From Stop-Loss Recovery to Annual Result
A stop-loss recovery changes the insurer’s claims cost, while the reinsurance premium is a separate cost of purchasing protection. To calculate the insurer’s annual result, both effects must be included: subtract the reinsurance premium from the insurer’s premium income, and subtract the recovery from aggregate losses before charging the retained losses against that income.
The calculation here concerns a realized annual result for a hypothetical insurer. It is not an expected profit calculation: no probability distribution, mortality assumption, or interest rate is needed. Unless stated otherwise, ignore expenses, investment income, taxes, and other cash flows. The result is the amount remaining from premium income after the reinsurance premium and retained annual losses.
Here \(A\) is the annual aggregate attachment, measured in the same currency units as losses, and \(u\) is the maximum amount recoverable under the treaty. In this tutorial, “limit \(u\)” means the maximum recovery, not the maximum total annual loss. Thus, if the limit binds, the insurer retains losses above \(A+u\) as well as losses below the attachment.
The Accounting Steps
The calculation can be organized into four steps. First determine the year’s aggregate gross losses. Next apply the attachment and limit to find the recovery. Then subtract the recovery from gross losses to obtain retained losses. Finally calculate net premium income after reinsurance and subtract retained losses.
Add the covered losses for the year, or use the stated annual aggregate \(S\).
Apply the attachment first, then cap the amount payable at the treaty limit \(u\).
Calculate \(S-R\) and \(P-C\) separately.
Subtract retained losses from net premium income, and check the calculation using gross losses and the recovery.
The last expression is algebraically equivalent to \(P-C-(S-R)\). It is useful as an accounting check: start with gross premium income, subtract the reinsurance premium and gross losses, then add the recovery. Both methods must give the same result.
Why the Recovery Formula Works
A stop-loss treaty pays nothing when annual losses do not exceed the attachment. Once losses exceed \(A\), the amount above \(A\) is potentially recoverable. The insurer cannot recover more than the limit \(u\). These three conditions produce the piecewise recovery below.
If \(S\leq A\), then \(S-A\leq0\), so the treaty pays zero. If \(A<S<A+u\), the amount above the attachment is positive but less than \(u\), so the recovery is \(S-A\). If \(S\geq A+u\), then \(S-A\geq u\), so the limit caps the payment at \(u\). This is exactly \(\min(\max(S-A,0),u)\).
The retained-loss expression also has a useful interpretation. Before the attachment, the insurer retains all losses. In the layer between \(A\) and \(A+u\), the recovery offsets losses above \(A\), leaving retained losses at \(A\). Beyond \(A+u\), the treaty is exhausted and additional losses are again retained.
Worked Calculations
Worked Example: Recovery Within the Stop-Loss Layer
A hypothetical insurer receives gross premium income of $1,250,000. Its annual aggregate gross losses are $940,000. A stop-loss treaty attaches at $800,000, has a recovery limit of $300,000, and costs the insurer a reinsurance premium of $90,000. Calculate the recovery, retained losses, and annual result. Assume no expenses, investment income, taxes, or other cash flows. This is a realized annual scenario; no interest rate, mortality assumption, or loss distribution is assumed.
Method. Apply the attachment and limit to the annual aggregate, then calculate the result using net premium income and retained losses.
Recovery. Losses exceed the attachment by \(940{,}000-800{,}000=140{,}000\). The recovery before applying the limit is $140,000, which is below the $300,000 limit. Therefore, \(R=\min(140{,}000,300{,}000)=\$140{,}000\).
Retained losses. Subtract the recovery from gross losses: \(940{,}000-140{,}000=\$800{,}000\).
Net premium income. Subtract the treaty premium from gross premium income: \(1{,}250{,}000-90{,}000=\$1{,}160{,}000\).
Annual result. Subtract retained losses: \(1{,}160{,}000-800{,}000=\$360{,}000\).
Check and mistake avoided. Using gross losses directly gives \(1{,}250{,}000-90{,}000-940{,}000+140{,}000=\$360{,}000\). The recovery must be credited, either by reducing losses to retained losses or by adding it in the gross-loss accounting. Omitting it would incorrectly produce $220,000.
Worked Example: The Aggregate Recovery Limit Binds
A hypothetical insurer has gross premium income of $2,400,000 and annual aggregate gross losses of $1,950,000. Its stop-loss treaty attaches at $1,400,000, limits recoveries to $350,000, and requires a reinsurance premium of $220,000. Calculate the annual result, excluding expenses, investment income, taxes, and other cash flows. No interest rate, mortality assumption, or loss distribution is assumed; the stated annual losses are realized amounts.
Method. Compare the loss excess above attachment with the recovery limit before calculating retained losses.
Losses above attachment. \(1{,}950{,}000-1{,}400{,}000=\$550{,}000\). The potential recovery is $550,000, but the treaty limit is $350,000. Therefore, \(R=\min(550{,}000,350{,}000)=\$350{,}000\).
Retained losses. \(1{,}950{,}000-350{,}000=\$1{,}600{,}000\). The limit has been exhausted, so the insurer retains losses above the treaty’s upper boundary of \(1{,}400{,}000+350{,}000=\$1{,}750{,}000\), as well as the losses below attachment.
Net premium income. \(2{,}400{,}000-220{,}000=\$2{,}180{,}000\).
Annual result. \(2{,}180{,}000-1{,}600{,}000=\$580{,}000\).
Check and mistake avoided. Gross-loss accounting gives \(2{,}400{,}000-220{,}000-1{,}950{,}000+350{,}000=\$580{,}000\). Using the full $550,000 excess as the recovery would ignore the limit and overstate the result by $200,000.
Worked Example: Aggregate Several Losses Before Applying the Treaty
A hypothetical insurer receives $1,800,000 in gross premium income. Four covered losses during the year are $120,000, $190,000, $260,000, and $210,000. The insurer purchases stop-loss protection attaching at $600,000, with a maximum recovery of $150,000, for a reinsurance premium of $75,000. Find the annual result. Ignore expenses, investment income, taxes, and all other cash flows. No interest rate, mortality assumption, or loss distribution is used.
Method. Stop-loss protection applies to the annual aggregate, so add the losses first; do not apply the annual attachment separately to each loss.
Aggregate gross losses. \(120{,}000+190{,}000+260{,}000+210{,}000=\$780{,}000\).
Recovery. The amount above attachment is \(780{,}000-600{,}000=\$180{,}000\). Applying the $150,000 limit gives \(R=\min(180{,}000,150{,}000)=\$150{,}000\).
Retained losses. \(780{,}000-150{,}000=\$630{,}000\).
Net premium income. \(1{,}800{,}000-75{,}000=\$1{,}725{,}000\).
Annual result. \(1{,}725{,}000-630{,}000=\$1{,}095{,}000\).
Check and mistake avoided. The alternative calculation is \(1{,}800{,}000-75{,}000-780{,}000+150{,}000=\$1{,}095{,}000\). Applying the $600,000 attachment to each individual loss would be incorrect: the treaty is defined on the annual aggregate, not on each loss separately.
Interpreting the Result and Avoiding Errors
A positive annual result means that, within the stated simplified accounting, net premium income exceeded retained losses. A negative result means retained losses exceeded net premium income. Neither result by itself establishes whether the treaty was worthwhile overall: the calculation describes the realized year, while a broader assessment could involve other years, expenses, and the value of reduced risk.
The reinsurance premium and recovery have different roles. The premium is paid for the coverage and reduces the insurer’s income whether or not the treaty pays a recovery in that particular year. The recovery is determined by the realized annual losses and the treaty terms. Do not subtract the treaty premium from losses, and do not treat it as part of the recovery.
| Quantity | Role in annual result | Typical error |
|---|---|---|
| Gross premium income \(P\) | Income before the reinsurance premium | Starting with net premium income and subtracting \(C\) again |
| Reinsurance premium \(C\) | Cost subtracted from \(P\) | Subtracting it from losses or treating it as a recovery |
| Recovery \(R\) | Reduces gross losses to retained losses | Omitting it, or exceeding the treaty limit |
| Retained losses \(S-R\) | Loss amount charged against net premium income | Subtracting both retained and gross losses |
A reliable check is to write both versions of the result and confirm that they agree:
This identity also diagnoses double-counting. If you use retained losses \(S-R\), do not add \(R\) again. If you use gross losses \(S\), add the recovery \(R\) separately. Use one approach consistently.
Check Your Understanding
For each hypothetical scenario, ignore expenses, investment income, taxes, and other cash flows. Treat losses as realized annual amounts; no interest rate, mortality assumption, or probability distribution is needed.
- An insurer has gross premium income of $900,000, annual gross losses of $640,000, a stop-loss attachment of $500,000, a recovery limit of $200,000, and a reinsurance premium of $55,000. Find the recovery, retained losses, and annual result. (Answer: Recovery $140,000; retained losses $500,000; annual result $345,000.)
- Gross premium income is $1,600,000, annual gross losses are $1,480,000, the attachment is $1,000,000, the recovery limit is $250,000, and the reinsurance premium is $90,000. Find the recovery and annual result. (Answer: Recovery $250,000; annual result $280,000.)
- A hypothetical portfolio has annual losses of $120,000, $190,000, $260,000, and $210,000. Gross premium income is $1,000,000; the stop-loss attachment is $600,000, the recovery limit is $150,000, and the reinsurance premium is $75,000. Find the annual aggregate, retained losses, and annual result. (Answer: Aggregate $780,000; retained losses $630,000; annual result $295,000.)
- An insurer has gross premium income of $2,050,000 and annual aggregate losses of $2,300,000. The treaty attaches at $1,700,000, has a maximum recovery of $400,000, and costs $180,000. Find the recovery and annual result. (Answer: Recovery $400,000; annual result \(-\$30,000\).)
- Gross premium income is $1,400,000, annual aggregate losses are $1,050,000, the attachment is $900,000, the recovery limit is $250,000, and the treaty premium is $110,000. Find the annual result with the treaty and compare it with the result without reinsurance, assuming the same losses and ignoring all other cash flows. (Answer: With treaty, $390,000; without reinsurance, $350,000; the realized result is $40,000 higher with the treaty.)