Premium Weights and Changing Policy Mix

Why Portfolio Rate Levels Need Weights

A rate level for an individual class, territory, or other rating group describes that group. A portfolio contains a distribution of policies across groups, however, so its overall rate level depends both on the group rate levels and on how much premium each group represents. An unweighted average gives a small group the same influence as a large one, which generally does not describe the portfolio.

The key is to define the weights and the basis for comparison before averaging. In particular, premium weights must be measured on a consistent reference basis. If group rate levels differ, changing the distribution of policies across groups can change the portfolio average even if no group’s rate level changes.

Definition: Let $j=1,\ldots,k$ index rating groups. Let $E_j$ be the exposure count or exposure amount in group $j$, and let $b_j$ be the premium per exposure at a common reference rate level. The group’s reference premium is $B_j=E_jb_j$. If its rate-level index is $r_j$, its premium at that level is $B_jr_j$. The reference premium weight is $w_j=B_j/\sum_{h=1}^{k}B_h$, so the portfolio’s weighted average rate-level index is $\bar r=\sum_{j=1}^{k}w_jr_j$. The weights are nonnegative and sum to 1.

The reference rate level is the same for all groups; it is the basis against which the indices $r_j$ are measured. An index of 1.08 means a rate level 8% above that reference, while 0.95 means a level 5% below it. The reference premium weights reflect the groups’ relative premium volumes at the reference rates. They do not give every group equal influence, nor do they change merely because the group indices differ.

This calculation assumes that the group definitions and reference premiums are appropriate for the portfolio being measured, that the rates can be represented by group indices on a common basis, and that the exposure weights represent the selected mix. It is a premium-weighted average of rate-level indices—not an average of claim costs, nor necessarily the ratio of actual premiums across two periods with different exposure distributions.

Calculating a Portfolio Average

When group exposures and reference premiums per exposure are available, first calculate each group’s reference premium. Divide each reference premium by the portfolio total to obtain its weight, then multiply each weight by that group’s rate-level index. Summing these products gives the portfolio index.

B_j=E_jb_j,\qquad w_j=\frac{B_j}{\sum_{h=1}^{k}B_h},\qquad \bar r=\sum_{j=1}^{k}w_jr_j

The same result can be checked by totaling the group premiums at the indexed rates and dividing by total reference premium:

\bar r = \frac{\sum_{j=1}^{k}B_jr_j}{\sum_{j=1}^{k}B_j}

This equivalent form gives a short justification for the weighted-average formula. Since $w_j=B_j/\sum_hB_h$, substituting into the weighted sum gives $\sum_j w_jr_j=\sum_jB_jr_j/\sum_hB_h$. Thus the weighted index is exactly the total indexed premium divided by total reference premium. The result is also bounded by the smallest and largest group indices when all weights are nonnegative: $\min_j r_j\leq\bar r\leq\max_j r_j$.

Worked Example: Combining Three Group Rate Levels

A hypothetical equipment insurer has three groups. Group A has 100 exposure units and a reference premium of $500 per unit; Group B has 50 units at $800; and Group C has 25 units at $1,200. Their rate-level indices are 1.08, 0.96, and 1.15, respectively. Calculate the portfolio average index using reference premium weights, and check the result using total premiums.

Method. Calculate each group’s reference premium, derive its share of the reference premium total, and weight its index. This method avoids treating groups with different premium volumes as equally important.

Step 1: Reference premiums. Group A has $100\times \$500=\$50{,}000$. Group B has $50\times \$800=\$40{,}000$. Group C has $25\times \$1{,}200=\$30{,}000$. The total reference premium is $\$50{,}000+\$40{,}000+\$30{,}000=\$120{,}000$.

Step 2: Weights. The weights are $50{,}000/120{,}000=5/12$ for A, $40{,}000/120{,}000=1/3$ for B, and $30{,}000/120{,}000=1/4$ for C. They sum to $5/12+4/12+3/12=1$.

Step 3: Weighted index. The contributions are $(5/12)(1.08)=0.4500$, $(1/3)(0.96)=0.3200$, and $(1/4)(1.15)=0.2875$. Therefore $\bar r=0.4500+0.3200+0.2875=1.0575$, or 105.75% of the reference level.

Check. Indexed premiums are $\$50{,}000(1.08)=\$54{,}000$, $\$40{,}000(0.96)=\$38{,}400$, and $\$30{,}000(1.15)=\$34{,}500$. Their total, $\$126{,}900$, divided by $\$120{,}000$, is 1.0575. The check agrees with the weighted calculation.

How a Changing Policy Mix Changes the Aggregate

Suppose group rate-level indices stay fixed, but the distribution of policies changes. The reference premium weights may then change because groups have different exposure counts and different reference premiums per exposure. Since the portfolio index is a weighted average, a shift toward a group with a lower index can reduce the aggregate; a shift toward a group with a higher index can increase it.

This does not mean that any group’s rates have changed. It means that the portfolio being summarized has a different composition. For a meaningful comparison of composition scenarios, use the same group reference premiums and group indices in each scenario, and recalculate the weights from each scenario’s exposure distribution.

Worked Example: A Shift Toward a Lower-Indexed Group

A hypothetical service-contract portfolio has two groups. Group L has a reference premium of $400 per exposure and an index of 1.10. Group H has a reference premium of $1,000 per exposure and an index of 0.90. Initially there are 80 exposures in L and 20 in H. Later, the mix is 40 exposures in L and 60 in H. The group indices and reference premiums are unchanged. Calculate the portfolio index in each mix and compare them.

Method. For each mix, calculate reference premium weights from exposures and reference premiums. The common error to avoid is averaging the two group indices equally even though the groups do not contribute equal reference premium.

Initial mix. Group L’s reference premium is $80\times \$400=\$32{,}000$. Group H’s is $20\times \$1{,}000=\$20{,}000$, for a total of $\$52{,}000$. The weights are $32/52=8/13$ for L and $20/52=5/13$ for H. Thus

\bar r_{\text{initial}} =\frac{8}{13}(1.10)+\frac{5}{13}(0.90) =\frac{8.80+4.50}{13} =\frac{13.30}{13} =1.02308

Later mix. Group L’s reference premium is $40\times \$400=\$16{,}000$, and group H’s is $60\times \$1{,}000=\$60{,}000$. The total is $\$76{,}000$, giving weights $16/76=4/19$ and $60/76=15/19$. Therefore

\bar r_{\text{later}} =\frac{4}{19}(1.10)+\frac{15}{19}(0.90) =\frac{4.40+13.50}{19} =\frac{17.90}{19} =0.94211

Comparison and check. The portfolio index falls from 1.02308 to 0.94211, although neither group index changed. For the initial mix, indexed premium is $32{,}000(1.10)+20{,}000(0.90)=\$35{,}200+\$18{,}000=\$53{,}200$, and $\$53{,}200/\$52{,}000=1.02308$. For the later mix, it is $16{,}000(1.10)+60{,}000(0.90)=\$17{,}600+\$54{,}000=\$71{,}600$, and $\$71{,}600/\$76{,}000=0.94211$. The change follows from the shift toward group H, which has the lower index.

Choosing Weights for a Rate Change

A related calculation arises when each group has a proposed percentage change from its current rate. To calculate the overall percentage change in premium for the current portfolio, weight each group’s change by its current premium. Here the current premium is the appropriate base because the percentage changes apply to those current premiums.

Result: If $P_j$ is current premium in group $j$, and $c_j$ is the proposed percentage change expressed as a decimal, the overall percentage change for the current portfolio is $\sum_j P_jc_j/\sum_jP_j$. This assumes the changes apply to the stated current premiums and the calculation holds the current group premium distribution fixed.

Worked Example: Aggregating Proposed Group Changes

A hypothetical specialty insurer has current premiums of $120,000 in Group A and $80,000 in Group B. A proposed change is +5% in A and −2% in B. Find the overall percentage change for the current portfolio, then verify it by calculating proposed premium.

Method. Weight each proposed percentage change by current premium. The common mistake is taking a simple average of +5% and −2%, which would assign equal weight to groups with unequal current premium.

Step 1: Current premium weights. Total current premium is $\$120{,}000+\$80{,}000=\$200{,}000$. The weights are $120{,}000/200{,}000=0.60$ for A and $80{,}000/200{,}000=0.40$ for B.

Step 2: Weighted change. The contributions are $0.60(0.05)=0.030$ and $0.40(-0.02)=-0.008$. The total change is $0.030-0.008=0.022$, or an increase of 2.2%.

Step 3: Proposed-premium check. Group A’s proposed premium is $\$120{,}000(1.05)=\$126{,}000$. Group B’s is $\$80{,}000(0.98)=\$78{,}400$. Total proposed premium is $\$204{,}400$, an increase of $\$4{,}400$ from $\$200{,}000$. Since $\$4{,}400/\$200{,}000=0.022$, the increase is 2.2%. The simple average, $(5\%-2\%)/2=1.5\%$, is not correct for these unequal premium volumes.

Common Traps and Practical Checks

The words “premium weight” do not by themselves identify a weight. Specify the premium basis in the calculation: reference premiums for a portfolio rate-level index, or current premiums for aggregating percentage changes applied to current premiums. Mixing these bases can produce a plausible-looking but incorrect result.

Taking an unweighted mean. Use an unweighted mean only when the groups have equal relevant premium weights, or when the question explicitly asks for the simple average. Otherwise, calculate the premium contributions first.
Using the wrong premium basis. For a portfolio index relative to reference rates, weight by reference premium. For proposed changes applied to current premium, weight by current premium.
Forgetting the group’s base premium per exposure. Exposure counts alone are not premium weights when $b_j$ differs by group. Calculate $E_jb_j$ before normalizing.
Using inconsistent group indices. The indices must refer to a common reference level. An index from one basis cannot be combined with an index from another without first putting them on a consistent basis.
Assuming the aggregate is a group-rate measure alone. The aggregate also reflects the mix used to construct its weights. Report the weighting basis along with the result.

A useful arithmetic check is that the weights sum to 1 and the weighted average lies between the smallest and largest group indices. When dollar premiums are available, independently calculate total indexed premium divided by total reference premium. These checks can reveal normalization errors, a missing group, or a wrong basis.

Key takeaway: A portfolio average rate level is a weighted average, not generally a simple average. Define the group indices and premium basis, calculate consistent weights, and recognize that a changed policy mix can alter the aggregate even when every group rate level is unchanged.

Check Your Understanding

For each problem, identify the relevant premium basis before calculating. Amounts are hypothetical.

A portfolio has 60 exposures with a reference premium of $300 each and 20 exposures with a reference premium of $900 each. The group indices are 1.04 and 0.98. Calculate the reference-premium-weighted portfolio index. (Answer: 1.0100, or 101.00% of reference.)
Two groups have current premiums of $75,000 and $25,000. Proposed changes are +4% and −8%, respectively. Calculate the overall percentage change for the current portfolio and verify by finding the proposed premium total. (Answer: +1.0%; proposed premium is $101,000.)
Two groups have reference premiums of $20,000 and $60,000, with indices 0.90 and 1.05. Calculate the weighted portfolio index and check it using indexed premium. (Answer: 1.0125, or 101.25% of reference; indexed premium is $81,000.)
A hypothetical portfolio has group L and group H, with reference premiums per exposure of $500 and $1,000 and indices 0.80 and 1.20. First the exposures are 90 in L and 10 in H; later they are 30 in L and 70 in H. Calculate both portfolio indices and explain the direction of the change. (Answer: initial index $48/55=0.8727$; later index $96/85=1.1294$; the aggregate increases as the mix shifts toward the higher-index group H.)
Current premiums are $40,000 in one group and $160,000 in another. The respective proposed changes are +3% and −1%. Calculate the portfolio percentage change and proposed premium total. (Answer: −0.2%; proposed premium is $199,600.)

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