Why Sensitivity Analysis Matters
A catastrophe model produces results conditional on inputs: the exposure represented, the hazard assumptions, the vulnerability functions, and the financial terms applied to modeled damage. Sensitivity analysis asks a controlled question: how would the output change if one specified input, or a defined set of inputs, were different? This is useful for understanding which assumptions drive a result and whether a decision depends on a particular modeling choice.
A sensitivity is not, by itself, a probability statement about what will happen. For example, recalculating loss after increasing a hazard rate by 20% describes the model response to that change. It does not say that the rate has a 20% chance of being higher. Uncertainty analysis instead represents uncertainty in inputs or model structure, often with ranges or probability distributions. Keeping these questions separate prevents a scenario result from being misread as a forecast.
Choose the Output and Control the Comparison
The output should match the question. Expected annual loss (EAL) summarizes the average modeled annual loss. An occurrence exceedance probability (OEP) at threshold \(x\) is the probability that the largest event loss in a year exceeds \(x\). An aggregate exceedance probability (AEP) instead concerns the total loss in a year. A change can have a modest effect on EAL but a substantial effect on the tail of an exceedance curve, or vice versa.
A useful comparison begins by fixing a baseline, naming the input being changed, and holding all other assumptions constant. For each scenario, record the revised input and recalculate the same output using the same definitions, loss basis, and threshold. If several assumptions change together, label the result a combined scenario; do not attribute the full output change to any one input.
For a simple model with event type \(j\) occurring at annual rate \(\lambda_j\) and deterministic loss \(L_j\) per event, assume independent Poisson event counts. The expected annual loss is the sum of each event type's expected loss contribution. The OEP formula follows from the probability that no event in any qualifying type occurs.
For OEP, “qualifying” means an event loss strictly greater than the stated threshold \(x\). A Poisson count with rate \(\lambda\) has probability \(e^{-\lambda}\) of zero events. Independent qualifying event types have a combined rate equal to the sum of their rates, so the probability of no qualifying event is the exponential of the negative combined rate.
Worked Example: Separating Input Effects on Expected Loss
Worked Example: Separating Input Effects on Expected Loss
A hypothetical insured portfolio has an insured value of $20.00 million. A modeled event type occurs at an annual Poisson rate of 0.0400 and produces a damage ratio of 25% of insured value. Assume the financial module pays the full modeled damage, with no deductible or limit. Calculate baseline EAL, then test (a) insured value increased by 10%, (b) event rate increased by 25%, (c) damage ratio increased by 20% relative, and (d) all three changes together. Identify the method and a common mistake to avoid.
Method. Calculate the event loss as insured value times damage ratio, then multiply by the annual event rate. For the controlled one-at-a-time sensitivities, change only the named input.
Baseline. Event loss is \(20.00\times0.25=\$5.00\) million. Therefore, \( \operatorname{EAL}=0.0400\times\$5.00=\$0.200\) million, or $200,000 per year.
(a) Insured value increases by 10%. The revised value is \(20.00\times1.10=\$22.00\) million. Event loss is \(22.00\times0.25=\$5.50\) million, so EAL is \(0.0400\times5.50=\$0.220\) million.
(b) Event rate increases by 25%. The revised rate is \(0.0400\times1.25=0.0500\). Loss remains $5.00 million, so EAL is \(0.0500\times5.00=\$0.250\) million.
(c) Damage ratio increases by 20% relative. The revised ratio is \(0.25\times1.20=0.30\), not 0.45. Event loss is \(20.00\times0.30=\$6.00\) million, and EAL is \(0.0400\times6.00=\$0.240\) million.
(d) All three changes together. Event loss is \(22.00\times0.30=\$6.60\) million, and the rate is 0.0500. EAL is \(0.0500\times6.60=\$0.330\) million. This is also the baseline EAL multiplied by \(1.10\times1.25\times1.20=1.65\): \(0.200\times1.65=\$0.330\) million.
Check and common mistake. The combined increase is 65%, not 55%, because the three factors multiply. Adding the percentages would miss the interaction between changed exposure and changed damage ratio. The direct recalculation confirms the combined result.
Why the Exceedance Curve May Move Differently
EAL weights each event loss by its rate and adds the contributions. An exceedance curve instead asks which event losses cross each threshold. Consequently, a rate change affects thresholds already crossed by that event type, while a severity change may cause the event to cross new thresholds. This threshold effect is why an OEP curve should be recalculated, rather than inferred from the percentage change in EAL.
Worked Example: Updating OEP After Rate and Loss Changes
A hypothetical model has two independent Poisson event types. Type A has annual rate 0.0600 and loss $4.00 million; Type B has rate 0.0200 and loss $10.00 million. In a combined scenario, both rates increase by a factor of 1.25 and both losses increase by a factor of 1.10. Calculate baseline and scenario EAL, and OEP above $9.00 million. The method is to update each rate and loss, then identify only the event types whose losses strictly exceed the threshold.
Baseline EAL. Type A contributes \(0.0600\times4.00=\$0.240\) million. Type B contributes \(0.0200\times10.00=\$0.200\) million. Thus baseline EAL is \(0.240+0.200=\$0.440\) million.
Scenario EAL. The revised rates are \(0.0600\times1.25=0.0750\) and \(0.0200\times1.25=0.0250\). The revised losses are \(4.00\times1.10=\$4.40\) million and \(10.00\times1.10=\$11.00\) million. The contributions are \(0.0750\times4.40=\$0.330\) million and \(0.0250\times11.00=\$0.275\) million. Scenario EAL is \(0.330+0.275=\$0.605\) million.
Baseline OEP above $9.00 million. Type A's $4.00 million loss does not exceed $9.00 million. Type B's $10.00 million loss does, so the qualifying rate is 0.0200. The OEP is \(1-e^{-0.0200}=0.0198\), to four decimals.
Scenario OEP above $9.00 million. The revised Type A loss of $4.40 million still does not exceed the threshold. The revised Type B loss of $11.00 million does, so the qualifying rate is 0.0250, not the sum of both revised rates. The OEP is \(1-e^{-0.0250}=0.0247\), to four decimals.
Check and common mistake. Counting both scenario event types would use rate \(0.0750+0.0250=0.1000\) and produce \(1-e^{-0.1000}=0.0952\). That is wrong for the $9.00 million threshold because Type A does not qualify. The calculation also checks that the scenario loss and rate changes were both applied to EAL.
Financial Assumptions Can Change Which Losses Exceed a Threshold
Financial terms are applied to modeled damage to obtain the payment measure of interest. For a per-occurrence deductible \(d\) and per-occurrence limit \(u\), a simple payment on gross event loss \(L\) is \(Y=\min(\max(L-d,0),u)\). This expression assumes the limit caps the insurer's payment after the deductible. Actual contract wording may define attachment, limit, or aggregation differently, so the model should follow the stated terms.
Worked Example: Testing a Per-Occurrence Limit
A hypothetical policy has per-occurrence deductible $2.00 million and limit $10.00 million. Two independent Poisson event types have annual rates 0.0300 and 0.0100, with gross event losses $8.00 million and $16.00 million. Calculate insurer EAL and OEP above $12.00 million. Then increase only the limit to $15.00 million and recalculate. The method is to apply the financial terms to each event loss before calculating the outputs.
Baseline payments. For the $8.00 million event, the loss after deductible is \(8.00-2.00=\$6.00\) million, below the $10.00 million limit, so payment is $6.00 million. For the $16.00 million event, loss after deductible is \(16.00-2.00=\$14.00\) million, capped at $10.00 million.
Baseline EAL. The contributions are \(0.0300\times6.00=\$0.180\) million and \(0.0100\times10.00=\$0.100\) million. Insurer EAL is $0.280 million.
Baseline OEP above $12.00 million. Neither payment exceeds $12.00 million: the largest payment is exactly $10.00 million. Therefore, the qualifying rate is zero and OEP is \(1-e^0=0.0000\).
Revised limit of $15.00 million. The first payment remains $6.00 million. The second becomes \(\min(14.00,15.00)=\$14.00\) million. Revised EAL is \(0.0300\times6.00+0.0100\times14.00=0.180+0.140=\$0.320\) million. Only the second payment now exceeds $12.00 million, so revised OEP is \(1-e^{-0.0100}=0.0100\), to four decimals.
Check and common mistake. The limit is applied to the insurer's payment after the deductible, not directly to the gross loss. The increased limit changes both EAL and the OEP at $12.00 million, but a threshold below $6.00 million would not show the same OEP change: both event types would qualify under either limit.
Interpreting Sensitivities Without Overclaiming
A well-designed sensitivity comparison identifies the baseline, the changed assumption, the held-fixed assumptions, the output measure, and any threshold convention. This makes it possible to explain what the result does and does not establish. For example, a higher modeled EAL under a higher event rate establishes the model's response to that rate change; it does not establish that the original rate was wrong or quantify the probability of the alternative.
If a scenario changes exposure, hazard, vulnerability, and financial terms simultaneously, report it as a combined scenario. To understand separate effects, run controlled changes as well. The combined result may not equal the sum of individual dollar changes when inputs interact multiplicatively or when deductibles, limits, or exceedance thresholds create nonlinear effects.
State the portfolio, loss basis, assumptions, and output measure being compared.
Give the changed input and its scale, such as a relative percentage change, a percentage-point change, or a replacement assumption.
Apply the model's exposure, hazard, vulnerability, and financial logic to the alternative assumptions before computing EAL or exceedance probabilities.
Describe the output difference as conditional on the scenario. Do not call it a probability or a forecast unless the analysis supplies the required probabilistic basis.
Check Your Understanding
For each question, state or use the specified assumptions. Probabilities are rounded to four decimals.
- A hypothetical portfolio has insured value $15.00 million, damage ratio 20%, and annual event rate 0.0500. If insured value increases by 8% with the other inputs fixed and all damage is paid, what are baseline and revised EAL? (Answer: $0.150 million and $0.162 million.)
- Two independent event types have rates 0.0400 and 0.0150 and losses $3.00 million and $12.00 million. What is baseline EAL and OEP above $10.00 million? (Answer: $0.300 million; 0.0149.)
- In Question 2, increase only the second event type's rate by 40%. What are revised EAL and OEP above $10.00 million? (Answer: $0.372 million; 0.0208.)
- A single event type has annual rate 0.0250, insured value $30.00 million, and damage ratio 20%; all damage is paid. If the damage ratio increases by 15% relative, what is the revised EAL and its absolute increase from baseline? (Answer: revised EAL $0.1725 million; increase $0.0225 million.)
- Two independent event types have rates 0.0200 and 0.0100 and gross losses $9.00 million and $20.00 million. A per-occurrence deductible of $3.00 million and limit of $12.00 million apply. If only the deductible is reduced to $2.00 million, with limit unchanged, what is revised insurer EAL and OEP above $11.00 million? (Answer: $0.260 million; 0.0100.)