Deriving the Excess-of-Loss Layer Payment Formula

Why a Layer Payment Needs Both an Attachment and a Cap

An excess-of-loss layer protects against the part of an eligible loss that reaches a specified level. The attachment determines where the reinsurer’s payment begins; the layer width determines how much the reinsurer can pay before the layer is exhausted. A correct recovery calculation must account for both boundaries.

For this tutorial, let $X$ be a nonnegative loss amount on the basis specified by a hypothetical contract, such as one risk or one occurrence. Assume the reinsurer covers 100% of the layer, with no deductible within the layer, coinsurance, exclusions, or separate aggregate cap. The formula below calculates the layer payment from $X$; contract terms that change the covered loss or the reinsurer’s share must be applied separately.

Definition: An excess-of-loss layer with attachment $A$ and width $L$ is often described as “$L$ excess of $A$.” The layer begins at $A$, ends at $A+L$, and has maximum payment $L$. For eligible loss $X$, its recovery is the amount of loss in the interval from $A$ to $A+L$.

The attachment is not itself the layer width. For example, a hypothetical layer of $80,000 excess of $120,000 begins paying above $120,000 and is exhausted at $200,000. Its maximum recovery is $80,000, not $200,000.

Deriving the Layer Payment

The amount of loss above attachment $A$, before applying the layer cap, is $(X-A)^+$, where $z^+=\max(z,0)$. This amount is zero if the loss does not reach the attachment. If it does reach the attachment, the amount above $A$ grows with the loss. However, the reinsurer cannot pay more than the width $L$, so the recovery is that positive part capped at $L$.

R(X)=\min\big((X-A)^+,L\big)

The same recovery can be written as the difference between the amount above the attachment and the amount above the top of the layer:

$$ R(X)=(X-A)^+-(X-A-L)^+ $$

To see why the difference formula works, consider the three possible positions of $X$. If $X\leq A$, neither positive part is greater than zero. If $A<X<A+L$, the loss is above the attachment but not above the top of the layer, so only the first positive part is nonzero. If $X\geq A+L$, both positive parts are nonzero and their difference is exactly $L$. Thus the expression gives zero below the layer, the loss within the layer, and the full layer width above the layer.

Theorem: For a 100% participating excess-of-loss layer with attachment $A$ and width $L$, the recovery on eligible loss $X\geq 0$ is $$ R(X)=\min\big((X-A)^+,L\big)=(X-A)^+-(X-A-L)^+. $$ Equivalently, $$ R(X)= \begin{cases} 0, & X\leq A,\\ X-A, & A<X<A+L,\\ L, & X\geq A+L. \end{cases} $$

At the exact boundaries, the formulas agree: $R(A)=0$, and $R(A+L)=L$. Whether a loss exactly equals the attachment or the top of the layer therefore causes no ambiguity in the payment.

Worked Applications

Worked Example: Checking Losses Below, Within, and Above a Layer

A hypothetical reinsurer covers a layer of $40,000 excess of $100,000 on each eligible claim. Calculate the recovery for losses of $75,000, $126,000, and $155,000. Verify each result using the layer boundaries.

Method. Here $A=\$100{,}000$ and $L=\$40{,}000$, so the top of the layer is $A+L=\$140{,}000$. Use $R(X)=\min((X-A)^+,L)$. The common mistake to avoid is treating $140,000, the exhaustion point, as the layer width.

For $X=\$75{,}000$, the loss is below the attachment: \[ (X-A)^+=(\$75{,}000-\$100{,}000)^+=0. \] Therefore $R(X)=\min(0,\$40{,}000)=\$0$. This is consistent with the piecewise rule because $\$75{,}000\leq A$.

For $X=\$126{,}000$, the amount above attachment is \[ X-A=\$126{,}000-\$100{,}000=\$26{,}000. \] Since $26,000 is less than the $40,000 width, the recovery is \[ R(X)=\min(\$26{,}000,\$40{,}000)=\$26{,}000. \] The loss falls within the layer because $\$100{,}000<\$126{,}000<\$140{,}000$.

For $X=\$155{,}000$, the amount above attachment is \[ X-A=\$155{,}000-\$100{,}000=\$55{,}000. \] The layer cannot pay more than its $40,000 width, so \[ R(X)=\min(\$55{,}000,\$40{,}000)=\$40{,}000. \] The loss exceeds the $140,000 exhaustion point, so the layer is fully paid.

Check. The recoveries are $0, $26,000, and $40,000. They are, respectively, no greater than zero below attachment, equal to the loss portion inside the layer, and capped at the layer width above it.

Worked Example: Separating the Recovery From the Loss Retained Outside the Layer

A hypothetical occurrence loss of $468,000 is subject to a 100% layer of $300,000 excess of $125,000. Calculate the recovery and the portion of the loss that remains outside this layer. Assume there are no other recoveries or contractual adjustments.

Method. The attachment is $A=\$125{,}000$, the width is $L=\$300{,}000$, and the top of the layer is \[ A+L=\$125{,}000+\$300{,}000=\$425{,}000. \] The common mistake to avoid is subtracting the attachment from the loss and reporting the full excess, without applying the layer cap.

The amount above attachment is \[ X-A=\$468{,}000-\$125{,}000=\$343{,}000. \] Apply the cap: \[ R(X)=\min(\$343{,}000,\$300{,}000)=\$300{,}000. \] The loss exceeds the top of the layer, so this is a full-layer recovery.

The part below the layer is $125,000. The part above the layer is \[ X-(A+L)=\$468{,}000-\$425{,}000=\$43{,}000. \] Together, these two amounts are the loss outside the layer: \[ \$125{,}000+\$43{,}000=\$168{,}000. \]

Check. The recovery plus the loss outside the layer reconciles to the original loss: \[ \$300{,}000+\$168{,}000=\$468{,}000. \] The recovery does not exceed the $300,000 width, and the loss above the top is not included in this layer’s payment.

Worked Example: Inferring a Loss From a Partial Layer Recovery

A hypothetical 100% excess-of-loss layer has a width of $90,000 and attaches at $210,000. A claim produces a layer recovery of $62,000. Assuming the recovery is calculated directly from one eligible loss and there are no other contract terms affecting payment, determine the loss amount. Then calculate the recovery on a second eligible loss of $355,000.

Method. First find the layer’s top: \[ A+L=\$210{,}000+\$90{,}000=\$300{,}000. \] A $62,000 recovery is positive and below the full width of $90,000. Therefore it is a partial recovery, and the loss lies strictly within the layer. The common mistake to avoid is treating a recovery equal to the full width and a recovery below the width as if both identify a unique loss in the same way.

For a partial recovery, the middle case of the piecewise formula gives $R(X)=X-A$. Thus \[ \$62{,}000=X-\$210{,}000, \] so \[ X=\$210{,}000+\$62{,}000=\$272{,}000. \] This is consistent with the partial-layer condition: \[ \$210{,}000<\$272{,}000<\$300{,}000. \]

For the second loss, $X=\$355{,}000$, which exceeds the $300,000 top. Its amount above attachment is \[ \$355{,}000-\$210{,}000=\$145{,}000. \] Capping at the $90,000 width gives \[ R(X)=\min(\$145{,}000,\$90{,}000)=\$90{,}000. \]

Check. The first inferred loss is within the layer and produces the stated $62,000 recovery. The second loss exceeds the layer, so its recovery is exactly the $90,000 maximum. A full-width recovery alone would establish only that $X\geq \$300{,}000$, not one unique loss amount.

Formula Checks and Common Traps

The positive-part difference is useful because it makes the two boundaries visible in a single expression. The first term, $(X-A)^+$, counts the amount above the attachment. The second, $(X-A-L)^+$, removes any amount above the top of the layer. What remains is only the loss allocated to the layer.

Identify the attachment and width.
Write down $A$ and $L$ separately. The top of the layer is $A+L$, not $L$.

Place the loss relative to both boundaries.
Compare $X$ with $A$ and $A+L$. This identifies whether the recovery is zero, partial, or full.

Calculate and cap the excess.
Find $(X-A)^+$, then take the smaller of that amount and $L$.

Check the result.
The recovery must satisfy $0\leq R(X)\leq L$. If calculating the loss outside the layer as well, verify that it plus the recovery equals $X$.

A frequent incorrect calculation is to pay $\min(X,L)$. For a $40,000 excess of $100,000 layer and a $75,000 loss, that calculation gives $\min(\$75{,}000,\$40{,}000)=\$40{,}000$, even though the loss never reaches the $100,000 attachment. The correct calculation is \[ \min\big((\$75{,}000-\$100{,}000)^+,\$40{,}000\big)=\$0. \] Another error is to subtract the attachment but forget the cap. For a $155,000 loss under that same layer, $X-A=\$55{,}000$, but the payment is $40,000, not $55,000. Both the zero-payment boundary and the maximum-payment boundary must be respected.

Key takeaway: An excess-of-loss layer pays the loss above its attachment, but never more than its width: $$ R(X)=\min\big((X-A)^+,L\big). $$ Equivalently, subtract the excess above the layer’s top: $R(X)=(X-A)^+-(X-A-L)^+$. Always distinguish the attachment $A$, the width $L$, and the exhaustion point $A+L$.

Check Your Understanding

For each hypothetical 100% layer, assume the stated loss is the eligible loss and that no other contract terms alter the recovery.

A layer of $30,000 excess of $70,000 covers a loss of $94,000. What is the recovery? (Answer: $24,000)
A layer of $50,000 excess of $140,000 covers a loss of $250,000. What is the recovery, and is the layer exhausted? (Answer: $50,000; yes, the layer is exhausted)
A layer of $80,000 excess of $200,000 covers a loss of $263,000. Calculate the recovery and check that the loss is within the layer. (Answer: $63,000; the loss is within the layer)
A layer of $125,000 excess of $500,000 covers a loss of $590,000. What is the recovery, and how much of the loss is outside the layer? (Answer: $90,000 recovery; $500,000 outside the layer)
A layer of $45,000 excess of $155,000 pays $31,000 on one eligible loss. What is the loss amount? (Answer: $186,000)
A layer of $45,000 excess of $155,000 pays its full width on an eligible loss. What can be concluded about the loss amount, and can the exact loss be determined? (Answer: $X\geq \$200{,}000$; the exact loss cannot be determined from the recovery alone)

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