Why a Consistent Comparison Matters
A monitoring result is only meaningful if realized losses and expected losses describe the same exposure and claim basis. Comparing losses from one period with an expectation built for another, using different exposure units, or mixing adjusted claims with unadjusted claims can create an apparent deviation that is only a measurement mismatch.
The basic comparison uses three related measures: frequency (claims per exposure unit), severity (loss per claim), and pure premium (loss per exposure unit). Rate-level changes require special care: they change premium, but do not by themselves change the dollar amount of a claim. This tutorial is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
An exposure unit might be a vehicle-year, policy-year, or another unit appropriate to the coverage. Use the same exposure definition for observed and expected results. Likewise, define which claim amounts are included—for example, paid losses or incurred losses—and apply any coverage, trend, or other adjustments consistently. Do not silently change from one loss basis to another partway through the calculation.
Relating Frequency, Severity, and Pure Premium
When \(N>0\), pure premium can be decomposed into frequency times severity. This identity provides both a calculation method and a diagnostic: a higher pure premium can result from higher claim frequency, higher severity, or both.
The same relationship holds for expected values when expected pure premium is defined as expected frequency times expected severity. Therefore, with a consistent basis, the ratio of observed to expected pure premium equals the product of the observed-to-expected frequency and severity ratios.
This ratio identity is not a separate statistical assumption: it follows by substituting the definitions. For example, observed pure premium is \(L/E=(N/E)(L/N)\), while expected pure premium is the product of expected frequency and expected severity. Dividing one product by the other gives the displayed result. If there are no observed claims, observed severity \(L/N\) is undefined, although observed pure premium remains defined; in that case report the frequency and pure-premium comparison without assigning an average severity.
Worked Example: Separating Frequency and Severity Deviations
A hypothetical book has 1,200 vehicle-years. The expected frequency is 0.080 claims per vehicle-year and expected severity is $5,000. During the period, 102 claims produce $561,000 of losses on the same claim basis. Compare observed and expected frequency, severity, and pure premium.
Method. First calculate expected claims and losses, then calculate the three observed measures using the same 1,200 vehicle-years.
Expected claims are \(1{,}200\times0.080=96\). Expected losses are \(96\times\$5{,}000=\$480{,}000\), so expected pure premium is \(\$480{,}000/1{,}200=\$400\) per vehicle-year.
Observed frequency is \(102/1{,}200=0.085\) claims per vehicle-year. Observed severity is \(\$561{,}000/102=\$5{,}500\) per claim. Observed pure premium is \(\$561{,}000/1{,}200=\$467.50\) per vehicle-year.
The frequency ratio is \(0.085/0.080=1.0625\), or 6.25% above expected. The severity ratio is \(\$5{,}500/\$5{,}000=1.10\), or 10% above expected. The pure-premium ratio is \(\$467.50/\$400=1.16875\), or 16.875% above expected.
Check. The product of the frequency and severity ratios is \(1.0625\times1.10=1.16875\), matching the pure-premium ratio. The method avoids attributing the whole deviation to frequency when severity also differs.
Aligning Claim Amounts to a Common Basis
Realized claims may need adjustment before comparison with an expectation. The adjustment should correspond to a stated difference in basis, such as a coverage change or a specified loss trend. Apply the adjustment to the claim amounts it is intended to change; do not apply it automatically to claim counts or exposure. If the adjustment changes only severity, frequency is unchanged.
For a multiplicative severity adjustment, let \(A\) be the claim-basis adjustment factor and let \(T\) be the trend factor from the historical loss period to the comparison period. Then adjusted losses are \(L^*=L A T\). If a constant annual severity trend rate \(g\) applies for \(t\) years, \(T=(1+g)^t\). This formula assumes the stated trend is applicable to the losses being adjusted and that the time interval and compounding convention are correct. It does not justify applying the same trend to frequency unless a frequency trend assumption is separately specified.
Worked Example: Adjusting Historical Losses Before Comparison
A hypothetical portfolio has 1,000 exposure units and 80 claims. Historical incurred losses total $720,000. For this comparison, claims require a 5% claim-basis adjustment and then two years of 4% annual severity trend. Expected frequency on the comparison basis is 0.075 and expected severity is $10,000. Calculate adjusted observed results and compare them with expected pure premium.
Method. Apply the claim-basis factor and trend to losses, while retaining the observed claim count and exposure because the stated adjustments affect severity only.
The claim-basis factor is \(A=1.05\). The two-year trend factor is \(T=(1.04)^2=1.0816\). The combined factor is \(1.05\times1.0816=1.13568\). Adjusted losses are \(\$720{,}000\times1.13568=\$817{,}689.60\).
Observed frequency remains \(80/1{,}000=0.080\). Adjusted severity is \(\$817{,}689.60/80=\$10{,}221.12\) per claim. Adjusted observed pure premium is \(\$817{,}689.60/1{,}000=\$817.6896\) per exposure unit.
Expected claims are \(1{,}000\times0.075=75\). Expected losses are \(75\times\$10{,}000=\$750{,}000\), giving expected pure premium of \(\$750\) per exposure unit. The observed-to-expected frequency ratio is \(0.080/0.075=1.0667\). The severity ratio is \(\$10{,}221.12/\$10{,}000=1.022112\). The pure-premium ratio is \(\$817{,}689.60/\$750{,}000=1.0902528\), about 9.03% above expected.
Check. \(1.0666667\times1.022112\approx1.0902528\), consistent with the pure-premium ratio. The result avoids trending claim frequency when only severity trend was specified.
Rate Level and the Loss Comparison
A rate-level change affects the premium denominator in a loss-ratio comparison. It is not a claim adjustment: increasing rates by a factor does not multiply the realized loss amount by that factor. If the purpose is to compare loss ratios at a common rate level, adjust premium to that level and compare it with losses measured on the appropriate common claim basis.
For a simple example of rate-level normalization, suppose earned premium at the old rate level is \(P\), and the applicable rate-level factor to the chosen comparison level is \(R\). Then on-level premium is \(P R\). This simple multiplication is appropriate only when \(R\) is the correct aggregate factor for the premium being normalized; it does not address differences in portfolio composition or the timing of individual rate changes.
Worked Example: Normalizing Premium Without Changing Losses
A hypothetical block has 2,000 exposure-years. Earned premium at its historical rate level is $1,600,000. The on-level factor to the comparison rate level is 1.10. Current-basis realized losses are $1,080,000, and expected pure premium at that basis is $500 per exposure-year. Compare actual and expected loss ratios at the common rate level.
Method. Normalize premium to the comparison rate level. Keep the realized losses unchanged because the stated adjustment is a rate-level change, not a loss adjustment.
On-level premium is \(\$1{,}600{,}000\times1.10=\$1{,}760{,}000\). Expected losses are \(2{,}000\times\$500=\$1{,}000{,}000\). The expected loss ratio is \(\$1{,}000{,}000/\$1{,}760{,}000=0.56818\), or 56.82%.
The actual loss ratio at the common rate level is \(\$1{,}080{,}000/\$1{,}760{,}000=0.61364\), or 61.36%. Realized losses are \(1{,}080{,}000/1{,}000{,}000=1.08\), or 8% above expected. The same 8% relationship appears in the actual-to-expected loss-ratio comparison because both ratios use the same on-level premium denominator.
Check. Using unadjusted historical premium would give \(\$1{,}080{,}000/\$1{,}600{,}000=67.50\%\), which is not the actual loss ratio at the comparison rate level. The correction is to adjust the premium denominator, not multiply losses by 1.10.
A Practical Comparison Sequence
A disciplined sequence makes the basis choices visible and helps prevent arithmetic errors. Record the observed and expected values, the adjustments applied, and the resulting units. If any input is not available on a comparable basis, state the limitation rather than implying that the comparison is fully aligned.
Specify the coverage, group, observation period, and exposure unit. Use the same scope for observed and expected quantities.
State which claims and loss amounts are included. Apply only the claim-basis adjustments and trend assumptions needed to align those amounts with the expectation.
Compute frequency as claims divided by exposure, severity as losses divided by claims when claims are positive, and pure premium as losses divided by exposure.
Multiply expected frequency by exposure to obtain expected claims, then multiply expected claims by expected severity to obtain expected losses.
For loss-ratio comparisons, adjust the premium denominator to the selected rate level. Do not treat a rate-level factor as a claim-cost factor.
Common Traps and Interpretation
- Mixing exposure units. A rate per policy-year cannot be compared directly with a result per vehicle-year unless the exposure bases are reconciled.
- Adjusting claims inconsistently. If losses are trended but the expected severity is already at the historical level, the comparison is misaligned. Identify the target period for both sides.
- Applying severity trend to frequency by default. A severity trend changes loss per claim; it does not change the number of claims unless a separate frequency assumption says so.
- Multiplying losses by the rate change. Rate level affects premium. Adjust the premium denominator for an on-level loss ratio, while keeping claims on their stated loss basis.
- Interpreting a deviation as a cause. An observed-to-expected ratio describes a difference, not its explanation. Frequency and severity ratios help locate the component of the aggregate difference, but do not establish why it occurred.
For an operational review, preserve enough detail to reproduce the comparison: exposure definition, claim count, loss basis, adjustment factors, expected assumptions, rate-level factor where relevant, and rounding. These details make it possible to distinguish a real result from a basis mismatch. The comparison alone does not separate changes in portfolio composition from rate effects; those require a separate analysis.
Check Your Understanding
For each problem, keep the stated exposure, claim, and premium bases distinct.
- A hypothetical program has 600 exposure units, expected frequency 0.12, and expected severity $4,000. It records 78 claims and $343,200 of losses on the same basis. Find observed frequency, severity, pure premium, and the pure-premium ratio to expected. (Answer: observed frequency 0.1300; severity $4,400; pure premium $572; ratio 1.1917.)
- A hypothetical account group has 800 exposure units, 50 claims, and historical losses of $600,000. Apply a 1.03 claim-basis factor and one year of 3% severity trend. Expected frequency is 0.06 and expected severity is $12,000. Find adjusted losses and the observed-to-expected pure-premium ratio. (Answer: adjusted losses $636,540; ratio 1.1051.)
- A block has $900,000 earned premium at its historical rate level, a rate-level factor of 1.08, 1,000 exposure units, and $540,000 of losses already on the comparison claim basis. Expected pure premium is $500. Find the actual and expected loss ratios at the comparison rate level and the percentage by which losses exceed expected. (Answer: actual loss ratio 55.56%; expected loss ratio 51.44%; losses are 8.00% above expected.)
- A hypothetical portfolio has 400 exposure-years, 36 claims, and adjusted losses of $198,000. Expected frequency is 0.08 and expected severity is $6,000. Calculate the observed frequency, severity, and pure premium, then identify whether frequency or severity is above expectation. (Answer: frequency 0.0900, 12.50% above expected; severity $5,500, 8.33% below expected; pure premium $495, 3.125% above the expected $480.)