Why an Event Loss Table Is Not Yet an OEP Curve
An event loss table lists modeled events, their losses, and their occurrence rates. The rates describe how frequently event types occur over time; they are not, by themselves, annual probabilities that the portfolio exceeds a loss threshold. To produce an occurrence exceedance probability (OEP) curve, first add the annual rates of all event types whose losses exceed the threshold, then convert that combined rate to the probability of at least one such event occurring during a year.
An OEP curve describes the distribution of the largest individual event loss in a year. It asks whether at least one event exceeds a specified loss amount. It does not add together losses from separate events in the year. The event loss table is treated as supplying one modeled loss value for each event type; uncertainty in losses conditional on an event is outside the calculation below.
The annual occurrence rate \(\lambda_j\) is an expected count per year and can be greater than 1. It is not a probability and should not be interpreted as one. For example, a rate of 0.30 means an expected 0.30 occurrences per year for that event type under the model. The Poisson conversion accounts for the possibility of multiple occurrences and makes the result an annual probability between 0 and 1.
If a table reports rates per a period other than one year, convert them to annual rates before summing. For example, a rate of 6 occurrences per 100 model-years corresponds to an annual rate of \(6/100=0.06\). The units of all rates in the sum must match.
Why the Poisson Formula Holds
Assume each event type \(j\) occurs according to a Poisson count \(N_j\) with mean \(\lambda_j\) per year, and assume the counts for different event types are independent. For a fixed threshold \(x\), only event types with \(x_j>x\) matter. A Poisson count with mean \(\lambda_j\) has probability \(\exp(-\lambda_j)\) of being zero. Independence therefore gives the probability that none of the qualifying event types occurs:
The event “at least one event exceeds \(x\)” is the complement of “no event exceeds \(x\).” Taking one minus the last expression gives \(1-\exp[-\Lambda(x)]\), the OEP formula. The independent Poisson assumption is material: if event occurrences are dependent or the rates represent probabilities for mutually exclusive annual scenarios rather than Poisson occurrence rates, this conversion need not apply.
For an event loss table with fixed losses, the exceedance rate is a step function of the threshold. It changes only when the threshold passes one of the listed losses. Tied losses must be treated together: all event types at a tied loss enter or leave the strict-exceedance sum at the same threshold.
A Practical Calculation Sequence
Convert every occurrence rate to a common annual basis before combining event types.
For a strict exceedance at \(x\), include only event losses satisfying \(x_j>x\). Do not include losses equal to \(x\).
The result \(\Lambda(x)\) is an annual exceedance rate, not yet an annual probability.
Calculate \(1-\exp[-\Lambda(x)]\), then report the probability to the requested precision.
As the threshold increases, fewer event types qualify, so the exceedance rate and OEP cannot increase.
Worked Applications
Worked Example: Calculating OEP at Several Loss Thresholds
A hypothetical catastrophe model produces the following event loss table. Losses are in millions of dollars, and each rate is an annual occurrence rate.
| Event | Loss ($ millions) | Annual rate |
|---|---|---|
| A | 12 | 0.08 |
| B | 8 | 0.15 |
| C | 5 | 0.20 |
| D | 3 | 0.30 |
| E | 1 | 0.50 |
Calculate the OEP at thresholds of $7 million and $8 million, using strict exceedance.
Method and assumption. Treat the event-type counts as independent Poisson counts with the stated annual rates. At the $7 million threshold, events A and B qualify because \(12>7\) and \(8>7\). Their combined rate is
Thus the annual probability of at least one qualifying event is
At the $8 million threshold, event B has loss exactly $8 million and does not strictly exceed the threshold. Only event A qualifies, so
Check. The OEP falls from 0.2055 at $7 million to 0.0769 at $8 million, as expected when the threshold rises. The common mistake is to include event B at the $8 million threshold; under the stated strict convention, equality is not exceedance.
Worked Example: Tied Losses and the Step Shape
A hypothetical model has event losses of $10 million at rate 0.04, $7 million at rates 0.10 and 0.06, $4 million at rate 0.20, and $2 million at rate 0.40. Find the OEP at $6 million and at $7 million, then describe the curve around the tied $7 million losses.
Method. Combine the two rates associated with the $7 million loss only when both event types satisfy the threshold test. At $6 million, the $10 million and both $7 million events qualify. The rate is
At $7 million, neither $7 million event strictly exceeds the threshold. Only the $10 million event qualifies:
For thresholds just below $7 million but at least $4 million, the qualifying rate is \(0.04+0.10+0.06=0.20\). At $7 million and above but below $10 million, it is 0.04. The curve therefore has a downward step at the tied loss level of $7 million; both tied event types leave the strict-exceedance sum together.
Check. The OEP at $7 million is lower than the OEP just below $7 million, and the tied events account for the full change in rate. A common error is to keep one of the tied events in the sum or remove only one of them at the threshold.
Worked Example: Converting Model-Period Rates and Selecting a Threshold
A hypothetical event catalog reports occurrence counts over 100 model-years. Its event loss and count pairs are: $25 million and 3 occurrences; $18 million and 5; $12 million and 12; $9 million and 20; and $4 million and 40. Find the OEP at $10 million, then find the smallest threshold on the listed loss grid for which OEP is at most 10%.
Step 1: Convert counts to annual rates. Divide each count by 100 model-years. The annual rates, in descending loss order, are \(0.03,\ 0.05,\ 0.12,\ 0.20,\ 0.40\).
Step 2: Calculate the OEP at $10 million. The $25 million, $18 million, and $12 million events exceed $10 million. Their combined annual rate is
Step 3: Apply the 10% target. An OEP no greater than 0.10 requires
At a threshold of $12 million, only the $25 million and $18 million events strictly exceed it. Their rate is \(0.03+0.05=0.08\), giving \(1-\exp(-0.08)=0.0769\), which meets the target. At any threshold just below $12 million, the $12 million event also qualifies, making the rate \(0.08+0.12=0.20\) and the OEP 0.1813. Therefore the smallest threshold on the listed loss grid that meets the target is $12 million.
Check. The target rate 0.1053605 lies between the qualifying rates 0.08 and 0.20 on either side of the $12 million loss step. The common mistake is to use the raw 100-year count as an annual probability or to linearly interpolate across the discrete step.
Interpretation and Common Traps
An OEP point is a probability for a year, not a recurrence interval and not an event rate. A rate of 0.20 per year converts to an OEP of \(1-\exp(-0.20)=0.1813\), not 0.20. The difference reflects the fact that a Poisson process can produce zero, one, or multiple occurrences in a year.
- Do not sum losses. OEP is based on whether an individual event exceeds the threshold. Multiple smaller events do not combine to exceed it for this calculation.
- Do not sum probabilities as if rates were probabilities. Sum the qualifying annual rates first, then use the Poisson complement formula.
- Respect the threshold convention. With strict exceedance, a loss equal to \(x\) is excluded. If a problem instead defines exceedance as “at least \(x\),” use \(x_j\geq x\) consistently.
- Check units and scope. Convert rates to a common time basis, and use event losses on the same basis as the threshold. A rate per 100 model-years is not an annual rate until divided by 100.
- State the model limitation. The formula relies on independent Poisson event-type counts. If the supplied values are scenario probabilities or event types are dependent, the table alone does not justify this formula.
The resulting curve is nonincreasing in the threshold, is constant between distinct event-loss levels, and can jump at a listed loss. These properties provide useful checks when constructing or reviewing an OEP curve from an event loss table. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
Use strict exceedance and the independent Poisson model unless a question states otherwise. Probabilities are rounded to four decimals.
- A hypothetical event table has losses of $15 million at annual rate 0.06, $9 million at 0.14, and $3 million at 0.25. Find the OEP at $8 million. (Answer: 0.1813)
- Two event types each have loss $6 million and annual rates 0.03 and 0.07. A third has loss $11 million and rate 0.02. Find the OEP at exactly $6 million. (Answer: 0.0198)
- A hypothetical catalog records 8 occurrences of a $20 million event and 12 occurrences of a $10 million event over 200 model-years. Find the OEP at $9 million. (Answer: 0.0952)
- Event losses and annual rates are $30 million at 0.04, $16 million at 0.05, $10 million at 0.09, and $5 million at 0.20. What is the OEP at $10 million, and does it meet a 10% maximum? (Answer: 0.0861; yes)
- Event types with losses $22 million, $14 million, and $8 million have annual rates 0.02, 0.06, and 0.18. Find the smallest listed-loss threshold whose OEP is at most 5%. (Answer: $14 million; OEP \(=0.0198\))