500\). Thus the total-premium ratio is \(55{,}500/76{,}000=0.7303\) to four decimal places. Exposure volume is unchanged, and \(1.0000(0.9421)(0.7751)\) is approximately \(0.7303\), with any tiny discrepancy due to rounding.
Check and trap. The period 1 mix shifts strongly toward Group A, whose period 1 premium is lower. That shift reduces the average premium. It would be incorrect to infer that all group rates rose because Group A’s premium rose: Group B’s premium fell from $500 to $450, which is why the fixed-mix index is below 1.
Worked Example: Three Groups and Unequal Rate Changes
For a hypothetical portfolio, period 0 exposures in Groups A, B, and C are 50, 30, and 20, with premiums of $100, $200, and $400. Period 1 exposures are 40, 40, and 20, with premiums of $110, $180, and $440. Calculate the fixed-mix rate index, mix effect, and total-premium ratio.
Method. Calculate the weighted average at the initial mix, the later premiums at that same mix, and the actual later average.
Both periods have 100 exposures. Period 0 shares are \(0.50\), \(0.30\), and \(0.20\). The period 0 average premium is \((0.50)(100)+(0.30)(200)+(0.20)(400)=50+60+80=190\). The fixed-mix period 1 average is \((0.50)(110)+(0.30)(180)+(0.20)(440)=55+54+88=197\). The fixed-mix rate index is \(197/190=1.0368\) to four decimal places.
Period 1 shares are \(0.40\), \(0.40\), and \(0.20\). The actual period 1 average is \((0.40)(110)+(0.40)(180)+(0.20)(440)=44+72+88=204\). The mix effect is \(204/197=1.0355\) to four decimal places.
Period 0 total premium is \(100(190)=\$19{,}000\), and period 1 total premium is \(100(204)=\$20{,}400\). The direct total-premium ratio is \(20{,}400/19{,}000=1.0737\) to four decimal places. The product \(1.0000(1.0368)(1.0355)\) also gives approximately \(1.0737\).
Check and trap. Group B’s premium decreases, while Groups A and C increase. The aggregate fixed-mix rate index is still above 1 because the weighted changes produce a net increase. Do not treat the index as a claim that every group’s rate rose by 3.68%; it is an aggregate measured at the period 0 weights.
Interpretation and Practical Checks
A fixed-mix comparison is useful when raw average premiums are being compared across periods with different group distributions. It gives a more focused answer to the question, “How would average premium have changed if the mix of exposures had remained as it was?” It does not explain why the mix changed, and it does not establish that the compared group premiums are otherwise equivalent.
The mix effect can be above, below, or equal to 1. It is above 1 when the period 1 distribution, evaluated at period 1 premiums, produces a higher average than the reference mix; it is below 1 when it produces a lower average. It equals 1 when those two weighted averages are equal, which can happen even if individual group shares change.
- Check the exposure basis. The same type of exposure and group definitions must be used in both periods.
- Keep volume separate. Exposure-volume change is \(E_1/E_0\), not part of either average-premium factor.
- Use the chosen weights consistently. In this decomposition, period 0 shares weight the rate comparison and period 1 shares determine the actual later average.
- Reconcile to total premium. The product of the volume, rate, and mix factors must equal \(P_1/P_0\), apart from rounding.
If group definitions change, a group has no comparable exposure in one period, or premiums are on different bases, the weighted comparison may be misleading or undefined. Resolve comparability before interpreting the factors. The comparison is a descriptive decomposition, not by itself a judgment about adequacy or a complete on-leveling calculation.
Check Your Understanding
For each problem, use period 0 shares for the fixed-mix rate index and period 1 premiums for the mix effect. Ratios in the answers are rounded to four decimal places where needed.
- A hypothetical portfolio has period 0 exposures of 40 and 60 in Groups A and B, with premiums of $100 and $300. Period 1 exposures are 30 and 90, with premiums of $110 and $330. Find the exposure-volume ratio, fixed-mix rate index, mix effect, and total-premium ratio. (Answer: \(1.2000\), \(1.1000\), \(1.1364\), and \(1.5000\), respectively.)
- In a hypothetical two-group program, period 0 has 100 exposures in each group, at premiums of $250 and $500. Period 1 has 80 and 120 exposures, at premiums of $275 and $550. Find the exposure-volume ratio, fixed-mix rate index, mix effect, and total-premium ratio. (Answer: \(1.0000\), \(1.1000\), \(1.0667\), and \(1.1733\), respectively.)
- A hypothetical portfolio has period 0 exposures of 60 and 40, with premiums of $100 and $300. Period 1 exposures are 20 and 80, with premiums of $120 and $270. Calculate the mix effect using the stated decomposition, and explain why it is not the same as comparing period 1 shares at period 0 premiums. (Answer: \(1.3333\); the stated mix effect uses period 1 premiums, whereas comparing period 1 shares at period 0 premiums gives a separate mix-first factor of \(260/180=1.4444\).)
- A hypothetical insurer reports total premiums of $48,000 on 160 exposures in period 0 and $54,000 on 180 exposures in period 1. What are the total-premium ratio, exposure-volume ratio, and average-premium ratio? Can the rate and mix effects be separated from these totals alone? (Answer: \(1.1250\), \(1.1250\), and \(1.0000\), respectively; no, group exposures and group premiums in both periods are needed to separate rate and mix effects.)