Separating Within-Group and Mix Effects

Why Separate Within-Group and Mix Effects?

An overall loss cost can change because claim experience changes within groups, because the portfolio shifts toward groups with different loss costs, or both. For example, an insurer might write a larger share of its exposures in a group that already has a higher loss cost, even if every group’s own frequency and severity remain unchanged. Separating these effects helps explain what drove the change rather than treating an overall average as if it described every group.

The decomposition requires the same group definitions in both periods. The groups must be mutually exclusive and collectively exhaustive, and exposures, claim counts, and losses must be measured on consistent bases. In particular, frequency and severity should refer to the same claim and loss scope, such as ultimate claim counts and ultimate losses. The methods below are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For period $t$ and fixed group $g$, let $E_{tg}$ be exposures, $N_{tg}$ be claims, and $L_{tg}$ be losses. Define frequency $f_{tg}=N_{tg}/E_{tg}$, severity $s_{tg}=L_{tg}/N_{tg}$ when $N_{tg}>0$, and group pure premium $c_{tg}=L_{tg}/E_{tg}=f_{tg}s_{tg}$. Let $w_{tg}=E_{tg}/\sum_g E_{tg}$ be the group’s share of exposures. The overall pure premium is $C_t=\sum_g w_{tg}c_{tg}$, measured as loss per exposure.

The identity $c_{tg}=f_{tg}s_{tg}$ follows by multiplying claims per exposure by losses per claim: $(N_{tg}/E_{tg})(L_{tg}/N_{tg})=L_{tg}/E_{tg}$. If a group has zero claims, its observed severity is undefined, but its observed pure premium is still zero. A projected severity for such a group requires a separately selected estimate; do not divide by zero or treat an undefined observed severity as zero.

An Exact Decomposition

Write the change in overall pure premium as a within-group effect plus an exposure-mix effect. Define midpoint values $\bar{w}_g=(w_{1g}+w_{0g})/2$ and $\bar{c}_g=(c_{1g}+c_{0g})/2$. Then the following decomposition is exact:

C_1-C_0 = \sum_g \bar{w}_g(c_{1g}-c_{0g}) + \sum_g \bar{c}_g(w_{1g}-w_{0g}).

The first sum is the within-group effect: it measures group pure-premium changes using midpoint exposure shares. The second is the mix effect: it measures changes in exposure shares using midpoint group pure premiums. A positive effect increases the overall pure premium; a negative effect reduces it.

Justification: For each group, expand the two midpoint products: $\bar{w}_g(c_{1g}-c_{0g})+\bar{c}_g(w_{1g}-w_{0g})$. Substituting the midpoint definitions and distributing gives $w_{1g}c_{1g}-w_{0g}c_{0g}$; the cross-products cancel. Summing over groups gives $\sum_g w_{1g}c_{1g}-\sum_g w_{0g}c_{0g}=C_1-C_0$. Thus the two effects add exactly to the observed change.

The midpoint method divides the interaction between changing weights and changing group costs symmetrically. Another valid convention is to change group costs first using period 0 weights, then change weights using period 1 costs:

C_1-C_0 = \sum_g w_{0g}(c_{1g}-c_{0g}) + \sum_g c_{1g}(w_{1g}-w_{0g}).

That sequential decomposition is also exact, but it assigns the interaction to the second step. State which convention is used when reporting results; do not combine pieces from different conventions.

Separating Frequency and Severity Within Groups

A within-group pure-premium change can itself be attributed to frequency and severity. For one fixed group, let $\Delta f_g=f_{1g}-f_{0g}$, $\Delta s_g=s_{1g}-s_{0g}$, $\bar{f}_g=(f_{1g}+f_{0g})/2$, and $\bar{s}_g=(s_{1g}+s_{0g})/2$. Then:

c_{1g}-c_{0g} = \bar{s}_g\Delta f_g+\bar{f}_g\Delta s_g.

To verify the identity, expand the right-hand side: \[ \frac{s_{1g}+s_{0g}}{2}(f_{1g}-f_{0g}) + \frac{f_{1g}+f_{0g}}{2}(s_{1g}-s_{0g}) =f_{1g}s_{1g}-f_{0g}s_{0g}. \] The frequency contribution is $\bar{s}_g\Delta f_g$; the severity contribution is $\bar{f}_g\Delta s_g$. Weight each contribution by $\bar{w}_g$ to obtain its contribution to the overall within-group effect. These midpoint attributions are symmetric and add exactly. They are an accounting decomposition, not proof that frequency caused severity to change or vice versa.

Calculate period-specific group quantities.
For each fixed group, calculate frequency, severity, group pure premium, and exposure share using compatible data.

Check the overall pure premiums.
Calculate $C_0=\sum_g w_{0g}c_{0g}$ and $C_1=\sum_g w_{1g}c_{1g}$, then find $C_1-C_0$.

Calculate within-group and mix effects.
Use midpoint weights for group cost changes and midpoint costs for exposure-share changes. Verify that the two effects sum to the overall change.

Break down the within-group effect if needed.
Use midpoint severity and frequency for the within-group frequency and severity contributions, and check that they sum to the within-group effect.

Worked Decompositions

Worked Example: Frequency, Severity, and Mix in Two Groups

A hypothetical insurer tracks urban and rural exposures. In period 0, the exposure shares are 0.60 urban and 0.40 rural. Urban frequency is 0.10 claims per exposure and severity is $1,000; rural frequency is 0.05 and severity is $2,000. In period 1, the shares are 0.40 urban and 0.60 rural; urban frequency is 0.12 and severity is $1,100, while rural frequency is 0.06 and severity is $1,800. Using the exact midpoint method, calculate the overall pure-premium change and separate the within-group, mix, frequency, and severity effects.

Method and group pure premiums. Multiply frequency by severity in each group. Urban costs are $0.10(1{,}000)=\$100$ in period 0 and $0.12(1{,}100)=\$132$ in period 1. Rural costs are $0.05(2{,}000)=\$100$ and $0.06(1{,}800)=\$108$.

Check the total change. Period 0 overall cost is $0.60(\$100)+0.40(\$100)=\$100$. Period 1 overall cost is $0.40(\$132)+0.60(\$108)=\$52.80+\$64.80=\$117.60$. The increase is $\$117.60-\$100=\$17.60$ per exposure.

Within-group effect. The midpoint weights are $0.50$ for each group. The within-group effect is $0.50(\$132-\$100)+0.50(\$108-\$100)=\$16+\$4=\$20.00$.

Mix effect. The midpoint costs are $(\$100+\$132)/2=\$116$ urban and $(\$100+\$108)/2=\$104$ rural. The share changes are $-0.20$ urban and $+0.20$ rural. The mix effect is $(-0.20)(\$116)+(0.20)(\$104)=-\$23.20+\$20.80=-\$2.40$.

Frequency and severity contributions. For urban, the frequency contribution is $(\$1{,}000+\$1{,}100)/2(0.12-0.10)=\$1{,}050(0.02)=\$21$. The severity contribution is $(0.10+0.12)/2(\$1{,}100-\$1{,}000)=0.11(\$100)=\$11$. These sum to the urban cost change of $32. For rural, the contributions are $(\$2{,}000+\$1{,}800)/2(0.06-0.05)=\$19$ and $(0.05+0.06)/2(\$1{,}800-\$2{,}000)=-\$11$, summing to $8. Weighting each by its midpoint share of 0.50 gives an overall frequency contribution of $0.50(\$21)+0.50(\$19)=\$20$ and severity contribution of $0.50(\$11)+0.50(-\$11)=\$0$.

Check and trap avoided. The within-group effects add to $\$20+\$0=\$20$, and $\$20-\$2.40=\$17.60$, matching the direct change. The mix effect is negative because exposure shifted toward the group with the lower midpoint cost. Do not infer that a shift toward urban exposures necessarily raises cost without comparing the relevant group costs.

Worked Example: A Change Caused Only by Exposure Mix

A hypothetical portfolio has two fixed groups, with pure premiums of $80 for Group A and $140 for Group B in both periods. Group A’s exposure share falls from 0.50 to 0.25, so Group B’s share rises from 0.50 to 0.75. Calculate the overall change and identify its source.

Method. Since group costs do not change, both within-group cost changes are zero. Period 0 overall cost is $0.50(\$80)+0.50(\$140)=\$40+\$70=\$110$. Period 1 cost is $0.25(\$80)+0.75(\$140)=\$20+\$105=\$125$. The increase is $15 per exposure.

Mix calculation. Midpoint costs remain $80 and $140. Group A’s share change is $-0.25$, and Group B’s is $+0.25$. Thus the mix effect is $(-0.25)(\$80)+(0.25)(\$140)=-\$20+\$35=\$15$. The within-group effect is $0$, so the decomposition reproduces the $15 increase.

Check and trap avoided. The entire increase comes from a shift toward the higher-cost group; neither group’s own pure premium changed. Reporting the overall increase as evidence of worsening experience within the groups would confuse portfolio mix with group-level experience.

Worked Example: Aggregate Severity Is Not Exposure-Weighted Severity

A hypothetical portfolio has Groups A and B. In period 0 their exposure shares are 0.80 and 0.20, frequencies are 0.05 and 0.10, and severities are $1,000 and $3,000. In period 1 the exposure shares are 0.40 and 0.60, frequencies are 0.06 and 0.08, and severities are $1,100 and $3,300. Calculate the overall pure premiums, the exact decomposition, and aggregate severity in each period. Explain why an exposure-weighted average of group severities is not aggregate severity.

Group costs and overall costs. Period 0 group costs are $0.05(\$1{,}000)=\$50$ and $0.10(\$3{,}000)=\$300$, so $C_0=0.80(\$50)+0.20(\$300)=\$100$. Period 1 group costs are $0.06(\$1{,}100)=\$66$ and $0.08(\$3{,}300)=\$264$, so $C_1=0.40(\$66)+0.60(\$264)=\$26.40+\$158.40=\$184.80$. The increase is $84.80 per exposure.

Within and mix effects. Midpoint weights are 0.60 for A and 0.40 for B. Their cost changes are $66-50=\$16$ and $264-300=-\$36$, giving a within-group effect of $0.60(\$16)+0.40(-\$36)=\$9.60-\$14.40=-\$4.80$. Midpoint costs are $58 for A and $282 for B; share changes are $-0.40$ and $+0.40$. The mix effect is $(-0.40)(\$58)+(0.40)(\$282)=-\$23.20+\$112.80=\$89.60$. Together, $-\$4.80+\$89.60=\$84.80$.

Aggregate frequency and severity. Overall frequency is the exposure-weighted average of group frequencies: period 0 frequency is $0.80(0.05)+0.20(0.10)=0.06$; period 1 frequency is $0.40(0.06)+0.60(0.08)=0.072$. Aggregate severity is total losses divided by total claims. Since pure premium equals frequency times aggregate severity, it is $C_t$ divided by overall frequency: period 0 severity is $\$100/0.06=\$1{,}666.67$, and period 1 severity is $\$184.80/0.072=\$2{,}566.67$, rounded to the nearest cent.

Check and trap avoided. Exposure-weighting group severities would instead give $0.80(\$1{,}000)+0.20(\$3{,}000)=\$1{,}400$ in period 0, which is not the aggregate severity of $1,666.67. Claims are not distributed across groups in the same proportions as exposures: groups with greater frequency contribute more claims per exposure. Aggregate severity weights group severities by their shares of claims, not their shares of exposures.

Common Decomposition Errors

A frequent mistake is to calculate the within-group effect with period 0 weights and the mix effect with period 0 costs, then assume the two pieces must add to the total change. They generally do not: that calculation omits the interaction between the changing weights and changing group costs. In the first worked example, those mismatched pieces would be a within effect of $0.60(\$32)+0.40(\$8)=\$22.40$ and a mix effect of $(-0.20)(\$100)+(0.20)(\$100)=\$0$, totaling $22.40 instead of $17.60. Use the symmetric midpoint formula, or use a complete sequential formula whose weights and costs are paired consistently.

Other checks are equally important. Shares in each period should sum to 1, so their changes should sum to 0. Use the same exposure definition and the same loss and claim scope in both periods. Keep units consistent: if frequency is claims per policy-year, severity must be dollars per claim on the corresponding basis, so their product is dollars per policy-year. Finally, a fixed-group decomposition describes changes within the groups as defined; if group definitions change, the comparison also contains a classification change that this formula does not isolate.

Key takeaway: Calculate overall pure premium as an exposure-weighted average of fixed-group pure premiums. Use midpoint weights and midpoint costs to separate within-group and mix effects exactly, and use midpoint severity and frequency to divide the within-group effect further. Confirm that every set of components sums to the observed total change.

Check Your Understanding

For each hypothetical scenario, use consistent group definitions and show enough intermediate work to check your result.

Groups A and B have unchanged pure premiums of $40 and $100. Group A’s exposure share changes from 0.30 to 0.50. Calculate the overall pure-premium change and its within-group and mix components. (Answer: Overall change $=-\$12.00$ per exposure; within-group effect $=\$0$; mix effect $=-\$12.00$.)
Two groups have exposure shares changing from 0.70 and 0.30 to 0.50 and 0.50. Their pure premiums change from $50 to $60 and from $120 to $110, respectively. Calculate the overall change and midpoint within-group and mix effects. (Answer: Overall change $=\$14.00$ per exposure; within-group effect $=\$2.00$; mix effect $=\$12.00$.)
Two groups have fixed exposure shares of 0.50 each. Group A frequency and severity change from 0.04 and $500 to 0.05 and $600; Group B changes from 0.08 and $1,000 to 0.06 and $1,200. Find the overall pure-premium change and its midpoint frequency and severity contributions. (Answer: Overall change $=\$1.00$ per exposure; frequency contribution $=-\$8.25$; severity contribution $=\$9.25$.)
A portfolio has exposure shares 0.75 and 0.25, frequencies 0.04 and 0.12, and severities $1,000 and $2,000. Calculate overall frequency, pure premium, and aggregate severity. (Answer: Overall frequency $=0.0600$ claims per exposure; pure premium $=\$90.00$ per exposure; aggregate severity $=\$1{,}500.00$ per claim.)
Group A’s share changes from 0.60 to 0.40 and Group B’s from 0.40 to 0.60. Their pure premiums change from $30 to $36 and from $80 to $70. Calculate the total change and its midpoint within-group and mix components. (Answer: Overall change $=\$6.40$ per exposure; within-group effect $=-\$2.00$; mix effect $=\$8.40$.)

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