Observed Claim Outcomes Under Experience Rating

Why Observed Outcomes Need Careful Interpretation

Experience rating changes premium in response to loss experience. After a plan is introduced, insurers may observe changes in claim counts, average claim size, or reported losses. Those changes matter, but they do not automatically show that the plan changed underlying risk. The number of insured exposures may have changed; the covered risks may differ; and claims may be reported or managed differently.

A useful evaluation therefore separates frequency, severity, and reporting. Frequency measures claims relative to exposure. Severity measures loss per claim under a defined loss basis. Reporting describes how underlying claim events become recorded claims. Each measure answers a different question, and each depends on consistent definitions across the periods being compared.

Definition: For a period with $E$ units of exposure, $N$ observed claims, and aggregate loss amount $A$, the observed claim frequency is $N/E$, the observed average severity is $A/N$ when $N>0$, and the observed pure premium is $A/E$. Exposure might be policy-years or another measure appropriate to the coverage. All three calculations must use compatible periods, claim definitions, and loss measures.

For example, $A$ might be paid loss, incurred loss, or another specified amount. These bases are not interchangeable: paid loss reflects payments to date, while incurred loss may also include estimates of amounts still owed. A comparison should use the same basis on both sides. Likewise, a claim counted by report date is not necessarily comparable with one counted by occurrence date unless the observation rules align.

The Frequency–Severity Relationship

Frequency and severity combine to explain aggregate loss per exposure. This relationship is an accounting identity, not an assumption about the underlying distribution of losses.

\frac{A}{E} = \frac{N}{E}\frac{A}{N}, \qquad N>0.

The $N$ in the numerator of frequency cancels the $N$ in the denominator of severity, leaving $A/E$. Thus, with consistent data, pure premium is exactly frequency multiplied by severity. A change in pure premium can result from a change in either component, or both. The identity does not establish why either component changed.

Worked Example: Separating Claim Counts From Exposure

A hypothetical insurer compares two periods for one coverage. Before an experience-rating plan, it records 120 claims, $360,000 of incurred losses, and 1,000 policy-years of exposure. After introduction, it records 150 claims, $510,000 of incurred losses, and 1,500 policy-years. Calculate and compare frequency, severity, and pure premium. Use claims per policy-year and dollars per policy-year.

Method. Calculate each measure separately using the same incurred-loss basis in both periods.

Before. Frequency is $120/1{,}000=0.12$ claims per policy-year. Severity is $\$360{,}000/120=\$3{,}000$ per claim. Pure premium is $\$360{,}000/1{,}000=\$360$ per policy-year. The identity checks: $0.12\times \$3{,}000=\$360$.

After. Frequency is $150/1{,}500=0.10$ claims per policy-year. Severity is $\$510{,}000/150=\$3{,}400$ per claim. Pure premium is $\$510{,}000/1{,}500=\$340$ per policy-year. The identity checks: $0.10\times \$3{,}400=\$340$.

The raw claim count increased from 120 to 150, or $30/120=25\%$, but exposure increased by $500/1{,}000=50\%$. Exposure-adjusted frequency decreased by $0.02/0.12=16.67\%$. Severity increased by $\$400/\$3{,}000=13.33\%$, while pure premium decreased by $\$20/\$360=5.56\%$. These arithmetic comparisons do not, on their own, attribute the changes to the plan.

Common mistake and correction. Treating the 25% increase in claim count as a 25% increase in claim frequency ignores the 50% exposure increase. Frequency must be calculated as claims divided by exposure for each period. No rounding was needed for the displayed dollar amounts; percentage changes are rounded to two decimal places.

Exposure Mix Can Change the Crude Rate

Even when exposure units are measured consistently, the portfolio can shift between groups with different expected claim frequencies. A crude rate combines each group’s rate with its share of total exposure. It can therefore change because of within-group experience, a different group mix, or both.

One way to make a fair descriptive comparison is to calculate each period’s group-specific rates and apply the same set of exposure weights to both periods. If $w_j$ is the chosen fixed weight for group $j$, and $\hat\lambda_j$ is its observed frequency, the standardized frequency is $\sum_j w_j\hat\lambda_j$, where the weights sum to one. This removes the effect of changing group shares for the groups included; it does not remove all possible differences within groups.

Worked Example: Comparing Frequency Under a Changed Risk Mix

A hypothetical book has lower- and higher-risk groups. Before a plan change, lower-risk exposures are 800 policy-years with 40 claims, and higher-risk exposures are 200 policy-years with 40 claims. After the change, lower-risk exposures are 400 policy-years with 24 claims, and higher-risk exposures are 600 policy-years with 108 claims. Calculate the crude frequencies and compare standardized frequencies using the before-period exposure weights.

Method. First calculate group rates and crude rates. Then hold group weights fixed at 80% lower-risk and 20% higher-risk.

Before the change, the lower-risk rate is $40/800=0.05$, and the higher-risk rate is $40/200=0.20$ claims per policy-year. Total frequency is $(40+40)/(800+200)=80/1{,}000=0.08$.

After the change, the lower-risk rate is $24/400=0.06$, and the higher-risk rate is $108/600=0.18$. Total frequency is $(24+108)/(400+600)=132/1{,}000=0.132$. The crude increase is $(0.132-0.08)/0.08=65\%$.

Using the same 80%/20% weights in both periods, the standardized before rate is $0.80(0.05)+0.20(0.20)=0.04+0.04=0.08$. The standardized after rate is $0.80(0.06)+0.20(0.18)=0.048+0.036=0.084$. The standardized increase is $(0.084-0.08)/0.08=5\%$. The group mix changed substantially, so the crude comparison makes the increase look much larger than the comparison at fixed weights.

Check. Applying the before weights to the after rates gives 0.084 claims per policy-year, equivalent to 84 claims per 1,000 policy-years. The crude after rate is 132 per 1,000 because the after portfolio has a larger share of higher-risk exposure. Percentages are rounded to two decimal places.

Observed Claims Also Depend on Reporting

An underlying covered event is not always recorded as a claim. Reporting may depend on awareness, administrative practices, claim thresholds, or other features of the process. Experience rating can also affect incentives surrounding claim reporting or management. Consequently, a change in observed frequency may reflect a change in event incidence, the chance an event is reported, or both.

A simple adjustment is possible when an independently supported reporting probability is available. Suppose each underlying event has the same probability $r$ of being reported, regardless of its loss amount, and the reporting probability is known or estimated suitably. Then, in expectation, the observed claim count is $r$ times the underlying event count, and observed aggregate loss is $r$ times the aggregate loss from all events. Dividing both observed quantities by $r$ corrects for incomplete reporting under those assumptions.

E[N_{\text{obs}}]=rN_{\text{true}}, \qquad E[A_{\text{obs}}]=rA_{\text{true}}.

This is an expectation-based correction, not a guarantee that a finite sample will equal its adjusted value. If reporting probability varies with claim size, the aggregate-loss adjustment need not be valid as written. If $r$ is uncertain, that uncertainty should also be considered. Without credible information about reporting, it is safer to describe the observed outcomes than to present a corrected estimate as known truth.

Worked Example: Adjusting for a Different Reporting Rate

In two hypothetical periods, each has 1,000 policy-years. The first has 80 reported claims and $200,000 of reported incurred loss; an independent audit estimates that 80% of underlying events are reported. The second has 90 reported claims and $216,000 of reported incurred loss; its estimated reporting probability is 90%. Assume reporting is independent of event loss amount within each period. Calculate observed and reporting-adjusted frequency, severity, and pure premium.

Observed results. In the first period, frequency is $80/1{,}000=0.08$, severity is $\$200{,}000/80=\$2{,}500$, and pure premium is $\$200$ per policy-year. In the second, frequency is $90/1{,}000=0.09$, severity is $\$216{,}000/90=\$2{,}400$, and pure premium is $\$216$ per policy-year. Observed frequency rises by 12.5%, observed severity falls by 4%, and observed pure premium rises by 8%.

Adjusted results. In the first period, estimated underlying event count is $80/0.80=100$, and estimated aggregate loss is $\$200{,}000/0.80=\$250{,}000$. Adjusted frequency is $100/1{,}000=0.10$, severity is $\$250{,}000/100=\$2{,}500$, and pure premium is $\$250$ per policy-year.

In the second period, estimated underlying event count is $90/0.90=100$, and estimated aggregate loss is $\$216{,}000/0.90=\$240{,}000$. Adjusted frequency is $100/1{,}000=0.10$, severity is $\$240{,}000/100=\$2{,}400$, and pure premium is $\$240$ per policy-year. The reporting adjustment removes the apparent frequency increase in this example; it does not remove the severity decrease.

Check and limitation. The adjusted identity holds in each period: $0.10\times \$2{,}500=\$250$ and $0.10\times \$2{,}400=\$240$. These adjusted values rely on the stated reporting probabilities and loss-independent reporting assumption. Dollar amounts are reported to the cent when necessary; here they are whole dollars.

Interpreting Changes After Introduction

Observed outcomes combine several influences. The plan may be associated with changed prevention, claim handling, or reporting behavior, but exposure quantity and composition can also change. Random variation remains relevant, particularly for smaller books or rare, high-severity claims. The comparison is therefore a measurement result first; a causal conclusion requires a design and assumptions that separate the plan’s effect from other changes.

Before interpreting a difference, specify the outcome and its basis. For frequency, define what counts as a claim and select an exposure measure aligned with the coverage. For severity, define the loss amount and claim population. For reporting, distinguish observed claims from underlying events and state how reporting probability was assessed. Compare periods on consistent definitions, and examine relevant risk groups rather than relying only on an overall average.

These distinctions are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. A useful evaluation reports both the measured change and the limits on what that change can establish. For example, “reported frequency declined per policy-year” is a direct observation; “the plan prevented claims” is a causal interpretation that requires additional support.

Key takeaway: Compare claim outcomes per consistent exposure, separate frequency from severity, and check whether risk mix or reporting changed. Observed differences can reflect behavior, but they do not by themselves identify the plan’s causal effect.

Check Your Understanding

For each problem, use the stated exposure and outcome definitions. Treat any reporting adjustment as valid only under the assumptions given.

A hypothetical coverage has 72 claims over 900 policy-years before a change and 84 claims over 1,200 policy-years afterward. Calculate both frequencies and the percentage change in frequency. (Answer: 0.0800 and 0.0700 claims per policy-year; a 12.5% decrease.)
Before a plan change, 45 claims produce $135,000 of incurred loss over 750 policy-years. Afterward, 60 claims produce $198,000 over 900 policy-years. Calculate severity and pure premium in each period, then give the percentage change in pure premium. (Answer: severity is $3,000 then $3,300; pure premium is $180 then $220 per policy-year; pure premium increases by 22.22%.)
Before a change, lower-risk exposures are 600 with 18 claims, and higher-risk exposures are 400 with 80 claims. Afterward, the respective exposures and claim counts are 300 and 12, and 700 and 119. Calculate the crude frequency change and the standardized change using before-period weights. (Answer: crude frequency rises from 0.098 to 0.131, or 33.67%; standardized frequency falls from 0.098 to 0.092, or 6.12%.)
A hypothetical portfolio has 500 policy-years, 50 reported claims, and $200,000 of reported loss. An audit estimates a 62.5% reporting probability, independent of loss amount. Calculate adjusted claim frequency, severity, and pure premium. (Answer: adjusted count is 80; frequency is 0.1600 per policy-year, severity is $4,000, and pure premium is $640 per policy-year.)

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