Why Assess a New Territory Carefully?
A new territory may have only a short history, a small number of claims, or a mix of policies unlike the rest of the book. Its observed results can still provide useful evidence, but the figures need context before they are used in a rate indication. A claim count by itself does not show how much risk was exposed; aggregate losses by themselves do not show whether the amount reflects frequency, severity, or both.
A sound assessment starts by putting the territory experience on a defined basis, calculating a small set of measures, and asking whether the proposed portfolio complement represents comparable risks. The aim here is to organize those inputs and assumptions, not to derive a universal credibility weight. Aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus, this approach emphasizes what the data can support and what remains uncertain.
These measures answer different questions. Frequency describes how often claims occur per exposure unit. Severity describes the average loss conditional on a claim. Pure premium combines both effects as loss per exposure. The loss ratio compares losses with the premium earned on the same experience basis. A measure is not comparable merely because it has the same name: the exposure definition, loss treatment, valuation date, coverage, and premium basis must also be suitable.
Organize the Territory Experience
Before calculating, specify the territory boundaries, coverage, experience dates, and the exposure unit. Determine whether the data are paid or incurred, whether claims are sufficiently mature for the intended use, and whether large or catastrophe claims are included. Check that claim counts and losses refer to the same claim population. Premium should be earned for the same risks and period as the losses, with adjustments or exclusions made consistently when needed.
The portfolio complement is the broader or otherwise selected experience used to temper the territory’s own result. It should be relevant to the territory’s coverage, rating basis, and risk mix. A whole-book average may be a poor complement if it is dominated by different risks. Where practical, check whether the territory itself is included in the proposed complement; overlap means the two sources of experience are not independent and can make the blend less informative than it appears.
Set the territory, coverage, period, exposure unit, loss measure, and valuation basis before computing results.
Check exposure earning, claim counts, losses, and premium for completeness and consistent scope.
Compute frequency, severity, pure premium, and loss ratio where appropriate, then compare each with the corresponding complement measure.
Identify changes in mix, maturity, claim handling, large losses, or other factors that affect interpretation.
Worked Example: Measuring a Territory Against Its Portfolio Complement
A hypothetical new territory has 1,200 earned annual exposures, 18 claims, $144,000 of losses, and $210,000 of earned premium. Its proposed complement, for the same coverage and period, has 10,000 earned exposures, 120 claims, $900,000 of losses, and $1,500,000 of earned premium. Calculate and compare the four measures. Assume the losses and counts use the same stated valuation basis.
Method: exposure-normalized comparison. First calculate each measure for the territory:
Frequency is \(18/1{,}200=0.015\) claims per exposure. Average severity is \($144{,}000/18=$8{,}000\) per claim. Pure premium is \($144{,}000/1{,}200=$120\) per exposure. The loss ratio is \($144{,}000/$210{,}000=0.685714\), or \(68.57\%\) to two decimal places.
For the complement, frequency is \(120/10{,}000=0.012\) claims per exposure. Average severity is \($900{,}000/120=$7{,}500\). Pure premium is \($900{,}000/10{,}000=$90\) per exposure. Its loss ratio is \($900{,}000/$1{,}500{,}000=0.60\), or \(60.00\%\).
The territory’s observed frequency is \(0.015/0.012=1.25\), or \(25\%\) above the complement. Its severity is \($8{,}000/$7{,}500=1.0667\), about \(6.67\%\) above. The pure premium is \($120/$90=1.3333\), about \(33.33\%\) above. The observed loss ratio is also higher, by \(68.57\%-60.00\%=8.57\) percentage points.
Check: territory frequency multiplied by severity gives \(0.015\times$8{,}000=$120\), its pure premium. The corresponding complement calculation is \(0.012\times$7{,}500=$90\). The calculation is internally consistent. It does not, by itself, establish that the territory’s future rate should be increased by \(33.33\%\): the comparison still depends on data quality, risk mix, and the credibility assigned to the territory experience.
Common mistake avoided: comparing the territory’s $144,000 of losses directly with the complement’s $900,000 would mainly reflect their different sizes. Dividing each by its own earned exposure makes the pure premiums comparable.
Earned Exposure Is Not Always Policy Count
A policy count is an appropriate exposure measure only when the policies represent comparable earned durations under the selected unit. If policy terms are unequal, or if coverage starts or ends during the experience period, summing policy-years or another appropriate earned unit is usually more informative than counting policies. The exposure measure must match the frequency and premium basis being analyzed.
Worked Example: Converting Unequal Policy Durations to Exposure
A hypothetical territory has 400 policies in force for a full year, 300 policies in force for six months, and 200 policies in force for three months during a one-year experience period. It reports 9 claims, $81,000 of losses, and $102,000 of earned premium. Calculate frequency, severity, pure premium, and loss ratio. Treat a full policy-year as one exposure, and assume all data are complete and on the same coverage and valuation basis.
Method: duration-weighted earned exposure. The full-year policies contribute \(400\times1=400\) exposures. The six-month policies contribute \(300\times0.5=150\). The three-month policies contribute \(200\times0.25=50\). Total earned exposure is \(400+150+50=600\) policy-years.
Frequency is \(9/600=0.015\) claims per policy-year. Severity is \($81{,}000/9=$9{,}000\) per claim. Pure premium is \($81{,}000/600=$135\) per policy-year. The loss ratio is \($81{,}000/$102{,}000=0.7941176\), or \(79.41\%\) to two decimal places.
Check: \(0.015\times$9{,}000=$135\), agreeing with the direct pure-premium calculation. Also, \(600\times$170=$102{,}000\), so earned premium is $170 per policy-year; \($135/$170=79.41\%\), agreeing with the direct loss ratio.
Common mistake avoided: dividing 9 claims by 900 policies gives \(0.010\) claims per policy, but it treats every policy as a full year of exposure. The actual policy-years total 600, so that denominator understates the frequency per earned policy-year.
Choosing and Interpreting a Complement
A complement is a modeling choice, not a mechanical default. A defensible candidate should have similar coverage and claim definitions, a compatible exposure unit, and a reasonably comparable mix of rating characteristics. If the territory is new because its risks differ from the established portfolio, the established average may be an especially weak guide. A narrower, more comparable segment can be preferable, provided it has adequate data and its own limitations are understood.
Differences in results should be investigated rather than explained away. A frequency difference could reflect a genuine risk difference, exposure measurement, claim reporting, or a change in claim frequency over time. A severity difference could reflect limits, deductibles, inflation, claim mix, or large losses. A loss ratio difference could also arise because premium levels or premium mix differ. These are diagnostic possibilities; the measures alone do not identify the cause.
A credibility-weighted indication is often written as a blend of the territory estimate and a complement estimate. In this form, \(Z\) is the weight on the territory estimate and \(1-Z\) is the weight on the complement. The weight must be selected using a stated credibility approach and relevant assumptions; it is not determined just by observing that the territory has a particular number of policies or claims.
The blend has a simple justification when it is used as a weighted estimate: the territory contributes the fraction \(Z\) of the result, and the complement contributes the remaining fraction. The weights sum to one, since \(Z+(1-Z)=1\). Thus if both estimates are the same value \(m\), the blend is \(Zm+(1-Z)m=m\); if \(Z=0\), the estimate is entirely the complement, and if \(Z=1\), it is entirely the territory. This algebra explains the blend’s behavior, but does not establish how \(Z\) should be chosen.
Worked Example: Applying a Stated Weight to Pure Premium
A hypothetical territory has 800 earned exposures, 12 claims, and $120,000 of losses. A selected complement has 4,000 earned exposures, 56 claims, and $448,000 of losses. For this illustration, an independent process has already assigned \(Z=0.25\) to the territory pure-premium estimate. Calculate the blended pure premium. Do not attempt to infer \(Z\) from these counts.
Method: weighted pure-premium blend. The territory pure premium is \($120{,}000/800=$150\) per exposure. The complement pure premium is \($448{,}000/4{,}000=$112\) per exposure. The complement weight is \(1-0.25=0.75\).
The territory contribution is \(0.25\times$150=$37.50\) per exposure. The complement contribution is \(0.75\times$112=$84.00\). The blended pure premium is \($37.50+$84.00=$121.50\) per exposure.
Check: the result is between $112 and $150, as a weighted average with weights between zero and one must be. The weights sum to \(0.25+0.75=1\).
Common mistake avoided: using \(0.75\) as the territory weight would produce \($140.50\), reversing the stated weights. The given \(Z=0.25\) applies to the territory estimate, not the complement.
What the Assessment Can and Cannot Establish
The calculations summarize observed experience; they do not guarantee that the territory’s next period will resemble its past. Before relying on the results, consider whether exposure and claims are complete, whether losses are mature enough for the chosen measure, and whether data definitions are consistent. Also consider changes in underwriting, distribution, coverage, limits, deductibles, claim handling, or the territory’s composition. If one unusual large loss or catastrophe materially changes the result, make its treatment explicit rather than allowing it to be hidden inside an aggregate figure.
Keep the distinction between an observed measure and a rate indication clear. A higher territory pure premium than the complement is evidence of a difference in the observed period, not proof that a particular rate change is adequate or appropriate. Likewise, a low claim count may signal substantial uncertainty, but it does not specify a credibility weight without a selected method and its assumptions. A useful assessment reports both the calculated measures and the reasons they may or may not represent future experience.
Check Your Understanding
For each hypothetical situation, calculate the requested measure and note the stated basis.
- A territory has 500 earned annual exposures, 6 claims, and $54,000 of losses. What are its frequency, severity, and pure premium? (Answer: \(0.012\) claims per exposure; $9,000 per claim; $108 per exposure.)
- A territory has 200 full-year policies and 160 policies each earning half a year. It has 7 claims and $63,000 of losses. What is earned exposure and claim frequency? (Answer: 280 policy-years; \(0.025\) claims per policy-year.)
- A territory has $84,000 of losses and $120,000 of earned premium; its complement has $240,000 of losses and $400,000 of earned premium. What are the two loss ratios, and what is their difference in percentage points? (Answer: \(70.00\%\) and \(60.00\%\); 10.00 percentage points.)
- A territory pure premium is $132 per exposure, the complement pure premium is $108, and the stated territory weight is \(Z=0.40\). Calculate the blended pure premium. (Answer: $117.60 per exposure.)
- A proposed complement has the same coverage but uses a different exposure definition and includes a substantially different risk mix. Give two reasons it may be unsuitable without adjustment. (Answer: Its exposure-normalized results may not be comparable, and its different risk mix may not represent the new territory; either issue can make the complement a poor benchmark.)