Why Coverage Terms Matter to an ILF
An increased limits factor (ILF) compares expected covered losses at two policy limits. The comparison is meaningful only when the loss amounts reflect the coverage being priced. An exclusion can remove part of a reported loss; an occurrence definition can combine claims before applying a cap; and settlement wording can determine whether expenses erode the limit or are paid in addition to it. Each term can change the severity distribution used in the calculation.
The key sequence is: identify what the contract covers, apply the settlement terms that determine the covered payment, and then impose the relevant limit in the way the contract specifies. The terms may also change how claims are grouped. A reported gross loss is therefore not automatically the severity amount to use in an ILF.
This definition assumes that the same covered-loss basis is being compared at both limits. If claim frequency is unchanged and severity has the same distribution at both limits, the ratio of expected limited losses is also the ratio of expected losses for a fixed exposure. For a portfolio with different coverage terms or exposure mixes, those differences must be reflected in the corresponding expected losses rather than hidden inside a single severity calculation.
Translate Contract Terms into a Covered Amount
A useful way to organize the calculation is to treat the policy wording as a sequence of operations on the reported loss. Let \(X\) denote a reported ground-up loss. Coverage terms determine which parts are eligible; settlement provisions may then adjust the eligible amount; finally, the applicable limit caps the payment. The exact order depends on the contract. In particular, aggregation terms specify whether the limit is applied to each claim separately or to the combined covered amount for an occurrence.
Remove excluded damage or other amounts that are not eligible under the hypothetical policy wording. Do not treat an excluded component as covered merely because it appears in the reported total.
Determine how the contract treats items such as depreciation, salvage credits, or defense expenses. These terms can alter the amount subject to the limit or the total payment made by the insurer.
Establish whether the limit applies per claim, per occurrence, or to another defined unit. Combine amounts only when the wording requires it.
Use the same coverage and settlement basis at the base and higher limits, then divide the higher-limit expected payment by the base-limit expected payment.
The order matters because a cap does not turn an ineligible loss into a covered loss. In general, if \(C\) is the eligible amount and \(E\) is excluded, the capped covered payment is \(\min(C,L)\), not \(\min(C+E,L)\). Likewise, summing claims before capping is correct for an occurrence limit only when the policy defines the relevant claims as one occurrence and applies the limit to their combined amount.
Worked Example: An Exclusion Changes the Severity Basis
A hypothetical policy has a per-claim limit. In thousands of dollars, a reported claim has one of three outcomes: with probability 0.50, covered damage is $10 and excluded damage is $30; with probability 0.30, covered damage is $25 and excluded damage is $10; with probability 0.20, covered damage is $70 and excluded damage is $20. Calculate the ILF from a $20,000 base limit to a $50,000 limit. Assume the probabilities and coverage terms are the same at both limits.
Method. Use the covered component alone as \(Z\), then cap it at each limit. All amounts below are in thousands of dollars.
At the $20,000 base limit, the limited covered amounts by outcome are \(\min(10,20)=10\), \(\min(25,20)=20\), and \(\min(70,20)=20\). Therefore,
At the $50,000 limit, the limited covered amounts are \(\min(10,50)=10\), \(\min(25,50)=25\), and \(\min(70,50)=50\). Thus,
The ILF is \(22.5/15=1.5\). The expected limited covered amounts are $15,000 and $22,500, respectively, so their ratio checks to 1.5000.
Common mistake avoided. Using the reported totals of $40,000, $35,000, and $90,000 would include excluded damage. That would give expected capped amounts of \(0.50(20)+0.30(20)+0.20(20)=20\) at the base limit and \(0.50(40)+0.30(35)+0.20(50)=40.5\) at the higher limit, producing an incorrect ratio of \(40.5/20=2.025\). The calculation must cap eligible covered loss, not the reported total.
Why the Order and Aggregation Unit Matter
For a per-claim limit, the covered payment is calculated claim by claim. For an occurrence limit, the covered claims belonging to one occurrence are first combined as specified by the contract, and the occurrence cap is applied to the combined amount. The two approaches generally yield different payments because capping and summing are not interchangeable.
A short justification makes the distinction explicit. Suppose an occurrence contains covered claim amounts \(Z_1,\ldots,Z_n\), and the policy has one occurrence limit \(L\). The payment is \(\min(\sum_{j=1}^{n}Z_j,L)\). Under separate per-claim limits \(L\), the payment instead is \(\sum_{j=1}^{n}\min(Z_j,L)\). For nonnegative claim amounts,
To see why, if the sum is below \(L\), each \(Z_j\) is also below \(L\), so both sides equal the sum. If the sum exceeds \(L\), the left side is \(L\), while the right side is at least \(L\): either some claim is at least \(L\), or all claims are below \(L\) and their uncapped amounts sum to more than \(L\). Thus, treating an occurrence cap as separate per-claim caps can overstate payment.
Worked Example: A Limit Applied to an Occurrence
A hypothetical policy covers one event per exposure. With probability 0.50, the event produces one covered claim of $20,000. With probability 0.50, it produces two covered claims, of $30,000 and $25,000. Compare the expected occurrence-limited losses at a $40,000 base limit and a $60,000 limit. The same claims and probabilities apply at both limits.
Method. Add the covered claims within each event, then apply the occurrence limit once. For a one-claim event, the occurrence amount is $20,000. For a two-claim event, the combined amount is \(30,000+25,000=55,000\).
At the $40,000 limit, the two possible event payments are \(\min(20,000,40,000)=20,000\) and \(\min(55,000,40,000)=40,000\). The expected loss is
At the $60,000 limit, the payments are $20,000 and $55,000, so the expected loss is
The occurrence ILF is \(37,500/30,000=1.25\). As a check, the higher limit increases payment only in the two-claim event, from $40,000 to $55,000.
Common mistake avoided. Applying the $40,000 limit separately to each claim in the two-claim event gives \(30,000+25,000=55,000\), not the correct occurrence payment of $40,000. The corresponding separate-claim calculation would produce the same expected payment, $37,500, at both the $40,000 and $60,000 limits and an erroneous ILF of 1.00. The cap applies to the event total under the stated wording.
Settlement Terms Can Change What the Limit Caps
Settlement provisions may change the amount subject to the limit, or may specify payments that are outside the limit. The ILF must reflect that distinction. For example, if covered defense expenses erode a limit, they are added to indemnity before the total is capped. If defense expenses are paid in addition to the indemnity limit, they are not included in that cap. Other settlement adjustments, such as a salvage credit, may reduce the covered amount before the limit is applied. These are contract-dependent illustrations, not universal rules; the policy wording controls.
Worked Example: Defense Expenses Inside or Outside the Limit
A hypothetical policy has one claim per exposure, with indemnity \(X\) equal to $20,000 or $60,000 with probability 0.50 each. Defense expense is $10,000 for every claim. Compare an $80,000 limit with a $40,000 base limit under two settlement provisions. Under the first, defense expense erodes the limit; under the second, it is paid in addition to the indemnity limit. Assume no deductible and identical frequency at both limits.
Method, expenses inside the limit. The amount capped is indemnity plus defense expense. For the $40,000 limit, the two payments are \(\min(20,000+10,000,40,000)=30,000\) and \(\min(60,000+10,000,40,000)=40,000\). The expected payment is
At the $80,000 limit, the payments are \(\min(30,000,80,000)=30,000\) and \(\min(70,000,80,000)=70,000\). The expected payment is \(0.50(30,000)+0.50(70,000)=15,000+35,000=50,000\). The inside-limit ILF is \(50,000/35,000=1.4286\), rounded to four decimals.
Method, expenses outside the limit. Now the payment is defense expense plus the indemnity capped at the applicable limit. At $40,000, expected payment is
At $80,000, it is \(10,000+0.50(20,000)+0.50(60,000)=10,000+10,000+30,000=50,000\). The outside-limit ILF is \(50,000/40,000=1.2500\). Both calculations check against their underlying payments: the high-limit expected payment is $50,000, but the base-limit payment differs because defense expense is treated differently.
Common mistake avoided. Applying \(\min(X, L)\) and then forgetting defense expense understates total payment under either provision. Adding expense before capping when it is outside the limit is also incorrect. The specified settlement wording determines which expression to use.
Checks Before Using a Limited-Loss Amount
A coverage-term calculation is only as reliable as its interpretation of the contract and its loss data. Before calculating an ILF, confirm that the covered components, settlement basis, and aggregation unit are consistent at both limits. In particular, check whether the loss information already reflects an exclusion or adjustment; applying it a second time would distort the severity amounts.
| Term to check | Question for the calculation | Potential error |
|---|---|---|
| Exclusion | Which reported components are eligible for payment? | Capping the full reported loss, including excluded amounts |
| Aggregation | What claims constitute one capped unit? | Applying an occurrence limit separately to each claim |
| Expense treatment | Does the expense erode the limit or sit outside it? | Adding the expense on the wrong side of the cap |
| Settlement adjustment | Does the term reduce or otherwise alter the eligible amount? | Using a pre-adjustment severity when payment is based on an adjusted amount |
A further check is that, for a fixed covered-loss basis with nonnegative payments and a positive denominator, increasing the limit cannot decrease expected limited loss. This does not mean all policies have the same ILF: changing exclusions, settlement terms, or the unit of aggregation changes the payment amounts being compared.
Check Your Understanding
For each problem, identify the applicable loss basis before calculating the requested amount.
- A hypothetical per-claim policy has covered losses of $8,000, $30,000, and $90,000 with probabilities 0.40, 0.40, and 0.20. An excluded component is present in some reported files but is not included in these covered amounts. Find the ILF from $25,000 to $60,000. (Answer: \(27,200/18,200=1.4945\))
- A hypothetical occurrence has one $18,000 claim with probability 0.60, or two claims of $28,000 and $22,000 with probability 0.40. Find expected limited loss at a $35,000 occurrence limit. (Answer: \(0.60(18,000)+0.40(35,000)=24,800\))
- Indemnity is $15,000 or $55,000 with equal probabilities, and $5,000 of defense expense is paid inside the limit. Find expected payment at a $30,000 limit. (Answer: \(0.50(20,000)+0.50(30,000)=25,000\))
- Indemnity is $25,000 or $75,000 with equal probabilities. A $10,000 salvage credit is applied before the per-claim limit, and the resulting covered amounts are capped at $30,000. Find expected payment. (Answer: \(0.50(15,000)+0.50(30,000)=22,500\))
- A hypothetical event produces one $24,000 covered claim with probability 0.50, or two covered claims of $35,000 and $20,000 with probability 0.50. Find the occurrence ILF from $40,000 to $50,000. (Answer: base expected loss \(=0.50(24,000)+0.50(40,000)=32,000\); higher-limit expected loss \(=0.50(24,000)+0.50(50,000)=37,000\); ILF \(=37,000/32,000=1.1563\))
Coverage definitions and settlement provisions determine the severity amounts that enter an ILF calculation. Keeping the contract wording, aggregation unit, and limit application aligned is essential to producing a meaningful comparison, an approach aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.