Uncertainty Sources in Catastrophe Model Results

Why Catastrophe Model Results Are Uncertain

A catastrophe model result is conditional on many choices and data items: which properties are included, how events are represented, what hazard reaches each location, how damage responds to that hazard, and how policy terms convert damage to insured loss. A reported loss estimate is therefore not just a number. It is an output of a chain of assumptions and inputs, each of which may be uncertain.

It is useful to distinguish two broad ideas. Stochastic variability describes the different event outcomes that can occur from year to year under a specified model. Epistemic uncertainty describes imperfect knowledge about the exposures, event representation, parameters, or model structure used to produce the result. The distinction matters: a large simulated event count can describe annual variability under the assumed model, but it does not by itself resolve whether the model or its inputs are appropriate.

Where Uncertainty Enters the Catastrophe Model

A practical classification follows the order in which information is used. The categories below identify where to investigate; they are not mutually exclusive. For example, an uncertain location coordinate is an exposure-data issue, but its impact may be expressed through uncertainty in the hazard assigned to that location.

The event catalog deserves particular care. Once a catalog and its rates are fixed, random event occurrence creates annual loss variability. But uncertainty about whether the catalog includes all relevant events, or whether its rates are appropriate, is epistemic uncertainty about the representation of possible events. The same distinction applies to hazard and vulnerability: event-to-event or damage-to-damage variation can be stochastic within a specified model, while uncertainty about the model or its numerical parameters is epistemic.

Financial terms are often treated as deterministic once the contract is known. For a loss \(X\) subject to an ordinary per-loss deductible \(d\) and limit \(u\), the payment is \(Y^L=\min((X-d)_+,u)\), where \((z)_+=\max(z,0)\). If \(d\) and \(u\) are verified, they transform the modeled loss rather than introducing uncertainty by themselves. If the policy record is incomplete, or contract wording admits more than one interpretation, the financial module’s result can be uncertain even when the event damage estimate is unchanged.

Input, Parameter, and Model Uncertainty

The word “input” is broad: it can refer to observed exposure data, a selected event catalog, or values passed between model components. Parameter uncertainty is narrower. It concerns uncertain numerical values under a specified structure, such as an estimated event rate or a coefficient in a vulnerability curve. Model uncertainty concerns which structure is appropriate—for example, whether a hazard relationship has the chosen form or whether an important damage mechanism is represented at all.

These labels describe different questions, not competing explanations. If an event rate is estimated from limited information, that is parameter uncertainty. If analysts are unsure whether the catalog’s event-generation structure adequately represents possible events, that is model uncertainty. If the rate supplied to a run is also a selected input, it is reasonable to describe the uncertainty as input uncertainty as well. A useful report states the source and the level of the uncertainty, rather than relying on a label alone.

Event Catalog Uncertainty and Annual Variability

A stochastic event set can generate different annual totals even if its rates and event severities are held fixed. That is annual variability conditional on the catalog. Changing the event rates or the set of events represented changes the model inputs or assumptions; it is not simply another realized year from the same fixed event process.

For a Poisson event count \(N\) with annual rate \(\lambda\), the probability of at least one event in a year is \(P(N\geq1)=1-e^{-\lambda}\). This formula assumes a Poisson count with constant rate over the stated year. It describes occurrence probability conditional on the rate; uncertainty about the rate itself is not included unless a distribution or other treatment for that rate is specified.

A Variance Decomposition for Nested Uncertainty

A useful way to explain the relationship between annual variability and uncertainty in a model quantity is the law of total variance. Let \(L\) be annual loss and let \(\Theta\) represent an uncertain model state or parameter value. The model specifies a conditional distribution of \(L\) given \(\Theta\). Then:

The first term is the average conditional annual variance across model states. The second is the variance of conditional mean losses across those states. This separates variability within a specified state from uncertainty in the state-specific mean. It does not automatically quantify all model uncertainty: a decision-maker must define the states and how they are weighted. In particular, weights assigned to alternative structures need not be objective probabilities.

Why the formula holds. Write \(L-E[L]\) as \((L-E[L\mid\Theta])+(E[L\mid\Theta]-E[L])\), then square and take expectations. The expected cross-product is zero because \(E[L-E[L\mid\Theta]\mid\Theta]=0\). The first squared term has expectation \(E[\operatorname{Var}(L\mid\Theta)]\); the second has expectation \(\operatorname{Var}(E[L\mid\Theta])\). Their sum gives the stated decomposition.

Interpreting and Communicating Uncertainty

The purpose of classification is to make a modeled result interpretable. If a result changes, first identify what changed: the exposure record, event catalog, hazard output, vulnerability relationship, or financial terms. Then distinguish a change in the stipulated input or model from a different random annual outcome under unchanged assumptions. This prevents a simulated range from being presented as if it captured every uncertainty in the model.

A concise uncertainty statement should identify the result, the assumptions held fixed, and the uncertain elements included. For example, a range across simulated years under one fixed event catalog describes annual variability conditional on that catalog. A range across alternative event-rate estimates describes parameter uncertainty, if the model structure is fixed. A comparison across different hazard-model structures describes model uncertainty. The range alone does not say which interpretation is appropriate.

Finally, uncertainty can be attenuated or hidden by financial terms. In the first worked example, the limit was high enough that the exposure discrepancy changed payment directly. If both calculated amounts had exceeded the limit, the payment would have been the same even though the gross damage differed. An unchanged insured loss therefore does not prove that upstream exposure, hazard, or vulnerability estimates are certain; it may only show that the financial transformation masks their differences.