Loss Adjustment Expenses Under Reinsurance

Why Expense Treatment Matters

A reinsurance calculation can change materially depending on whether loss adjustment expenses count as part of the loss, sit outside the layer, or are not covered. The same indemnity amount and expense amount can therefore produce different recoveries under different treaty wording. The key is to identify the contract’s expense treatment before applying the attachment and limit.

This tutorial focuses on excess-of-loss calculations for a single covered occurrence. The examples use deterministic dollar amounts: no interest, mortality, or probability distribution assumptions apply. Any expense reimbursement method must be stated explicitly because treaty wording can differ; the phrase “expenses in addition” does not, by itself, specify how expenses are allocated.

Definition: Let $L$ be the covered indemnity loss for an occurrence and $E$ the eligible loss adjustment expense associated with it. Loss adjustment expense (LAE) is an expense of investigating, adjusting, or settling a claim. A treaty may include eligible LAE in the subject loss, reimburse some LAE in addition to the indemnity recovery, or exclude it. The contract wording determines which expenses are eligible and how they are treated.

In the examples, $A$ denotes the layer’s attachment point and $M$ its limit. The layer pays above $A$, up to $M$. For any amount $C$ defined by the treaty as subject to the layer, the recovery is:

R(C)=\min\bigl(\max(C-A,0),M\bigr).

This expression is only a calculation rule after $C$ has been identified. It does not decide whether $C$ includes expenses.

Three Expense Treatments

Treaty treatment	Amount used to calculate layer recovery	Effect on expenses
Expenses excluded	$C=L$	LAE does not generate recovery under this layer.
Expenses included within the layer	$C=L+E$	Eligible LAE can reach the attachment and erode the layer limit.
Expenses reimbursed in addition	First calculate the indemnity recovery from $L$.	If wording provides for proportional reimbursement, the expense share is paid in addition to the indemnity limit.

The third row needs particular care. One possible contract mechanism is that eligible expenses follow the reinsurer’s share of the indemnity loss, without eroding the layer limit. Under that explicitly assumed method, if the indemnity recovery is $R_L$ and $L>0$, expense reimbursement is $R_L E/L$. Other wording may set a different allocation, cap, or expense share. Never infer a reimbursement formula from a label alone.

Worked Calculations

Worked Example: Expenses Included Within the Layer

A hypothetical property treaty covers one occurrence with indemnity loss $L=\$690{,}000$ and eligible adjustment expense $E=\$150{,}000$. The layer attaches at $A=\$600{,}000$ and has limit $M=\$200{,}000$. The wording states that eligible expenses are included in the subject loss and are within the limit. Calculate the recovery and identify how much of the layer limit is used.

Method and assumptions. This is a deterministic single-occurrence calculation. The treaty treats indemnity and eligible expense together as the amount subject to the layer. No interest, mortality, or probability distribution assumptions apply.

First combine the indemnity and expense:

C=L+E=\$690{,}000+\$150{,}000=\$840{,}000.

The amount above the attachment is:

C-A=\$840{,}000-\$600{,}000=\$240{,}000.

The recovery cannot exceed the $\$200{,}000$ limit:

R(C)=\min(\max(\$240{,}000,0),\$200{,}000)=\$200{,}000.

The full limit is used. The amount above the layer’s maximum covered amount is $\$240{,}000-\$200{,}000=\$40{,}000$. As a check, if the expense were excluded, the recovery would instead be $\min(\$690{,}000-\$600{,}000,\$200{,}000)=\$90{,}000$. Thus including the expense increases recovery by $\$110{,}000$, but the limit prevents the recovery from increasing by the full $\$150{,}000$.

Common mistake avoided: Adding the full expense to a recovery already calculated on indemnity would give $\$90{,}000+\$150{,}000=\$240{,}000$. That exceeds the stated limit and treats expenses as paid in addition to the layer, contrary to the wording.

Worked Example: Expenses Reimbursed in Addition to the Indemnity Limit

A hypothetical liability treaty has indemnity loss $L=\$900{,}000$, eligible adjustment expense $E=\$90{,}000$, attachment $A=\$500{,}000$, and indemnity limit $M=\$250{,}000$. The wording says the layer recovery is based on indemnity loss and that eligible expenses are reimbursed in addition to the limit, in proportion to the reinsurer’s share of indemnity loss. Find the indemnity recovery, expense reimbursement, and total payment.

Method and assumptions. First apply the layer to $L$ alone. Then apply the stated proportional expense rule. The expense payment does not erode the indemnity limit. This is a deterministic calculation with no interest, mortality, or probability distribution assumptions.

The indemnity amount above the attachment is $\$900{,}000-\$500{,}000=\$400{,}000$. Applying the limit gives:

R_L=\min(\$400{,}000,\$250{,}000)=\$250{,}000.

The reinsurer’s share of indemnity loss is:

\frac{R_L}{L}=\frac{\$250{,}000}{\$900{,}000}=\frac{5}{18}.

Apply that share to eligible expense:

R_E=\frac{5}{18}\times \$90{,}000=\$25{,}000.

The total payment is $\$250{,}000+\$25{,}000=\$275{,}000$. The indemnity recovery uses the full $\$250{,}000$ limit; the $\$25{,}000$ expense reimbursement is additional under the stated terms. As a check, the expense reimbursement is less than the total expense, and its fraction of the expense equals the reinsurer’s $\frac{5}{18}$ share of indemnity loss.

Common mistake avoided: Adding $E$ to $L$ before applying the layer would calculate the wrong basis. The wording makes expenses additional and proportional, not part of the amount used to reach attachment.

Worked Example: Expenses Excluded From the Layer

A hypothetical commercial treaty covers indemnity loss $L=\$360{,}000$ for an occurrence, with associated adjustment expense $E=\$80{,}000$. Its layer attaches at $A=\$300{,}000$, has limit $M=\$200{,}000$, and explicitly excludes adjustment expenses. Calculate the recovery and the expense amount not recovered from this layer.

Method and assumptions. Only indemnity is subject to the layer. The example assumes no other contract reimburses the expense; it calculates only this layer’s payment. No interest, mortality, or probability distribution assumptions apply.

The subject amount is $C=L=\$360{,}000$, not $L+E$. The amount above attachment is $\$360{,}000-\$300{,}000=\$60{,}000$, which is below the $\$200{,}000$ limit. Therefore:

R(C)=\min(\$60{,}000,\$200{,}000)=\$60{,}000.

The layer pays $\$0$ toward the $\$80{,}000$ expense, so the full $\$80{,}000$ is unrecovered from this layer. As a check, the recovery does not exceed either the amount above attachment or the layer limit.

Common mistake avoided: Treating the expense as included would use $\$360{,}000+\$80{,}000=\$440{,}000$ and produce a $\$140{,}000$ recovery. That contradicts the explicit exclusion.

Why the Layer Formula Works

A layer pays nothing when the subject amount does not exceed its attachment. Once the subject amount exceeds $A$, the portion above attachment is $C-A$. Since a recovery cannot be negative, the amount before applying the limit is $\max(C-A,0)$. The layer cannot pay more than $M$, so the final recovery is the lesser of that excess and the limit:

R(C)=\min\bigl(\max(C-A,0),M\bigr).

This derivation also shows why the definition of $C$ is decisive. If wording makes expenses part of subject loss, they can help move $C$ above $A$. If expenses are excluded or reimbursed separately, they do not change the indemnity amount tested against the attachment.

Reading Wording and Avoiding Traps

Before calculating, translate the wording into a rule for the subject amount and, if applicable, a separate expense payment. A useful sequence is:

Identify eligible expense.
Determine which expense categories the contract recognizes. Do not assume every claim-handling cost is covered.

Classify the treatment.
Decide whether expense is included within the layer, excluded, or paid separately under a stated allocation method.

Apply attachment and limit to the correct amount.
Use $L+E$ only when the wording includes eligible expense in subject loss. Otherwise, calculate the indemnity layer recovery from $L$.

Calculate any separate expense payment.
Use the contract’s specified share, cap, or other rule. Add it to indemnity recovery only when the wording provides for an additional payment.

The most common exam trap is to recognize that expenses are covered but overlook whether they are within or in addition to the limit. Included expenses can consume limit that might otherwise pay indemnity. Expenses paid in addition may increase the reinsurer’s total payment beyond the indemnity limit. Excluded expenses do neither. These distinctions make treaty wording, not a generic assumption about LAE, the controlling input.

For a treaty calculation aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus, show the subject amount, attachment calculation, limit application, and any separate expense calculation. That makes the assumed treatment visible and makes it easier to check that no expense has been counted twice.

Check Your Understanding

For each problem, use only the expense treatment stated. Each scenario is hypothetical and deterministic; no interest, mortality, or probability distribution assumptions apply.

A treaty excludes LAE. For an occurrence, $L=\$280{,}000$, $E=\$45{,}000$, $A=\$250{,}000$, and $M=\$100{,}000$. Find the recovery from the layer. (Answer: $\$30{,}000$)
A layer includes eligible expenses within its limit. Given $L=\$410{,}000$, $E=\$70{,}000$, $A=\$450{,}000$, and $M=\$150{,}000$, calculate recovery and unused limit. (Answer: recovery $\$30{,}000$; unused limit $\$120{,}000$)
A treaty includes expenses within the layer. For $L=\$760{,}000$, $E=\$140{,}000$, $A=\$700{,}000$, and $M=\$160{,}000$, determine the subject amount, recovery, and amount by which the subject amount above attachment exceeds the limit. (Answer: subject amount $\$900{,}000$; recovery $\$160{,}000$; excess above the limit $\$40{,}000$)
A treaty pays indemnity recovery from $L$, then reimburses expense in the same proportion as the indemnity recovery divided by $L$. Let $L=\$1{,}200{,}000$, $E=\$150{,}000$, $A=\$800{,}000$, and $M=\$300{,}000$. Find the indemnity recovery, expense reimbursement, and total payment. (Answer: $\$300{,}000$; $\$37{,}500$; $\$337{,}500$)
For $L=\$540{,}000$, $E=\$100{,}000$, $A=\$500{,}000$, and $M=\$120{,}000$, compare (a) expenses included within the layer, (b) expenses excluded, and (c) expenses reimbursed in addition in proportion to indemnity recovery divided by $L$. (Answer: (a) $\$120{,}000$; (b) $\$40{,}000$; (c) indemnity $\$40{,}000$, expense $\$7{,}407.41$, total $\$47{,}407.41$)

Treaty treatment	Amount used to calculate layer recovery	Effect on expenses
Expenses excluded	\(C=L\)	LAE does not generate recovery under this layer.
Expenses included within the layer	\(C=L+E\)	Eligible LAE can reach the attachment and erode the layer limit.
Expenses reimbursed in addition	First calculate the indemnity recovery from \(L\).	If wording provides for proportional reimbursement, the expense share is paid in addition to the indemnity limit.

What You'll Learn