Why Event Catalogs Need Careful Interpretation
Catastrophe models often represent possible events in a stochastic event catalog. The catalog is not simply a list of disasters that happened in the past: it is a modeled set of possible events, with rates or weights intended to describe how often those events may occur. A simulation uses that information to generate many possible years of experience. Interpreting the results requires keeping three things separate: the modeled events, their annual rates, and the simulated years in which events occur.
That distinction matters in ratemaking. A catalog may include events that have not occurred in the historical record, while a historical record may omit rare events that remain relevant to future risk. The simulated years are model-generated outcomes, not additional observations of actual history. Their results depend on how the catalog was constructed and how its rates were used.
Rates, Probabilities, and Simulated Years
An annual event rate describes modeled frequency, not necessarily the probability of at least one occurrence in a year. Under a Poisson frequency model with annual rate \(\lambda\), the number of occurrences \(N\) in one year has a Poisson distribution with parameter \(\lambda\). Thus \(E[N]=\lambda\), and the probability of at least one occurrence is \(1-e^{-\lambda}\). These quantities are close when \(\lambda\) is small, but they are not identical.
If catalog events are modeled as independent Poisson occurrence processes, their counts can be simulated separately. The total number of occurrences in a year then has a Poisson distribution with rate equal to the sum of the event rates. Independence is an assumption, not a universal feature of catastrophe risk. A model may instead represent dependence, clustering, or other frequency structures; in those cases, the appropriate simulation follows that specified structure.
The number of simulated years controls how many model-generated outcomes are observed. It does not change the underlying annual rates. For example, simulating 100,000 years rather than 10,000 years generally makes estimates less affected by simulation noise, but does not make the modeled events intrinsically more frequent.
Worked Example: From Event Rates to Event-Containing Years
A hypothetical catastrophe catalog has three event groups, modeled as independent Poisson processes with annual rates 0.04, 0.03, and 0.02. The model simulates 50,000 years. Find the total annual rate, the probability a year contains at least one occurrence, and the expected counts of occurrences and event-containing years in the simulation. No interest or mortality assumption is relevant. The frequency distributions are Poisson with the stated rates.
Method. Add the independent Poisson rates to obtain the total rate, then use the Poisson probability of zero occurrences to find the probability of at least one. The expected occurrence count over simulated years is the number of years multiplied by the annual rate.
Total rate. The annual rate is \(0.04+0.03+0.02=0.09\).
Probability of at least one occurrence. For one simulated year, \(P(N\geq 1)=1-e^{-0.09}=0.0861\), rounded to four decimals.
Expected occurrence counts. The expected counts for the three groups are \(50{,}000(0.04)=2{,}000\), \(50{,}000(0.03)=1{,}500\), and \(50{,}000(0.02)=1{,}000\). The expected total number of occurrences is \(2{,}000+1{,}500+1{,}000=4{,}500\).
Expected event-containing years. The expected number of years with one or more occurrences is \(50{,}000(0.0861)\), or about 4,305 years. This is less than the expected 4,500 occurrences because a year can contain more than one occurrence.
Check and trap. The calculations are consistent: occurrences count event instances, while event-containing years count years, so these totals need not match. A common mistake is to treat the annual rate 0.09 as exactly a 9% chance of an event. The Poisson probability is 8.61%, not 9%.
What the Catalog Represents
Catalog design determines which modeled events are available to the simulation and how much weight each receives. Relevant design choices include the geographic domain, the definition and granularity of an event, the modeled range of event sizes, the annual rates or weights, and the period or conditions represented by the model. If a catalog omits a type of event or gives it an unsuitable rate, running more simulated years will not correct that limitation.
A rate can apply to a particular event record, a broader event group, or a weighted set of possible outcomes. The analyst must identify what the rate refers to before summing rates or interpreting simulated counts. In particular, a catalog with many event records does not necessarily describe more annual activity than a catalog with fewer records: its rate structure and event definitions matter.
A historical event set answers a different question. It records what was observed over a specified period, subject to the record’s coverage and completeness. The count of historical events divided by the number of years can be a descriptive historical frequency, but it is not automatically the model’s forward-looking rate. The modeled catalog may account for possible events absent from the observed period. Conversely, its completeness and assumptions must be evaluated rather than accepted merely because it is stochastic.
| Feature | Stochastic event catalog | Historical event set |
|---|---|---|
| What it contains | Modeled possible events and frequency weights | Events observed during a defined period |
| How to interpret frequency | According to the model’s stated rate or weight convention | As observed experience, subject to the record’s scope and completeness |
| Role of years | Simulated years are generated outcomes | Calendar years are part of the actual observation period |
| Main limitation | Results depend on catalog construction and assumptions | Rare events may be absent from a finite record |
Using Event Rates to Check Annual Loss Results
A simple catalog can also provide a check on simulated annual loss. Suppose each event type \(j\) occurs according to an independent Poisson process with annual rate \(\lambda_j\), and each occurrence produces a fixed loss \(L_j\). If \(N_j\) is the number of occurrences of type \(j\) during a year, total annual loss is \(S=\sum_j N_jL_j\). Linearity of expectation gives \(E[S]=\sum_j L_jE[N_j]\). Since \(E[N_j]=\lambda_j\), the expected annual loss is \(\sum_j\lambda_jL_j\). This calculation is a check under the stated assumptions, not a substitute for understanding the catalog’s definitions.
The result follows because each event type contributes its loss per occurrence multiplied by its expected number of occurrences. It does not require that different types have different losses, and linearity of expectation itself does not require independence. The Poisson assumptions establish the specified expected counts; if event rates vary by year, losses are random, or the model uses dependence structures, the expected-loss calculation must reflect those features.
Worked Example: Checking a Simulation Against Catalog Rates
A hypothetical catalog contains three independent Poisson event types. Their annual rates and fixed losses per occurrence are: type A, rate 0.02 and loss $40,000; type B, rate 0.03 and loss $120,000; type C, rate 0.05 and loss $50,000. A simulation covers 10,000 years and produces 195, 312, and 488 occurrences of types A, B, and C, respectively. Calculate the rate-based expected annual loss and the simulation estimate. No interest or mortality assumption is relevant; event losses are fixed at the stated amounts.
Method. First calculate the rate-based expectation by multiplying each annual rate by its loss. Then calculate total simulated loss by multiplying each observed event count by its loss, and divide by the number of simulated years.
Rate-based expectation. Type A contributes \(0.02($40{,}000)=$800\) per year. Type B contributes \(0.03($120{,}000)=$3{,}600\) per year. Type C contributes \(0.05($50{,}000)=$2{,}500\) per year. Therefore, the expected annual loss is \($800+$3{,}600+$2{,}500=$6{,}900\).
Simulation estimate. The simulated loss from type A is \(195($40{,}000)=$7{,}800{,}000\). From type B it is \(312($120{,}000)=$37{,}440{,}000\). From type C it is \(488($50{,}000)=$24{,}400{,}000\). Total simulated loss is \($7{,}800{,}000+$37{,}440{,}000+$24{,}400{,}000=$69{,}640{,}000\). Dividing by 10,000 years gives a simulation estimate of \($6{,}964\) per year.
Check and trap. The observed counts are near their rate-based expectations of 200, 300, and 500 occurrences, and the simulated estimate is near $6,900. It need not equal $6,900 exactly because the simulated counts vary. A common mistake is to divide total simulated loss by the number of event records rather than by the 10,000 simulated years; that would not produce an annual estimate.
Catalog Size, Weights, and Simulation Stability
A simulation estimate is based on the particular outcomes generated. For event type \(j\), the average simulated count per year can be compared with its catalog rate. A substantial difference may arise from random variation, especially for rare events, but it can also indicate a mismatch between the rate convention and the simulation procedure. Checking event counts is therefore a useful diagnostic before interpreting loss results.
More simulated years generally reduce the sampling variability of estimates when the simulation process is unchanged. They do not remove uncertainty about whether the catalog’s event definitions, geographic coverage, rates, or assumptions are appropriate. Nor do they turn modeled outcomes into historical observations. A very large simulation based on a poorly specified catalog can produce a precise estimate of the wrong modeled quantity.
Weights require particular care. A catalog may use event rates, sampling probabilities, or other weights, and these are not interchangeable. A sampled event’s share of records is not automatically its annual occurrence probability. Before using a count or rate, confirm the unit (per year or another period), the event definition, and whether the weight describes occurrence frequency or only how the simulation samples possible events.
Worked Example: Comparing a Historical Record With a Stochastic Catalog
A hypothetical historical record contains one event with a loss of $1,200,000 during 25 years. Separately, a stochastic catalog models one event type with a Poisson annual rate of 0.08 and a fixed loss of $550,000 per occurrence. In 100,000 simulated years, this event occurs 8,120 times. Compare the descriptive historical average loss per year with the catalog’s rate-based expected annual loss and its simulation estimate. No interest or mortality assumption is relevant; the catalog event loss is deterministic.
Method. Divide historical losses by historical years for the descriptive historical average. For the catalog, multiply the annual rate by loss per occurrence. For the simulation estimate, divide simulated total loss by simulated years.
Historical average. The observed total loss is $1,200,000, so the historical average is \($1{,}200{,}000/25=$48{,}000\) per year.
Catalog expectation. The rate-based expectation is \(0.08($550{,}000)=$44{,}000\) per year.
Simulation estimate. Total simulated loss is \(8{,}120($550{,}000)=$4{,}466{,}000{,}000\). Dividing by 100,000 years gives \($44{,}660\) per year. The simulated occurrence rate is \(8{,}120/100{,}000=0.0812\) per year, close to the catalog rate of 0.08.
Check and trap. The three figures are $48,000, $44,000, and $44,660 per year, respectively. They refer to different quantities: a finite historical average, the catalog’s model-based expectation, and one finite simulation’s estimate. It would be a mistake to treat the historical average as proof that the catalog rate is wrong, or to treat the simulated estimate as exactly equal to its theoretical expectation.
Practical Interpretation
When reviewing a stochastic catalog, make the assumptions visible before interpreting results. Confirm the period to which rates apply, whether rates describe occurrences or probabilities, and whether event records can occur more than once in a year. Check whether the simulation creates independent annual replicates or uses another dependence structure. Then assess whether the modeled event set and its weights are appropriate for the portfolio and purpose being studied.
A useful sequence is to state what an event record means, verify how its rate enters annual simulation, and compare simulated counts with their expected values. Only after those checks should an analyst draw conclusions from simulated annual experience. This discipline is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus because it tests whether a candidate can interpret the modeled frequency behind catastrophe estimates, rather than treating a catalog as a self-explanatory data file.
Determine what each catalog record represents and the time unit and meaning of its frequency weight.
For example, distinguish a Poisson annual rate from a direct annual probability, and note any dependence assumptions.
Keep occurrence counts, event-containing years, historical observations, and simulated years distinct.
Compare observed simulation counts with their expected levels, while allowing for random variation.
Check Your Understanding
For each problem, identify the frequency interpretation before calculating.
- A catalog has two independent Poisson event types with annual rates 0.01 and 0.05. What is the probability that a simulated year has at least one event? Give the probability to four decimals. (Answer: \(1-e^{-0.06}=0.0582\))
- A single event type has a Poisson annual rate of 0.12. In 100,000 simulated years, what are the expected number of occurrences and the expected number of years containing at least one occurrence? Round the counts to the nearest whole year or occurrence. (Answer: 12,000 occurrences; 11,308 event-containing years)
- Two independent Poisson event types have annual rates 0.01 and 0.04, with fixed losses of $200,000 and $50,000 per occurrence. Find the rate-based expected annual loss. (Answer: \(0.01($200{,}000)+0.04($50{,}000)=$4{,}000\) per year)
- A historical record contains two events in 40 years. A model uses a Poisson annual rate of 0.07 for its corresponding event type. Give the descriptive historical frequency and the model’s expected number of occurrences over 100 years; explain why the two quantities need not agree. (Answer: Historical frequency \(2/40=0.05\) per year; model expectation \(100(0.07)=7\) occurrences; they describe finite observed history and a model-based expectation, respectively.)
- A catalog has one fixed-loss event type with annual Poisson rate 0.02 and loss $300,000 per occurrence. A 5,000-year simulation produces 112 occurrences. Calculate the rate-based expected annual loss and the simulation estimate. (Answer: $6,000 per year; \(112($300{,}000)/5{,}000=$6{,}720\) per year)