From Portfolio Data to a Count Projection
A claim-frequency model is only as useful as the counts and exposure units supplied to it. A portfolio can have a plausible total count and still produce a misleading rate if a segment is omitted, policy-years are confused with policies, or claims are counted under different rules across segments. The practical task is therefore a sequence: validate the data, estimate rates, decide whether a common rate is adequate, and project counts using exposures on the same basis.
This workshop uses \(N_j\) for the claim count in segment \(j\), \(e_j\) for its exposure, and \(\lambda_j\) for its expected claim count per exposure unit. The exposure unit might be one vehicle-year or one policy-year; it must be defined in the problem and used consistently. A count is also defined consistently—for example, claims reported during a specified period, or claims ultimately attributable to that period. Those definitions are not interchangeable.
Before calculating, check that counts are nonnegative integers, exposures are positive and expressed in the same units, segment labels are mutually exclusive and collectively cover the intended portfolio, and the count definition and observation period match across segments. If exposures differ in duration, the denominator should reflect the applicable exposure time; a policy in force for half a year is not automatically one full policy-year.
Count Models and Exposure
A standard starting point is the Poisson model. Conditional on the relevant segment and exposure, the count is Poisson with mean proportional to exposure. In a common-rate model, independent segment counts satisfy \(N_j\sim\operatorname{Poisson}(e_j\lambda)\). Thus \(E[N_j]=e_j\lambda\) and \(\operatorname{Var}(N_j)=e_j\lambda\). The model assumes a stable rate within the modeled group, independent counts across the units or cells being modeled, and no unaccounted clustering or heterogeneity that materially changes the variance.
The exposure appears as a multiplier of the mean. Doubling exposure doubles the expected count; it does not double the frequency. For segmented data, a common-rate model says that after accounting for exposure, each segment has the same expected frequency. A segment-specific model instead uses \(\lambda_j\) for segment \(j\). The latter is appropriate only when the distinction is meaningful and the data support stable estimates.
For a Poisson count, the variance equals the mean. If residual variation is consistently greater than this relationship predicts, a negative binomial model is one common alternative. Under a frequently used parameterization, its conditional variance is \(\mu+\alpha\mu^2\), where \(\alpha>0\) allows extra variation beyond the Poisson variance. A negative binomial model is not justified merely because one small data set produces a large dispersion statistic; first investigate data quality, segment mix, and whether observations are genuinely comparable.
Why the Pooled Poisson Estimate Is Exposure-Weighted
Suppose independent counts \(N_j\) follow a common Poisson rate \(\lambda\), with known exposures \(e_j\). The likelihood is the product of the segment probabilities. Ignoring terms that do not depend on \(\lambda\), its log is \(\ell(\lambda)=(\sum_jN_j)\log\lambda-\lambda\sum_je_j+\text{constant}\). Differentiating gives \(\ell'(\lambda)=(\sum_jN_j)/\lambda-\sum_je_j\). Setting this equal to zero yields the maximum-likelihood estimate.
This is total claims divided by total exposure. It is also a weighted average of segment frequencies, with weights proportional to segment exposures: \(\widehat{\lambda}=\sum_j(e_j/\sum_ke_k)(N_j/e_j)\). The derivation explains why taking a simple average of segment rates is generally wrong: it gives a small segment the same influence as a large one, although the pooled Poisson likelihood does not.
Worked Example: Validate and Project a Three-Segment Portfolio
A hypothetical insurer records the following annual experience. Each exposure is one vehicle-year, and each count is the number of claims reported during the year under one consistent definition. Estimate segment and portfolio frequencies, then project counts at next year’s planned exposures using the historical segment rates.
| Segment | Historical exposure | Historical claims | Next-year exposure |
|---|---|---|---|
| Urban | 1,200 | 96 | 1,300 |
| Rural | 800 | 48 | 850 |
| Seasonal-use | 500 | 40 | 600 |
Step 1: Check the inputs. The exposures are positive and all use vehicle-years. The segment counts sum to \(96+48+40=184\), and exposures sum to \(1{,}200+800+500=2{,}500\). The segment definitions are presented as non-overlapping, so the totals do not double-count vehicles.
Step 2: Estimate each segment’s frequency. Urban frequency is \(96/1{,}200=0.0800\) claims per vehicle-year. Rural frequency is \(48/800=0.0600\). Seasonal-use frequency is \(40/500=0.0800\).
Step 3: Calculate the portfolio frequency. The pooled estimate is \(184/2{,}500=0.0736\) claims per vehicle-year. As a check, the exposure-weighted calculation gives \((1{,}200/2{,}500)(0.0800)+(800/2{,}500)(0.0600)+(500/2{,}500)(0.0800)=0.0384+0.0192+0.0160=0.0736\).
Step 4: Project the expected counts. Applying the stated assumption that each segment rate remains unchanged, urban expected claims are \(1{,}300(0.0800)=104\); rural expected claims are \(850(0.0600)=51\); and seasonal-use expected claims are \(600(0.0800)=48\). The total projected expected count is \(104+51+48=203\).
Check and method. The estimated method is to project within segments, preserving their distinct observed rates. The projection reconciles to 203, and its implied portfolio frequency is \(203/(1{,}300+850+600)=203/2{,}750\approx0.0738\). It is not required to equal the historical overall rate because the segment exposures have changed. The common mistake avoided is applying the historical portfolio rate to a future mix without checking whether segment rates differ.
Assessing Whether a Common Poisson Rate Fits
For \(J\) independent cells with fitted expected counts \(\widehat{\mu}_j\), calculate each Pearson contribution \((N_j-\widehat{\mu}_j)^2/\widehat{\mu}_j\), then add them. When the common rate is estimated from the same data, a conventional approximate degrees-of-freedom count is \(J-1\): one degree is used to estimate the common rate. The dispersion estimate is the statistic divided by those degrees of freedom. The chi-square approximation is less reliable with very small expected counts, so interpretation should account for sample size and cell structure.
Worked Example: Diagnose Dispersion in Comparable Cells
Five hypothetical, comparable cells each have one exposure unit. Their claim counts are 5, 8, 10, 12, and 15. Assume independent Poisson counts with one common frequency, and calculate the fitted rate, Pearson statistic, and dispersion estimate.
Step 1: Fit the common frequency. Total claims are \(5+8+10+12+15=50\), and total exposure is 5. Therefore \(\widehat{\lambda}=50/5=10\) claims per exposure unit. Each fitted expected count is \(1(10)=10\).
Step 2: Calculate the Pearson contributions. The squared deviations from 10 are \((5-10)^2=25\), \((8-10)^2=4\), \((10-10)^2=0\), \((12-10)^2=4\), and \((15-10)^2=25\). Their sum is \(25+4+0+4+25=58\). Dividing by the fitted expected count of 10 gives \(X^2_{\mathrm P}=58/10=5.8\).
Step 3: Estimate dispersion. There are five cells and one estimated common rate, so the approximate degrees of freedom are \(5-1=4\). The dispersion estimate is \(5.8/4=1.45\).
Interpretation and check. The estimate of 1.45 is above one, suggesting more variation than the fitted Poisson model’s mean-equals-variance assumption predicts. With only five cells, it is weak evidence on its own; review comparability and collect or examine more experience before choosing a more variable count model. The common mistake is misadding the squared deviations: they total 58, not 50, so the Pearson statistic is 5.8 and the dispersion estimate is 1.45.
Choosing Between a Common Rate and Segment Rates
A high dispersion statistic can arise because segments have different underlying frequencies, even if each segment individually follows a Poisson model. The practical response may be to model a relevant segment distinction rather than immediately adopt a negative binomial model. Conversely, an apparent segment difference based on very little experience may be noise. Use the portfolio context, the stability of segment definitions, fit diagnostics, and available credibility evidence together. A model should represent material, supportable differences rather than reproduce every observed fluctuation.
Under a common-rate Poisson model, fitted expected counts are \(\widehat{\mu}_j=e_j\widehat{\lambda}\). Under a segment-specific Poisson model, they are \(\widehat{\mu}_j=e_j\widehat{\lambda}_j\), with \(\widehat{\lambda}_j=N_j/e_j\). If the segment-specific rates are projected into a future portfolio, the projected total is the sum of each future segment exposure times its rate. This preserves the effect of changing portfolio mix.
Worked Example: Select a Segmented Model for a Changing Mix
A hypothetical insurer has two established vehicle-use classes. Historical experience is 60 claims on 1,000 exposure units in Class A and 60 claims on 500 exposure units in Class B. Planned next-period exposures are 1,200 in Class A and 400 in Class B. Compare a common-rate model with segment-specific Poisson rates, and explain which projection better reflects the observed class difference.
Step 1: Calculate historical segment rates and the common rate. Class A has \(60/1{,}000=0.0600\) claims per exposure unit; Class B has \(60/500=0.1200\). Under the common-rate model, total claims are \(120\), total exposure is \(1{,}500\), and the fitted common rate is \(120/1{,}500=0.0800\).
Step 2: Check the common-rate fit. The common model expects \(1{,}000(0.0800)=80\) claims in A and \(500(0.0800)=40\) in B. The Pearson statistic is \((60-80)^2/80+(60-40)^2/40=400/80+400/40=5+10=15\). There are two cells and one fitted common rate, giving one approximate degree of freedom. The difference is substantial relative to the common model’s expected variation and supports investigating a segment-specific model.
Step 3: Project with segment-specific rates. The segment-specific projection is \(1{,}200(0.0600)=72\) expected claims in A and \(400(0.1200)=48\) in B, for a total of 120. The changing mix is material: the future portfolio has fewer units in the higher-frequency class.
Step 4: Compare the common-rate projection. Applying the historical common rate to future total exposure gives \((1{,}200+400)(0.0800)=1{,}600(0.0800)=128\). The difference from 120 arises because the historical pooled rate reflects the historical mix, not the future mix.
Model choice and check. Given that the classes are established and their observed rates differ, the segment-specific projection is the more informative of these two calculations, subject to whether the rates are sufficiently stable for use. Under independent Poisson counts with these rates treated as fixed, the future total has mean 120 and variance 120, so its standard deviation is \(\sqrt{120}\approx10.95\) claims. The expected value 120 is not a guarantee of the realized count. The common mistake avoided is treating a historical portfolio-average rate as unchanged when the portfolio composition changes.
Interpreting Projections for Ratemaking and Liability Work
A projected expected count is an average under the selected assumptions, not a certain outcome. If independent segment counts are Poisson, their sum is Poisson with mean equal to the sum of their means; its variance equals that mean. If a model allows overdispersion or dependence, the uncertainty calculation changes. In either case, a plug-in projection also treats estimated rates as fixed, so its simple count variance does not include all uncertainty in the rate estimates.
For ratemaking, projected counts help quantify expected claim volume on a stated exposure basis. A rate comparison is meaningful only when the counts, exposure definition, segment mix, and time period are aligned. For liability work, the claim-count basis matters: a count of reported claims is not automatically an estimate of ultimate claims. If the target is ultimate count, use an ultimate-count estimate appropriate to the reporting information before treating it as the count basis for another calculation. This tutorial’s projections do not estimate reporting development.
- Use a common rate when evidence supports a stable frequency across the modeled cells after exposure is accounted for.
- Use segment rates when the distinctions are meaningful, sufficiently supported, and expected to remain relevant to the projection.
- Investigate overdispersion by checking data definitions, omitted segmentation, dependence, and model assumptions before changing the count distribution.
- State the projection basis clearly: exposure unit, claim-count definition, period, segment assumptions, and whether the result is expected or realized count.
Check Your Understanding
Work each problem using the stated count and exposure definitions. Treat projected rates as unchanged unless the question says otherwise.
- A hypothetical book has 900 policy-years and 54 claims in one class, and 600 policy-years and 48 claims in a second class. Calculate each class frequency and the pooled frequency. If next-period exposures are 1,000 and 750 policy-years, project the expected count using class rates. (Answer: Frequencies \(0.0600\) and \(0.0800\); pooled frequency \(102/1{,}500=0.0680\); projected count \(60+60=120\).)
- Four comparable cells each have one exposure unit and counts 4, 6, 8, and 10. Under a common Poisson rate, calculate the fitted frequency, Pearson statistic, and dispersion estimate using \(n-1\) degrees of freedom. (Answer: \(\widehat{\lambda}=7\); \(X^2_{\mathrm P}=20/7\approx2.8571\); dispersion estimate \((20/7)/3\approx0.9524\).)
- A portfolio has an estimated frequency of \(0.075\) claims per exposure unit. Under a Poisson projection with 2,400 future exposure units, find the expected count, variance, and standard deviation of the count. (Answer: Expected count \(180\); variance \(180\); standard deviation \(\sqrt{180}\approx13.42\) claims.)
- Two classes have historical data: Class A has 20 claims on 400 exposures and Class B has 30 claims on 600 exposures. Calculate the segment rates and common rate, then calculate the common-rate expected counts by class. What does this comparison indicate? (Answer: Both segment rates and the common rate are \(0.0500\); common-rate expectations are 20 and 30 claims, so these data show no observed rate difference.)
- Historical Class A has 40 claims on 800 exposures, and Class B has 48 claims on 400 exposures. Next period has 1,000 Class A exposures and 300 Class B exposures. Project using segment rates, then compare with a projection using the historical pooled rate. (Answer: Segment rates \(0.0500\) and \(0.1200\); segmented projection \(50+36=86\); pooled rate \(88/1{,}200\approx0.07333\); pooled projection \(1{,}300(88/1{,}200)\approx95.33\) claims.)