Comparing Rate Impact Across Policyholders

Why Compare Rate Changes Across Policyholders?

A portfolio average can make two proposed rate plans look alike even when their effects on individual policyholders are very different. One plan may distribute a modest change broadly; another may leave most policyholders nearly unchanged while imposing large increases and decreases on a small group. Comparing the distribution of policy-level changes helps reveal these patterns.

The comparison is useful after rate indications have been translated into proposed policy premiums. The object is not to judge a plan from one statistic, but to summarize its impact in ways that answer distinct questions: What is the average change on a stated weighting basis? Where is the middle of the distribution? How many policies, or how much premium, cross a material-change threshold?

Definition: For policy $j$, let $P_j^0$ be its current premium and $P_j^1$ its proposed premium, calculated on a consistent basis. Its percentage rate change is $$ r_j=\frac{P_j^1-P_j^0}{P_j^0} =\frac{P_j^1}{P_j^0}-1, $$ provided $P_j^0>0$. A positive value is an increase, a negative value is a decrease, and zero means no change. The change is often reported as $100r_j\%$.

“Consistent basis” matters. The current and proposed premiums should correspond to the same policy, exposure, coverage, and premium components, with only the proposed rate plan changing. If the comparison includes a change in coverage or exposure, the resulting premium difference does not isolate the rate impact. Policies with zero current premium also require a separate treatment: the percentage change is undefined, so they should not be assigned a percentage by dividing by zero.

Every summary depends on its weighting basis. An unweighted mean treats every policy as equally important. An exposure-weighted mean gives more influence to policies with more exposure. A current-premium-weighted mean gives more influence to policies contributing more current premium. These statistics answer different questions; name the basis whenever reporting a result.

Weighted Averages and Their Interpretation

Let $w_j$ be a nonnegative weight for policy $j$, with $\sum_j w_j>0$. The weighted mean percentage change is the sum of each policy’s change multiplied by its weight, divided by the total weight. When the weights are current premiums, the statistic also equals the modeled percentage change in total premium, provided exposures and the policy set are held fixed.

\bar r_w=\frac{\sum_j w_jr_j}{\sum_j w_j}.

The aggregate interpretation follows directly from the definition of $r_j$. If $w_j=P_j^0$, then $P_j^1=P_j^0(1+r_j)$. Therefore,

\frac{\sum_j P_j^1-\sum_j P_j^0}{\sum_j P_j^0} = \frac{\sum_j P_j^0r_j}{\sum_j P_j^0} =\bar r_w.

This identity is exact for the premiums being compared. It does not mean that an exposure-weighted average necessarily equals the aggregate premium change: that equality requires weights proportional to current premium. It also does not account for changes in policy count, coverage, or policyholder behavior after implementation.

Worked Example: Weighted Average and Material-Change Shares

A hypothetical insurer compares proposed premiums for four policies. The weights are each policy’s current premium. Calculate the premium-weighted average change, the weighted median, and the number and current-premium share of policies increasing by at least 10%.

Policy	Current premium	Proposed premium	Percentage change
A	$800	$880	10%
B	$1,200	$1,080	-10%
C	$600	$660	10%
D	$2,000	$1,900	-5%

Method and assumptions. Use $r_j=(P_j^1-P_j^0)/P_j^0$ and $w_j=P_j^0$. These are deterministic hypothetical policy premiums; no interest rate, mortality assumption, or stochastic loss distribution is used.

Step 1: Verify the changes. Policy A changes by $(880-800)/800=80/800=10\%$. B changes by $(1{,}080-1{,}200)/1{,}200=-120/1{,}200=-10\%$. C changes by $60/600=10\%$. D changes by $-100/2{,}000=-5\%$.

Step 2: Calculate the weighted mean. Total current premium is $800+1{,}200+600+2{,}000=4{,}600$. The weighted change amount is $800(0.10)+1{,}200(-0.10)+600(0.10)+2{,}000(-0.05)=80-120+60-100=-80$. Thus $\bar r_w=-80/4{,}600=-0.0173913\ldots$, or -1.74% rounded to two decimal places.

Step 3: Find the weighted median. In ascending order, the changes and weights are $-10\%$ with $1,200, $-5\%$ with $2,000, and $10\%$ with a combined $1,400. Half the total weight is $4{,}600/2=2{,}300$. Cumulative weight first reaches or exceeds $2,300 at $-5\%$, so the weighted median is -5%.

Step 4: Count large increases and measure their premium share. A and C increase by at least 10%, so 2 of 4 policies qualify. Their current premium is $800+600=1{,}400$, and their share of current premium is $1{,}400/4{,}600=0.3043$, or 30.43%.

Check and common mistake. Proposed total premium is $880+1{,}080+660+1{,}900=4{,}520$, a decrease of $80 from $4,600. The aggregate change is $-80/4{,}600=-1.74\%$, agreeing with the weighted mean. The unweighted mean is $(10-10+10-5)/4=1.25\%$; reporting it as the aggregate premium change would be wrong because policies have different current premiums.

Weighted Percentiles

A percentile gives a location in the distribution rather than an average. For policy-level changes, a weighted percentile answers: what is the first change level at which at least a specified share of the total weight has accumulated? The convention below is a discrete inverse-cumulative definition; other interpolation conventions exist, so the convention should be stated when results must be compared.

Definition: Given changes $r_j$, nonnegative weights $w_j$, a positive total weight $\sum_j w_j>0$, and $0<p\leq1$, sort the changes from smallest to largest. The weighted $p$th percentile is the first value $r$ whose cumulative weight is at least $p\sum_jw_j$. If several policies have the same change, their weights are combined at that value.

The requirement that total weight be positive is essential. If all weights are zero, the target weight $p\sum_jw_j$ is also zero, so the cumulative-weight rule would not define a meaningful percentile. When the weights are policy counts, the percentile describes the policy distribution; when they are current premiums, it describes the premium-weighted distribution. A weighted median is the weighted 50th percentile under this convention.

Worked Example: Weighted Percentiles and Tail Shares

For five hypothetical policies, the current premiums are $1,000, $2,000, $3,000, $2,000, and $2,000. Their proposed percentage changes are respectively $-20\%$, $-5\%$, $0\%$, $12\%$, and $30\%$. Find the premium-weighted mean, weighted median, weighted 80th and 90th percentiles, and the number and premium share of increases greater than 15%.

Method and assumptions. Use current premiums as weights and the discrete weighted-percentile definition. Premiums and changes are hypothetical and deterministic; no interest rate, mortality assumption, or stochastic loss distribution is involved.

Step 1: Compute the weighted mean. Total weight is $1{,}000+2{,}000+3{,}000+2{,}000+2{,}000=10{,}000$. The weighted change amount is $1{,}000(-0.20)+2{,}000(-0.05)+3{,}000(0)+2{,}000(0.12)+2{,}000(0.30)=-200-100+0+240+600=540$. The mean is $540/10{,}000=0.054$, or 5.40%.

Step 2: Accumulate weights in change order. The ordered changes and cumulative weights are: $-20\%$, $1,000; $-5\%$, $3,000; $0\%$, $6,000; $12\%$, $8,000; and $30\%$, $10,000.

Step 3: Locate the percentiles. The 50th-percentile target is $0.50(10{,}000)=5{,}000$, first reached at $0\%$, so the weighted median is 0%. The 80th-percentile target is $8,000, first reached at $12\%$, so the weighted 80th percentile is 12%. The 90th-percentile target is $9,000, first reached at $30\%$, so the weighted 90th percentile is 30%.

Step 4: Measure the large-increase tail. Only the policy with a 30% change exceeds 15%. Thus 1 of 5 policies qualifies, and its current premium share is $2{,}000/10{,}000=20\%$.

Check and common mistake. The percentile target is a cumulative weight, not a percentage of the numerical change. For example, the 80th-percentile target is $8,000, not “80% of 12%.” Because cumulative weight is exactly $8,000 at 12%, the stated first-at-least convention assigns the percentile to 12%.

Comparing Plans Beyond Their Portfolio Averages

A fair comparison applies each candidate plan to the same policy set and uses the same baseline premiums, weights, thresholds, and percentile convention. The average measures a center of mass, while percentiles and threshold counts reveal spread and tail impact. No single summary replaces the others: a plan can have a stable mean but still have large changes concentrated among particular policies.

Worked Example: Two Plans With the Same Average but Different Tails

A hypothetical insurer evaluates two plans on the same five policies, each with a current premium of $1,000. Plan A produces changes of $-10\%$, $-5\%$, $0\%$, $5\%$, and $10\%$. Plan B produces $-30\%$, $-10\%$, $0\%$, $10\%$, and $30\%$. Compare their premium-weighted averages, 80th percentiles, mean absolute changes, and counts of changes exceeding 20% in magnitude.

Method and assumptions. Each policy’s weight is its current premium, so the five equal weights make the weighted summaries coincide with the corresponding equally weighted summaries. The inputs are hypothetical deterministic changes; no interest rate, mortality assumption, or stochastic loss distribution is used.

Step 1: Compare averages. Plan A’s changes sum to $-10-5+0+5+10=0$ percentage points. Plan B’s sum is $-30-10+0+10+30=0$. Each plan’s weighted mean is therefore $0/5=0\%$. In dollar terms, each plan has zero net change across five $1,000 premiums.

Step 2: Compare the 80th percentiles. For either plan, total weight is $5,000 and the 80th-percentile target is $4,000. Plan A’s ordered cumulative weights are $1,000, $2,000, $3,000, $4,000, and $5,000, so its 80th percentile is the fourth change, 5%. Plan B’s fourth change is 10%, so its weighted 80th percentile is 10%.

Step 3: Compare average absolute changes. For Plan A, the absolute changes sum to $10+5+0+5+10=30$ percentage points; the mean absolute change is $30/5=6\%$. For Plan B, they sum to $30+10+0+10+30=80$ percentage points; the mean absolute change is $80/5=16\%$.

Step 4: Count changes exceeding 20% in magnitude. Plan A has none. Plan B has two: $-30\%$ and $30\%$. Thus Plan B has 2 of 5 policies, or 40%, beyond the threshold; their current premiums total $2,000, or 40% of the portfolio’s $5,000 current premium.

Check and common mistake. Both plans have the same aggregate average, but Plan B has a higher 80th percentile, a larger mean absolute change, and more changes beyond the threshold. Concluding that the plans have the same policyholder impact from their equal averages would conceal this difference.

Practical Choices and Exam Traps

A concise impact summary should state the population being compared, the definition of policy-level change, the weighting basis, the percentile convention, and whether threshold boundaries are strict or inclusive. For example, “at least 10%” means $r_j\geq0.10$, while “greater than 10%” means $r_j>0.10$. “At most a 10% decrease” should be translated carefully into the intended inequality before counting.

Use both counts and weighted shares when the question concerns customer reach and financial concentration. A policy count answers how many policies are affected; a premium-weighted share answers how much current premium belongs to those policies. A small number of high-premium policies can produce a low count but a substantial premium share, or the reverse.

Mixing weighting bases: Do not call an exposure-weighted mean a premium-weighted average or interpret it as aggregate premium change unless the weights are proportional to current premium.
Using a mean as a distribution summary: A mean alone does not describe dispersion, percentiles, or how many policies have large changes.
Changing the denominator: Compute each policy’s percentage change relative to its own current premium, not the proposed premium.
Leaving conventions unstated: Percentile interpolation and threshold inclusivity can change reported results, especially with discrete policy-level data.
Ignoring zero current premiums: A percentage change is undefined when $P_j^0=0$; handle such records separately rather than forcing a ratio.

Key takeaway: Report a policy-level impact distribution, not just a portfolio average. A weighted mean summarizes the center on a stated basis; weighted percentiles and threshold counts expose where changes fall and how much of the portfolio they affect.

Check Your Understanding

For each hypothetical case, identify the requested summary using the stated weights and convention.

Two policies have current premiums of $500 and $1,500 and changes of 8% and $-4\%$. What is the current-premium-weighted average change? (Answer: $-1.00\%$.)
Four policies have changes $-12\%$, $0\%$, $6\%$, and $18\%$, with weights $1,000, $2,000, $3,000, and $4,000. Under the first-cumulative-weight-at-least convention, what is the weighted 75th percentile? (Answer: $18\%$.)
Five policies have changes $-10\%$, $-9\%$, $0\%$, $15\%$, and $25\%$. How many, and what percentage of policies, have decreases of at least 10% in magnitude? (Answer: 1 policy, or 20%.)
Three policies have changes $-20\%$, $5\%$, and $10\%$, with weights $2,000, $1,000, and $1,000. Find the weighted average change and state whether it indicates an aggregate increase or decrease on this weighting basis. (Answer: $-6.25\%$, a decrease.)
Two policies have changes $-5\%$ and $5\%$, with weights $0 and $2,000. What is the weighted 50th percentile under the stated definition? What if both weights were zero? (Answer: $5\%$ when total weight is $2,000; undefined when total weight is zero.)

What You'll Learn