Diagnosing Catastrophe Model Output With a Worked Portfolio

Introduction

A catastrophe model output is not just a set of loss numbers. It is a chain of calculations: exposure data feeds event-level damage and insured loss, event rates determine annual frequency, and the resulting annual loss distribution produces exceedance probabilities. When a result looks unusual, checking that chain in order is more useful than treating the final number as an isolated answer.

This tutorial uses a small hypothetical portfolio to demonstrate that diagnostic approach. The focus is reconciliation: identifying whether a surprising result is consistent with the exposure, the event rates, and the model components that produced it. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: An event-loss summary reports modeled loss for a specified event, often by location or coverage. An annual aggregate loss is the sum of all modeled losses during one year. The occurrence exceedance probability (OEP) at threshold $x$ is the annual probability that at least one event loss exceeds $x$. The aggregate exceedance probability (AEP) at $x$ is the annual probability that total annual loss exceeds $x$. These are different questions: AEP allows losses from multiple events to accumulate.

Start With the Event-Loss Components

An event-level result should be traceable to the locations exposed to that event and the model components applied to them. At a basic level, gross damage at a location is its exposed value multiplied by the modeled damage ratio. Insured loss may then differ from gross damage because the financial terms—such as deductibles, limits, and policy conditions—change the payment. A comparison is meaningful only when the reported figures use the same loss basis and financial terms.

A useful first reconciliation is to calculate each location's contribution and sum those contributions. If the event total does not match, investigate the inputs and transformations in sequence: location identifiers, exposure values and units, event footprint, damage ratios, and financial terms. A portfolio total alone can conceal an offsetting error, so retain the location-level detail where possible.

Worked Example: Reconciling a Single Event Total

A hypothetical wind event affects three insured locations. The exposure file lists values of $4.00 million, $3.00 million, and $2.00 million, with modeled damage ratios of 25%, 40%, and 10%, respectively. The event summary reports gross damage of $3.00 million. Reconcile the result and identify an exposure input that could explain the difference.

Method. Calculate gross damage by location as exposure value multiplied by damage ratio, then sum. This checks the event loss before deductibles or limits.

Location 1 damage is $4.00 \times 0.25 = 1.00$ million dollars. Location 2 damage is $3.00 \times 0.40 = 1.20$ million dollars. Location 3 damage is $2.00 \times 0.10 = 0.20$ million dollars.

1.00+1.20+0.20=2.40\text{ million dollars}

The reported total is $3.00 million, which exceeds the recalculated total by $3.00-2.40=0.60$ million dollars. If location 2's exposure were $4.50 million rather than $3.00 million, its damage would be $4.50 \times 0.40=1.80$ million dollars. The resulting total would be $1.00+1.80+0.20=3.00$ million dollars, matching the report. This does not prove that the exposure value is wrong; it identifies a specific input to check.

Check and common mistake. The recalculated $2.40 million is below the reported $3.00 million by exactly the additional $0.60 million implied by the possible exposure discrepancy. A common mistake is to apply policy deductibles or limits to this comparison even though both figures are stated as gross damage. That would compare different loss bases and would not resolve the discrepancy.

Check Annual Frequency Against Annual Loss Results

For a simple frequency-severity model, let $N$ be the number of events in a year and $Y_j$ the loss from event $j$. Annual aggregate loss is $S=\sum_{j=1}^{N}Y_j$. If $N$ has a Poisson distribution with mean $\lambda$, and each event has the same fixed loss $y$, then $S=yN$. This simplified model is useful for checking whether annual outputs are consistent with the event rate and event loss.

E[S]=\lambda y

The mean follows by conditioning on the event count: given $N=n$, the aggregate is $ny$, so $E[S]=E[Ny]=yE[N]=y\lambda$. This check applies under the stated Poisson and fixed-severity assumptions. With variable event severity, the corresponding compound Poisson mean is $\lambda E[Y]$; neither formula should be used with a rate or severity measured on a different time basis.

Worked Example: Checking an Annual Aggregate Summary

A hypothetical catastrophe model has a Poisson event count with annual rate $\lambda=0.4$. Every event produces a net loss of $2.50 million. Calculate the expected annual aggregate loss and the probability that annual aggregate loss exceeds $2.50 million. Compare those calculations with a reported mean of $1.00 million and reported exceedance probability of 0.0616.

Method. Use the compound Poisson model with fixed event severity. Since each event costs $2.50 million, annual aggregate loss exceeds $2.50 million exactly when there are at least two events.

The expected annual aggregate loss is

E[S]=\lambda y=0.4(2.50)=1.00\text{ million dollars}.

For $N\sim\operatorname{Poisson}(0.4)$, the probability of at least two events is

P(S>2.50)=P(N\geq 2) =1-P(N=0)-P(N=1).

Using $P(N=n)=e^{-0.4}0.4^n/n!$, the zero-event probability is $e^{-0.4}=0.6703$, and the one-event probability is $e^{-0.4}(0.4)=0.2681$. Therefore,

$$ P(S>2.50)=1-0.6703-0.2681=0.0616. $$

Both reported figures are consistent with the stated model. The mean is $1.00 million and the exceedance probability is 0.0616.

Check and common mistake. The strict threshold matters: one event gives exactly $2.50 million, which does not exceed $2.50 million. Counting one event as an exceedance would give $P(N\geq1)=1-e^{-0.4}=0.3297$, an incorrect answer for the question asked. If a model instead reported a mean of $1.25 million with this same $2.50 million event loss, the implied Poisson rate would be $1.25/2.50=0.5$, suggesting a frequency input or time-basis discrepancy to investigate.

Use OEP and AEP Relationships as Diagnostics

For Poisson event counts, the OEP has a direct connection to the rate of events whose individual losses exceed a threshold. If the annual event rate is $\lambda$, and an event loss $Y$ has a specified severity distribution, then the rate of events with $Y>x$ is $\lambda P(Y>x)$. The number of such events in a year is Poisson with that mean. The probability of at least one is one minus the probability of none.

\operatorname{OEP}(x)=1-\exp\bigl(-\lambda P(Y>x)\bigr)

For a portfolio with several independent Poisson event types, add the rates of all types whose individual losses exceed $x$, then apply the same at-least-one-event calculation. AEP requires a different calculation because multiple losses can accumulate, including losses from events that are each individually below $x$. Under the same annual horizon and consistent loss basis, annual aggregate loss is at least as large as the largest individual event loss. Consequently, the AEP at a common threshold cannot be smaller than the OEP.

Worked Example: Diagnosing OEP and AEP at One Threshold

A hypothetical portfolio has two independent Poisson event types. Type X occurs at annual rate 0.10 and always causes a $4 million loss. Type Y occurs at annual rate 0.20 and always causes a $2 million loss. Calculate OEP and AEP at a threshold of $3 million, then check a model output that reports OEP of 0.0952 and AEP of 0.1110.

Method. For OEP, count only event types whose individual event loss exceeds $3 million. For AEP, include annual combinations of events whose losses sum to more than $3 million.

Only type X has an individual loss exceeding $3 million. The annual count of type X events is Poisson with mean 0.10, so

\operatorname{OEP}(3)=P(N_X\geq1)=1-e^{-0.10}=0.0952.

For AEP, an annual total exceeds $3 million if at least one type X event occurs, or if no type X event occurs and at least two type Y events occur. These cases are disjoint. The probability of no type X event is $e^{-0.10}=0.9048$. For type Y, the probability of at least two events is

P(N_Y\geq2)=1-e^{-0.20}(1+0.20) =1-(0.8187)(1.20)=0.0175.

Independence gives

\begin{aligned} \operatorname{AEP}(3) &=P(N_X\geq1)+P(N_X=0)P(N_Y\geq2)\\ &=0.0952+(0.9048)(0.0175)\\ &=0.1110. \end{aligned}

The reported OEP and AEP are consistent with the calculations to four decimal places. AEP is larger because two type Y events can combine to exceed the threshold even though neither event does so alone.

Check and common mistake. Here AEP exceeds OEP, as required when both use the same threshold and annual horizon. A common mistake is to calculate OEP using the probability that total annual loss exceeds $3 million; that is the AEP calculation, not the OEP calculation.

A Practical Diagnostic Sequence

The order of checks matters. Confirming the meaning and basis of an output before recalculating it prevents a correct calculation from being compared with the wrong model quantity. The following sequence is useful when event summaries, annual statistics, or curves appear inconsistent.

Confirm the output definition.
Identify whether the number is event loss, annual aggregate loss, OEP, or AEP. Confirm the threshold convention, such as loss greater than $x$ versus loss at least $x$.

Check the exposure and event components.
Reconcile location values, units, event participation, damage, and any financial terms. Keep gross damage and insured loss distinct.

Check frequency and time basis.
Confirm the event rate uses the intended annual horizon and that event counts are not confused with annual probabilities or rates for a different period.

Recalculate a simple aggregate check.
Under a stated frequency-severity model, compare the reported mean and exceedance probabilities with calculations from the event rate and severity assumptions.

Check relationships across outputs.
At a common threshold and horizon, AEP should not be below OEP. A mismatch can indicate different definitions, bases, periods, or model components rather than a rounding difference.

This sequence does not establish that a particular input is erroneous; it narrows down where the discrepancy may have entered. For example, an event loss that fails a location-level reconciliation points toward exposure, footprint, damage, or financial-module inputs. An event summary that reconciles but an annual mean that does not points toward frequency, severity aggregation, or the annual output definition.

Common Diagnostic Traps

Mixing loss bases. Gross damage and insured loss are not interchangeable. Check whether deductibles and limits have been applied before comparing totals.
Mixing event and annual quantities. An event rate is not the probability of at least one event. Under a Poisson model, that probability is $1-e^{-\lambda}$.
Misreading the exceedance threshold. For discrete losses, $P(S>x)$ and $P(S\geq x)$ may differ materially.
Comparing mismatched curves. OEP and AEP should be compared only at the same loss basis, threshold, time horizon, and portfolio scope.
Treating a discrepancy as proof of one cause. A reconciliation can identify plausible causes to investigate, but the output alone may not distinguish an exposure error from a model configuration or data-processing issue.

Key takeaway: Diagnose catastrophe output from the bottom up. Reconcile event losses to exposure and model components, reconcile annual results to frequency and severity, and then check that exceedance measures obey their definitions and relationships.

Check Your Understanding

Work each problem from the stated assumptions, keeping event losses distinct from annual aggregate losses.

A hypothetical event affects three locations. Their exposure values are $2.00 million, $1.50 million, and $0.50 million, and their damage ratios are 20%, 40%, and 20%. Calculate total gross damage. A report gives $1.40 million; state the difference from your calculation. (Answer: Gross damage is $1.10 million; the report is $0.30 million higher.)
Annual event count is Poisson with rate 0.30, and every event causes a fixed $4.00 million net loss. Calculate expected annual aggregate loss and the probability annual aggregate loss exceeds $4.00 million. (Answer: Expected loss is $1.20 million; exceedance probability is 0.0369.)
A single event type has annual Poisson rate 0.08 and always causes $7 million. A second type has rate 0.12 and always causes $3 million. Calculate OEP at a threshold of $5 million. (Answer: $1-e^{-0.08}=0.0769$.)
Independent event types X and Y have Poisson annual rates 0.10 and 0.20, with fixed losses of $4 million and $2 million, respectively. Calculate AEP at $3 million, allowing losses from multiple events to accumulate. (Answer: $1-e^{-0.10}+e^{-0.10}[1-e^{-0.20}(1+0.20)]=0.1110$.)
A model reports OEP of 0.14 and AEP of 0.09 at the same loss threshold, annual horizon, and loss basis. Is this consistent with their definitions? Name one class of issue to investigate. (Answer: No; AEP cannot be below OEP under those common conditions. Investigate inconsistent definitions, thresholds, horizons, or model output configuration.)

What You'll Learn