Why Measure Net Catastrophe Protection?
A catastrophe model may estimate a portfolio’s gross loss: the covered and uncovered loss amount before reinsurance recoveries. Management also needs to know the net loss retained after the applicable treaty responds. The difference is not a single percentage reduction. It depends on event size, the contract’s attachment and limit, annual accumulation, and any exclusions or other terms.
Measuring the net distribution means applying the contract to each modeled event or annual scenario, then examining the resulting distribution. This identifies not only how much recovery is expected, but also the conditions under which a loss remains substantially with the insurer. All products and data below are hypothetical. Net loss means loss after treaty recoveries, before reinsurance premium, expenses, tax, or timing effects.
For multiple events, an occurrence treaty applies its terms to each event separately. An annual aggregate excess-of-loss treaty instead applies to the total eligible loss for the contract year. For annual gross event losses \(X_1,\ldots,X_m\), an annual aggregate layer attaching at \(A\) with limit \(L\) has recovery $$ R_{\mathrm{agg}}=\min\left(\left(\sum_{j=1}^{m}X_j-A\right)^+,L\right). $$ The treaty wording determines which losses enter the total and whether special terms, such as reinstatements, apply. The simple formula above assumes one annual aggregate layer, no reinstatements, and all listed losses are eligible.
From Gross Scenarios to Net Distributions
The calculation must be performed scenario by scenario. If each scenario \(s\) has gross loss \(x_s\) and probability \(p_s\), apply the contract to obtain \(n_s=x_s-R(x_s)\), carrying the same probability to the net outcome. The expected gross and net losses are then \(\sum_s p_sx_s\) and \(\sum_s p_sn_s\), respectively. A recovery may change the location and shape of the distribution, not merely its mean.
Worked Example: Measuring an Occurrence Layer’s Tail Protection
A hypothetical insurer models one catastrophe event with gross loss of $600, $1,800, or $4,000 thousand, with probabilities 0.60, 0.30, and 0.10, respectively. The event is eligible for a layer attaching at $800 thousand with a $2,000 thousand limit. Calculate the scenario net losses, expected losses, and 90% VaR and TVaR before and after reinsurance. Use the quantile definition \(\operatorname{VaR}_q=\inf\{x:F(x)\geq q\}\) and \(\operatorname{TVaR}_q=(1-q)^{-1}\int_q^1\operatorname{VaR}_p\,dp\).
Method. Apply \(R(x)=\min((x-800)^+,2{,}000)\) separately to each event loss. All amounts in the calculations are thousands of dollars.
At gross loss 600, the loss is below attachment: \(R(600)=0\), so net loss is \(600-0=600\).
At gross loss 1,800, the amount above attachment is \(1{,}800-800=1{,}000\), below the limit. Thus \(R(1{,}800)=1{,}000\) and net loss is \(1{,}800-1{,}000=800\).
At gross loss 4,000, the amount above attachment is \(4{,}000-800=3{,}200\), which exceeds the $2,000 limit. Thus \(R(4{,}000)=2{,}000\) and net loss is \(4{,}000-2{,}000=2{,}000\).
The gross expected loss is \(0.60(600)+0.30(1{,}800)+0.10(4{,}000)=360+540+400=1{,}300\). The net expected loss is \(0.60(600)+0.30(800)+0.10(2{,}000)=360+240+200=800\). The expected recovery is \(1{,}300-800=500\), also equal to \(0.60(0)+0.30(1{,}000)+0.10(2{,}000)=500\). These calculations reconcile.
Gross cumulative probabilities are 0.60 at 600, 0.90 at 1,800, and 1.00 at 4,000. Therefore gross \(\operatorname{VaR}_{0.90}=1{,}800\). Net cumulative probabilities are 0.60 at 600, 0.90 at 800, and 1.00 at 2,000, so net \(\operatorname{VaR}_{0.90}=800\). The upper 10% of the probability distribution consists exactly of the highest-loss scenario; hence gross \(\operatorname{TVaR}_{0.90}=4{,}000\) and net \(\operatorname{TVaR}_{0.90}=2{,}000\). Expected loss, VaR, and TVaR are all in thousands of dollars.
Check. In every scenario, net loss is no greater than gross loss, and gross loss minus net loss equals the calculated recovery. Common mistake avoided: subtracting the attachment from every loss. Losses below $800 receive no recovery; the attachment is not a blanket credit.
Why the Distribution Changes
The transformation from gross to net loss explains why the treaty can reduce a tail measure without eliminating severe retained losses. For one eligible event under a per-occurrence layer, substitute the recovery formula into \(N=X-R(X)\). The resulting piecewise relationship is:
Below attachment, the insurer retains the whole loss. While the layer is paying, the retained loss is held at the attachment. Beyond exhaustion, each additional dollar of gross loss is again retained because the treaty has reached its limit. In particular, the largest gross losses do not disappear from the net distribution.
Confirm the event definition, covered peril, eligible losses, and applicable period. A scenario outside the treaty’s scope cannot be assumed to generate recovery.
Calculate recovery separately for each event under an occurrence layer, or to the eligible annual total under an aggregate layer. Respect every applicable attachment and limit.
For each event or annual scenario, subtract the treaty recovery from gross loss. Keep any exclusions and losses beyond the limit in the retained amount.
Use each scenario’s original probability to calculate the net distribution and the requested expected or tail measures. Check that recoveries do not exceed gross loss or treaty limits.
For a standard layer with no recovery-related adjustments, \(0\leq N\leq X\) scenario by scenario. Consequently the net distribution’s quantiles cannot exceed the corresponding gross quantiles. This follows because \(N\leq X\) for every outcome: whenever gross loss is at most \(x\), net loss is also at most \(x\), so \(F_N(x)\geq F_X(x)\). The comparison depends on applying the same scenarios and probabilities and on defining net loss as above; premium, reinstatement costs, or other financial effects are not included in this loss-only result.
Multiple Events and Annual Exhaustion
A per-occurrence recovery is calculated separately for each covered event. An annual aggregate recovery is based on cumulative eligible losses across the contract year, so an early event can use part of the aggregate protection before a later event occurs. The event sequence is useful for monitoring remaining protection, even though the simple annual formula depends on the final eligible total.
Worked Example: Tracking an Annual Aggregate Layer Across Events
A hypothetical insurer has three eligible catastrophe losses during one contract year: $1,300, $1,800, and $3,900 thousand. It buys an annual aggregate layer attaching at $2,000 thousand with a $3,000 thousand limit. Assume no reinstatement and no other contract adjustment. Find the final annual recovery and net loss, and show when the layer begins paying and is exhausted.
Method. Accumulate eligible gross losses and calculate the change in cumulative recovery after each event. The final eligible gross total is \(1{,}300+1{,}800+3{,}900=7{,}000\).
After event 1, cumulative gross loss is 1,300, below the 2,000 attachment, so cumulative recovery is 0. After event 2, cumulative gross loss is \(1{,}300+1{,}800=3{,}100\). Cumulative recovery is \(\min(3{,}100-2{,}000,3{,}000)=1{,}100\). Thus event 2 adds 1,100 of recovery.
After event 3, cumulative gross loss is 7,000. Cumulative recovery is \(\min(7{,}000-2{,}000,3{,}000)=3{,}000\). Event 3 therefore adds \(3{,}000-1{,}100=1{,}900\) of recovery. The aggregate limit is exhausted when cumulative eligible loss reaches \(2{,}000+3{,}000=5{,}000\); the final event takes the annual total past that point.
The event-level retained amounts using these cumulative increments are \(1{,}300-0=1{,}300\), \(1{,}800-1{,}100=700\), and \(3{,}900-1{,}900=2{,}000\). Their sum is \(1{,}300+700+2{,}000=4{,}000\). Checking by the annual total gives \(7{,}000-3{,}000=4{,}000\), the same net loss.
Check. Total recovery is $3,000 thousand, exactly the aggregate limit, and net annual loss plus recovery equals gross annual loss: \(4{,}000+3{,}000=7{,}000\). Common mistake avoided: applying the annual aggregate attachment anew to each event. That would treat one annual contract as though it were three separate occurrence layers.
Protection Gaps: Exclusions, Attachments, and Exhaustion
A net loss can remain high for distinct reasons, and each points to a different protection gap:
- Exclusion or ineligible loss: the loss does not enter the treaty calculation, so no recovery is available under that treaty.
- Below attachment: the loss is eligible but does not exceed the attachment; the insurer retains it.
- Above exhaustion: the loss exceeds attachment plus limit; the insurer retains the excess beyond the exhausted layer.
- Aggregate already used: earlier eligible losses have consumed some or all of an annual aggregate limit, leaving less protection for later events.
The distinction matters in reviewing catastrophe scenarios. A loss excluded by peril wording is not fixed by increasing an occurrence limit, while an exhausted layer may call for a higher limit or additional protection. An attachment gap may instead be deliberate retention. The modeled event definition must also match the contract’s definition of an occurrence; otherwise, losses grouped differently by the model and contract may produce misleading recovery estimates.
Worked Example: Separating Excluded Loss from Layer Exhaustion
A hypothetical portfolio has three event losses: eligible wildfire loss of $3,800 thousand, flood loss of $2,500 thousand excluded by the treaty, and eligible earthquake loss of $5,000 thousand. The per-occurrence treaty covers wildfire and earthquake, attaches at $1,000 thousand, and has a $2,000 thousand limit. Calculate each recovery, the total gross and net losses, and the retained protection gaps.
Method. Apply the occurrence layer only to eligible events. For wildfire, the amount above attachment is \(3{,}800-1{,}000=2{,}800\), so recovery is \(\min(2{,}800,2{,}000)=2{,}000\). Net wildfire loss is \(3{,}800-2{,}000=1{,}800\).
The $2,500 flood loss is excluded, so recovery is 0 and net flood loss is $2,500. For earthquake, the amount above attachment is \(5{,}000-1{,}000=4{,}000\), so recovery is \(\min(4{,}000,2{,}000)=2{,}000\). Net earthquake loss is \(5{,}000-2{,}000=3{,}000\).
Gross total loss is \(3{,}800+2{,}500+5{,}000=11{,}300\). Total recovery is \(2{,}000+0+2{,}000=4{,}000\), and total net loss is \(1{,}800+2{,}500+3{,}000=7{,}300\). The reconciliation is \(11{,}300-4{,}000=7{,}300\).
The flood loss is retained because it is excluded. Each eligible event also has a retained amount after recovery: $1,800 thousand for wildfire and $3,000 thousand for earthquake. For each, $1,000 thousand is the attachment-level retention; the remaining $800 thousand for wildfire and $2,000 thousand for earthquake is above the layer’s exhaustion point of \(1{,}000+2{,}000=3{,}000\). Common mistake avoided: treating every retained dollar as an exclusion. Here, the flood is excluded, while the eligible-event retentions reflect the attachment and exhausted limit.
Common Measurement Traps
- Subtracting expected recovery from a tail scenario. Expected recovery is an average, not the recovery for every event. Apply treaty terms to each scenario before calculating tail measures.
- Confusing the attachment with the amount paid. For an eligible loss above attachment, recovery is the loss above attachment, capped at the limit—not the entire loss and not simply the limit in every case.
- Ignoring exhaustion. A layer with attachment \(a\) and limit \(u\) stops increasing its recovery once loss reaches \(a+u\). Higher gross loss then increases net loss dollar for dollar.
- Resetting an annual aggregate attachment at every event. Aggregate terms accumulate over the stated period; occurrence terms apply separately to each defined event.
- Using inconsistent event or coverage definitions. Confirm that modeled losses are eligible under actual treaty wording before presenting modeled recoveries as protection.
Check Your Understanding
For each problem, assume the stated losses are in thousands of dollars unless specified otherwise. Apply only the treaty terms given.
- A covered event loss is $400, $1,100, or $2,200 with probabilities 0.20, 0.50, and 0.30. The occurrence layer attaches at $500 and has a $1,200 limit. What are the three net losses? (Answer: $400, $500, and $1,000.)
- A hypothetical event loss is $700 with probability 0.50, $1,700 with probability 0.30, and $3,500 with probability 0.20. A covered occurrence layer attaches at $700 and has a $1,500 limit. Calculate expected gross and expected net loss. (Answer: Expected gross loss $1,560; expected net loss $960.)
- Four eligible annual event losses are $800, $1,300, $2,100, and $1,800. An annual aggregate layer attaches at $2,500 and has a $2,500 limit, with no reinstatement. Find cumulative annual gross loss, recovery, and net loss. (Answer: Gross loss $6,000; recovery $2,500; net loss $3,500.)
- A treaty covers two perils but excludes flood. It attaches at $1,000 per occurrence and has a $2,500 limit. Eligible losses are $3,200 and $5,000; the excluded flood loss is $2,500. Find total recovery and total net loss. (Answer: Recovery $4,700; net loss $6,000.)
- Three eligible events have losses of $900, $2,600, and $4,100. Compare net annual loss under (a) a per-occurrence layer attaching at $500 with a $2,000 limit and (b) a single annual aggregate layer attaching at $2,500 with a $3,000 limit. Assume no reinstatement. (Answer: Occurrence recoveries are $400, $2,000, and $2,000, so net annual loss is $3,200; aggregate recovery is $3,000, so net annual loss is $4,600.)