Why the Level of Aggregation Matters
A claim is a unit of loss experience; a policy can have no claims, one claim, or several claims. Ratemaking data may therefore describe dollars at the claim level while requiring totals at the policy level, or may need both. The arithmetic is simple, but the units being added and counted must remain clear: a dollar total is not a claim count, and a policy is not a claim.
This tutorial treats the claim-level paid amounts and case reserves as already summarized for the claims in scope. The hypothetical examples use dollar amounts at a common valuation date, with no discounting, trending, or other adjustment. The purpose is to aggregate those amounts correctly—not to determine how raw claim transactions are combined or how claims are assigned to experience periods.
Keep the three dollar measures separate. Paid loss records amounts paid; case reserve records the amount held for future payments on the claims; incurred loss combines paid loss and case reserve. Aggregating one measure does not change what the others represent. In particular, a claim with no payment yet can still have a positive incurred amount because it has a case reserve.
From Claim Totals to Policy Totals
The basic operation is a sum within each policy. First group the claim-level amounts by policy; then total paid loss, case reserves, and incurred loss separately. The arithmetic also provides a direct check: if every claim satisfies \(I_j=P_j+R_j\), then the policy totals must satisfy \(I_{\text{policy}}=P_{\text{policy}}+R_{\text{policy}}\).
Determine which claim-level records and dollar measures belong in the calculation. Use the supplied claim-level amounts and the stated valuation basis.
For each policy, list its claims. Keep the claims distinct even when a claim has zero paid loss or zero case reserve.
Add paid amounts, case reserves, and incurred amounts separately. Record the number of claims separately from these dollar totals.
Check incurred equals paid plus case reserves for each claim and for each policy. If the total covers all policies, also check that policy totals add to the portfolio total.
Worked Example: Roll Up Two Policies
A hypothetical insurer has the following claim-level amounts for two policies. All amounts are in dollars. For each policy, determine total paid loss, total case reserves, total incurred loss, and claim count; then reconcile the overall totals.
| Policy | Claim | Paid loss | Case reserve |
|---|---|---|---|
| A | A1 | $1,200 | $300 |
| A | A2 | $500 | $0 |
| B | B1 | $0 | $2,400 |
| B | B2 | $900 | $600 |
Method. Compute incurred loss for each claim as paid loss plus case reserve, then sum each measure within each policy. No interest, mortality, or probability model is involved; this is a deterministic aggregation of the stated claim-level amounts.
For Policy A, claim A1 has incurred loss \(1{,}200+300=1{,}500\), and claim A2 has incurred loss \(500+0=500\). Thus paid loss is \(1{,}200+500=1{,}700\), case reserves are \(300+0=300\), and incurred loss is \(1{,}500+500=2{,}000\). The claim count is 2.
For Policy B, claim B1 has incurred loss \(0+2{,}400=2{,}400\), and claim B2 has incurred loss \(900+600=1{,}500\). Thus paid loss is \(0+900=900\), case reserves are \(2{,}400+600=3{,}000\), and incurred loss is \(2{,}400+1{,}500=3{,}900\). The claim count is 2.
Across both policies, paid loss is \(1{,}700+900=2{,}600\), case reserves are \(300+3{,}000=3{,}300\), and incurred loss is \(2{,}000+3{,}900=5{,}900\). There are \(2+2=4\) claims. The check is \(2{,}600+3{,}300=5{,}900\), which agrees with total incurred loss. The claim with zero paid loss is still one of Policy B's claims.
Common mistake avoided: Dropping claim B1 because it has no paid loss would understate Policy B's incurred loss by $2,400 and its claim count by one. Paid loss and claim count are different measures.
Why the Reconciliation Works
The policy-level identity follows directly from adding the claim-level identities. It does not require assumptions about how large losses are distributed or how claims develop; it requires only that the same claims and valuation basis are used for paid loss, case reserves, and incurred loss.
The third line uses the distributive property of finite sums: adding each claim's paid amount and reserve and then summing is the same as summing all paid amounts and all reserves separately. This also explains why a reconciliation difference is a useful warning. It may indicate omitted or misclassified amounts, inconsistent scope, or an arithmetic error; it should not be erased by forcing the totals to agree.
Claim Counts and Average Loss
A claim count describes the number of claims in scope, not the number of policies and not the number of nonzero dollar entries. A policy with three claims has count three even if one claim has no paid amount. Across a portfolio, the total claim count is the sum of the claim counts for the policies, provided each claim is assigned to exactly one policy in the defined scope.
When an average incurred loss per claim is required, divide total incurred loss by total claim count. If policy-level average losses are available instead, combine them using claim counts as weights. A simple unweighted average of policy averages generally gives the wrong portfolio result when policies have different numbers of claims.
Worked Example: Include a Claim With No Payment
A hypothetical small commercial portfolio has these claim-level amounts. Determine the paid, reserve, and incurred totals and the average incurred loss per claim for each policy and for the portfolio. Amounts are in dollars.
| Policy | Claim | Paid loss | Case reserve |
|---|---|---|---|
| M | M1 | $400 | $100 |
| M | M2 | $0 | $900 |
| M | M3 | $1,600 | $0 |
| N | N1 | $700 | $0 |
| N | N2 | $300 | $200 |
Method. Add paid loss and case reserve claim by claim, aggregate by policy, and divide incurred totals by the corresponding claim counts. These are hypothetical fixed amounts at one valuation date, so no adjustment or probability assumption is applied.
For Policy M, the claim incurred amounts are \(400+100=500\), \(0+900=900\), and \(1{,}600+0=1{,}600\). Paid loss totals \(400+0+1{,}600=2{,}000\); case reserves total \(100+900+0=1{,}000\); incurred loss totals \(500+900+1{,}600=3{,}000\). The count is 3, so average incurred loss per claim is \(3{,}000/3=\$1{,}000\).
For Policy N, incurred amounts are \(700+0=700\) and \(300+200=500\). Paid loss totals \(700+300=1{,}000\); case reserves total \(0+200=200\); incurred loss totals \(700+500=1{,}200\). The count is 2, so average incurred loss per claim is \(1{,}200/2=\$600\).
For the portfolio, paid loss is \(2{,}000+1{,}000=3{,}000\), case reserves are \(1{,}000+200=1{,}200\), incurred loss is \(3{,}000+1{,}200=4{,}200\), and claim count is \(3+2=5\). Therefore average incurred loss per claim is \(4{,}200/5=\$840\). The reconciliation check is \(3{,}000+1{,}200=4{,}200\).
Common mistake avoided: Counting only claims with positive paid amounts would give Policy M a count of 2, despite its three claims. A zero-paid claim with a reserve remains a claim and contributes $900 of incurred loss.
Weighted Aggregation Across Policies
Policy totals can be added to produce portfolio totals because each policy contributes its own amounts. Averages require more care. The portfolio average incurred loss per claim is based on all claims, so a policy with more claims contributes more weight. The following example shows why simply averaging policy-level averages can be misleading.
Worked Example: Calculate a Portfolio Average Correctly
Two hypothetical policies have the following claim-level paid amounts and case reserves, in dollars. Calculate each policy's incurred total and average incurred loss per claim, then calculate the portfolio average. Compare it with the unweighted average of the two policy averages.
| Policy | Claim | Paid loss | Case reserve |
|---|---|---|---|
| R | R1 | $200 | $100 |
| R | R2 | $700 | $200 |
| S | S1 | $100 | $150 |
| S | S2 | $400 | $50 |
| S | S3 | $500 | $150 |
| S | S4 | $800 | $550 |
Method. Aggregate claim incurred amounts by policy, then calculate the portfolio average as total incurred loss divided by total claim count. The hypothetical amounts are already on a common basis; there is no additional adjustment.
For Policy R, incurred amounts are \(200+100=300\) and \(700+200=900\). Its incurred total is \(300+900=1{,}200\), and its count is 2, giving an average of \(1{,}200/2=\$600\).
For Policy S, incurred amounts are \(100+150=250\), \(400+50=450\), \(500+150=650\), and \(800+550=1{,}350\). Its incurred total is \(250+450+650+1{,}350=2{,}700\), and its count is 4, giving an average of \(2{,}700/4=\$675\).
The portfolio incurred total is \(1{,}200+2{,}700=3{,}900\), and the portfolio claim count is \(2+4=6\). Its average incurred loss per claim is \(3{,}900/6=\$650\). The unweighted mean of the policy averages would be \((600+675)/2=\$637.50\), which is not the portfolio average because the policies have different claim counts.
As a check, total paid loss is \(900+(100+400+500+800)=2{,}700\), and total case reserves are \(300+(150+50+150+550)=1{,}200\). Their sum, $3,900, matches portfolio incurred loss.
Common Aggregation Errors
- Confusing the unit of count. The number of policies is not the number of claims. Sum policy claim counts to obtain the total claim count.
- Excluding zero-paid claims. A claim may have a case reserve even when paid loss is zero. Do not use positive paid amounts as a substitute for the claim count.
- Mixing dollar measures. Paid loss, case reserves, and incurred loss answer different questions. Keep their totals distinct and check the incurred identity.
- Averaging policy averages without weights. Use total incurred loss divided by total claims, or use policy averages weighted by their claim counts.
- Adding amounts with inconsistent scope. The claim set and valuation basis must match across paid loss, case reserves, incurred loss, and claim count. A reconciliation difference should prompt investigation.
Check Your Understanding
For each hypothetical situation, keep dollar totals and claim counts separate, and check the incurred-loss identity where possible.
- A policy has two claims: one with paid loss $250 and case reserve $50, and another with paid loss $0 and case reserve $700. Find policy paid loss, case reserves, incurred loss, and claim count. (Answer: paid $250; reserves $750; incurred $1,000; 2 claims.)
- Policy U has claims with incurred amounts $400, $800, and $1,000. Policy V has claims with incurred amounts $600 and $700. Find each policy total and count, then the portfolio total, count, and average incurred loss per claim. (Answer: U $2,200 and 3 claims; V $1,300 and 2 claims; portfolio $3,500, 5 claims, and $700 per claim.)
- A policy has claim-level paid amounts $1,100, $0, and $500, with case reserves $100, $900, and $0, respectively. Find the claim incurred amounts, policy incurred total, and average incurred loss per claim. (Answer: incurred amounts $1,200, $900, and $500; total $2,600; average $866.67 per claim.)
- Policy A has 2 claims with average incurred loss $400. Policy B has 6 claims with average incurred loss $700. Calculate the portfolio average incurred loss per claim and compare it with the unweighted average of the two policy averages. (Answer: portfolio average \((2(400)+6(700))/8=\$625\); unweighted average $550.)
- A portfolio has total paid loss $4,800, total case reserves $1,200, and 8 claims across 3 policies. Find total incurred loss and average incurred loss per claim. Can the number of claims per policy be determined from these totals alone? (Answer: incurred loss $6,000; average $750 per claim; no, the totals do not determine how the 8 claims are distributed among the 3 policies.)