How Coinsurance Changes Aggregate Excess-Loss Cover
An aggregate deductible determines where a layer of protection begins. Coinsurance determines what fraction of losses in that layer is paid by the reinsurer. A payment limit, if present, restricts how much of the layer can be covered or how much the reinsurer can pay. These contract terms must be identified separately before calculating an expected payment.
Let \(S\) be the nonnegative aggregate loss over the contract period, and let \(d\geq 0\) be the aggregate attachment point. For this tutorial, \(c\) is the ceded share of covered excess, where \(0\leq c\leq 1\). The amount retained by the ceding company is the aggregate loss minus the ceded payment. All examples use hypothetical losses and undiscounted payments for one period.
A coinsurance percentage is not applied to the entire aggregate loss. It applies only above the attachment point. For example, if the aggregate loss is $100,000, the attachment is $60,000, and the ceded share is 40%, the ceded payment is 40% of $40,000, or $16,000—not 40% of $100,000.
Choosing the Correct Meaning of a Payment Limit
A contract may specify a maximum covered excess amount or a maximum ceded payment. These are different limits when \(c<1\). State which convention applies before using a formula.
| Contract wording interpreted as | Ceded payment | Effect of the limit |
|---|---|---|
| Maximum covered excess amount \(u\) | \(C=c\min((S-d)_+,u)\) | The coinsurance share is applied to no more than \(u\) of excess loss. |
| Maximum ceded payment \(K\) | \(C=\min(c(S-d)_+,K)\) | The ceded payment itself cannot exceed \(K\). |
For instance, with a 50% ceded share and a $20,000 limit, a covered-excess limit of $20,000 produces a maximum payment of $10,000. A $20,000 ceded-payment limit instead permits payments up to $20,000. Treating these limits as interchangeable can materially change the expected payment.
Why the Capped-Payment Formula Works
The capped formulas follow from a pointwise identity, so they remain valid for discrete, continuous, or mixed aggregate loss distributions. For any nonnegative \(S\), attachment \(d\), and covered-excess limit \(u\geq0\),
To verify the identity, consider the possible values of \(S\). If \(S\leq d\), both positive-part terms are zero. If \(d<S<d+u\), the second term is zero and the first is \(S-d\), which is less than \(u\). If \(S\geq d+u\), the difference is \((S-d)-(S-d-u)=u\). These cases give exactly the capped excess in every outcome.
Taking expectations of both sides gives \(E[\min((S-d)_+,u)]=T(d)-T(d+u)\). Multiplying by the ceded share \(c\) gives the expected payment under a covered-excess limit. For a cap \(K\) on the payment itself, the identity \(\min(cX,K)=c\min(X,K/c)\) for \(X\geq0\) and \(c>0\) reduces the calculation to the same result with effective excess limit \(K/c\). If \(c=0\), the ceded payment is zero regardless of the cap.
Record the attachment \(d\), ceded share \(c\), and whether any limit applies to covered excess or to the ceded payment.
For each possible aggregate loss, subtract the attachment, apply the relevant cap, and then apply the ceded share if the cap is on covered excess.
Weight each ceded payment by its probability. Find expected retained loss from \(E[R]=E[S]-E[C]\), or calculate retained loss outcome by outcome as a check.
Worked Calculations
Worked Example: Coinsurance Without a Payment Limit
A hypothetical portfolio has aggregate loss \(S\) equal to $0, $40,000, $100,000, or $160,000 with probabilities 0.20, 0.30, 0.30, and 0.20, respectively. A reinsurer pays 40% of aggregate loss above an attachment of $60,000, with no payment limit. Find the expected ceded payment and expected retained loss.
Method and contract interpretation. This is an uncapped coinsured excess layer, so \(C=0.40(S-60{,}000)_+\). The probabilities sum to \(0.20+0.30+0.30+0.20=1.00\).
Step 1: Find the ceded payment for each outcome. At losses of $0 and $40,000, the aggregate loss does not exceed $60,000, so the payment is $0. At $100,000, excess loss is \(100{,}000-60{,}000=40{,}000\), giving a payment of \(0.40(40{,}000)=$16{,}000\). At $160,000, excess loss is \(160{,}000-60{,}000=100{,}000\), giving a payment of \(0.40(100{,}000)=$40{,}000\).
Step 2: Take the expectation.
Step 3: Find and check expected retention. The expected aggregate loss is \(E[S]=0.20(0)+0.30(40{,}000)+0.30(100{,}000)+0.20(160{,}000)=$74{,}000\). Therefore \(E[R]=74{,}000-12{,}800=$61{,}200\). Directly, the retained amounts at the four outcomes are $0, $40,000, $84,000, and $120,000; their expectation is \(0.30(40{,}000)+0.30(84{,}000)+0.20(120{,}000)=$61{,}200\), as required.
Common mistake avoided: applying 40% to the full $100,000 or $160,000 loss. Only the part above $60,000 is subject to coinsurance.
Worked Example: A Limit on Covered Excess
A hypothetical aggregate loss \(S\) is $30,000, $80,000, $130,000, or $200,000 with probabilities 0.25, 0.25, 0.30, and 0.20. A reinsurer pays 60% of excess loss above $50,000, subject to a maximum covered excess amount of $50,000. Calculate the expected ceded payment and expected retained loss.
Method and contract interpretation. The $50,000 cap is on covered excess, so \(C=0.60\min((S-50{,}000)_+,50{,}000)\). The given probabilities sum to 1.
Step 1: Calculate capped excess and ceded payment. For \(S=$30,000\), excess and payment are both zero. For \(S=$80,000\), excess is $30,000, below the cap, so the payment is \(0.60(30{,}000)=$18{,}000\). For \(S=$130,000\), excess is $80,000, of which only $50,000 is covered, so the payment is \(0.60(50{,}000)=$30{,}000\). At \(S=$200,000\), excess is $150,000, but the same $50,000 cap applies, giving a $30,000 payment.
Step 2: Take the expectation.
Step 3: Calculate expected retained loss. The expected aggregate loss is \(0.25(30{,}000)+0.25(80{,}000)+0.30(130{,}000)+0.20(200{,}000)=$106{,}500\). Thus \(E[R]=106{,}500-19{,}500=$87{,}000\). Checking directly, retained amounts are $30,000, $62,000, $100,000, and $170,000; weighting them gives \(7{,}500+15{,}500+30{,}000+34{,}000=$87{,}000\).
Common mistake avoided: multiplying the uncapped excess by 60% at the two largest losses. The contract first limits covered excess to $50,000, so the ceded payment cannot exceed \(0.60(50{,}000)=$30,000\).
Worked Example: A Limit on the Ceded Payment
A hypothetical aggregate loss \(S\) is $20,000, $60,000, $100,000, or $160,000 with probabilities 0.20, 0.30, 0.30, and 0.20. Above an attachment of $40,000, the ceded share is 50%, but the maximum ceded payment is $25,000. Find the expected ceded payment and expected retained loss.
Method and contract interpretation. Here the limit applies to the payment, not to covered excess: \(C=\min(0.50(S-40{,}000)_+,25{,}000)\). The probabilities sum to 1.
Step 1: Calculate the uncapped payment and apply the limit. At $20,000, there is no excess and the payment is $0. At $60,000, excess is $20,000 and the uncapped payment is \(0.50(20{,}000)=$10{,}000\), below the cap. At $100,000, excess is $60,000 and the uncapped payment is $30,000, reduced to $25,000 by the payment limit. At $160,000, excess is $120,000 and the uncapped payment is $60,000, also reduced to $25,000.
Step 2: Take the expectation.
Step 3: Calculate and check expected retention. The expected aggregate loss is \(0.20(20{,}000)+0.30(60{,}000)+0.30(100{,}000)+0.20(160{,}000)=$84{,}000\), so expected retention is \(84{,}000-15{,}500=$68{,}500\). Directly, retained amounts are $20,000, $50,000, $75,000, and $135,000. Their expectation is \(4{,}000+15{,}000+22{,}500+27{,}000=$68{,}500\).
Common mistake avoided: treating the $25,000 payment cap as a $25,000 cap on excess. A $25,000 excess cap would produce a maximum ceded payment of only \(0.50(25{,}000)=$12{,}500\), a different contract.
Exam Checks and Common Traps
For a discrete aggregate loss distribution, a reliable calculation is to make a row for every possible value of \(S\), determine the payment using the contract wording, and then multiply by its probability. Check that the probabilities sum to 1 and that no payment exceeds the applicable cap. If expected retention is requested, the identity \(E[R]=E[S]-E[C]\) provides a useful independent check.
A frequent error is to apply coinsurance before subtracting the attachment, using \(cS\) instead of \(c(S-d)_+\). Another is to cap at the attachment itself rather than cap the excess above it. Finally, do not assume that the word “limit” identifies which amount is limited: verify whether the limit is on covered excess or on the ceded payment. These distinctions are particularly important when the ceded share is less than 100%.
Check Your Understanding
For each problem, identify the cap convention before calculating. Amounts are in dollars.
- A hypothetical aggregate loss is $0 with probability 0.25, $50 with probability 0.25, and $100 with probability 0.50. A reinsurer pays 50% above an attachment of $40, with no limit. Find expected ceded payment and expected retained loss. (Answer: Expected ceded payment $16.25; expected retained loss $46.25.)
- Aggregate loss is $20 with probability 0.20, $50 with probability 0.30, $90 with probability 0.30, and $150 with probability 0.20. A 80% ceded share applies above $30, with a $40 limit on covered excess. Find expected ceded payment and expected retained loss. (Answer: Expected ceded payment $20.80; expected retained loss $55.20.)
- Aggregate loss is $10 with probability 0.40, $50 with probability 0.40, and $100 with probability 0.20. The ceded share is 60% above $20, subject to a maximum ceded payment of $18. Find expected ceded payment and expected retained loss. (Answer: Expected ceded payment $10.80; expected retained loss $33.20.)
- Aggregate loss is $40 with probability 0.20, $80 with probability 0.50, and $140 with probability 0.30. The reinsurer pays 25% above $50, with a limit of $60 on covered excess. Find expected ceded payment and expected retained loss. (Answer: Expected ceded payment $8.25; expected retained loss $81.75.)
- Aggregate loss is $50 with probability 0.50 and $120 with probability 0.50. Above an attachment of $20, a 50% ceded share is subject to a $20 maximum ceded payment. Find expected ceded payment, and state what the expectation would be if instead $20 limited covered excess. (Answer: Payment cap: $17.50; covered-excess cap: $10.00.)