Two Curves, Two Annual Loss Quantities
Catastrophe models can report more than one annual exceedance probability curve for the same portfolio. The key difference is not the probability scale or the threshold units. It is which annual loss quantity is compared with the threshold. An occurrence curve considers the largest modeled event loss in the year. An aggregate curve considers the sum of all modeled event losses in the year.
For a chosen threshold, the occurrence exceedance probability (OEP) answers: “What is the probability that at least one event in a year produces a loss above this amount?” The aggregate exceedance probability (AEP) answers: “What is the probability that total loss from all events in a year is above this amount?” The losses should be measured on a consistent basis—for example, both gross of reinsurance or both net of the same reinsurance arrangements.
Here an “event” means an event as represented in the catastrophe model and its event definition. It is not automatically identical to a policy claim, a location loss, or a calendar-day loss. Losses must first be assigned to modeled events under the chosen event definition and financial terms. That event-level loss is the input to the OEP maximum; the AEP adds the event-level losses over the year.
How the Annual Loss Quantities Differ
| Feature | Occurrence exceedance probability | Aggregate exceedance probability |
|---|---|---|
| Annual quantity compared with \(x\) | Largest event loss, \(M\) | Sum of event losses, \(S\) |
| Exceedance condition | At least one event loss exceeds \(x\) | Total loss from all events exceeds \(x\) |
| Can several smaller events together exceed \(x\)? | No, unless one event itself exceeds \(x\) | Yes |
| Meaning of a curve point | Probability that the year’s largest event loss exceeds the threshold | Probability that the year’s total loss exceeds the threshold |
The two curves use a common loss scale, such as dollars or thousands of dollars, but that does not make their thresholds interchangeable in interpretation. A threshold of $10 million on an OEP curve concerns the largest single event loss. The same threshold on an AEP curve concerns the total from all events during the year. If two events each cause $6 million of loss, the aggregate loss is $12 million even though neither event loss exceeds $10 million.
Under the definitions above, all event losses are nonnegative, and both quantities refer to the same year and loss basis. Since a sum of nonnegative values cannot be less than the largest value, \(S\ge M\) in every year. Therefore, at any common threshold, aggregate exceedance probability cannot be less than occurrence exceedance probability.
This is a useful consistency check, not a rule that the probabilities must be different. If a year can contain at most one event, then \(S=M\), so the two probabilities are equal. If multiple events can occur, aggregate exceedance probability may be larger because several losses can add up to cross a threshold.
Worked Example: OEP From Poisson Event Rates
Worked Example: OEP From Poisson Event Rates
A hypothetical catastrophe portfolio has three modeled event classes. Their annual event counts are independent Poisson random variables. Class A has rate 0.12 events per year and a fixed event loss of $9 million; Class B has rate 0.08 and a fixed loss of $4 million; Class C has rate 0.05 and a fixed loss of $1 million. Find the OEP at thresholds of $3 million and $8 million.
Method and assumptions. For an OEP threshold \(x\), count only events whose individual losses exceed \(x\). Under the stated independent Poisson assumptions, the number of such qualifying events is Poisson with mean equal to the sum of their rates. The year exceeds the threshold in the OEP sense exactly when there is at least one qualifying event.
At \(x=\$3\) million, Classes A and B qualify because $9 million and $4 million are both strictly greater than $3 million. Class C does not qualify. The qualifying rate is \(0.12+0.08=0.20\) per year. The probability of at least one qualifying event is one minus the probability of zero:
Thus the OEP at $3 million is 0.1813, or 18.13% to four decimal places. At \(x=\$8\) million, only Class A qualifies. Its rate is 0.12, so
Thus the OEP at $8 million is 0.1131, or 11.31% to four decimal places. The probabilities decrease as the threshold rises, as an exceedance curve should. The common mistake avoided here is to include Class C at the $3 million threshold just because it has a nonzero event rate; its $1 million event loss does not exceed that threshold.
The Poisson calculation can be justified directly. If \(N_x\) is the number of events whose losses exceed \(x\), and \(N_x\) is Poisson with mean \(\Lambda_x\), then \(P(N_x=0)=e^{-\Lambda_x}\). The event that the annual maximum exceeds \(x\) is exactly the event \(N_x\ge1\). Taking the complement gives
This expression requires the Poisson count assumption and the correct rate of events exceeding the threshold. It is not a general formula for every event-frequency model. In particular, it does not by itself calculate the AEP: that requires comparing the sum of all annual event losses with the threshold.
Worked Example: One Annual Scenario Set, Two Curves
Worked Example: One Annual Scenario Set, Two Curves
A hypothetical model summarizes all possible annual outcomes in the following mutually exclusive scenarios. In a no-event year, probability 0.50, there is no loss. With probability 0.20, one event causes $8 million of loss. With probability 0.10, one event causes $12 million. With probability 0.10, two events each cause $6 million. With probability 0.10, two events cause $4 million and $5 million. Find the OEP and AEP at a threshold of $10 million.
Method. For each scenario, calculate the largest event loss \(M\) and the annual aggregate \(S\). Then add the probabilities of scenarios satisfying \(M>\$10\) million for OEP, and separately add those satisfying \(S>\$10\) million for AEP.
The scenario probabilities sum to \(0.50+0.20+0.10+0.10+0.10=1.00\), so the listed outcomes form a complete probability distribution. In the no-event, single-$8-million, and two-event $4-million-plus-$5-million scenarios, neither the largest event loss nor the aggregate exceeds $10 million. In the single-$12-million scenario, both quantities exceed $10 million. In the two-$6-million-event scenario, the largest event is $6 million, but aggregate loss is \(6+6=\$12\) million.
Only the single-$12-million scenario contributes to OEP, so
The single-$12-million scenario and the two-$6-million-event scenario contribute to AEP, so
The OEP is 0.1000 and the AEP is 0.2000. The aggregate probability is higher because the two moderate event losses combine to exceed the annual threshold, although neither event individually exceeds it. This result also checks against \(P(M>10)\le P(S>10)\).
Worked Example: Building Both Curves From Event Outcomes
Worked Example: Building Both Curves From Event Outcomes
A hypothetical portfolio has two independent event opportunities in a year. Event A occurs with probability 0.20 and, if it occurs, causes a $7 million loss. Event B occurs with probability 0.30 and, if it occurs, causes a $5 million loss. If an event does not occur, its loss is zero. Calculate OEP and AEP at thresholds of $5 million, $7 million, and $10 million.
Method and assumptions. Independence gives the probability of each joint outcome by multiplying the event and non-event probabilities. For each outcome, record the largest event loss and the sum. The threshold comparison is strict: a loss equal to the threshold does not exceed it.
The four outcome probabilities are:
These probabilities sum to \(0.56+0.14+0.24+0.06=1.00\). The neither outcome has \(M=0,S=0\); A only has \(M=7,S=7\); B only has \(M=5,S=5\); and both has \(M=7,S=12\), with losses in millions of dollars.
At a $5 million threshold, OEP includes A only and both, since their largest losses are $7 million. Thus \(P(M>5)=0.14+0.06=0.20\). AEP includes A only and both as well: the B-only aggregate is exactly $5 million and does not exceed the threshold. Thus \(P(S>5)=0.14+0.06=0.20\).
At a $7 million threshold, no largest event loss is strictly greater than $7 million, so \(P(M>7)=0\). The aggregate exceeds $7 million only when both events occur, giving \(P(S>7)=0.06\).
At a $10 million threshold, no individual event exceeds the threshold, so \(P(M>10)=0\). The aggregate exceeds $10 million only in the both-events outcome, where it is $12 million, so \(P(S>10)=0.06\). The results are summarized below.
| Threshold (millions) | OEP | AEP |
|---|---|---|
| $5 | 0.2000 | 0.2000 |
| $7 | 0.0000 | 0.0600 |
| $10 | 0.0000 | 0.0600 |
A common mistake is to call the $12 million combined loss in the both-events outcome a $12 million occurrence loss. The correction is to preserve the event boundary: the largest event loss is $7 million, while the annual aggregate is $12 million. At $10 million, that outcome contributes to AEP but not OEP.
Reading and Using the Curves Carefully
The distinction matters because the curves answer different risk questions. OEP describes the annual chance of a sufficiently large individual modeled event; AEP describes the annual chance that all event losses together reach a specified level. The appropriate quantity depends on the decision being assessed. For example, an analysis of a per-occurrence protection threshold may need an occurrence view, while an analysis concerned with total annual loss needs an aggregate view. The assumed financial basis and event definition must remain clear in either case.
At a fixed threshold and consistent basis, use these checks:
- Each curve should be nonincreasing as the threshold rises: exceeding a higher threshold implies exceeding every lower threshold.
- For nonnegative event losses, AEP should be at least as large as OEP at the same threshold.
- If only one event can occur in a year, the annual maximum and annual total are equal, so the curves coincide.
- Different probabilities at the same threshold can result from multiple events combining in the annual total; that difference does not mean the thresholds have different units.
- Check whether the comparison is strict exceedance (\(>x\)) or a different convention. In this tutorial, a loss equal to \(x\) is not an exceedance.
When an output reports an “annual exceedance probability” without making its loss quantity explicit, do not infer from the phrase alone whether it is OEP or AEP. Identify whether the reported annual variable is the largest event loss or the sum of event losses. A clear interpretation states both the probability and the quantity—for example, “the probability that annual aggregate loss exceeds $10 million”—rather than reporting only “the probability of exceeding $10 million.”
Check Your Understanding
For each problem, identify whether the requested quantity is occurrence or aggregate exceedance probability, and apply the threshold to the correct annual loss quantity.
- A hypothetical model has independent Poisson event classes: Class R has rate 0.03 per year and fixed event loss $12 million; Class S has rate 0.07 and fixed event loss $4 million. Find OEP at $5 million. (Answer: \(1-e^{-0.03}=0.0296\).)
- A hypothetical annual scenario set has probability 0.40 of no events, 0.25 of one $9 million event, 0.20 of two $4 million events, and 0.15 of one $11 million event. Find OEP and AEP at a threshold of $7 million. (Answer: OEP \(=0.25+0.15=0.4000\); AEP \(=0.25+0.20+0.15=0.6000\).)
- Two independent event opportunities have losses $6 million with probability 0.10 and $3 million with probability 0.40, respectively, and zero otherwise. Find OEP and AEP at thresholds of $5 million and $6 million. (Answer: At $5 million, OEP \(=0.1000\) and AEP \(=0.1000\); at $6 million, OEP \(=0.0000\) and AEP \(=0.0400\).)
- Independent Poisson event classes have rates 0.12 and 0.09 per year, with fixed event losses $10 million and $7 million, respectively. Find OEP at $6 million and state a lower bound for AEP at the same threshold. (Answer: OEP \(=1-e^{-(0.12+0.09)}=0.1894\); AEP is at least \(0.1894\).)
- A hypothetical year has two events with losses $8 million and $5 million. State the largest event loss and aggregate loss, then determine which curve records an exceedance at a $10 million threshold. (Answer: Largest event loss \(=\$8\) million and aggregate loss \(=\$13\) million; no OEP exceedance, but an AEP exceedance.)