OEP Versus AEP Through Worked Examples

What OEP and AEP Measure

A catastrophe program can produce several covered events in one year. A single event may stay below a chosen loss threshold while the combined losses from multiple events exceed it. This distinction matters when interpreting catastrophe model output and pricing or evaluating contracts with different event and annual-aggregate terms. The comparison of OEP and AEP is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Both measures describe exceedance probabilities over a year, but they summarize annual losses differently. To compare them correctly, use the same portfolio, coverage basis, loss measure, and threshold. For example, comparing gross occurrence loss with net annual aggregate loss would confound the distinction between OEP and AEP with the effect of reinsurance.

Definition: Let \(L_j\) be the covered loss from event \(j\) during a year. The occurrence exceedance probability (OEP) at threshold \(x\) is the probability that the largest event loss in the year exceeds \(x\): \(P(\max_j L_j>x)\). The aggregate exceedance probability (AEP) at \(x\) is the probability that the sum of all event losses in the year exceeds \(x\): \(P(\sum_j L_j>x)\). If there are no events, define the largest event loss and the annual aggregate loss as zero.

The event loss \(L_j\) should be the loss assigned to one modeled event after applying the relevant coverage terms and aggregation rules. It is not necessarily the loss to one policy or one location. The annual aggregate includes every event in the year on the chosen basis; the OEP calculation retains only the largest event loss.

Why AEP Is at Least OEP

Suppose event losses are nonnegative and OEP and AEP use the same events and loss basis. The annual aggregate \(S=\sum_j L_j\) is at least as large as the largest event loss \(M=\max_j L_j\). Therefore, whenever the largest event exceeds \(x\), the annual aggregate also exceeds \(x\).

M \leq S \quad\Longrightarrow\quad \{M>x\}\subseteq\{S>x\} \quad\Longrightarrow\quad P(M>x)\leq P(S>x).

This establishes \( \text{AEP}(x)\geq \text{OEP}(x) \) for a common nonnegative-loss basis. The inequality can be strict when multiple losses combine to exceed the threshold even though no individual event does. It can be equality when the same years exceed under both definitions. The result should not be applied blindly if the two calculations use different netting, coverage, or event-definition conventions.

For independent event types, there is also a convenient OEP calculation. If type \(i\) occurs with probability \(p_i\) and produces a fixed event loss \(c_i\), then, for a fixed threshold \(x\), only types with \(c_i>x\) can cause OEP exceedance. Independence gives:

\operatorname{OEP}(x) =1-\prod_{i:c_i>x}(1-p_i).

The product is the probability that none of the event types whose loss exceeds \(x\) occurs. Subtracting that probability from one gives the chance that at least one such event occurs. This formula requires independent occurrence indicators for the event types; if their occurrences are dependent, their joint probabilities are needed instead. It does not by itself calculate AEP, because AEP depends on the distribution of the sum.

Calculate Both Measures from Simulated Years

A simulated-year data set gives a direct comparison. For each year, identify the event losses, calculate the maximum event loss, and calculate the annual total. With equally weighted simulated years, the empirical exceedance probability is the number of years exceeding the threshold divided by the number of simulated years. A strict exceedance uses \(>\), not \(\geq\).

Worked Example: Eight Simulated Years and a Threshold of 100

A hypothetical catastrophe model produces eight equally weighted simulated years. Losses below are in thousands of dollars. Calculate empirical OEP and AEP at \(x=100\) thousand dollars. Treat a year with no events as having maximum and total loss zero.

Method and assumptions. Each listed row is one complete year, and each year has probability \(1/8\) in this empirical distribution. No interest or mortality assumption is relevant: the modeled amounts are annual catastrophe losses, and the calculation does not discount them. For each row, compare the maximum event loss and the annual total separately with 100.

Simulated year	Event losses (thousands)	Maximum event loss	Annual total
1	No events	0	0
2	45	45	45
3	70, 20	70	90
4	55, 55	55	110
5	120	120	120
6	35, 40, 30	40	105
7	80, 15	80	95
8	25	25	25

OEP calculation. The maximum event loss exceeds 100 only in year 5. In year 4, the largest event is 55, and in year 6 it is 40; neither year has an event above the threshold. Thus:

\widehat{\operatorname{OEP}}(100)=\frac{1}{8}=0.1250.

AEP calculation. The annual total exceeds 100 in years 4, 5, and 6. The totals in those years are \(55+55=110\), \(120\), and \(35+40+30=105\), respectively. Therefore:

\widehat{\operatorname{AEP}}(100)=\frac{3}{8}=0.3750.

Check and common mistake avoided. The AEP exceeds the OEP, as expected: years 4 and 6 exceed in aggregate even though no event in either year exceeds 100. A common error is to count only years with an event above the threshold for both measures; that incorrectly gives AEP \(1/8\). The correction is to sum all event losses within each year before testing for AEP exceedance.

Independent Event Types: Occurrence Versus Sum

When a model specifies event-type occurrence probabilities, calculate OEP from the chance of at least one individually large event. To calculate AEP, enumerate the combinations of events that make the sum exceed the threshold, taking care not to count overlapping combinations twice.

Worked Example: Small Events Combine Above the Threshold

A hypothetical annual model has three independent event types. Type A occurs with probability 0.10 and causes a fixed loss of 120 thousand dollars; type B occurs with probability 0.20 and causes 65 thousand dollars; type C occurs with probability 0.30 and causes 50 thousand dollars. Each type can occur at most once in the year. Calculate OEP and AEP at \(x=100\) thousand dollars.

Method and assumptions. Occurrence indicators are mutually independent Bernoulli variables with the stated probabilities, and event losses are fixed when an event occurs. No interest or mortality assumption is relevant. AEP uses the sum of the losses from all types that occur.

OEP. Only type A has an event loss greater than 100. Types B and C have losses below the threshold, even if considered separately. Consequently:

\operatorname{OEP}(100)=P(\text{A occurs})=0.10.

AEP. Any occurrence of A makes the annual total at least 120, so that case has probability 0.10. If A does not occur, the annual total exceeds 100 only when both B and C occur, because \(65+50=115\). By independence, the probability of this additional case is:

P(\text{A absent, B and C present}) =(1-0.10)(0.20)(0.30) =0.90(0.06) =0.054.

These two cases are disjoint: the first has A present, and the second has A absent. Adding them gives:

\operatorname{AEP}(100)=0.10+0.054=0.1540.

Check and common mistake avoided. The AEP is 0.1540, greater than the OEP of 0.1000 because B and C can combine to exceed the threshold. A tempting but incorrect calculation is \(0.10+(0.20)(0.30)=0.16\). It counts the B-and-C case even when A occurs, although those years are already included in the 0.10 probability for A; the corrected calculation restricts the additional case to A being absent.

Poisson Event Counts and Multiple Subthreshold Losses

A Poisson model makes it possible for a year to have multiple events of the same modeled type. If the annual count \(N\) has a Poisson distribution with parameter \(\lambda\), then \(P(N=n)=e^{-\lambda}\lambda^n/n!\). When each event has a fixed loss, the annual aggregate is the event count multiplied by that loss. The following calculation assumes no event-size variability.

Worked Example: A Poisson Year with Identical Event Losses

A hypothetical model assigns a Poisson annual count with parameter \(\lambda=0.5\) to a type of event. Every occurrence produces a fixed loss of 60 thousand dollars. Calculate OEP and AEP at \(x=100\) thousand dollars.

Method and assumptions. The event count \(N\) is Poisson with parameter 0.5, and every event loss is exactly 60 thousand dollars. Conditional on the count, the annual total is \(60N\) thousand dollars. The event loss distribution has no severity variation, and no interest or mortality assumption is relevant.

OEP. Every individual event loss is 60, which is not greater than 100. Even if many events occur, no single event crosses this threshold. Therefore:

\operatorname{OEP}(100)=0.

AEP. The annual total exceeds 100 when \(60N>100\), which requires \(N\geq2\). The probability of this is one minus the probabilities of zero and one event:

\begin{aligned} \operatorname{AEP}(100) &=P(N\geq2)\\ &=1-P(N=0)-P(N=1)\\ &=1-e^{-0.5}-e^{-0.5}(0.5)\\ &=1-e^{-0.5}(1.5)\\ &=1-(0.6065307)(1.5)\\ &=1-0.9097960\\ &=0.0902. \end{aligned}

Check and common mistake avoided. AEP is 0.0902 although OEP is zero: two or more 60-thousand-dollar events produce an annual total above 100. The relevant cutoff is two events because one event totals 60 and two total 120. Do not use \(N\geq1\) for AEP; that would include years whose annual total does not exceed the threshold.

Interpretation and Practical Checks

The examples use different probability representations, but the distinction is always the same: OEP tests the largest event in a year, while AEP tests the sum across the year. Before interpreting a result, confirm the following:

Event definition: Check how losses are grouped into events. A multi-location event loss may be one occurrence, not several independent events.
Threshold convention: Apply the stated strict or inclusive comparison consistently. \(L>x\) and \(L\geq x\) can produce different probabilities when losses can equal \(x\).
Annual aggregation: Sum all event losses assigned to the same year for AEP; do not sum losses across different years.
Consistent basis: Use the same gross or net treatment, coverage terms, and units for both measures.
Dependence: Do not multiply nonoccurrence probabilities unless the assumed event occurrences are independent. Simulated years can preserve dependencies represented in the simulation, while separate marginal probabilities alone do not specify their joint behavior.

An OEP curve and an AEP curve answer related but different questions about modeled annual losses. The AEP curve includes the effect of multiple events accumulating in one year, so it can show exceedance at thresholds that no individual event reaches. Keeping the event maximum separate from the annual sum is the essential calculation check.

Check Your Understanding

For each problem, use the stated strict exceedance convention \(>x\). Loss amounts are in thousands of dollars unless indicated otherwise.

Two independent event types can each occur at most once per year. Type A causes 90 with probability 0.20, and type B causes 40 with probability 0.25. At \(x=100\), calculate OEP and AEP. (Answer: OEP \(=0.0000\); AEP \(=0.0500\).)
Four equally likely simulated years have event losses \([110]\), \([70,40]\), \([60,30]\), and no events, respectively. At \(x=100\), calculate empirical OEP and AEP. (Answer: OEP \(=0.2500\); AEP \(=0.5000\).)
Three independent event types have fixed losses and annual occurrence probabilities: A has loss 150 and probability 0.04, B has loss 70 and probability 0.15, and C has loss 45 and probability 0.20. At \(x=100\), calculate OEP and AEP. (Answer: OEP \(=0.0400\); AEP \(=0.0688\).)
The annual number of events is Poisson with parameter \(\lambda=1\), and each event causes a fixed loss of 55. At \(x=100\), calculate OEP and AEP. (Answer: OEP \(=0.0000\); AEP \(=1-2e^{-1}=0.2642\).)
Three independent event types have fixed losses 100, 50, and 50, with occurrence probabilities 0.10, 0.20, and 0.20, respectively. At \(x=100\), use strict exceedance and calculate OEP and AEP. (Answer: OEP \(=0.0000\); AEP \(=0.0360\).)

What You'll Learn