Why Test the Components?
A projected loss cost can look reasonable in total while one of its components is not. For example, an overly low selected claim frequency and an overly high selected severity can offset one another in the product. The total may then agree with an aggregate benchmark even though the underlying assumptions tell an implausible story about how claims arise and how large they are.
Testing frequency and severity assumptions means checking whether each selection is plausible on its own, whether their combined effect is consistent with relevant aggregate evidence, and whether the result is unduly sensitive to reasonable alternatives. These checks support judgment; they do not prove that a selection is correct. A discrepancy may reflect a biased assumption, a change in risk mix, random variation, or differences in data basis.
The earlier material on combining developed frequency and severity establishes the component approach. Here the emphasis is on testing its assumptions, including with experience not used to set those assumptions. The checks are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
What the Frequency-Severity Product Assumes
Let \(N\) be the number of claims for a risk or exposure unit, and let \(X_j\) be the loss amount for claim \(j\). The aggregate loss is \(S=\sum_{j=1}^{N}X_j\), with \(S=0\) when \(N=0\). If claim amounts have a common expected value \(\mu\) and are independent of the claim count, then expected aggregate loss is expected claim count times expected severity. For an exposure base with expected frequency \(f=E[N]\) per unit, the expected loss cost per unit is \(f\mu\).
The key step is \(E[S\mid N]=N\mu\). Given \(N\), there are \(N\) claims, each with expected loss \(\mu\), so their expected total is \(N\mu\). Taking the expectation over possible claim counts gives the result. Independence of count and severity, or at least a constant conditional mean severity \(E[X_j\mid N]\), is needed for this simple product.
This justification concerns expected loss, not the realized loss in a particular period. Actual claims fluctuate. Also, the product can be misleading if the severity average is conditional on a different population than the frequency estimate, if claim definitions differ, or if severity changes with the number of claims. For instance, a high-claim-count group might have a different risk mix and a different average claim size. The selections must use compatible periods, exposures, claim definitions, limits, and loss bases before their product is interpreted.
A Practical Testing Sequence
Specify the exposure unit, coverage, claim definition, valuation and development basis, and whether losses are limited or unlimited. Put observations and selections on a consistent basis before comparing them.
Compare selected frequency and severity with relevant historical levels, trends, and segment experience. Ask whether each change has an identifiable explanation rather than relying only on the final product.
Multiply the unrounded component selections and compare the resulting loss cost with an independent aggregate indication or other suitable benchmark. Investigate meaningful differences; do not mechanically alter a component just to eliminate them.
Apply the selections to a period not used to choose them, where possible, and compare projected counts, severity, and total loss with outcomes. Then vary important assumptions to see which ones drive the result.
Component Comparisons and Aggregate Checks
A component comparison is clearest when expressed as a ratio to a stated baseline. A frequency ratio above 1.00 means selected frequency is higher than the baseline; a severity ratio above 1.00 means selected severity is higher. Their product is the relative change in loss cost, provided both ratios refer to the same basis. This is a diagnostic comparison, not a separate estimate of actual future claims.
Worked Example: Testing Two Component Changes Against an Aggregate Benchmark
A hypothetical portfolio has a baseline frequency of 0.080 claims per exposure unit and baseline severity of $1,200. Selections for a proposed rating period are 0.084 claims per exposure unit and $1,260. An independent aggregate benchmark indicates an 8% increase from the baseline loss cost. Compare the component result with the benchmark.
Step 1: Calculate the baseline loss cost. The baseline frequency and severity refer to the same exposure and loss basis, so the indicated baseline is:
Step 2: Calculate the component ratios. Selected frequency is \(0.084/0.080=1.05\), a 5% increase. Selected severity is \(\$1{,}260/\$1{,}200=1.05\), also a 5% increase.
Step 3: Combine the components. Using the unrounded component ratios, the implied relative loss-cost change is \(1.05\times1.05=1.1025\), or an increase of 10.25%. The selected loss cost is \(0.084\times \$1{,}260=\$105.84\) per exposure unit. Checking against baseline, \(\$105.84/\$96.00=1.1025\).
Step 4: Compare with the aggregate benchmark. An 8% increase implies \(\$96.00\times1.08=\$103.68\) per exposure unit. The component result exceeds that benchmark by \(\$105.84-\$103.68=\$2.16\) per unit, or by \(1.1025/1.08-1=0.0208333\), approximately 2.08% relative to the benchmark.
Interpretation and check. The component selections imply a larger increase than the aggregate benchmark, so the difference merits investigation. It does not, by itself, show which assumption is wrong: the aggregate benchmark may use a different basis, or one or both components may need further support. The direct loss-cost calculation agrees with the ratio calculation. The method avoids a common mistake: rounding each component ratio early and then treating the product of rounded values as exact.
Back-Testing Against Later Experience
A back-test applies a selected model to a period whose outcomes were not used to set the assumptions, then compares projected with observed experience. A useful test examines components as well as total loss: projected claim count against observed claim count, projected average severity against observed severity, and projected total loss against observed total loss. A total-only comparison can conceal offsetting component errors.
The comparison must account for exposure volume. A selected frequency is a rate, not a claim count; multiply it by the period's exposure to obtain expected claims. Likewise, severity should be compared using the same coverage, limit, development, and trend basis as the selection. If the test period is immature, its observed losses may not yet be comparable with ultimate selections. A single back-test is also subject to random fluctuation, so multiple periods and operational explanations can be more informative than one favorable or unfavorable result.
Worked Example: Separating a Frequency Miss From a Severity Match
A hypothetical insurer selected a frequency of 0.060 claims per exposure unit and severity of $1,500 for a defined coverage. It then evaluates a completed test period with 20,000 exposure units. The period has 1,260 developed claims and $1,890,000 of developed losses, all on the same basis as the selections. Compare projected and observed components and total loss.
Step 1: Project the claim count. Expected claims are \(20{,}000\times0.060=1{,}200\).
Step 2: Project the loss. Expected loss is \(1{,}200\times\$1{,}500=\$1{,}800{,}000\). Equivalently, the selected loss cost is \(0.060\times\$1{,}500=\$90\) per exposure unit, and \(20{,}000\times\$90=\$1{,}800{,}000\).
Step 3: Calculate observed frequency and severity. Observed frequency is \(1{,}260/20{,}000=0.063\) claims per exposure unit. Observed severity is \(\$1{,}890{,}000/1{,}260=\$1{,}500\) per claim.
Step 4: Compare the components. The observed-to-selected frequency ratio is \(0.063/0.060=1.05\), or 5% above selection. The severity ratio is \(\$1{,}500/\$1{,}500=1.00\), matching selection. Observed total loss is \(1{,}260\times\$1{,}500=\$1{,}890{,}000\), which is \(1.05\) times projected loss, also 5% above selection.
Interpretation and check. The aggregate miss is attributable to the frequency component in this test period, not to average severity. The observed average multiplied by observed frequency gives \(0.063\times\$1{,}500=\$94.50\) per exposure unit; \(20{,}000\times\$94.50=\$1{,}890{,}000\). The count, severity, and total reconcile. This diagnosis does not establish that selected frequency is biased: the observed count could reflect ordinary fluctuation or a change not captured by the model. The method avoids the trap of calling the whole miss a severity problem simply because the total loss is high.
Sensitivity Checks
A sensitivity check changes one or more assumptions while holding the rest of the model fixed, then measures the effect on the projected loss cost. One-at-a-time tests show the effect of each assumption in isolation. Combined scenarios show how assumptions can reinforce or offset one another. Neither is a probability statement unless the scenarios have been assigned a probability model.
When frequency and severity changes are multiplicative, their combined percentage effect is not generally the sum of their percentage effects. For example, a 3% frequency increase and an 8% severity increase produce a combined factor of \(1.03\times1.08=1.1124\), a 11.24% increase, not 11%. This cross-effect is small for small changes, but it should be included in calculations.
Worked Example: Sensitivity to Trend Selections
A hypothetical portfolio has current frequency of 0.050 claims per exposure unit and severity of $2,000, for a current loss cost of $100. For a one-year projection, the selected annual frequency trend is 2% and the selected annual severity trend is 4%. Test the central projection and a range of trend assumptions: frequency trends of 1% and 3%, and severity trends of 3% and 5%.
Step 1: Calculate the central projection. Projected frequency is \(0.050\times1.02=0.051\). Projected severity is \(\$2{,}000\times1.04=\$2{,}080\). The projected loss cost is \(0.051\times\$2{,}080=\$106.08\), a 6.08% increase from $100.
Step 2: Vary frequency alone. Holding severity trend at 4%, a 1% frequency trend gives \(0.050\times1.01=0.0505\) and loss cost \(0.0505\times\$2{,}080=\$105.04\). A 3% frequency trend gives \(0.050\times1.03=0.0515\) and loss cost \(0.0515\times\$2{,}080=\$107.12\).
Step 3: Vary severity alone. Holding frequency trend at 2%, a 3% severity trend gives severity \(\$2{,}000\times1.03=\$2{,}060\) and loss cost \(0.051\times\$2{,}060=\$105.06\). A 5% severity trend gives severity \(\$2{,}000\times1.05=\$2{,}100\) and loss cost \(0.051\times\$2{,}100=\$107.10\).
Step 4: Combine low and high scenarios. Under the low-low scenario, projected frequency is 0.0505 and severity is $2,060, giving \(0.0505\times\$2{,}060=\$104.03\). Under the high-high scenario, projected frequency is 0.0515 and severity is $2,100, giving \(0.0515\times\$2{,}100=\$108.15\).
Interpretation and check. The tested scenarios range from $104.03 to $108.15 per exposure unit, compared with the central $106.08. Each combined result is the product of its own frequency and severity, and the low and high endpoints are not sums of component percentage changes. The range shows the projection's sensitivity to these trend choices; it does not say which scenario will occur. The method avoids changing both assumptions without showing how much each change contributes.
Interpreting Results and Avoiding Traps
Testing is most useful when it leads to a question that can be investigated: Is the frequency movement consistent with changes in exposure mix or claim handling? Is the severity movement consistent with the selected trend and the loss basis? Was the back-test period mature, and were its claims developed consistently? Are the component estimates based on compatible groups? Such questions help distinguish a model issue from a data or comparison issue.
- Do not treat agreement as validation. A plausible total can result from offsetting frequency and severity errors. Inspect the components as well as their product.
- Do not treat one period as conclusive. A back-test is evidence, not proof. Consider exposure volume, maturity, random variation, and whether the period is representative.
- Do not compare unlike bases. Differences in claim definition, limits, development, trend, exposure, or mix can create apparent residuals that are not assumption failures.
- Do not add percentage changes when a product is required. Calculate the product of factors, retaining full precision until the final reported result.
- Do not force components to match an aggregate benchmark without explanation. Reconciliation may be appropriate, but a mechanical adjustment can hide the source of the discrepancy.
Check Your Understanding
For each problem, keep the frequency and severity bases consistent and show the component reasoning before interpreting the total.
- A selected frequency rises from 0.040 to 0.042 claims per exposure unit, while selected severity rises from $900 to $945. Calculate each component ratio and the combined loss-cost change. (Answer: Frequency ratio 1.05; severity ratio 1.05; combined ratio 1.1025, or a 10.25% increase.)
- A test period has 12,000 exposure units, 744 developed claims, and $1,339,200 of developed losses. Selections are 0.060 claims per exposure unit and $1,800 severity. Calculate projected claims, projected loss, observed frequency, observed severity, and the observed-to-projected total ratio. (Answer: 720 projected claims; $1,296,000 projected loss; observed frequency 0.062; observed severity $1,800; total ratio 1.0333, or 3.33% above projection.)
- A current loss cost is $75. Frequency is projected to increase by 4% and severity by 6%. Calculate projected loss cost and its percentage change. (Answer: \( \$75\times1.04\times1.06=\$82.68\), an increase of 10.24%.)
- A selected frequency of 0.030 is tested at 0.029 and 0.032. Severity is held at $2,500. Calculate the two resulting loss costs and their range around the selected loss cost. (Answer: Selected $75.00; low $72.50, which is $2.50 or 3.33% below selected; high $80.00, which is $5.00 or 6.67% above selected.)
- A back-test has observed claim count 5% above projection, observed severity 4% below selection, and the same exposure basis. Calculate the aggregate observed-to-projected loss ratio using the component factors, then state why the result does not alone identify a biased assumption. (Answer: \(1.05\times0.96=1.008\), or 0.8% above projected total; the near-match can reflect offsetting component differences and does not establish whether either difference is systematic or due to variation or basis.)