Why an Integrated Decision Is Needed
A catastrophe portfolio decision is rarely resolved by one model output. A proposed account may have an attractive premium but breach a geographic accumulation cap; reinsurance may reduce a stressed loss but carry a material cost; and the portfolio may satisfy an individual-event limit while failing a multi-event stress. The decision should therefore connect exposure, pricing, event scenarios, treaty terms, and available capital.
The hypothetical case below uses dollar amounts in millions. Accumulation values are insured exposure, while event losses, premiums, expenses, and reinsurance costs are also stated in millions of dollars. The case assumes the modeled event losses and expected catastrophe costs are supplied estimates; it does not treat a scenario stress as an expected annual loss.
Case Data and Decision Sequence
The insurer has three geographic zones. Its accumulation limits and current insured exposures are shown below. Three proposed accounts, Cedar, Delta, and Elm, would add exposure in Zones A, B, and C, respectively.
| Zone | Current exposure | Accumulation limit | Proposed exposure |
|---|---|---|---|
| A | $22 million | $30 million | Cedar: $10 million |
| B | $17 million | $25 million | Delta: $8 million |
| C | $15 million | $20 million | Elm: $5 million |
The proposed accounts have the following quoted premiums, modeled expected catastrophe costs, expenses, and target underwriting margins. For this case, the figures for each account are assumed to scale linearly with the accepted share of its proposed exposure. That is a simplifying assumption for the exercise; actual rating and expense functions may not be linear.
| Account | Premium | Expected catastrophe cost | Expenses | Target margin |
|---|---|---|---|---|
| Cedar | $4.00 million | $1.40 million | $0.90 million | $0.50 million |
| Delta | $2.80 million | $1.25 million | $0.85 million | $0.40 million |
| Elm | $2.50 million | $0.80 million | $0.70 million | $0.35 million |
The insurer also considers two occurrence reinsurance layers: a primary layer of $6 million excess of $4 million, and an additional layer of $2 million excess of $10 million. The primary layer pays losses in the band above $4 million through $10 million; the additional layer pays losses in the band above $10 million through $12 million. The additional layer costs $0.70 million. The capital stress limit for the sum of the two specified net event losses is $22 million.
Worked Example: Apply Accumulation Limits Before Accepting Accounts
Worked Example: Apply Accumulation Limits Before Accepting Accounts
Determine how much of each proposed account may be accepted without exceeding the zone limits. Assume partial acceptance is available and exposure is divisible. Identify the accepted share of each account.
Method. For each zone, calculate remaining capacity as the limit less current exposure. Then compare the proposed exposure with that capacity. Do not use reinsurance recovery to increase the underwriting accumulation limit: the limit applies to the stated gross insured exposure.
Zone A. Remaining capacity is \(30-22=8\) million. Cedar proposes $10 million, so the accepted amount is \(\min(10,8)=8\) million. The accepted share is \(8/10=0.80\), or 80%. Ending exposure is \(22+8=30\) million, exactly the limit.
Zone B. Remaining capacity is \(25-17=8\) million. Delta proposes $8 million, so all $8 million can be accepted. Ending exposure is \(17+8=25\) million, exactly the limit.
Zone C. Remaining capacity is \(20-15=5\) million. Elm proposes $5 million, so all $5 million can be accepted. Ending exposure is \(15+5=20\) million, exactly the limit.
Check. The resulting exposures are $30 million, $25 million, and $20 million, each equal to but not greater than its respective cap. The method avoids the common mistake of accepting Cedar in full: \(22+10=32\) million would exceed Zone A's limit by $2 million.
Scale the Underwriting Economics Consistently
When only part of a submission is accepted, the associated premium and cost estimates must be scaled using the stated assumption. Cedar is accepted at 80%; Delta and Elm are accepted at 100%. The arithmetic below separates the premium available from the expected catastrophe cost and expenses, then compares the result with the target margin.
Worked Example: Calculate the Accepted Portfolio's Underwriting Contribution
Calculate the accepted premium, expected catastrophe cost, expenses, and target margin for the three accounts. Then calculate the contribution remaining after the cost of the additional reinsurance layer. Treat the reinsurance premium as an additional cost; no expected recovery credit is provided in the case data.
Cedar at 80%. Premium is \(4.00(0.80)=3.20\) million. Expected catastrophe cost is \(1.40(0.80)=1.12\) million. Expenses are \(0.90(0.80)=0.72\) million. Target margin is \(0.50(0.80)=0.40\) million.
Delta at 100%. Premium is $2.80 million, expected catastrophe cost is $1.25 million, expenses are $0.85 million, and target margin is $0.40 million.
Elm at 100%. Premium is $2.50 million, expected catastrophe cost is $0.80 million, expenses are $0.70 million, and target margin is $0.35 million.
Portfolio totals. Accepted premium is \(3.20+2.80+2.50=8.50\) million. Expected catastrophe cost is \(1.12+1.25+0.80=3.17\) million. Expenses are \(0.72+0.85+0.70=2.27\) million. Target margin is \(0.40+0.40+0.35=1.15\) million.
Before reinsurance cost, the contribution after expected catastrophe costs and expenses is \(8.50-3.17-2.27=3.06\) million. After the $0.70 million reinsurance premium, it is \(3.06-0.70=2.36\) million. This exceeds the aggregate target margin by \(2.36-1.15=1.21\) million.
Check. The calculation counts the additional treaty premium once and does not subtract modeled recoveries from expected catastrophe costs, because no expected recovery estimate is supplied. It shows an underwriting contribution under the case assumptions; it is not a complete profit forecast.
Derive the Recovery Formula From the Layer Terms
For a gross event loss \(L\), the recovery from a layer attaching at \(a\) with limit \(b\) is the portion of the loss above \(a\), capped at \(b\). In this case, the primary layer attaches at $4 million and has a $6 million limit, so it is exhausted when the loss reaches \(4+6=10\) million.
If \(L\leq 4\), then \(\max(L-4,0)=0\), so there is no recovery. If \(4<L<10\), the amount above attachment is \(L-4\), which is less than $6 million and is paid in full. If \(L\geq10\), the amount above attachment is at least $6 million, so the recovery is capped at $6 million. These three cases establish the formula and explain why subtracting the attachment alone is not enough for losses above the layer's exhaustion point.
The additional layer attaches at $10 million and has a $2 million limit. Its recovery is therefore:
When both layers apply to the same event, total recovery is \(R_1(L)+R_2(L)\), and retained event loss is \(L-R_1(L)-R_2(L)\). These formulas assume the specified layers respond to the same covered loss, with no exclusions, reinstatement issue, or other contractual adjustment. Those terms would need to be considered if present.
Worked Example: Calculate Net Event Stresses and Capital Impact
Worked Example: Calculate Net Event Stresses and Capital Impact
The current portfolio has modeled gross losses of $12 million for a hurricane and $14 million for a wildfire. The accepted accounts add $4 million and $1 million, respectively, to the hurricane scenario, and $2 million and $1 million, respectively, to the wildfire scenario. Use the two reinsurance layers to calculate the aggregate net stress and assess the $22 million capital limit.
Hurricane gross loss. The total is \(12+4+2+1=19\) million. Primary recovery is \(\min\{\max(19-4,0),6\}=\min(15,6)=6\) million. Additional-layer recovery is \(\min\{\max(19-10,0),2\}=\min(9,2)=2\) million. Net hurricane loss is \(19-6-2=11\) million.
Wildfire gross loss. The total is \(14+2+1=17\) million. Primary recovery is \(\min\{\max(17-4,0),6\}=\min(13,6)=6\) million. Additional-layer recovery is \(\min\{\max(17-10,0),2\}=\min(7,2)=2\) million. Net wildfire loss is \(17-6-2=9\) million.
Aggregate stress. Under the stated measure, the two net event losses are summed: \(11+9=20\) million. This is \(22-20=2\) million below the capital stress limit, so the selected portfolio passes this particular stress test.
Capital impact and check. With only the primary layer, the hurricane net loss would be \(19-6=13\) million and the wildfire net loss would be \(17-6=11\) million; the aggregate would be \(13+11=24\) million. Adding the upper layer reduces the stress by \(24-20=4\) million, exactly the sum of its $2 million recovery on each event. The result is consistent with the layer limits and the stated aggregate stress definition.
Make the Integrated Management Decision
The calculation supports accepting Cedar only up to $8 million of exposure, accepting Delta and Elm as proposed, and purchasing the additional $2 million excess of $10 million layer, subject to confirming that its contract terms match the modeled recoveries. The accumulation results are at the caps, so they leave no capacity for further exposure in these zones without a separate limit decision or a reduction in existing exposure.
The selected portfolio has $2.36 million of contribution after the stated expected catastrophe costs, expenses, and reinsurance premium, compared with a $1.15 million aggregate target margin. Its two-event net stress is $20 million, below the $22 million limit. These results support the decision on the supplied assumptions, but they do not establish that every possible event combination is within appetite. Management should consider the completeness of the scenario set, uncertainty in modeled losses and exposure, and whether reinsurance terms and counterparty considerations are reflected appropriately.
The distinction between an expected loss and a stress loss matters. Expected catastrophe cost is used here in the account-level underwriting calculation. The event scenarios are used in the stated capital test. Substituting one for the other, or treating the two-event stress as an annual expected amount, would change the meaning of the calculations.
Common Exam Trap: Ignore the Layer Limit or Net Before Applying the Cap
A common error is to calculate the primary recovery as \(L-4\) for any loss above $4 million. For the hurricane loss of $19 million, that incorrect calculation gives \(19-4=15\) million of primary recovery and a supposed primary-layer retained loss of \(19-15=4\) million. It is wrong because the primary layer is limited to $6 million: its actual recovery is \(\min(15,6)=6\) million. The additional layer then pays $2 million, leaving \(19-6-2=11\) million net.
Another error is to treat reinsurance as extra accumulation capacity. Reinsurance affects covered net losses under the treaty; it does not alter the gross exposure totals used here to test the geographic limits. For example, Zone A's accepted exposure remains \(22+8=30\) million regardless of the modeled recovery.
Check Your Understanding
For each problem, apply the stated assumptions and show the intermediate calculations before checking the final answer.
- A zone has current exposure of $13 million, an accumulation limit of $18 million, and a proposed account of $7 million. What amount can be accepted, and what is the ending exposure? (Answer: Accept $5 million; ending exposure is $18 million.)
- A hypothetical account has premium $2.40 million, expected catastrophe cost $0.90 million, expenses $0.60 million, and target margin $0.30 million. What is its contribution above target before reinsurance cost? (Answer: $0.60 million.)
- A $4 million excess of $3 million occurrence layer applies separately to two event losses of $11 million and $5 million. Calculate each recovery and the sum of net event losses. (Answer: Recoveries are $4 million and $2 million; net losses are $7 million and $3 million, totaling $10 million.)
- A portfolio has two gross event losses of $15 million and $9 million. A $5 million excess of $5 million layer applies to each event. The capital stress limit is $14 million. Calculate the aggregate net stress and determine whether it passes. (Answer: Recoveries are $5 million and $4 million; net stress is $10 million + $5 million = $15 million, so it fails by $1 million.)
- For the two losses in question 4, add a $2 million excess of $10 million layer, also applying separately to each event. Recalculate the aggregate net stress against the same $14 million limit. (Answer: The upper-layer recoveries are $2 million and $0; total net stress is $8 million + $5 million = $13 million, so it passes by $1 million.)