Weighting Rate Levels Across Policies

Why Rate Levels Need Weights

A portfolio can contain policies with different rate levels. If one group has a rate increase and another has a decrease, the portfolio’s average rate level depends not just on the size of each change, but on how much premium each group represents. A change affecting a large premium base should generally have more influence than the same change affecting a small one.

The relevant weights reflect the portfolio’s exposure to each rate level. When policies have different base premiums, exposure counts alone may not represent that exposure: a group of policies with larger base premiums contributes more to total premium at a common rate level. The calculation must therefore match the weighting basis to the question.

Definition: A rate-level index is a factor expressing a group’s rates relative to a specified common reference level. An index of $1.06$ means that the group’s rates are 6% above the reference level; an index of $0.97$ means they are 3% below it. The average rate level for a portfolio is the aggregate premium at the groups’ rate levels divided by the premium those same exposures would produce at the common reference level.

This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. It assumes that each group’s rate-level index applies to the premium basis being compared and that the mix of exposures and coverages is held fixed for the calculation. If the exposure or coverage mix changes, that is a separate issue from weighting the rate levels.

The Weighted Average Formula

Let group $i$ have rate-level index $r_i$, exposure amount $E_i$, and base premium per exposure $b_i$ at the common reference level. Its premium at that reference level is $B_i=E_i b_i$. The average rate level is:

\bar{r} = \frac{\sum_i B_i r_i}{\sum_i B_i} = \frac{\sum_i E_i b_i r_i}{\sum_i E_i b_i}.

When every exposure has the same base premium, $b_i=b$ for all groups, the common factor cancels. In that case, the formula simplifies to an exposure-weighted average:

\bar{r} = \frac{\sum_i E_i r_i}{\sum_i E_i}.

Thus, exposure weighting is appropriate when exposure units represent equal premium at the reference level. If the groups have different base premiums per exposure, use their reference-level premiums $B_i$ as weights. Policy counts are appropriate only when those counts accurately represent the relevant exposure and base-premium volume.

Why the Formula Works

The formula follows from adding premiums. At the reference level, group $i$ would generate $B_i$. At its actual rate level, it generates $B_i r_i$, because the rate-level index multiplies the reference-level premium. Summing across groups gives total premium at the groups’ rate levels, $\sum_i B_i r_i$, and total premium at the reference level, $\sum_i B_i$. Their ratio is therefore the aggregate rate-level index:

\text{aggregate rate level} = \frac{\text{premium at the groups' rate levels}} {\text{premium at the common reference level}} = \frac{\sum_i B_i r_i}{\sum_i B_i}.

Equivalently, define $w_i=B_i/\sum_j B_j$. These weights are nonnegative and sum to 1, so $\bar r=\sum_i w_i r_i$. The result is a weighted average of the group indices, with each group’s weight equal to its share of reference-level premium.

If actual premium $P_i$ is known for each group and it was generated at index $r_i$, then $P_i=B_i r_i$, so $B_i=P_i/r_i$. The portfolio index can also be calculated as:

\bar{r} = \frac{\sum_i P_i}{\sum_i P_i/r_i}.

This form is useful when the observed premiums are at the groups’ different rate levels, but it requires the correct index for each group and a consistent reference basis.

A Practical Calculation Sequence

Set the common reference level.
Express each group’s rate level as an index relative to the same reference. An index of $1.00$ is the reference itself.

Find each group’s weight.
Use exposure amounts if each exposure has the same reference-level premium. Otherwise calculate the reference-level premium for each group.

Calculate premium at the group rate levels.
Multiply each reference-level premium weight by its rate-level index.

Divide and check.
Divide total premium at the group rate levels by total reference-level premium. Confirm that the result lies between the smallest and largest group indices.

Worked Example: Equal Base Premium per Exposure

A hypothetical insurer has two groups of earned exposures. Group A has 600 exposures at a rate level 10% above the reference level; Group B has 400 exposures at a rate level 5% below it. Assume that every exposure has the same premium at the reference level. Find the portfolio’s average rate level, then compare it with the simple average of the percentage changes.

Method. Since base premium per exposure is equal, exposure counts are valid weights. The indices are $r_A=1.10$ and $r_B=0.95$.

The exposure-weighted premium at the group rate levels, measured in units of the common base premium per exposure, is:

$$ 600(1.10)+400(0.95)=660+380=1{,}040. $$

The corresponding total at the reference level is $600+400=1{,}000$ units. Therefore:

\bar r=\frac{1{,}040}{1{,}000}=1.04.

The average rate level is 1.04, or 4% above the reference. The unweighted average of the two rate changes would be $(10\%-5\%)/2=2.5\%$, which does not account for the groups’ different exposure volumes. The weighted result is checked: 1.04 lies between 0.95 and 1.10.

Common mistake avoided: averaging the two percentage changes equally. Each group does not represent the same number of exposures, so each should not receive equal weight.

Worked Example: Different Base Premiums per Exposure

A hypothetical portfolio has 300 exposures in Group A, each with a reference-level premium of $500, and 200 exposures in Group B, each with a reference-level premium of $900. Group A is at index 1.08 and Group B at index 0.96. Calculate the average rate level. Explain why an exposure-weighted average is not appropriate.

Method. The groups have different reference-level premiums per exposure, so use reference-level premium as the weight.

Group A’s reference-level premium is $300\times \$500=\$150{,}000$, and its premium at its group rate level is $\$150{,}000\times1.08=\$162{,}000$. Group B’s reference-level premium is $200\times \$900=\$180{,}000$, and its premium at its group rate level is $\$180{,}000\times0.96=\$172{,}800$.

Total premium at the group rate levels is $\$162{,}000+\$172{,}800=\$334{,}800$. Total premium at the reference level is $\$150{,}000+\$180{,}000=\$330{,}000$. Thus:

\bar r = \frac{\$334{,}800}{\$330{,}000} = 1.014545\ldots \approx 1.0145.

The portfolio’s average rate level is approximately 1.0145, or 1.45% above the reference level. The exposure-weighted calculation would instead be $[300(1.08)+200(0.96)]/500=1.032$, or 3.2% above the reference. That result incorrectly treats all 500 exposures as if they had the same reference-level premium.

Check. The calculated index lies between 0.96 and 1.08. Also, applying 1.014545… to the total reference premium gives $\$330{,}000\times1.014545\ldots=\$334{,}800$, matching the group-premium total.

Common mistake avoided: using exposure counts as weights when the exposures have materially different premiums at the common reference level.

Worked Example: Weighting Compounded Rate Changes

A hypothetical book has 1,000 exposures in Group A and 500 in Group B. All exposures have a reference-level premium of $240. Group A first receives a 5% increase and later a 4% increase. Group B is at an index of 0.98 relative to the same reference. Calculate the portfolio average rate level. Compare it with the simple average of the groups’ cumulative percentage changes.

Method. Compound the sequential changes for Group A to obtain its index, then weight the group indices by reference-level premium.

Group A’s index is $1.05\times1.04=1.092$, a cumulative increase of 9.2%. Its reference-level premium is $1{,}000\times\$240=\$240{,}000$, and premium at its rate level is $\$240{,}000\times1.092=\$262{,}080$.

Group B’s reference-level premium is $500\times\$240=\$120{,}000$, and premium at its rate level is $\$120{,}000\times0.98=\$117{,}600$. Total reference-level premium is $\$240{,}000+\$120{,}000=\$360{,}000$, and total premium at the group rate levels is $\$262{,}080+\$117{,}600=\$379{,}680$. Therefore:

\bar r = \frac{\$379{,}680}{\$360{,}000} = 1.054666\ldots \approx 1.0547.

The portfolio average is approximately 5.47% above the reference level. The unweighted average of the two cumulative percentage changes is $(9.2\%-2.0\%)/2=3.6\%$. It is wrong here for two reasons: the groups have different exposure volumes, and the sequential changes for Group A must be compounded before its group index is weighted.

Check. The reference-level-premium shares are $\$240{,}000/\$360{,}000=2/3$ and $\$120{,}000/\$360{,}000=1/3$. Thus $(2/3)(1.092)+(1/3)(0.98)=1.054666\ldots$, confirming the result.

Common mistake avoided: taking an unweighted mean of the rate changes or adding sequential percentage changes instead of first calculating the cumulative index.

Common Traps and Interpretation

A simple arithmetic average of rate changes is correct only in special cases, such as when all groups have equal reference-level premium weights. In general, it silently assumes that every group represents the same premium volume. The correct weighted result is not necessarily the average of the stated rate changes, even when there are only two groups.

Using policy counts without checking policy size. A count gives each policy equal weight; it does not give each policy equal premium weight. Use it only when it represents the intended exposure basis and the base premiums are equal or otherwise appropriately accounted for.
Weighting by premium at the wrong level. The denominator of the weighted-average formula is premium at the common reference level. If using observed premium at group-specific levels, first convert it to the reference basis, for example $B_i=P_i/r_i$.
Averaging rate changes rather than indices. A rate change of $-5\%$ corresponds to index $0.95$, not index $-0.05$. The rate-level index is the multiplicative factor used to scale premium.
Forgetting sequential compounding. If a group experiences successive changes, calculate its cumulative index by multiplying the successive factors before applying the group weight.
Ignoring the comparison basis. Every index must be relative to the same reference level. A weighted average of indices defined against different references has no consistent interpretation.

Key takeaway: The portfolio average rate level is total premium at the groups’ rate levels divided by total premium for the same exposures at a common reference level. Use exposure weights only when exposures represent equal reference-level premium; otherwise weight by reference-level premium.

Check Your Understanding

For each problem, calculate the average rate-level index on the stated basis. Round final indices to four decimal places.

A hypothetical portfolio has 250 equal-base-premium exposures at index 1.06 and 750 at index 0.98. Find the exposure-weighted average index and compare it with the unweighted average. (Answer: weighted index $1.0000$; unweighted index $1.0200$.)
Group A has 100 exposures with a reference-level premium of $400 each and index 1.05. Group B has 50 exposures with a reference-level premium of $1,000 each and index 0.97. Find the premium-weighted average index. (Answer: $1.0056$.)
A hypothetical portfolio has 400 exposures at index 1.12, each with a reference-level premium of $200, and 600 exposures at index 0.94, each with a reference-level premium of $350. Find the average index. (Answer: $0.9897$.)
Group A has 2,000 equal-base-premium exposures and receives sequential changes of +3% and −2%. Group B has 1,000 such exposures at index 1.07. Find the portfolio average index. (Answer: $1.0296$.)
A group’s actual premium is $44,000 at index 1.10, and another group’s actual premium is $34,200 at index 0.95. Using actual premiums and their indices, find the portfolio average rate-level index. (Answer: $1.0289$.)

What You'll Learn