Applying Surplus Share Before Excess-of-Loss

Why the Order of Reinsurance Matters

When a policy is covered by both surplus share and excess-of-loss reinsurance, the contracts do not necessarily respond to the same amount. A surplus share treaty cedes a proportion of the policy’s insured amount, and therefore a corresponding proportion of a loss when the loss is allocated proportionally. An excess-of-loss layer may then cover part of the amount that remains after that cession. The calculation must follow the order stated in the contracts.

This tutorial considers the sequence surplus share first, then excess-of-loss. It assumes that the excess-of-loss contract applies to the post-surplus retained loss. Contract wording controls in practice: if a layer instead applies to gross loss or to another defined amount, that amount must be used.

Definition: Let $S$ be a policy’s sum insured, $R$ the insurer’s retention (one surplus line), and $n$ the number of additional lines the surplus treaty can accept. The treaty’s maximum cession is $nR$. Under proportional loss allocation, the surplus cession proportion of the policy is $$ c=\frac{\min\bigl(\max(S-R,0),\,nR\bigr)}{S}, \qquad S>0. $$ For a gross policy loss $L$, the surplus recovery is $cL$, and the post-surplus retained loss is $Z=L-cL=L(1-c)$.

Calculate the Surplus Share Cession

A surplus treaty’s line capacity is a limit on the amount of insurance ceded, not a percentage of the claim chosen independently of the policy amount. The amount ceded is the lesser of (1) the sum insured above the insurer’s retention and (2) the treaty’s capacity. Dividing that cession by the policy sum insured gives the cession proportion $c$.

If $S\leq R$, the amount above retention is zero, so the treaty cedes nothing. If $R<S\leq R+nR$, the treaty can accept all the amount above the retention. If $S>R+nR$, its capacity is exhausted: some insurance above the retention remains with the insurer. In all cases, under the stated proportional allocation assumption, the loss is split using the same proportion $c$.

\text{surplus recovery}=cL,\qquad \text{post-surplus retained loss}=Z=L(1-c).

The excess-of-loss calculation then uses $Z$, not the original gross loss $L$, when the layer is contractually written on the post-surplus retained amount. Let $A$ be the layer’s attachment and $M$ its maximum recovery (the layer limit). The layer pays only the part of $Z$ above $A$, up to $M$.

Definition: For an excess-of-loss layer attaching at $A$ with a maximum recovery $M$, applied to post-surplus retained loss $Z$, the recovery and final retained loss are $$ \text{excess-of-loss recovery}=\min\bigl(\max(Z-A,0),M\bigr), $$ $$ \text{final retained loss}=Z-\text{excess-of-loss recovery}. $$ The layer pays nothing if $Z\leq A$, pays $Z-A$ if $A<Z<A+M$, and pays its full limit $M$ if $Z\geq A+M$.

Why the Formulas Work

The surplus formula follows from proportional allocation. The treaty accepts a specified amount of insurance, namely the lesser of the policy amount above $R$ and its capacity $nR$. Its share of the policy is therefore that accepted amount divided by $S$, which is $c$. If each part of the loss is allocated in proportion to the insured amount, the treaty’s share of a loss $L$ is $cL$. Subtracting that recovery from the gross loss gives $L-cL=L(1-c)$.

For the excess-of-loss formula, first compare $Z$ with the attachment $A$. Only $Z-A$ can be above the attachment, and if $Z\leq A$, the recovery is zero. The treaty cannot pay more than its limit $M$. Combining these conditions gives the smaller of the amount above attachment and the limit, with the amount above attachment set to zero when it is negative.

Determine the surplus capacity.
Multiply the retention per line $R$ by the number of treaty lines $n$ to find the maximum cession $nR$.

Find the policy cession proportion.
Limit the amount above retention to the treaty capacity, then divide the ceded amount by the policy sum insured $S$.

Allocate the gross loss.
Multiply $L$ by the cession proportion to obtain the surplus recovery; subtract it from $L$ to obtain $Z$.

Apply the excess-of-loss layer.
Use $Z$ as the layer’s subject amount, calculate the amount above attachment, and cap the recovery at the layer limit.

Reconcile the allocation.
Check that gross loss equals the surplus recovery plus the excess-of-loss recovery plus the final retained loss.

Worked Applications

Worked Example: A Policy Loss Within the Surplus Treaty’s Capacity

A hypothetical policy has a sum insured of $180,000. The insurer retains $30,000 per line, and a surplus share treaty accepts four additional lines. A proportional loss is $72,000. The post-surplus excess-of-loss layer attaches at $25,000 and has a $20,000 limit. Calculate both recoveries and the final retained loss. Assume the layer applies to the post-surplus retained amount.

Method. Calculate the amount ceded from the policy sum insured, convert it to a cession proportion, and then apply the excess-of-loss layer to the resulting retained loss.

The surplus treaty capacity is $4(30{,}000)=\$120{,}000$. The amount of insurance above the retention is $180{,}000-30{,}000=\$150{,}000$. Thus the cession is the lesser of $150,000 and $120,000:

\text{ceded amount}=\min(150{,}000,120{,}000)=120{,}000, \qquad c=\frac{120{,}000}{180{,}000}=\frac{2}{3}.

The surplus recovery is $72{,}000(2/3)=\$48{,}000$. The post-surplus retained loss is $72{,}000-48{,}000=\$24{,}000$. Because $24,000 is below the $25,000 attachment, the excess-of-loss recovery is $0. The final retained loss is therefore $24,000.

Check. The allocations sum to the gross loss: $48{,}000+0+24{,}000=\$72{,}000$. The common mistake avoided here is treating all $150,000 above the retention as ceded, even though the treaty can accept only $120,000 of insurance.

Worked Example: Applying the Layer to the Correct Retained Amount

A hypothetical policy has a sum insured of $300,000, a $50,000 retention per line, and a surplus treaty capacity of four lines. A proportional loss is $120,000. After the surplus share cession, an excess-of-loss layer applies to the retained amount, attaching at $25,000 with a $20,000 limit. Find the surplus recovery, the excess-of-loss recovery, and the final retained loss.

Method. First cap the surplus cession at the treaty’s line capacity. Then apply the layer to the amount remaining with the insurer.

The maximum surplus cession is $4(50{,}000)=\$200{,}000$. The policy amount above the retention is $300{,}000-50{,}000=\$250{,}000$, so the cession is capped at $200,000. The cession proportion is $200{,}000/300{,}000=2/3$. Therefore, the surplus recovery is $120{,}000(2/3)=\$80{,}000$, leaving $120{,}000-80{,}000=\$40{,}000$.

The amount above the excess-of-loss attachment is $40{,}000-25{,}000=\$15{,}000$. This is less than the $20,000 limit, so the excess-of-loss recovery is $15,000. The final retained loss is $40{,}000-15{,}000=\$25{,}000$.

Check. $80{,}000+15{,}000+25{,}000=\$120{,}000$. A tempting but incorrect calculation would use $(300{,}000-50{,}000)/300{,}000=5/6$ as the cession proportion, giving a $100,000 surplus recovery. That ignores the four-line capacity. The correct proportion is $200{,}000/300{,}000=2/3$, and the layer must then be applied to $40,000, not to the gross loss.

Worked Example: A Layer That Pays Its Full Limit

A hypothetical policy has a $240,000 sum insured. The insurer’s retention is $60,000 per line, and the surplus share treaty accepts three lines. The proportional policy loss is $100,000. The excess-of-loss cover applies after the surplus cession, attaches at $10,000, and has an $8,000 limit. Calculate the two recoveries and the final retention.

Method. Compute the policy’s surplus share first, then compare the retained loss with the layer’s attachment and exhaustion point.

The treaty capacity is $3(60{,}000)=\$180{,}000$, while the amount above the retention is $240{,}000-60{,}000=\$180{,}000$. Thus the cession proportion is $180{,}000/240{,}000=3/4$. The surplus recovery is $100{,}000(3/4)=\$75{,}000$, leaving a post-surplus retained loss of $100{,}000-75{,}000=\$25{,}000$.

The layer begins at $10,000 and is exhausted at $10{,}000+8{,}000=\$18{,}000$. Since $25,000 exceeds $18,000, the layer pays its full $8,000 limit. The final retained loss is $25{,}000-8{,}000=\$17{,}000$.

Check. $75{,}000+8{,}000+17{,}000=\$100{,}000$. The insurer retains both the amount below the attachment and the amount above the layer’s exhaustion point; the limit does not make the insurer’s final retention equal to the attachment.

Common Exam Traps

Ignoring line capacity. The entire sum insured above the retention is not necessarily ceded. Cap it at $nR$ before calculating the cession proportion.
Using the wrong denominator. Under the proportional allocation assumption, divide the amount ceded by the policy sum insured $S$, not by the amount above the retention.
Applying the layer to gross loss. If the contract says the layer applies to the post-surplus retained amount, calculate $Z$ first and use $Z$ as the layer’s subject amount.
Forgetting the attachment or the limit. The layer does not pay the first $A$ of its subject amount, and it cannot pay more than $M$.
Subtracting recoveries twice. The excess-of-loss recovery reduces $Z$. Do not subtract the surplus recovery from $Z$ again; it has already been deducted in forming $Z$.

Key takeaway: Follow the treaty sequence and identify the amount each contract covers. For surplus share followed by excess-of-loss on the post-surplus amount, calculate $c$, then $cL$, then $Z=L-cL$, and finally the layer recovery from $Z$. Reconcile the recoveries and final retention to the gross loss.

Check Your Understanding

For each problem, assume the stated policy loss is allocated proportionally to the sum insured and that the excess-of-loss layer applies to the post-surplus retained loss. Calculate in the order specified.

A $200,000 policy has a $40,000 retention and a three-line surplus treaty. A proportional loss is $50,000. The post-surplus layer attaches at $15,000 and has a $10,000 limit. Find both recoveries and the final retention. (Answer: surplus recovery $30,000; excess-of-loss recovery $5,000; final retention $15,000.)
A $150,000 policy has a $30,000 retention and a four-line surplus treaty. A proportional loss is $25,000. The post-surplus layer attaches at $8,000 and has a $12,000 limit. Find both recoveries and the final retention. (Answer: surplus recovery $20,000; excess-of-loss recovery $0; final retention $5,000.)
A $400,000 policy has an $80,000 retention and a two-line surplus treaty. A proportional loss is $150,000. The post-surplus layer attaches at $50,000 and has a $25,000 limit. Find both recoveries and the final retention. (Answer: surplus recovery $60,000; excess-of-loss recovery $25,000; final retention $65,000.)
A $250,000 policy has a $50,000 retention and a three-line surplus treaty. A proportional loss is $100,000. The post-surplus layer attaches at $30,000 and has a $20,000 limit. Find both recoveries and the final retention. (Answer: surplus recovery $60,000; excess-of-loss recovery $10,000; final retention $30,000.)
A $180,000 policy has a $60,000 retention and a two-line surplus treaty. A proportional loss is $90,000. The post-surplus layer attaches at $20,000 and has a $5,000 limit. Find both recoveries and the final retention. (Answer: surplus recovery $60,000; excess-of-loss recovery $5,000; final retention $25,000.)

What You'll Learn