Per-Risk Excess-of-Loss Versus Stop-Loss Protection

Why the Level of Protection Matters

Two reinsurance covers can respond to the same portfolio in very different ways because they assess loss at different levels. Per-risk excess-of-loss applies its attachment and limit separately to each insured risk. Stop-loss applies its attachment and limit to the portfolio's aggregate loss over a stated period, commonly a year. As a result, the number and size of individual claims matter directly to per-risk recovery, while stop-loss recovery depends on the aggregate total.

This distinction is important when comparing protection or calculating retained losses. A portfolio with a few large claims may produce substantial per-risk recoveries, while a portfolio with many smaller claims can have the same aggregate loss but a different per-risk recovery. Conversely, a portfolio may cross a stop-loss attachment even though no individual loss crosses the per-risk attachment.

Definition: Let $x_j$ be the loss on risk $j$, for $j=1,\ldots,n$, and let $S=\sum_{j=1}^n x_j$ be the aggregate loss. A per-risk excess-of-loss cover with attachment $d$ and limit $u$ pays, on risk $j$, $$ R_j=\min\bigl((x_j-d)_+,u\bigr), \qquad (z)_+=\max(z,0). $$ Its total recovery is $\sum_{j=1}^n R_j$. A stop-loss cover with aggregate attachment $A$ and limit $L$ pays $$ R_{\mathrm{SL}}=\min\bigl((S-A)_+,L\bigr). $$

The notation “$d$ xs $u$” means an attachment of $d$ and a limit of $u$ on the layer above it. For example, “$10$ xs $20$” pays at most $20$ on a loss, and only the part above $10$ is eligible for payment. The same calculation applies to a stop-loss cover, but there the loss being tested is the aggregate $S$, not an individual claim.

These formulas describe the covered loss payments, assuming the stated losses are eligible under the contract and that no other terms alter the calculation. The limit is the maximum payment for each covered unit: for per-risk cover it applies to each risk, while for stop-loss it applies to the aggregate period. Do not multiply a stop-loss limit by the number of risks.

Same Aggregate Loss, Different Claim Patterns

The comparison is clearest when two portfolios have the same total loss but distribute it differently. The stop-loss payment must be the same for both if they have the same aggregate loss and the same stop-loss terms. Their per-risk recoveries need not be the same.

Worked Example: Equal Annual Totals, Unequal Per-Risk Recoveries

A hypothetical insurer has two possible annual claim patterns. Portfolio A has losses of $40$ and $30$. Portfolio B has losses of $20,20,15,$ and $15$. A per-risk cover is $10$ xs $20$. A stop-loss cover is $50$ xs $20$. Calculate each cover's recovery under each portfolio and the insurer's retained aggregate loss.

Method. Apply the per-risk formula to each listed loss separately. For stop-loss, first total the losses and then apply its aggregate attachment and limit.

Portfolio A, per-risk recovery. The loss of $40$ produces $\min(40-10,20)=20$. The loss of $30$ produces $\min(30-10,20)=20$. Total per-risk recovery is $20+20=40$.

Portfolio B, per-risk recovery. Each loss of $20$ produces $\min(20-10,20)=10$, giving $10+10=20$. Each loss of $15$ produces $\min(15-10,20)=5$, giving $5+5=10$. Total per-risk recovery is $20+10=30$.

Stop-loss recovery for either portfolio. Portfolio A totals $40+30=70$. Portfolio B totals $20+20+15+15=70$. Thus, in either case, the stop-loss pays $\min(70-50,20)=20$.

The original aggregate loss is $70$ in both portfolios. After per-risk reinsurance, retained loss is $70-40=30$ for A and $70-30=40$ for B. After stop-loss reinsurance, retained loss is $70-20=50$ for either portfolio. These results check: each retained amount equals the original total less the applicable total recovery. The common mistake is to infer that equal aggregate losses must produce equal per-risk recoveries; the claim-by-claim calculations show why that is false.

Why the Two Calculations Differ

For per-risk cover, the operation $(x_j-d)_+$ is applied before the individual recoveries are summed. For stop-loss, losses are summed first, and the attachment is applied once to that sum. In general, these operations do not commute: applying an attachment to each part and then adding is not the same as applying an attachment once to the total.

\sum_{j=1}^n \min\bigl((x_j-d)_+,u\bigr) \quad\text{is generally not equal to}\quad \min\bigl((\sum_{j=1}^n x_j-A)_+,L\bigr).

The formulas themselves justify the distinction. In the per-risk expression, each $x_j$ is tested against $d$; a loss at or below $d$ contributes zero, and a loss above $d$ contributes no more than $u$. In the stop-loss expression, the only loss tested against $A$ is $S=\sum_jx_j$, and the total payment cannot exceed $L$. Therefore, rearranging losses while keeping their sum fixed leaves stop-loss recovery unchanged, but can change the sum of per-risk recoveries.

Identify the unit covered.
For per-risk excess-of-loss, the unit is each individual risk. For stop-loss, it is the aggregate loss over the contract period.

Apply the attachment at that level.
Subtract the per-risk attachment from each risk loss, or subtract the aggregate attachment once from the annual total. A negative excess is zero.

Apply the corresponding limit.
Cap each individual per-risk recovery separately, or cap the single stop-loss recovery at its aggregate limit.

Calculate retained loss.
Subtract the total recovery from the original aggregate loss, making sure that recoveries from different covers are not mixed.

How Protection Responds to Claim Patterns

Per-risk cover is designed to respond to losses that are unusually large relative to the individual risk attachment. If losses are spread across many risks and each remains below the attachment, the cover can pay nothing even when the portfolio total is substantial. Stop-loss responds instead to an adverse total for the covered period, so the count and distribution of individual claims do not affect its payment once the aggregate is known.

The following example shows that a stop-loss recovery can be positive even when the per-risk cover pays nothing. It also illustrates why comparing contracts requires keeping their attachment points and limits separate.

Worked Example: Aggregate Protection Without a Per-Risk Recovery

A hypothetical insurer has five claims of $14$ each. Its per-risk cover is $15$ xs $20$, and its stop-loss cover is $60$ xs $10$. Find the aggregate loss, each type of recovery, and the retained aggregate loss under each cover considered separately.

Aggregate loss. There are five claims of $14$, so $S=5(14)=70$.

Per-risk recovery. Every claim is less than the per-risk attachment of $15$. Each recovery is $\min((14-15)_+,20)=0$. Summing five zero recoveries gives a total per-risk recovery of $0$.

Stop-loss recovery. Apply the aggregate attachment to $70$: $\min(70-60,10)=10$. This equals the stop-loss limit, so the insurer retains $70-10=60$ after this cover.

If only the per-risk cover is considered, retained aggregate loss is $70-0=70$. If only the stop-loss cover is considered, retained aggregate loss is $60$. The result is consistent with the respective attachments: no individual claim exceeds $15$, but the aggregate exceeds $60$ by $10$. The common mistake is to decide that per-risk cover must respond because $70$ exceeds $15$; the per-risk attachment is tested against each $14$ claim, not against their sum.

The reverse pattern is also possible: an individual claim can exceed a per-risk attachment even when the aggregate remains below a stop-loss attachment. Thus, neither cover's attachment can be substituted for the other's. A useful comparison asks two separate questions: Are individual losses large enough to cross the per-risk attachment? and Is the aggregate large enough to cross the stop-loss attachment?

Worked Example: One Large Loss Below the Stop-Loss Attachment

A hypothetical insurer reports three losses of $25,8,$ and $2$. A per-risk cover is $10$ xs $15$; a stop-loss cover is $40$ xs $10$. Calculate the recovery and retained aggregate loss under each cover separately.

Aggregate loss. The total is $25+8+2=35$.

Per-risk recovery. The loss of $25$ pays $\min(25-10,15)=15$, which exhausts the per-risk limit. The loss of $8$ is below the attachment and pays $0$. The loss of $2$ also pays $0$. Total per-risk recovery is $15+0+0=15$, leaving $35-15=20$ retained.

Stop-loss recovery. The aggregate $35$ is below the attachment $40$, so the recovery is $\min((35-40)_+,10)=0$. Retained aggregate loss is $35-0=35$.

The arithmetic checks against both contract structures: the per-risk payment cannot exceed its $15$ limit on the one claim, and the stop-loss payment is zero because the aggregate does not reach $40$. The common mistake is to treat the claim of $25$ as if it alone determined the stop-loss recovery. Stop-loss tests the total $35$, not the largest individual loss.

Common Comparison Traps

Aggregating too early. Do not add all claims before calculating per-risk recovery. Calculate each covered risk's payment, then sum those payments.
Applying the stop-loss attachment repeatedly. A stop-loss attachment applies once to the aggregate for the stated period, not once to every claim.
Confusing a limit with a maximum loss. In a layer $d$ xs $u$, $u$ is the maximum recovery, not the maximum amount of loss. A covered loss can be greater than $d+u$, while the payment remains capped at $u$.
Comparing only the attachment numbers. Per-risk and aggregate attachments apply to different quantities, so their numerical sizes alone do not identify which cover is more protective.
Subtracting recovery from the wrong amount. Retained aggregate loss equals aggregate loss less total recovery. For per-risk cover, first sum the recoveries across risks; for stop-loss, use its one aggregate recovery.

Key takeaway: Per-risk excess-of-loss responds to the size of each individual loss, whereas stop-loss responds to the portfolio's aggregate loss. When claim patterns differ, calculate each contract at its own level before comparing total recoveries and retained loss.

Check Your Understanding

For each problem, calculate at the level specified by the contract. Treat covers separately unless the question explicitly asks for a comparison.

A portfolio has claims of $9,20,35,$ and $50$. Per-risk cover is $12$ xs $18$, and stop-loss cover is $80$ xs $25$. Find total recovery under each cover. (Answer: per-risk recovery $44$; stop-loss recovery $25$.)
Two hypothetical portfolios each have aggregate loss $60$: Portfolio A has five claims of $12$, and Portfolio B has one claim of $60$. Per-risk cover is $10$ xs $25$; stop-loss cover is $40$ xs $15$. Compare the recoveries under each cover. (Answer: per-risk recoveries are $10$ for A and $25$ for B; stop-loss recovery is $15$ for each.)
A portfolio has claims of $18,26,$ and $55$. Per-risk cover is $20$ xs $30$; stop-loss cover is $70$ xs $20$. Find the retained aggregate loss under each cover separately. (Answer: per-risk recovery $36$ and retained loss $63$; stop-loss recovery $20$ and retained loss $79$.)
A portfolio consists of seven claims of $9$ each. Per-risk cover is $10$ xs $15$, and stop-loss cover is $50$ xs $8$. Calculate the total loss and each recovery. (Answer: total loss $63$; per-risk recovery $0$; stop-loss recovery $8$.)
A portfolio has claims of $20,20,$ and $5$. Per-risk cover is $8$ xs $12$, and stop-loss cover is $35$ xs $15$. Calculate each recovery and the retained aggregate loss under each cover separately. (Answer: total loss $45$; per-risk recovery $24$ and retained loss $21$; stop-loss recovery $10$ and retained loss $35$.)

What You'll Learn