Why Combining Scenarios Requires Care
An internal operational loss model summarizes observed events, but its data may contain few observations of severe or unusual events. A scenario estimate can provide information about those risks. Combining the two sources is not simply a matter of adding their estimated losses: the scenario may describe a risk already represented in the internal data. A defensible combination begins by deciding whether the scenario adjusts an existing model component or represents a distinct supplemental event.
The event boundary, time horizon, and loss measure must match before estimates are combined. For example, an annual frequency estimate for a defined event type cannot be added directly to a one-time loss estimate. Nor should a scenario's annual percentile loss be treated as though it were an average loss. Write down what each estimate represents, then select a treatment that preserves that meaning.
Parameter Adjustments for the Same Event Population
Suppose internal data and a scenario assessment both concern the frequency of the same defined event type. One transparent approach is to combine the two annual frequency estimates using a documented weight:
Here, \(\lambda_{\text{int}}\) is the internal-model frequency estimate, \(\lambda_{\text{scen}}\) is the scenario-based frequency estimate for the same event population and horizon, and \(w\) is a selected scenario weight. The weight is a modeling or governance choice, not a universal statistical constant. It should be documented with its rationale. This weighted estimate does not claim to be a uniquely correct estimate or a formal credibility calculation.
The same idea can be applied to a severity parameter, provided both inputs describe the same quantity. For example, if both estimates are conditional mean severities, a chosen weighted mean can adjust the model's severity mean. The severity distribution family and other parameters must then be handled consistently. A scenario's 99th percentile is not a conditional mean, so it cannot be inserted into this formula as though it were one.
When a parameter changes, recalculate model outputs using the adjusted parameter. Do not add the scenario estimate as a separate loss on top of the model unless it represents events outside the model's existing event population.
Worked Example: Adjusting a Severity Mean
An insurer's internal model has annual event count \(N\sim\operatorname{Poisson}(4)\). Each event has an exponential severity with mean \(25{,}000\) dollars. A scenario assessment concerns the same event type and estimates a conditional mean severity of \(40{,}000\) dollars. The modeling committee selects \(w=0.30\) for the scenario estimate. Assume event severities are independent and identically distributed, independent of \(N\), and remain exponential after adjustment. Find the adjusted mean severity, expected annual aggregate loss, and aggregate variance. Treat the scenario as a parameter adjustment, not an additional event source.
Method. Blend the two estimates of the same conditional mean, then apply the compound Poisson formulas. This avoids adding scenario losses for events already represented by the internal frequency model.
Adjusted severity mean. The internal weight is \(1-w=0.70\), so \(0.70(25{,}000)=17{,}500\) dollars. The scenario contribution is \(0.30(40{,}000)=12{,}000\) dollars. Therefore, \(17{,}500+12{,}000=29{,}500\) dollars.
Expected annual aggregate loss. For a compound Poisson model, the expected aggregate is the frequency parameter times the mean severity: \(4(29{,}500)=118{,}000\) dollars.
Aggregate variance. For an exponential severity with mean \(\mu\), \(E[X^2]=2\mu^2\). Thus \((29{,}500)^2=870{,}250{,}000\) dollars squared, \(E[X^2]=2(870{,}250{,}000)=1{,}740{,}500{,}000\) dollars squared, and the aggregate variance is \(4(1{,}740{,}500{,}000)=6{,}962{,}000{,}000\) dollars squared.
Check. The adjusted mean, \(29{,}500\), lies between the two input means. The aggregate expectation is four times that mean, as required by the model. The variance uses the adjusted exponential distribution, not the original severity mean.
Supplemental Components for Distinct Events
If a scenario describes a distinct event population that is excluded from the internal model, it can be represented as a supplemental component. For instance, a model might represent routine processing errors with a compound frequency-severity model, while a separately defined scenario represents a rare, single-site outage. The event definitions must make that separation credible; different labels alone do not establish non-overlap.
For an internal compound Poisson component with annual frequency parameter \(\lambda\) and event severity \(X\), let \(S_{\text{int}}\) be annual aggregate loss. If a separate scenario component has annual loss \(S_{\text{scen}}\), then \(S_{\text{total}}=S_{\text{int}}+S_{\text{scen}}\). If the two components are independent, their means and variances add. Independence is an assumption, not an automatic consequence of using separate model components.
Worked Example: Adding a Distinct Single-Event Scenario
A hypothetical internal model represents routine errors with \(N\sim\operatorname{Poisson}(3)\) events per year, each producing a fixed loss of \(20{,}000\) dollars. A separate scenario represents a major outage, defined so that it is not included in the routine-error data. The outage occurs at most once per year, with probability \(0.02\), and produces a fixed loss of \(1{,}000{,}000\) dollars if it occurs. Assume the two components are independent. Calculate the total expected annual loss and variance.
Method. Calculate each component's moments and add them under the stated independence assumption. The scenario is modeled as a Bernoulli annual event, not as another routine-error claim.
Internal component. Each routine event costs \(20{,}000\) dollars. Its expected annual aggregate loss is \(3(20{,}000)=60{,}000\) dollars. For a compound Poisson sum, its variance is \(\lambda E[X^2]\). Here \(E[X^2]=(20{,}000)^2=400{,}000{,}000\) dollars squared, giving \(3(400{,}000{,}000)=1{,}200{,}000{,}000\) dollars squared.
Scenario component. Let \(B\) equal 1 if the outage occurs and 0 otherwise, so the scenario loss is \(1{,}000{,}000B\), with \(P(B=1)=0.02\). Its expected loss is \(1{,}000{,}000(0.02)=20{,}000\) dollars. Its variance is \((1{,}000{,}000)^2(0.02)(0.98)=19{,}600{,}000{,}000\) dollars squared.
Total. Independence gives an expected annual loss of \(60{,}000+20{,}000=80{,}000\) dollars and variance \(1{,}200{,}000{,}000+19{,}600{,}000{,}000=20{,}800{,}000{,}000\) dollars squared.
Check. The outage component contributes \(20{,}000\) dollars to expected loss, not \(1{,}000{,}000\) dollars, because the large loss occurs with probability \(0.02\). If the outage were already included in the routine-error component, adding this Bernoulli component would count it twice.
Why the Compound Poisson Formulas Hold
The moment calculations above follow from conditioning on the event count. Assume \(N\sim\operatorname{Poisson}(\lambda)\), independent of identically distributed severities \(X_1,X_2,\ldots\), and define \(S=\sum_{j=1}^{N}X_j\), with \(S=0\) when \(N=0\). Write \(m_1=E[X]\) and \(m_2=E[X^2]\), both assumed finite. Given \(N\), the conditional mean is \(E[S\mid N]=Nm_1\), so taking expectations gives \(E[S]=E[N]m_1=\lambda m_1\).
For the variance, conditional independence of severities gives \(\operatorname{Var}(S\mid N)=N\operatorname{Var}(X)\). The law of total variance then yields \(\operatorname{Var}(S)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)m_1^2\). A Poisson variable has both mean and variance \(\lambda\); therefore,
This result depends on the stated compound Poisson assumptions. It does not automatically apply to a fixed event count, dependent severities, or a different frequency distribution. When adding components, variance additivity also requires independence; otherwise a covariance term is needed.
Checking for Overlap Before Combining
A practical overlap review compares the scenario definition with the internal model's event boundary and data. At a minimum, check whether they share the same cause, event type, location or system, loss components, and time horizon. Also check whether scenario events or related costs are already present in the internal data, including costs recorded under a different event label.
If the scenario and internal losses concern the same population, adjust an existing parameter or otherwise revise the existing component. If the scenario is genuinely outside that population, model it separately and state the assumptions needed to combine the components. If the boundary cannot be resolved, do not quietly treat the estimates as independent and additive; document the ambiguity and the chosen treatment.
A simple overlap register can make the decision auditable:
| Question | Why it matters |
|---|---|
| Do both estimates describe the same event type and cause? | If yes, a supplemental component may duplicate events already modeled. |
| Do the time horizons match? | An annual frequency and a multi-year scenario estimate cannot be compared without conversion. |
| Are the loss measures the same? | A percentile, conditional mean, and total event loss are different quantities. |
| Are scenario-type events already in the internal records? | Existing observations may mean the scenario is not a separate component. |
| What dependence assumption is being made? | It affects the distribution of total loss and, when components are dependent, its variance. |
Worked Example: Preventing Frequency Double Counting
An internal model uses a Poisson annual frequency for a defined payment-processing event. Four years of comparable exposure contain five such events, giving an internal estimate of \(5/4=1.25\) events per year. A scenario assessment for the same event definition and exposure basis gives \(2.25\) events per year. The model uses a fixed conditional severity of \(40{,}000\) dollars. The committee assigns \(w=0.40\) to the scenario frequency estimate. Find the adjusted expected annual loss and compare it with the result of incorrectly adding the two frequencies.
Method. Because the scenario and internal records describe the same event population, blend their frequency estimates. Do not treat the scenario frequency as a second event stream.
Adjusted frequency. The internal weight is \(1-0.40=0.60\). Its contribution is \(0.60(1.25)=0.75\). The scenario contribution is \(0.40(2.25)=0.90\). Thus \(\lambda^*=0.75+0.90=1.65\) events per year.
Adjusted expected loss. With fixed severity \(40{,}000\) dollars, the expected annual loss is \(1.65(40{,}000)=66{,}000\) dollars.
Overlap trap. Adding the rates would give \(1.25+2.25=3.50\) events per year and an expected loss of \(3.50(40{,}000)=140{,}000\) dollars. That calculation incorrectly treats two estimates of the same event population as separate event streams.
Check. The adjusted rate \(1.65\) lies between \(1.25\) and \(2.25\), and the adjusted expected loss is \(1.65\) times the fixed severity. The data and scenario are not counted as two populations.
Common Traps and Practical Controls
- Adding estimates that refer to the same events. A scenario rate for an event type already modeled internally usually calls for an adjustment, not a second frequency component.
- Blending unlike quantities. Do not combine a mean severity with a percentile loss as if both were means. Translate the scenario into a compatible measure or keep it as a separately defined scenario calculation.
- Changing a mean but not the implied distribution. If the severity family is retained, state that assumption and recalculate the moments used by the aggregate model.
- Assuming independence without a basis. Separate labels do not prove independent causes. If components may co-occur or share a driver, simple variance addition may be wrong.
- Omitting units and horizon. Express frequency per the same exposure period and severity in consistent monetary units before applying a formula.
A useful record of the combination states the event boundary, the source and meaning of each input, the selected adjustment weight or supplemental distribution, the overlap check, and the assumptions used to calculate aggregate results. This makes it possible to distinguish a data-informed parameter change from an additional modeled risk.
Check Your Understanding
For each problem, identify the treatment and compute the requested result using the stated assumptions.
- An internal frequency estimate is 1.8 events per year and a scenario estimate for the same event type is 2.6. Use scenario weight \(w=0.25\) and fixed severity of \(30{,}000\) dollars. Find the adjusted frequency and expected annual loss. (Answer: 2.0 events per year; $60,000.)
- An internal model has Poisson frequency 2.5 per year and exponential severity with mean \(12{,}000\) dollars. A scenario estimates a conditional mean of \(20{,}000\) dollars for the same event type. Use \(w=0.25\), retain the exponential family, and find the adjusted expected annual loss. (Answer: adjusted mean severity $14,000; expected annual loss $35,000.)
- An internal model has Poisson frequency 1.2 per year and fixed severity \(10{,}000\) dollars. A distinct, non-overlapping event occurs at most once per year with probability 0.05 and loss \(200{,}000\) dollars. Assuming independence, find the combined expected annual loss. (Answer: $22,000.)
- Five internal events occurred over four comparable years. A scenario rate of 2.25 per year describes those same events. Use \(w=0.40\) and fixed severity \(40{,}000\) dollars. Find the adjusted expected annual loss and the expected loss from incorrectly adding the two rates. (Answer: adjusted $66,000; incorrect additive result $140,000.)
- An internal severity model has mean \(50{,}000\) dollars. A scenario reports that annual aggregate loss exceeds \(500{,}000\) dollars at its 99th percentile, but gives no event frequency or conditional mean severity. Can the \(500{,}000\) figure be blended directly with the internal severity mean? State what must be clarified before choosing a combination method. (Answer: No; a 99th-percentile annual aggregate loss is not a conditional mean severity. Clarify its event boundary, horizon, and loss measure, and obtain a compatible parameter estimate or model it as an explicitly defined aggregate scenario.)