Why Exposure Mix Matters
An aggregate pure premium can change even when the expected loss cost for every exposure type stays exactly the same. The reason is composition: if a larger share of total exposure comes from types with higher loss costs, the portfolio’s aggregate expected loss per exposure rises. If the share shifts toward lower-cost types, it falls.
This distinction matters when comparing periods, territories, or hypothetical portfolios. An aggregate change alone does not show whether the underlying type-specific loss costs changed. It may instead reflect a different distribution of exposure among types. To make a fair comparison, the exposure base and the type definitions must be consistent.
The exposure share \(w_j\) is between zero and one, and the shares sum to one. It is a share of the selected exposure base, not necessarily a share of policies. For example, if the analysis uses vehicle-years, the mix must be calculated from vehicle-years. Counts based on another unit cannot be substituted without a reasoned conversion.
The Exposure-Weighted Aggregate
Because \(P_j=L_j/E_j\), the losses for type \(j\) can be written as \(L_j=E_jP_j\). Adding the losses across types and dividing by total exposure gives the aggregate pure premium:
Thus, aggregate pure premium is a weighted average of the type-specific pure premiums, with exposure shares as the weights. It is not generally the simple average of the type-specific values. The simple average would give every type equal influence, even if one type accounts for most of the portfolio and another accounts for very little.
If type-specific pure premiums remain fixed while the mix changes from shares \(w_j^{(0)}\) to \(w_j^{(1)}\), then the change in aggregate pure premium is:
The weights in each mix sum to one, so the changes in weights sum to zero. A shift in share toward a high-cost type must be offset by a shift away from one or more other types. The aggregate effect depends on both the size of each share shift and the differences in type-specific pure premiums.
Worked Example: A Shift Between Two Exposure Types
A hypothetical insurer groups a selected portfolio into two exposure types. Type A has an expected pure premium of $120 per exposure, and type B has an expected pure premium of $300 per exposure. In the first period, the portfolio has 750 type A exposures and 250 type B exposures. In the second period, it has 500 of each. Assuming the type-specific pure premiums are unchanged, calculate both aggregate pure premiums and the percentage increase.
Method and setup. Calculate the expected losses by type, divide total expected losses by total exposures in each period, and then compare the results. The common mistake to avoid is taking an unweighted average of $120 and $300.
First period. Expected losses for type A are \(750\times \$120=\$90{,}000\). Expected losses for type B are \(250\times \$300=\$75{,}000\). Total expected losses are \(\$90{,}000+\$75{,}000=\$165{,}000\), and total exposure is \(750+250=1{,}000\). Therefore:
Second period. Type A expected losses are \(500\times \$120=\$60{,}000\), and type B expected losses are \(500\times \$300=\$150{,}000\). Total expected losses are \(\$210{,}000\); total exposure remains \(1{,}000\). Thus, \(P_{\mathrm{agg}}^{(1)}=\$210{,}000/1{,}000=\$210\).
The dollar increase is \(\$210-\$165=\$45\). The percentage increase, using the first-period aggregate as the denominator, is \(\$45/\$165=0.272727\ldots\), or 27.27% to two decimal places. As a check, the first-period shares are 75% and 25%, giving \(0.75(\$120)+0.25(\$300)=\$165\); the second-period shares are 50% and 50%, giving \(0.50(\$120)+0.50(\$300)=\$210\). The result rises solely because the higher-cost type has a larger share.
Why the Weighted Formula Holds
The weighting is a consequence of how total losses accumulate, not an arbitrary averaging convention. For each type, its pure premium multiplied by its exposure gives its losses on the stated basis. Total losses are therefore the sum of these type-level amounts. Dividing that sum by total exposure produces the aggregate cost per exposure.
By the definition of pure premium, \(L_j=E_jP_j\) for each type.
Total losses are \(\sum_j L_j=\sum_j E_jP_j\).
Since \(E=\sum_j E_j\), the aggregate is \(\sum_j E_jP_j/E\).
Each \(E_j/E\) is \(w_j\), so the aggregate is \(\sum_j w_jP_j\). Also, \(\sum_j w_j=1\).
This derivation uses a common exposure unit and a common loss basis. If one type’s exposure count represents a different unit, or if the loss amounts are measured on a different basis, the products cannot be added and interpreted as a consistent aggregate pure premium without first reconciling those differences.
Worked Example: Three Types and a Mix-Only Change
For a hypothetical product, three exposure types have expected pure premiums of $80, $160, and $400 per exposure. In period 1, the exposure counts are 1,000, 600, and 400, respectively. In period 2, they are 600, 1,000, and 400. Each period has 2,000 exposures, and the type-specific pure premiums are unchanged. Calculate the aggregate pure premium in both periods, the percentage change, and the contribution of the mix shift.
Method and setup. Compute the weighted average using exposure shares, then verify it by calculating expected losses. To find the mix contribution, compare the change in shares for each type and multiply each share change by that type’s pure premium.
Period 1. The shares are \(1{,}000/2{,}000=0.50\), \(600/2{,}000=0.30\), and \(400/2{,}000=0.20\). The weighted pure premium is:
As a loss check, expected losses are \(1{,}000(\$80)=\$80{,}000\), \(600(\$160)=\$96{,}000\), and \(400(\$400)=\$160{,}000\). They total \(\$336{,}000\), and \(\$336{,}000/2{,}000=\$168\).
Period 2. The shares are \(600/2{,}000=0.30\), \(1{,}000/2{,}000=0.50\), and \(400/2{,}000=0.20\). The aggregate is:
The loss check is \(\$48{,}000+\$160{,}000+\$160{,}000=\$368{,}000\); dividing by 2,000 exposures gives $184. The aggregate increase is \(\$184-\$168=\$16\), or \(\$16/\$168=0.095238\ldots\), which is 9.52% to two decimal places.
The share changes are \(-0.20\), \(+0.20\), and \(0\). Their effects are \((-0.20)(\$80)=-\$16\), \((+0.20)(\$160)=+\$32\), and \(0(\$400)=\$0\). The net is \(+\$16\), matching the aggregate change. The common mistake here is to use the simple mean \((\$80+\$160+\$400)/3=\$213.33\) for either period. That ignores the actual exposure shares and matches neither aggregate.
Standardizing to a Common Mix
A direct comparison of aggregate pure premiums combines two possible sources of change: different type-specific costs and different exposure shares. Standardizing to a common mix helps separate them. To calculate the pure premium at a chosen standard mix, apply the same set of weights to the relevant type-specific pure premiums. The standard weights must sum to one and must refer to the same type definitions and exposure base as the costs.
For example, if the goal is to compare costs at the first period’s mix, use the first period’s exposure shares for both periods’ type-specific pure premiums. If those type-specific premiums are identical across periods, both standardized aggregates will be identical, even if the unstandardized aggregates differ. If type-specific premiums differ, standardization holds composition constant while reflecting those within-type differences.
Standardization does not change the observed mix or the actual observed aggregate. It answers a different comparison question: what would the aggregate be if the selected type-specific pure premiums were combined using a common set of exposure shares? The choice of standard mix should fit the purpose of the analysis and be stated clearly.
Worked Example: Reweighting to a Reference Mix
A hypothetical portfolio has two types with expected pure premiums of $100 for type A and $250 for type B. The reference portfolio has 800 type A exposures and 200 type B exposures. A later portfolio has 400 type A exposures and 600 type B exposures. Assume the type-specific pure premiums have not changed. Calculate the later portfolio’s observed aggregate and its aggregate at the reference mix, then interpret the difference.
Method and setup. First calculate the later observed aggregate using its own shares. Then calculate the standardized value using the reference shares for the same type costs. The trap is to call the standardized result the later portfolio’s observed pure premium.
Observed later aggregate. The later shares are \(400/1{,}000=0.40\) for type A and \(600/1{,}000=0.60\) for type B. Thus, \(0.40(\$100)+0.60(\$250)=\$40+\$150=\$190\) per exposure. In loss terms, expected losses are \(\$40{,}000+\$150{,}000=\$190{,}000\) for 1,000 exposures, also giving $190.
Aggregate at the reference mix. The reference shares are \(0.80\) and \(0.20\). Reweight the unchanged type costs: \(0.80(\$100)+0.20(\$250)=\$80+\$50=\$130\) per exposure.
The later observed aggregate is $190, while its value at the reference mix is $130. The $60 difference is the effect of applying the later, higher share of type B to these costs. Since the type-specific costs were assumed constant, the standardized value does not imply that the later portfolio actually had the reference exposure mix.
Interpretation and Common Traps
Mix analysis is useful for interpreting changes in an overall pure premium, but it requires care. A mix shift can raise an aggregate even without any increase in costs within the types. Conversely, a favorable shift toward lower-cost types can make the aggregate fall even if type-specific costs have increased. Looking only at the total can conceal these different effects.
- Using a simple average. The correct weights are exposure shares. Equal weighting is appropriate only when the types have equal exposure volumes.
- Using the wrong denominator. Shares must be calculated from the exposure base used for the pure premium, not automatically from policy counts, premiums, or another volume measure.
- Mixing measurement bases. Type-specific losses and exposures must be consistent across types and periods before their pure premiums can be combined.
- Calling a mix effect a within-type change. An aggregate difference alone does not establish that type-specific loss costs changed.
- Confusing percentage points and percent change. A shift in a type’s share is a percentage-point change; the aggregate percentage change is the dollar change divided by the original aggregate.
Check Your Understanding
Solve each problem using exposure shares or total expected losses. Each scenario is hypothetical; assume the stated type-specific pure premiums are on a consistent basis.
- Two types have pure premiums of $90 and $270. Their exposure shares change from 70% and 30% to 40% and 60%. Calculate both aggregate pure premiums and the percentage increase. (Answer: $144 initially, $198 after the shift, a 37.50% increase.)
- Three types have pure premiums of $50, $150, and $350. A 1,000-exposure portfolio has counts of 400, 400, and 200; a second portfolio has counts of 200, 500, and 300. Find both aggregate pure premiums and the percentage change. (Answer: $150 initially, $190 after the shift, a 26.67% increase.)
- Type A has a pure premium of $120 and type B has a pure premium of $240. A current portfolio is 25% type A and 75% type B. Find its aggregate pure premium, then calculate the aggregate at a standard mix of 60% type A and 40% type B. (Answer: $210 at the current mix and $168 at the standard mix.)
- Three types have pure premiums of $80, $200, and $500. Their shares move from 50%, 30%, and 20% to 20%, 50%, and 30%. Calculate the aggregate at each mix and the percentage change. (Answer: $200 initially, $266 after the shift, a 33.00% increase.)
- A portfolio’s type-specific pure premiums are $100 for type A and $300 for type B. A 1,000-exposure portfolio has total expected losses of $190,000. If those costs are unchanged, infer type B’s share and calculate the aggregate at a reference mix with 80% type A and 20% type B. (Answer: type B is 45% of exposure; the reference-mix aggregate is $140.)