Benchmarking the Selected Rate Change

Why Benchmark a Selected Rate Change?

A selected rate change is an actuarial judgment supported by indications, assumptions, and other relevant evidence. Benchmarking provides a separate reasonableness check: it asks how that selection compares with a clearly identified reference, such as an internal plan, a historical implemented change, or an external market measure. A benchmark is context, not a substitute for the indication analysis.

A difference can prompt useful questions. Are the figures measured over the same period? Do they describe the same coverage and population? Is one a change in filed rates while the other is a change in premium actually collected? A numerical gap alone does not answer those questions or establish that the selected rate change is wrong.

Definition: Let $s$ be the selected rate change, expressed as a decimal, and let $b$ be a benchmark rate change, also expressed as a decimal. For example, an increase of 6% is written as $0.06$. The percentage-point gap is $100(s-b)$ percentage points. The selected and benchmark rate-change factors are $1+s$ and $1+b$, respectively.

The percentage-point gap answers, “How many percentage points apart are the changes?” A second comparison, the relative difference in rate-change factors, answers, “How much larger is the selected factor than the benchmark factor?” These are different measures and should not be presented as though they were interchangeable.

\text{Percentage-point gap}=100(s-b)

\text{Relative factor difference}=\frac{1+s}{1+b}-1

For a positive benchmark, a positive percentage-point gap means the selected increase is larger than the benchmark increase. If either change is a decrease, the factor comparison can be especially useful: a less negative change has a higher rate-change factor. The relative factor difference is meaningful when $1+b\ne 0$; ordinary insurance rate changes have positive factors, so this condition is normally satisfied.

Make the Comparison Comparable

Before calculating a gap, define what each number measures. A sound comparison specifies the line or coverage, the business included, the effective dates, the time period, and whether the figure is an indicated, selected, filed, implemented, or collected change. The benchmark also needs a stated source and a reason it is relevant. A number labelled “market change” is not self-explanatory.

Timing matters. A cumulative change over several years cannot be compared directly with a one-year change. Convert the cumulative factor to an annualized factor when the comparison calls for an annual rate. If a cumulative factor $C$ applies over $n$ years and an equivalent constant annual rate is appropriate, its annualized rate is $C^{1/n}-1$. This conversion assumes compounding at a constant equivalent annual rate; it does not claim that actual changes occurred evenly in each year.

The comparison also needs a sensible interpretation. Historical implemented changes describe past actions, not necessarily future adequacy. An internal budget may reflect a management objective rather than an actuarial estimate. An external benchmark may cover a different mix of insurers, products, or policy terms. Each can still be informative, but its limitations must be stated.

Comparison rule: Compare like with like where possible. If periods, definitions, or populations differ, either convert them on a justified basis or explain why the resulting comparison is only directional. Do not treat a benchmark gap as evidence of a specific cause unless the supporting data establish that cause.

Worked Comparisons

Worked Example: Comparing a Selection With an Internal Plan

A hypothetical insurer selects an overall rate increase of 8.4% for a specified coverage and effective period. Its internal operating plan assumes a 6.0% increase for the same coverage and period. A hypothetical review guideline calls for investigation when the gap exceeds 2.0 percentage points. Compare the selection with the plan and assess whether review is triggered.

Method: matched-period comparison. Both values refer to the same coverage and period, so no timing conversion is needed. Write the selected change and benchmark as decimals: $s=0.084$ and $b=0.060$.

First calculate the percentage-point gap:

100(s-b)=100(0.084-0.060)=100(0.024)=2.4\text{ percentage points}.

The selected rate-change factor is $1+s=1.084$. The plan factor is $1+b=1.060$. Their relative factor difference is:

\frac{1.084}{1.060}-1 =\frac{0.024}{1.060} =0.0226415\ldots =2.264\%.

The gap of 2.4 percentage points exceeds the hypothetical 2.0-point review guideline by $2.4-2.0=0.4$ percentage points. Therefore, the guideline calls for investigation. It does not direct the actuary to reduce the selection to 6.0%; the plan is a comparison point, not automatically the correct rate change.

Check: Applying each factor to a hypothetical base rate of $100 gives $108.40 and $106.00. The difference is $2.40 per $100 of base rate, consistent with a 2.4-percentage-point gap. The factor comparison is approximately 2.264% relative to the plan factor. The common mistake this method avoids is describing the 2.4-point gap as a 2.4% relative difference.

Worked Example: Annualizing an External Benchmark

A hypothetical external market summary reports a cumulative rate-change factor of 1.1025 over two years for a broadly similar coverage. The insurer’s selected change is a one-year increase of 6.2%. Assume, solely for this comparison, that an equivalent constant annual benchmark is useful. Calculate the annualized benchmark and compare it with the selection.

Method: convert both measures to one-year changes before comparing. The cumulative factor is $C=1.1025$, and the period is $n=2$ years. Its equivalent constant annual factor is:

C^{1/n}=1.1025^{1/2}=1.05,

because $1.05^2=1.1025$. Therefore the annualized benchmark change is $1.05-1=0.05$, or 5.0%.

The one-year selected change is 6.2%, so the percentage-point gap is $6.2\%-5.0\%=1.2$ percentage points. The respective factors are 1.062 and 1.050. The relative factor difference is:

\frac{1.062}{1.050}-1 =\frac{0.012}{1.050} =0.0114286\ldots =1.143\%.

Thus the selected factor is 1.143% above the annualized benchmark factor, while the changes themselves differ by 1.2 percentage points. The comparison is directional because “broadly similar coverage” does not establish that the external summary has the same population or measurement definition.

Check: Compounding the annualized benchmark for two years returns $1.05^2=1.1025$, the stated cumulative factor. The selected change exceeds the annualized benchmark by 1.2 points. The common mistake this method avoids is comparing 6.2% directly with the two-year cumulative change of 10.25%, which uses different time units.

Worked Example: Reading a Set of Benchmarks

A hypothetical insurer selects a 6.6% increase. Three reference points for the same one-year period and stated coverage are an internal plan of 6.0%, a historical implemented change of 3.0%, and a hypothetical external estimate of 7.1%. A review guideline flags any individual gap greater than 1.5 percentage points. Calculate the gaps and decide what the guideline indicates.

Method: compare the selection separately with each benchmark. Do not average the references into a new rate change: they represent different kinds of evidence, and no weighting rule has been supplied.

Against the internal plan, the gap is $6.6\%-6.0\%=0.6$ percentage points. Against the historical change, it is $6.6\%-3.0\%=3.6$ percentage points. Against the external estimate, it is $6.6\%-7.1\%=-0.5$ percentage points. The negative gap means the selection is 0.5 percentage points below that benchmark.

Compare the absolute size of each gap with the 1.5-point guideline: $0.6\le 1.5$, $3.6>1.5$, and $0.5\le 1.5$. Only the historical comparison triggers review under this rule. The selection is therefore broadly near two references but materially above the historical value. That pattern motivates checking whether the historical reference remains relevant; it does not establish that the selection should be changed.

Check: The signed gaps are $+0.6$, $+3.6$, and $-0.5$ percentage points, respectively, and only one absolute gap exceeds 1.5 points. The common mistake this method avoids is averaging the three references and treating that average as a justified selected rate change.

Why the Two Gap Measures Differ

The factor comparison follows directly from how a rate change acts on a starting rate. If the same hypothetical base rate is $R$, the selected rate is $R(1+s)$ and the benchmark rate is $R(1+b)$. Their dollar difference is $R[(1+s)-(1+b)]=R(s-b)$. Dividing that difference by the benchmark rate $R(1+b)$ gives the relative difference in factors:

\frac{R(1+s)-R(1+b)}{R(1+b)} =\frac{R(s-b)}{R(1+b)} =\frac{1+s}{1+b}-1.

This derivation assumes the same positive base rate $R$ is being compared. It explains why the relative factor difference uses the benchmark factor in its denominator, while the percentage-point gap simply subtracts the rate changes. For small changes, the two numbers may be close, but they are not identical by definition.

When Does a Difference Merit Investigation?

A benchmark gap is a screening signal. Whether it warrants further work depends on the benchmark’s relevance, the size and direction of the gap, the uncertainty in the underlying figures, and the consequences of being wrong. A stated tolerance can make the review process more consistent, but a tolerance is a governance rule—not a statistical proof that smaller gaps are immaterial or larger gaps are unreasonable.

A practical review can proceed in a disciplined sequence:

Define the comparison.
Record the selected change, benchmark source, coverage, business scope, and effective period. Confirm that each percentage refers to a rate change rather than another premium measure.

Align the basis.
Convert periods when the conversion is justified, and disclose material differences in scope or definition that cannot be reconciled.

Measure and interpret the gap.
Report the signed percentage-point gap. Use a factor comparison when relative scale is useful, and label each measure clearly.

Decide whether to investigate.
Apply any stated review tolerance, then assess whether the benchmark is sufficiently relevant and reliable to support follow-up. A flag prompts examination; it does not dictate a new selection.

The follow-up should be proportionate to the evidence. For example, confirm the benchmark’s dates and definition, verify that the arithmetic and units are correct, and consider whether the reference covers a comparable risk population. If those checks explain the gap, document the explanation. If they do not, the difference may justify revisiting assumptions or the evidence supporting the selection.

Common Benchmarking Traps

Confusing percentage points with percent. A change from 5% to 7% is a 2-percentage-point difference; relative to 5%, it is a 40% increase in the change value. State which comparison is intended.
Mixing time periods. A two-year cumulative change is not directly comparable with a one-year change. Annualize only when an equivalent annual rate is appropriate and explain the conversion.
Treating a benchmark as an indication. A budget, historical result, or external estimate is a reference, not automatically the actuarially indicated or selected change.
Ignoring definition and scope. Similar labels do not guarantee that two figures cover the same business or measure the same stage of a rate change.
Making a causal claim from an aggregate gap. A difference identifies something to examine; by itself it does not explain why the figures differ.

Key takeaway: A useful benchmark comparison aligns definitions and periods, reports the gap in clearly labelled units, and interprets the result in light of the benchmark’s limitations. A material difference can justify investigation without determining the final selection.

Check Your Understanding

For each problem, state the relevant comparison and interpret the result. Treat all scenarios and data as hypothetical.

A selected one-year increase is 7.3%; an internal plan for the same period is 5.8%. Find the percentage-point gap and the relative factor difference, to two decimal places. (Answer: 1.5 percentage points; $(1.073/1.058)-1=1.42\%$.)
An external benchmark is a cumulative factor of 1.0609 over two years. A selected one-year increase is 5.4%. Annualize the benchmark and find the percentage-point gap. (Answer: annualized benchmark $=1.03-1=3.0\%$; selected change is 2.4 percentage points higher.)
A selected change is a 2.0% decrease, and the benchmark is a 4.0% decrease. Find the signed percentage-point gap and relative factor difference. Explain which factor is higher. (Answer: $+2.0$ percentage points; $(0.98/0.96)-1=2.08\%$; the selected factor is higher.)
A 6.6% selection is compared with three same-period benchmarks: 4.5%, 5.8%, and 7.1%. A hypothetical rule flags absolute gaps greater than 1.5 percentage points. Which comparisons are flagged? (Answer: gaps are $+2.1$, $+0.8$, and $-0.5$ percentage points; only the 4.5% benchmark comparison is flagged.)
A selected one-year increase is 6.2%. An external reference reports a cumulative factor of 1.1025 over two years. Under a constant-equivalent annualization assumption, calculate the annual benchmark, the percentage-point gap, and the relative factor difference. (Answer: annual benchmark $=5.0\%$; gap $=1.2$ percentage points; relative factor difference $=(1.062/1.05)-1=1.14\%$.)

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