Catastrophe Loss Estimates and Model Outputs

What Catastrophe Model Outputs Describe

A catastrophe model may report losses for individual events, a probability that losses exceed selected amounts, and an expected annual loss. These outputs answer different questions. An event loss describes the modeled financial consequence of one event; an annual aggregate loss combines the losses from all events in a year; and an exceedance probability summarizes how often a loss threshold is crossed under the model’s assumptions. Keeping the units and time basis clear is essential when using these results in a ratemaking or reinsurance estimate.

The calculations below use hypothetical model outputs. Unless stated otherwise, amounts are gross insured losses in dollars, event rates are expected occurrences per year, and no discounting is applied. A model output is conditional on its specified exposure and loss basis: it should not be mixed with losses on a different basis, such as net losses after reinsurance.

Definition: An event loss is the modeled loss associated with one specified catastrophe event, on a stated financial basis. Annual aggregate loss is the sum of losses from all modeled events occurring during one year. Expected annual loss (EAL, also commonly called annual average loss or AAL) is the expected value of annual aggregate loss under the model. An exceedance probability is the probability that a specified loss measure is greater than a stated threshold over a specified time period.

Event Rates and Expected Annual Loss

A catastrophe event catalog can pair each event loss with an annual occurrence rate. The rate $\lambda_j$ is an expected number of occurrences per year, not necessarily a probability. For a model with event types $j=1,\ldots,n$, loss $L_j$, and annual rate $\lambda_j$, the expected annual loss is

\operatorname{EAL}=\sum_{j=1}^{n}\lambda_j L_j.

This formula applies when the catalog rates represent expected event counts per year and the listed loss is the relevant loss per occurrence. It does not require that only one event can occur in a year. For example, if event types occur according to independent Poisson processes, their expected annual losses add even though several events may occur in the same year. If a model instead supplies a probability-weighted set of mutually exclusive annual scenarios, use those scenario probabilities and annual losses; do not treat scenario probabilities as event rates.

The justification follows from expectation. If $N_j$ is the number of events of type $j$ during a year, and each such event has loss $L_j$, annual loss is $S=\sum_j N_jL_j$. Therefore

\operatorname{E}[S] =\sum_j \operatorname{E}[N_jL_j] =\sum_j L_j\operatorname{E}[N_j] =\sum_j \lambda_jL_j,

because the annual rate is $\operatorname{E}[N_j]=\lambda_j$. This derivation concerns the mean only. It does not determine the distribution of annual loss, which is needed to calculate aggregate exceedance probabilities.

Worked Example: From an Event Catalog to EAL and Occurrence Exceedance

A hypothetical model lists three event types on a gross-loss basis: event A has annual rate 0.04 and loss $12 million; event B has annual rate 0.10 and loss $4 million; event C has annual rate 0.25 and loss $1 million. Assume independent Poisson occurrence counts for the event types. Calculate EAL and the probability that at least one event in a year causes a loss greater than $3 million.

Method: rate-weighted event losses. Multiply each event loss by its annual rate:

0.04(12)=0.48,\qquad 0.10(4)=0.40,\qquad 0.25(1)=0.25

The products are in millions of dollars per year. Thus,

\operatorname{EAL}=0.48+0.40+0.25=1.13\text{ million dollars per year}.

For the probability of at least one event loss greater than $3 million, events A and B qualify; C does not. Their combined annual rate is $0.04+0.10=0.14$. Under the independent Poisson assumption, the count of qualifying events is Poisson with mean 0.14. The probability of zero qualifying events is $e^{-0.14}$, so the probability of one or more is

1-e^{-0.14}=1-0.8694=0.1306.

Therefore, the requested probability is 0.1306, or 13.06%. As a check, EAL is the sum of all three event contributions, while the exceedance calculation includes only event types whose individual losses exceed the $3 million threshold. The common mistake is to add 0.04 and 0.10 and report 0.14 as a probability; 0.14 is the expected count, whereas the probability of at least one occurrence is $1-e^{-0.14}$.

Occurrence and Aggregate Exceedance Probabilities

An exceedance probability is meaningful only after specifying both the loss measure and the time period. Two common measures are the largest event loss in a year and total loss over a year. Their exceedance curves are not interchangeable.

Definition: The occurrence exceedance probability (OEP) at threshold $x$ is $P(M>x)$, where $M$ is the maximum event loss in a year. The aggregate exceedance probability (AEP) at threshold $x$ is $P(S>x)$, where $S$ is the sum of all event losses in a year. Both are annual probabilities, but AEP includes the combined effect of multiple events, while OEP records only the largest single event.

If event types follow independent Poisson processes, and the event types with individual loss greater than $x$ have rates $\lambda_1,\ldots,\lambda_k$, then the annual OEP is

P(M>x)=1-\exp\left(-\sum_{j=1}^{k}\lambda_j\right).

This follows because the total number of events exceeding $x$ is Poisson with mean $\sum_j\lambda_j$, and OEP is one minus the probability of zero such events. The formula applies to exceedance by an individual event. It is not an AEP formula: several events, each below $x$, can combine to make annual aggregate loss exceed $x$.

A model may estimate AEP by simulating many years, calculating $S$ for each simulated year, and counting the fraction with $S>x$. Alternatively, if the frequency and severity distributions are specified in a tractable form, that probability can be calculated from the resulting annual aggregate distribution. A single event catalog with rates and losses can give EAL directly, but generally does not supply enough information to determine the full AEP curve unless assumptions about event frequency and annual aggregation are also given.

Worked Example: Annual Aggregate Exceedance with Poisson Frequency

A hypothetical flood portfolio has a fixed loss of $2 million for each qualifying flood event. The annual number of events $N$ follows a Poisson distribution with mean $\lambda=0.20$. Find EAL, the probability annual aggregate loss is greater than zero, and the probability it is greater than $2 million.

Method: compound frequency-severity model. Annual aggregate loss is $S=2N$ million dollars. The expected annual aggregate loss is

\operatorname{E}[S]=2\operatorname{E}[N]=2(0.20)=0.40\text{ million dollars}.

Annual loss is greater than zero exactly when $N\geq 1$. For a Poisson variable, $P(N=0)=e^{-0.20}=0.8187$, so

P(S>0)=P(N\geq 1)=1-e^{-0.20}=0.1813.

Annual loss exceeds $2 million exactly when there are at least two events, since one event produces precisely $2 million and does not exceed that threshold. Therefore,

P(S>2)=P(N\geq 2) =1-P(N=0)-P(N=1) =1-e^{-0.20}-0.20e^{-0.20} =0.0175.

The results are EAL of $400,000, a 0.1813 probability of positive loss, and a 0.0175 probability of aggregate loss greater than $2 million. The calculation is checked against the threshold wording: $S=2$ does not satisfy $S>2$. Treating the aggregate threshold as an occurrence threshold, or counting exactly one event as an exceedance, would give the wrong answer.

Return Periods: A Useful Summary, Not a Schedule

For an annual exceedance probability $p$, the reciprocal $1/p$ is often called the return period in years. For example, an annual probability of 0.02 corresponds to a 50-year return period. This is a long-run frequency summary, not a prediction that an event occurs exactly once every 50 years or at regular 50-year intervals. Under independent, identically distributed annual outcomes, the probability of at least one exceedance in $T$ years is $1-(1-p)^T$. This expression depends on the annual independence and stationarity assumptions; changes in exposure, climate, or model basis can make those assumptions inappropriate.

The return period must also be tied to the curve being reported. A 50-year OEP loss is a threshold with 2% annual probability of at least one event loss exceeding it. A 50-year AEP loss is a threshold with 2% annual probability of annual aggregate loss exceeding it. These thresholds need not be equal.

Connecting Model Losses to Treaty Recoveries

A catastrophe model can also support an estimate of expected reinsurance recovery, provided its loss basis matches the treaty definition and the model output contains enough event-level detail. For a simple occurrence excess-of-loss treaty with attachment $a$ and limit $u$, recovery on one event loss $L$ is

R(L)=\min\bigl((L-a)_+,u\bigr), \qquad (z)_+=\max(z,0).

The attachment is the amount of loss retained before the layer responds, and the limit is the maximum recovery for that occurrence. If event type $j$ has annual rate $\lambda_j$, expected annual recovery under this simple event-based calculation is $\sum_j\lambda_jR(L_j)$. This is a mean estimate, not the distribution of recoveries or the probability that the treaty is exhausted. The calculation assumes that the listed loss is the treaty-defined loss per occurrence and that no other terms alter the recovery. Policy sublimits, event definitions, hours clauses, annual aggregate limits, and reinstatements can change the result and must be handled using their actual terms.

Worked Example: Estimating Expected Occurrence-Treaty Recovery

A hypothetical property portfolio has three modeled event types. Event A has annual rate 0.05 and gross treaty-defined event loss $8 million; event B has rate 0.15 and loss $4 million; event C has rate 0.30 and loss $1 million. A simple occurrence excess-of-loss treaty attaches at $2 million and has a $5 million limit. Calculate EAL gross, expected annual recovery, and expected annual net loss. Assume the stated losses are the applicable losses for this treaty and ignore all other treaty terms.

Method: apply the occurrence recovery function to each event, then weight by rate. For event A, the loss above attachment is $8-2=6$ million dollars, but the recovery is capped at $5 million:

R(8)=\min(8-2,5)=5\text{ million dollars}.

For event B, the loss above attachment is $4-2=2$ million, below the limit:

R(4)=\min(4-2,5)=2\text{ million dollars}.

Event C is below the attachment:

R(1)=\min\bigl((1-2)_+,5\bigr)=0.

Gross EAL is

0.05(8)+0.15(4)+0.30(1) =0.40+0.60+0.30 =1.30\text{ million dollars}.

Expected annual recovery is

0.05(5)+0.15(2)+0.30(0) =0.25+0.30 =0.55\text{ million dollars}.

Expected annual net loss is gross EAL less expected recovery:

1.30-0.55=0.75\text{ million dollars}.

As a check, rate-weighting the event-level net losses gives $0.05(8-5)+0.15(4-2)+0.30(1-0)=0.15+0.30+0.30=0.75$ million dollars. The common mistake is to subtract the $2 million attachment from the annual aggregate loss; the attachment applies separately to each occurrence in this treaty, so recovery must be calculated event by event before rate-weighting.

Interpreting the Outputs Carefully

Model results are useful only when their meanings remain attached to them. Before applying a number, confirm whether it represents gross, ceded, or net loss; whether it is per event or per year; and whether its probability is an occurrence or aggregate exceedance probability. Also check whether the loss is stated in dollars or thousands, whether event rates are annual frequencies or probabilities, and whether the treaty responds to the same definition of loss shown by the model.

A common exam and practice error is to infer a complete risk distribution from EAL alone. EAL is one average; different loss distributions can have the same mean but very different probabilities of severe outcomes. Similarly, an event loss at a stated return period cannot automatically be treated as annual aggregate loss at that return period. Use the model output that matches the question, and state the assumptions that connect one measure to another.

Key takeaway: Event losses and event rates can be combined to estimate EAL, while exceedance probabilities require a clearly defined loss measure and time period. For treaty estimates, calculate the recovery on each applicable event loss first, then aggregate or rate-weight the recoveries on a consistent basis.

Check Your Understanding

For each hypothetical problem, identify the loss measure and time basis before calculating.

A model lists event X with annual rate 0.03 and loss $10 million, and event Y with annual rate 0.12 and loss $2 million. Find EAL in millions of dollars. (Answer: $0.54 million per year)
Independent Poisson event types with combined annual rate 0.08 each cause a loss greater than a stated threshold. Find the annual probability of at least one threshold-exceeding event, to four decimals. (Answer: 0.0769)
A portfolio has a fixed $3 million loss per event and Poisson event count with mean 0.10 per year. Find the probability annual aggregate loss is greater than $3 million, to four decimals. (Answer: $1-e^{-0.10}(1+0.10)=0.0047$)
An occurrence treaty attaches at $1 million and has a $4 million limit. A modeled event has treaty-defined loss $7 million and annual rate 0.06. Find its expected annual recovery contribution. (Answer: $0.06\min(7-1,4)=\$0.24$ million per year)
A model reports a 2% annual OEP at $20 million. State the corresponding reciprocal return-period summary and explain what it does not mean. (Answer: 50 years; it does not mean an exceedance occurs regularly once every 50 years.)

What You'll Learn