What Paid Loss Measures
A paid-loss total answers a specific question: how much indemnity did the insurer pay during the stated period? It is a dollar measure built from payment transactions. It is not, by itself, a measure of how many claims occurred, how many claims were closed, or how much remains unpaid.
The transaction period matters. A claim may involve payments in several periods, and a claim reported during one period may not be paid until another. For a paid-loss calculation, include the indemnity payment transactions assigned to the specified period under the data’s payment-date convention. Do not substitute the claim’s report date or the date of the event that caused the loss.
The word signed is important when the transaction file contains reversals. If a payment of $900 is entered and then fully voided by a transaction of -$900 during the same period, the two entries contribute $0 net paid for that period. The negative entry does not mean that the insurer made a negative cash payment; it records an adjustment to the transaction history.
For the calculations here, each ledger entry is assumed to be an indemnity transaction, amounts are nominal dollars, and no discounting or probability model is applied. Entries for other types of amounts should not be included unless the requested measure explicitly includes them. For example, claim-handling expenses and recoveries are not indemnity payments and are outside the measure calculated here.
Summing Transactions for a Period
Let \(z_j\) be the signed amount of transaction \(j\), and let \(t_j\) be its assigned payment date. For a period from \(a\) through \(b\), including both endpoints, paid indemnity is:
Here, \(\mathbf{1}(a\le t_j\le b)\) equals 1 when transaction \(j\) falls in the stated period and 0 otherwise. Thus, a transaction outside the period contributes nothing to this period’s total. The notation also makes clear that the calculation is over transactions, not over distinct claim identifiers.
The formula follows directly from the meaning of a period total. Each eligible transaction contributes its amount once; each ineligible transaction contributes zero. Adding these contributions gives the total amount recorded as paid during the period. If a transaction file is grouped by claim, summing each claim’s eligible transactions and then summing across claims produces the same result as summing all eligible transactions directly, provided no transaction is omitted or duplicated.
Confirm that the question asks for paid indemnity, rather than a count of claims or a balance of unpaid amounts.
Use the payment date assigned to each transaction and include only dates inside the stated period.
Add positive payments and any applicable negative reversals or corrections recorded in the period.
Exclude non-indemnity entries and amounts that are still unpaid, then verify the arithmetic.
Worked Calculations
Worked Example: A Monthly Ledger With a Reversal
A hypothetical insurer is calculating paid indemnity for March. Its ledger contains the following entries. Dates shown are the assigned payment transaction dates, and every amount is an indemnity transaction.
| Claim | Transaction date | Signed amount |
|---|---|---|
| A | March 3 | $1,200 |
| B | February 26 | $600 |
| A | March 21 | $800 |
| C | March 14 | $900 |
| C | March 18 | -$900 |
| B | March 10 | $450 |
| B | April 2 | $250 |
Method. Apply the March cutoff first, then sum the signed amounts. The February 26 entry and April 2 entry are outside March, so neither is included. The March entries are $1,200, $800, $900, -$900, and $450.
The arithmetic is \(1{,}200+800=2{,}000\); \(2{,}000+900=2{,}900\); \(2{,}900-900=2{,}000\); and \(2{,}000+450=2{,}450\). Therefore, March paid indemnity is $2,450.
The $900 payment and its $900 reversal cancel in the March total. The common mistake this avoids is including every ledger entry associated with these claims regardless of date: that would wrongly add the February and April transactions to March.
Worked Example: Paid Amounts Versus Outstanding Amounts
A hypothetical insurer reviews three claims as of June 30. For this calculation, payment dates are in the second quarter, and all stated payments are indemnity.
| Claim | Second-quarter payments | Unpaid amount noted at June 30 |
|---|---|---|
| R-14 | $2,400 and $1,100 | $1,300 |
| R-15 | $500 | $0 |
| R-16 | $0 | $900 |
Method. Sum only the payment transactions made during the second quarter. For R-14, the paid amount is \(2{,}400+1{,}100=3{,}500\). For R-15, it is $500. R-16 has no payment transaction in the period, so its paid amount is $0.
The total is \(3{,}500+500+0=\) $4,000. The unpaid amounts, \(1{,}300+0+900=\) $2,200, do not enter this paid-loss total. In particular, R-16’s $900 unpaid amount does not become paid merely because it appears in the claim information at June 30.
The common mistake is to add the unpaid amounts to the period’s cash payments. That answers a different question. This example does not calculate an incurred amount; it separates the period’s paid transactions from amounts that remain unpaid.
Worked Example: Multiple Payments on the Same Claim
A hypothetical equipment insurer wants paid indemnity for January from this transaction extract. The extract contains only indemnity payment entries, and the transaction date determines inclusion.
| Claim | Transaction date | Amount |
|---|---|---|
| E-7 | January 4 | $600 |
| E-7 | January 25 | $400 |
| E-8 | January 12 | $1,250 |
| E-9 | January 31 | $800 |
| E-9 | February 2 | $300 |
Method. Include the four January entries and exclude the February 2 entry. Claim E-7 contributes \(600+400=1{,}000\); E-8 contributes $1,250; and E-9 contributes $800 for January. The total is \(1{,}000+1{,}250+800=\) $3,050.
There are four January transactions but only three claim identifiers represented in those transactions. The requested dollar total is $3,050; neither the number of transaction rows nor the number of represented claims is a substitute for that amount. The common mistake is to count the two E-7 payment rows as two claims, or to include E-9’s February payment in January just because it belongs to a claim with a January payment.
Interpreting and Checking a Paid-Loss Total
A paid-loss figure depends on both the amounts and the measurement rules used to select them. A clear calculation therefore states the period, the payment-date convention, and the scope of transactions. If the data include signed adjustments, the calculation should preserve their signs. If the question specifies indemnity, entries that represent something else should not be mixed into the sum.
It is useful to reconcile totals in two ways when the data are available: first by adding eligible transactions directly, and then by summing eligible transactions within each claim and adding those claim subtotals. These totals should agree. A discrepancy can signal a missing row, a duplicate transaction, a date-cutoff error, or an incorrect treatment of a reversal.
Do not infer claim counts from a paid-loss total. One claim may have several payments, and a claim can have no payment in the period. Conversely, a transaction extract can have multiple rows for the same claim. Payment amounts describe dollars transacted, not the number of claims represented.
Common Exam Trap
Suppose a problem asks for paid indemnity during a quarter and gives a claim with a $1,000 payment in the quarter, a $400 payment in the following quarter, and $700 still unpaid at quarter-end. A tempting but wrong calculation is \(1{,}000+400+700=\)$2,100. It includes a later payment and an unpaid amount, neither of which is a payment transaction in the stated quarter.
The correction is to apply the quarter cutoff to payment dates and sum only eligible transactions: the correct paid indemnity is $1,000. The later $400 belongs to the later payment period, and the $700 is not paid. A quick final check is to ask: “Was this indemnity transaction paid within the period?”
Check Your Understanding
For each problem, assume the entries are indemnity transactions and use the stated payment dates. Amounts are nominal dollars; no interest or probability model is assumed.
- A hypothetical insurer’s April ledger shows $750 on April 2, $1,250 on April 20, and -$250 on April 20 for a reversal. A $500 payment is dated March 31. What is April paid indemnity? (Answer: $1,750)
- For a hypothetical claim, $900 is paid on September 30, $400 is paid on October 1, and $600 remains unpaid on September 30. What is paid indemnity for the third quarter? (Answer: $900)
- A hypothetical transaction file has two May payments of $300 and $700 on claim K, one May payment of $1,100 on claim L, and a $250 payment on claim K dated June 1. What is May paid indemnity, and how many claim identifiers have May payments? (Answer: $2,100 and 2 claim identifiers)
- A hypothetical June ledger contains a $1,600 payment and a -$300 correction for claim M, plus a $500 payment for claim N dated July 1. What is June paid indemnity? (Answer: $1,300)
- A hypothetical quarter-end summary lists $2,200 paid during the quarter, $800 paid in the next quarter, and $1,500 unpaid at quarter-end. What is paid indemnity for the stated quarter, and what amount should be excluded from that total because it is unpaid? (Answer: $2,200; exclude $1,500)