From an Event Footprint to an Insured Loss
A catastrophe event affects locations differently. A location’s insured value alone does not determine its loss: the event’s hazard intensity at that location and the exposure’s vulnerability also matter. Policy terms then transform physical damage into an insurer payment. A sound event-level calculation keeps these stages separate so that assumptions and errors can be identified.
This tutorial follows one hypothetical event through those stages. It calculates damage and policy payment for the event as a whole, but does not construct an annual loss distribution. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
The event footprint assigns an intensity to each location, either directly or by a stated interpolation rule. A vulnerability function maps that intensity to a damage ratio, usually between zero and one. The ratio is the modeled fraction of the insured value damaged under the event. The calculation must use consistent exposure values and units: for example, an intensity measured in units of acceleration must be compared with a vulnerability curve defined on the same intensity scale.
For the examples below, assume the insured values and event intensities are known, the stated vulnerability relationship applies to each exposure, and no other coverage features apply unless specified. Damage is calculated before policy terms. A “limit” is treated as a maximum insurer payment under the stated coverage, not a maximum ground-up damage amount.
A Consistent Calculation Sequence
Location-level terms and occurrence-level terms operate on different amounts. A location deductible and limit are applied separately to the damage at each location. The resulting location payments are then added. If the contract specifies an occurrence deductible and limit on that sum, those terms are applied once to the event total. The policy wording determines the order; the sequence below is an explicit assumption, not a universal rule for every contract.
Identify the affected locations, their insured values, and the event being modeled. Do not add losses from different events at this stage.
Use the event footprint or a stated interpolation method to obtain an intensity for each location.
Read or interpolate the damage ratio from the vulnerability function, then multiply that ratio by the location’s insured value.
Calculate each location payment under its own deductible and limit, then sum those payments for the event.
Apply the occurrence deductible and limit once to the aggregate location payment, if the stated policy wording calls for that order.
Worked Event Calculations
Worked Example: Multiple Locations With Location and Occurrence Terms
A hypothetical wind event affects three insured buildings. The event intensities and deterministic damage ratios are given below. Each building has a $10,000 deductible and a $250,000 payment limit. After the three location payments are summed, the policy applies one $25,000 occurrence deductible and a $300,000 occurrence payment limit. Calculate the event payment.
| Location | Insured value | Damage ratio | Ground-up damage |
|---|---|---|---|
| A | $1,000,000 | 0.30 | To calculate |
| B | $600,000 | 0.20 | To calculate |
| C | $400,000 | 0.10 | To calculate |
Method. Calculate physical damage first, apply each location’s deductible and limit, and only then apply the occurrence terms to the sum. The stated policy sequence and all amounts are in dollars.
Location A. Ground-up damage is \(D_A=\$1{,}000{,}000(0.30)=\$300{,}000\). After the deductible, the amount is \(\$300{,}000-\$10{,}000=\$290{,}000\). The location limit caps payment at \(\$250{,}000\), so \(L_A=\$250{,}000\).
Location B. Ground-up damage is \(D_B=\$600{,}000(0.20)=\$120{,}000\). After the deductible, the amount is \(\$120{,}000-\$10{,}000=\$110{,}000\), below the location limit. Thus \(L_B=\$110{,}000\).
Location C. Ground-up damage is \(D_C=\$400{,}000(0.10)=\$40{,}000\). After the deductible, the amount is \(\$40{,}000-\$10{,}000=\$30{,}000\), also below the location limit. Thus \(L_C=\$30{,}000\).
The sum before occurrence terms is \(\$250{,}000+\$110{,}000+\$30{,}000=\$390{,}000\). Subtract the occurrence deductible: \(\$390{,}000-\$25{,}000=\$365{,}000\). The occurrence limit reduces that amount to \(\min(\$365{,}000,\$300{,}000)=\$300{,}000\). The event payment is therefore $300,000.
Check. The event payment does not exceed the $300,000 occurrence limit. Location A’s payment is capped at its location limit, while B and C each have their deductible subtracted. The common mistake avoided here is applying the occurrence deductible separately to each building; the wording says it applies once to the event total.
Worked Example: Interpolating Intensity and Vulnerability
A hypothetical earthquake footprint gives intensity at two grid points: \(0.4g\) and \(0.8g\). Location A lies one-quarter of the way from the first point to the second. Location B has an assigned intensity of \(0.5g\). The vulnerability curve is specified by the points \((0.2g,0.05)\), \((0.6g,0.25)\), and \((1.0g,0.65)\); use straight-line interpolation between adjacent points. A has insured value $800,000 and B has insured value $500,000. Each location has a $20,000 deductible and a $220,000 payment limit. The occurrence deductible is $30,000 and the occurrence payment limit is $240,000. Find the event payment.
Method. Interpolate the event intensity first, then interpolate the damage ratio. Apply those ratios to insured values before calculating policy payments.
For A, the intensity is \(I_A=0.4g+0.25(0.8g-0.4g)=0.5g\). For B, \(I_B=0.5g\) as given. Both intensities lie between \(0.2g\) and \(0.6g\). Across that interval the damage ratio rises from 0.05 to 0.25, a change of 0.20 over \(0.4g\). At \(0.5g\), the fraction of the interval traversed is \((0.5-0.2)/(0.6-0.2)=0.3/0.4=0.75\). Therefore each damage ratio is \(0.05+0.75(0.20)=0.20\).
A’s ground-up damage is \(\$800{,}000(0.20)=\$160{,}000\). After the deductible, it is \(\$160{,}000-\$20{,}000=\$140{,}000\), below the $220,000 limit, so \(L_A=\$140{,}000\). B’s damage is \(\$500{,}000(0.20)=\$100{,}000\). After its deductible, B’s payment is \(\$100{,}000-\$20{,}000=\$80{,}000\), also below the limit. Thus \(L_B=\$80{,}000\).
The location payments sum to \(\$140{,}000+\$80{,}000=\$220{,}000\). The occurrence deductible reduces this to \(\$220{,}000-\$30{,}000=\$190{,}000\). That is below the $240,000 occurrence limit, so the final event payment is $190,000.
Check. Both damage ratios are between the curve’s values 0.05 and 0.25, as expected for intensity \(0.5g\). The final payment is less than the location-payment sum because the occurrence deductible applies; the occurrence limit does not bind.
Stochastic Vulnerability and Policy Terms
A vulnerability model may describe a distribution of possible damage ratios rather than one fixed ratio. In that case, a simulated event realization has a particular damage amount and corresponding policy payment. If the quantity of interest is an expected payment, calculate the payment for each possible damage outcome and then average the payments using their probabilities.
This order matters because deductibles and limits are nonlinear: in general, the expected payment under policy terms is not the payment calculated from expected damage. In notation, for a payment function \(g\), \(E[g(D)]\) need not equal \(g(E[D])\). The following example shows the difference directly.
Worked Example: Expected Payment Under a Stochastic Damage Ratio
For one hypothetical location, event damage is $50,000 with probability 0.20, $200,000 with probability 0.50, and $400,000 with probability 0.30. The policy has a $50,000 deductible and a $200,000 payment limit. Calculate the expected insurer payment, and compare it with applying the terms to expected damage.
Method. Apply the deductible and limit to each outcome, then probability-weight the resulting payments. All outcomes belong to the same specified event scenario; the probabilities describe damage uncertainty conditional on that scenario.
For $50,000 damage, payment is \(\min(\$200{,}000,\max(\$50{,}000-\$50{,}000,0))=\$0\). For $200,000 damage, payment is \(\min(\$200{,}000,\$200{,}000-\$50{,}000)=\$150{,}000\). For $400,000 damage, the amount after the deductible is $350,000, which is capped at $200,000.
The expected payment is \(0.20(\$0)+0.50(\$150{,}000)+0.30(\$200{,}000)=\$0+\$75{,}000+\$60{,}000=\$135{,}000\).
For comparison, expected damage is \(0.20(\$50{,}000)+0.50(\$200{,}000)+0.30(\$400{,}000)=\$10{,}000+\$100{,}000+\$120{,}000=\$230{,}000\). Applying the terms to that amount gives \(\min(\$200{,}000,\$230{,}000-\$50{,}000)=\$180{,}000\), not $135,000.
Check. The expected payment, $135,000, is the probability-weighted average of the three actual policy payments. Applying terms to the $230,000 expected damage gives a different quantity and is the common mistake avoided.
Why the Formula Matches the Contract Sequence
The location payment formula follows directly from the stated deductible and limit. If damage does not exceed the deductible, the insurer pays zero. Otherwise, subtract the deductible. If the remaining amount exceeds the limit, payment cannot exceed the limit. These cases combine as \(\min\{u_j,\max(D_j-d_j,0)\}\).
The occurrence formula uses the same reasoning at the event level, but its input is the sum of location payments. The sum is the amount entering the occurrence layer under the assumed policy wording. The occurrence deductible removes the first \(d_{\mathrm{occ}}\) dollars of that amount, with a floor at zero; the occurrence limit caps the resulting payment. Thus:
This expression is valid only for the policy sequence stated here. If wording instead applies an occurrence limit to ground-up loss, includes a shared deductible in a different layer, or has special sublimits, the calculation must follow that wording. The formula is not a substitute for identifying what amount each term applies to.
Common Traps and Practical Checks
- Confusing damage with insured payment. Damage is calculated from insured value and damage ratio. Deductibles, limits, and other contract terms come afterward.
- Applying an occurrence deductible per location. If the contract says the deductible applies once per occurrence, first aggregate the appropriate location amounts and subtract it once.
- Using the wrong interpolation interval. Find the two curve points that surround the intensity, and interpolate within that interval rather than extending an unrelated segment.
- Applying policy terms to average damage. When damage is uncertain, calculate each outcome’s payment before taking an expectation.
- Adding unlike amounts. Keep track of whether each number is insured value, ground-up damage, location payment, or event payment. Also retain a consistent currency unit throughout.
A final check should confirm that no location payment exceeds its location limit, no payment is negative, and the event payment does not exceed the applicable occurrence limit. Compare the event payment with the sum of location payments as well: under the sequence used here, occurrence terms cannot increase that sum.
Check Your Understanding
For each hypothetical event, follow the stated calculation order and keep damage separate from insurer payment.
- A location has insured value $450,000 and a deterministic damage ratio of 0.24. Its deductible is $18,000 and its payment limit is $80,000, with no occurrence terms. Calculate its ground-up damage and payment. (Answer: ground-up damage $108,000; payment $80,000.)
- Two locations have ground-up damages of $170,000 and $90,000. Each has a $10,000 deductible; their location limits are $150,000 and $100,000, respectively. The location payments are subject to one $20,000 occurrence deductible and a $190,000 occurrence limit. Calculate the event payment. (Answer: $190,000.)
- A location lies 60% of the way from a footprint point with intensity \(0.3g\) to one with intensity \(0.8g\). Its insured value is $500,000. The vulnerability curve is linear from damage ratio 0.10 at \(0.2g\) to 0.40 at \(0.7g\). The location deductible is $25,000 and limit is $140,000; the occurrence deductible is $15,000 and limit is $130,000. Calculate the event payment, assuming this is the only affected location. (Answer: intensity \(0.6g\), damage ratio 0.34, ground-up damage $170,000, location payment $140,000, event payment $125,000.)
- For one location, damage is $40,000, $120,000, or $260,000 with probabilities 0.25, 0.50, or 0.25, respectively. The deductible is $30,000 and the payment limit is $150,000. Calculate the expected payment by applying terms to each outcome. (Answer: $85,000.)
- Two locations have ground-up damages of $70,000 and $55,000. Each has a $10,000 deductible and no binding location limit. The contract applies one $40,000 occurrence deductible to the sum of location payments, followed by a $75,000 occurrence limit. Calculate the payment. (Answer: $65,000.)