What a Policyholder Rate Impact Exhibit Shows
A rate indication describes the overall change supported by an analysis. A policyholder rate impact exhibit answers a different question: how would the proposed rates affect the policies in the modeled population? It can show whether most policies receive similar changes, whether a smaller group sees larger increases or decreases, and how the changes combine when premiums are weighted by their current size.
The exhibit is only interpretable when the old and proposed premiums use the same population, coverage basis, and rating assumptions. For example, if the current premium includes one coverage that the proposed premium omits, their difference is not a meaningful rate impact. The exhibit should also say whether its unit is a policy, policy term, or another defined record; a count of terms is not automatically a count of unique customers.
The percent change is calculated relative to the old premium, not the proposed premium. A move from $1,000 to $1,100 is a 10% increase because the $100 change is divided by $1,000. The reverse move from $1,100 to $1,000 is a decrease of about 9.09%, not 10%: the denominator is now $1,100.
Count-Weighted and Premium-Weighted Summaries
Two averages commonly appear in impact exhibits, and they answer different questions. The count-weighted average gives each policy equal weight. The premium-weighted average gives a policy weight in proportion to its old premium. For policies with positive old premiums, the latter reconciles exactly to the percentage change in total premium.
The count average describes the mean percentage change for a policy in the exhibit. The premium-weighted result describes the change in aggregate premium, relative to aggregate old premium. Neither is inherently a substitute for the other. A small policy with a large percentage increase can strongly affect the count average while contributing little to the aggregate premium change.
The reconciliation follows directly from the definition \(g_j=(P_j^{new}-P_j^{old})/P_j^{old}\). Multiplying both sides by \(P_j^{old}\) gives \(P_j^{old}g_j=P_j^{new}-P_j^{old}\). Summing over policies gives the total dollar change; dividing by total old premium gives the aggregate percentage change. This identity also explains why the premium weights must be old premiums: they are the denominators in the individual percentage changes.
Worked Example: Reconciling Policy Changes to Total Premium
A hypothetical book has four policy terms. Their current premiums are $1,000, $2,000, $3,000, and $4,000. The proposed changes are respectively −10%, 0%, +10%, and +20%. Calculate each proposed premium, the count-weighted average change, and the aggregate premium change. State the method and identify a common mistake.
Step 1: Calculate each dollar change and proposed premium. For the first policy, \(-10\%\times \$1{,}000=-\$100\), so the proposed premium is $900. For the second, the change is $0 and the proposed premium is $2,000. For the third, \(10\%\times \$3{,}000=\$300\), giving $3,300. For the fourth, \(20\%\times \$4{,}000=\$800\), giving $4,800.
Step 2: Calculate the count-weighted mean. Add the four percentage changes and divide by four:
Step 3: Calculate aggregate premium impact. Old premium totals \(1{,}000+2{,}000+3{,}000+4{,}000=\$10{,}000\). Proposed premium totals \(900+2{,}000+3{,}300+4{,}800=\$11{,}000\). The aggregate increase is $1,000, or \(\$1{,}000/\$10{,}000=10\%\).
Check and common mistake. The dollar changes are \(-\$100+\$0+\$300+\$800=\$1{,}000\), which agrees with the difference between the totals. Reporting 5% as the overall premium change would confuse the count average with the premium-weighted change. The correct results are a 5% count average and a 10% aggregate premium increase.
Summarizing Impact by Segment and Distribution
An exhibit may group policies by a useful segment or by ranges of percentage change. Segment summaries help compare the impact across existing groupings in the analysis. A distribution summary makes the range and concentration of customer impacts visible. A useful table can report the number of policies, count share, old premium, premium share, and average or total change in each group.
For a segment \(k\), calculate its aggregate percentage change as the segment’s total proposed premium minus its total old premium, divided by its total old premium. The overall result is not generally the simple average of segment changes: it is weighted by each segment’s old premium. Similarly, a distribution table’s count shares and premium shares are separate quantities.
A rate impact distribution is descriptive, not an explanation of why a particular policy changes. The table can show that policies are concentrated in an increase range, but it does not establish which rating factor or model component caused those changes. Keep the summary tied to the proposed-rate comparison and avoid interpreting it as a separate measure of rate adequacy.
Worked Example: Comparing Segment and Overall Changes
A hypothetical insurer models two segments. Segment A has 100 policies with average old premium of $800 and a proposed increase of 5%. Segment B has 20 policies with average old premium of $2,000 and a proposed decrease of 2%. Calculate each segment’s old and proposed premium, the total premium-weighted change, and the count-weighted average policy change. Name the method and the weighting trap.
Step 1: Segment A. Its old premium is \(100\times \$800=\$80{,}000\). Its dollar increase is \(5\%\times \$80{,}000=\$4{,}000\), so its proposed premium is $84,000.
Step 2: Segment B. Its old premium is \(20\times \$2{,}000=\$40{,}000\). Its dollar decrease is \(2\%\times \$40{,}000=\$800\), so its proposed premium is $39,200.
Step 3: Combine premium totals. Old premium is \(\$80{,}000+\$40{,}000=\$120{,}000\); proposed premium is \(\$84{,}000+\$39{,}200=\$123{,}200\). Thus the aggregate change is \(\$3{,}200/\$120{,}000=2.6667\%\), or 2.667% to three decimal places.
Step 4: Calculate the policy-count average. There are 120 policies, so the count-weighted average is \((100\times5\%+20\times(-2\%))/120=460\%/120=3.833\%\), to three decimal places.
Check and common mistake. The segment dollar changes, $4,000 and −$800, sum to $3,200, matching the total-premium difference. Simply averaging 5% and −2% would give 1.5%, which incorrectly gives each segment equal weight. Weighting by policy counts gives the count average, 3.833%; weighting by old premium gives the aggregate change, 2.667%.
Worked Example: Reading a Rate-Change Distribution
A hypothetical impact exhibit groups policies into four mutually exclusive change ranges. The counts, old premiums, and average change within each range are shown below. Calculate each group’s count share and premium share, then reconcile the count-weighted average with the aggregate premium change. Use the lower endpoint convention “at least 10%” for the last range.
| Change range | Policy count | Old premium | Average change |
|---|---|---|---|
| Less than −10% | 10 | $20,000 | −20% |
| −10% to less than 0% | 30 | $15,000 | −4% |
| 0% to less than 10% | 40 | $40,000 | +4% |
| At least 10% | 20 | $25,000 | +10% |
Step 1: Calculate totals and shares. There are \(10+30+40+20=100\) policies and \(\$20{,}000+\$15{,}000+\$40{,}000+\$25{,}000=\$100{,}000\) of old premium. The count shares are 10%, 30%, 40%, and 20%. The premium shares are 20%, 15%, 40%, and 25%.
Step 2: Calculate dollar impact by range. Multiply each old premium by its average change: \(\$20{,}000(-20\%)=-\$4{,}000\); \(\$15{,}000(-4\%)=-\$600\); \(\$40{,}000(4\%)=\$1{,}600\); and \(\$25{,}000(10\%)=\$2{,}500\). The total dollar change is \(-\$4{,}000-\$600+\$1{,}600+\$2{,}500=-\$500\).
Step 3: Reconcile both overall measures. The aggregate premium change is \(-\$500/\$100{,}000=-0.5\%\). The count average is \((10(-20\%)+30(-4\%)+40(4\%)+20(10\%))/100=0.4\%\). The measures differ because policies in the decrease ranges carry a different share of old premium than their share of policy count.
Check and common mistake. The proposed total premium is $99,500, equal to $100,000 less the $500 net decrease. Treating the count average of 0.4% as the aggregate premium result would be incorrect. This calculation also uses the stated average change within each range; the range label alone is not enough to determine the group’s dollar impact.
Construction Checks and Interpretation
A sound exhibit should be traceable from individual premiums to grouped results and from grouped results to the overall total. These checks help identify mistakes before a summary is interpreted:
- Reconcile totals. The sum of proposed premiums less the sum of old premiums must equal the sum of policy dollar changes, subject to any explicitly documented rounding.
- Reconcile groups. Counts and old premiums across mutually exclusive groups should sum to the corresponding exhibit totals. Group boundaries should neither overlap nor leave gaps.
- Keep measures distinct. Label a mean as count-weighted or premium-weighted. Report count shares and premium shares with their own denominators.
- Handle zero old premiums carefully. If \(P_j^{old}=0\), the individual percentage change is undefined. Do not divide by zero or silently include that policy in a percentage-average denominator. Report its dollar change separately or specify an appropriate treatment for the exhibit.
- Explain the calculation basis. Differences in term length, coverage, exposure, or premium components can make a comparison misleading unless both premiums are placed on the same stated basis.
- Control rounding. Sum unrounded amounts when possible, then round reported money to cents and rates to an appropriate number of decimals. Small differences between displayed rounded rows and totals should be labeled as rounding, not hidden.
A common trap is to compare the proposed and current total premiums for different sets of policies. That calculation mixes rate impact with changes in the underlying population. For a policyholder impact exhibit, hold the modeled policy population and basis fixed and recalculate its premium under each rate set. If the exhibit intentionally uses different populations, describe the difference and do not present the result as a pure rate change.
Check Your Understanding
For each problem, distinguish policy-count results from premium-weighted results and check the totals.
- A hypothetical three-policy book has old premiums of $500, $1,500, and $2,000, with proposed changes of +8%, −4%, and +2%. Find the count-weighted average and the aggregate premium change. (Answer: count-weighted average = 2.000%; aggregate change = +0.500%)
- A segment has 50 policies with average old premium $600 and a +6% change. A second segment has 10 policies with average old premium $1,800 and a −3% change. Calculate the overall premium-weighted change and the count-weighted average. (Answer: premium-weighted change = +2.625%; count-weighted average = +4.500%)
- An exhibit has 20 policies with $10,000 old premium and an average change of −5%, and 30 policies with $30,000 old premium and an average change of +4%. Find each group’s count share and premium share, and the total premium change. (Answer: count shares = 40% and 60%; premium shares = 25% and 75%; total change = +1.750%)
- A proposed-rate exhibit reports $250,000 old premium and $262,500 proposed premium across the same policies. Calculate the aggregate percentage change. Can the policy-count average be determined from these totals alone? (Answer: aggregate change = +5.000%; no, the policy-level changes or equivalent count-weighted information are also needed.)
- A policy has old premium $0 and proposed premium $240. What is its dollar change, and should it be assigned a percentage change of 100%? (Answer: dollar change = +$240; no, the percentage change is undefined because the old premium is zero.)