How an Industry-Loss Trigger Works
An industry-loss trigger bases a contractual payment on a defined estimate of losses across an insurance market or geographic area, rather than on the sponsoring insurer’s verified claims alone. The contract specifies the index, its source, the event and reporting period, and the payment formula. A sponsor may have a very different actual loss from the loss implied by the industry index. That difference is basis risk: the index payment may not match the sponsor’s need for funds.
A sponsor-specific share can be used to convert the industry estimate into an estimate of the sponsor’s own losses. However, contracts do not all use the share in the same place. Some define attachment and exhaustion on the industry index itself; others define them on the share-adjusted estimate. The contract wording controls which calculation applies. Applying the share to a payment formula that already uses industry-wide thresholds can materially misstate the payment.
Calculating a Layer Payment
For a layer defined on an index value \(Z\), where \(Z\) is either \(I\) or \(J\) as specified by the contract, a common proportional payment formula is:
If the index is at or below attachment, the payment is zero. Between attachment and exhaustion, the payment increases linearly. At or above exhaustion, the full limit is paid. The formula assumes a single index value, fixed terms, and a linear payout between the stated attachment and exhaustion points. It does not describe a contract with a different payout curve, a separate sponsor-loss condition, a reinstatement, or other bespoke terms.
The formula follows from the layer’s proportional design. At attachment, the fraction of the layer reached is 0; at exhaustion, it is 1. A straight line between those endpoints has slope \(1/(E-A)\), so the fraction reached at \(Z\) is \((Z-A)/(E-A)\). Capping that fraction below at 0 and above at 1 handles values outside the layer. Multiplying the resulting fraction by \(L\) gives the payment.
Determine the relevant industry loss estimate and verify the event, measurement period, currency, and index source.
Check whether attachment and exhaustion apply to the industry index \(I\) itself or to a sponsor-share-adjusted index \(J=qI\).
Use consistent units for the index, attachment, exhaustion, and limit; then calculate the capped fraction of the layer reached.
Use actual sponsor losses to assess the payment’s fit, not to replace the contractual index calculation unless the terms expressly require it.
Worked Calculations
Worked Example: A Share-Adjusted Index Layer
A hypothetical insurer’s contract uses an industry estimate of $8.0 billion and a sponsor share of 2.5%. The attachment is $120 million, the exhaustion point is $240 million, and the maximum payment is $60 million. All layer terms apply to the sponsor-share-adjusted index. Calculate the payment. The contract uses a single event estimate and a linear payment between attachment and exhaustion.
Method. Convert the industry estimate into the share-adjusted index, then apply the layer formula to that value.
First convert the percentage to a fraction: \(q=2.5/100=0.025\). The share-adjusted index is:
The index is within the layer because \(120<200<240\), with all amounts in millions. The fraction of the layer reached is:
Therefore, the payment is \((2/3)(\$60\text{ million})=\$40\text{ million}\). Check: the payment is positive and below the $60 million limit, as required for an index between attachment and exhaustion. This method avoids treating 2.5% as 2.5 rather than 0.025.
Worked Example: A Layer Written Directly on the Industry Index
A hypothetical risk-transfer contract defines its layer directly on an industry index of $7.0 billion. Its attachment is $5.0 billion, exhaustion is $9.0 billion, and limit is $80 million. The sponsor’s specified share is 1.2%, and its actual event losses are $110 million. Calculate the payment and compare the index-implied sponsor estimate with actual losses. Assume the contract has no additional sponsor-loss condition.
Method. Because the attachment and exhaustion are stated on the industry index, first calculate the payment using \(I\), not \(qI\).
The industry index lies within the layer. Its reached fraction is:
The payment is \(0.5(\$80\text{ million})=\$40\text{ million}\). Separately, the share-adjusted estimate is \(0.012(\$7{,}000\text{ million})=\$84\text{ million}\). The sponsor’s actual losses of $110 million exceed that estimate by $26 million. The contract still pays $40 million under the stated terms: actual losses do not replace the industry index in this calculation.
Check: the industry index is halfway between $5 billion and $9 billion, so half the limit is paid. Multiplying the payment by the sponsor share would wrongly apply the share to a payment already calculated under an industry-index layer.
Worked Example: Combining Regional Indices Before Applying the Layer
A hypothetical sponsor’s contract defines a share-adjusted index from two regions. The first region’s industry loss estimate is $4.0 billion and the sponsor share there is 1.5%; the second estimate is $3.0 billion and its share is 2.0%. The resulting index is subject to attachment of $90 million, exhaustion of $150 million, and a $30 million limit. Calculate the payment. Assume the contract defines the combined index as the sum of the two share-adjusted regional estimates.
Method. Apply each regional share to its own regional estimate, add those amounts, and then apply the layer once to the combined index.
For the first region, \(0.015(\$4{,}000\text{ million})=\$60\text{ million}\). For the second, \(0.020(\$3{,}000\text{ million})=\$60\text{ million}\). Thus:
The combined index is between attachment and exhaustion. Its reached fraction is:
The payment is \(0.5(\$30\text{ million})=\$15\text{ million}\). Check: $120 million is halfway through the $60 million-wide layer, so half of the $30 million limit is paid. Using one unweighted average share for both regions would not, in general, reproduce the specified sum.
Index Basis and Sponsor-Specific Shares
A sponsor-specific share is an estimate or contractual allocation; it is not necessarily the sponsor’s measured proportion of the final industry loss for a particular event. It can reflect the sponsor’s geographic footprint, lines of business, or expected market participation. A fixed share makes a broad industry index more relevant to that sponsor, but it cannot ensure that the sponsor’s actual claims move in proportion to the industry total.
Two contract designs that can look similar in a summary may produce different payments:
| Contract basis | Value compared with attachment and exhaustion | Payment calculation |
|---|---|---|
| Industry-index layer | Industry estimate \(I\) | Apply the layer formula to \(I\) |
| Share-adjusted layer | Sponsor index \(J=qI\) | Calculate \(J\), then apply the layer formula to \(J\) |
For example, if the contract sets thresholds in billions of industry losses, applying a sponsor share first changes the units and the trigger basis. Conversely, if thresholds are stated in sponsor-estimate dollars, using the industry total directly can make the trigger appear far more severe than intended. Always compare quantities expressed on the same basis and in the same units.
Basis Risk and Common Exam Traps
The index may diverge from sponsor losses because the sponsor’s insured locations or policy mix differ from the industry’s, because event damage is unevenly distributed, or because the index and the sponsor use different loss definitions, reporting dates, or adjustments. A share-adjusted index can reduce some mismatch, but it remains an estimate. Payment adequacy therefore depends on both the contractual formula and how closely the index tracks the sponsor’s exposure.
- Applying the share twice. If the contract’s attachment and exhaustion apply to \(J=qI\), calculate the payment from \(J\). Do not multiply that payment by \(q\) again unless the contract explicitly says to do so.
- Using the wrong index basis. A layer written on the industry total uses \(I\); a layer written on the sponsor estimate uses \(qI\). The share does not automatically belong in every industry-loss calculation.
- Ignoring attachment or the limit. The uncapped fraction \((Z-A)/(E-A)\) is not valid as a payment fraction outside the layer. The payment must be zero below attachment and no more than \(L\) above exhaustion.
- Substituting actual losses for the index. Actual losses are relevant to evaluating basis risk. They determine the payment only if the contract says they are part of the trigger or settlement condition.
- Mixing units or regional shares. Convert billions to millions consistently, and apply a regional share to its corresponding regional index before combining values when that is what the contract defines.
Check Your Understanding
For each problem, use the stated contract basis and assume a linear layer payment unless otherwise specified.
- A contract applies its layer to a share-adjusted index. The industry estimate is $6.0 billion, the sponsor share is 1.8%, attachment is $80 million, exhaustion is $140 million, and limit is $30 million. Calculate the payment. (Answer: $14 million)
- A layer is written directly on the industry index. The index is $3.6 billion, attachment is $4.0 billion, exhaustion is $6.0 billion, and limit is $50 million. The sponsor share is 2%. Calculate the payment and explain whether the share changes it. (Answer: $0; the index is below attachment, and the share is not applied to this industry-index layer.)
- A contract uses an industry-index layer with index $10 billion, attachment $7 billion, exhaustion $11 billion, and limit $100 million. The sponsor share is 0.7%, and the sponsor’s actual losses are $125 million. Calculate the payment and the share-adjusted estimate. (Answer: payment $75 million; share-adjusted estimate $70 million)
- A share-adjusted index is the sum of two regional estimates: $2.0 billion at a 1% share and $1.5 billion at a 2% share. The layer attaches at $40 million, exhausts at $90 million, and has a $25 million limit. Calculate the combined index and payment. (Answer: combined index $50 million; payment $5 million)
- A contract applies its layer to a sponsor-share-adjusted index. The industry estimate is $20 billion, the share is 0.4%, attachment is $50 million, exhaustion is $110 million, and the limit is $36 million. Actual sponsor losses are $125 million. Calculate the index payment and the difference between actual losses and the share-adjusted estimate. (Answer: payment $18 million; actual losses exceed the share-adjusted estimate of $80 million by $45 million)