From Event Intensity to Expected Damage
A catastrophe model connects a physical event to potential insured loss through several distinct quantities. At a particular location, a hazard model describes the intensity of the event, such as wind speed or ground shaking. A vulnerability function then describes how that intensity translates into damage to a specified type of property. Exposure value converts the damage ratio into an amount of physical damage.
Keeping these steps separate is essential. A wind speed is not a damage ratio, and a damage ratio is not a dollar loss until it is applied to an exposure value. This tutorial focuses on that mapping for a specified event and exposure; it does not apply policy terms or aggregate simulated events across a year.
The definition depends on the model’s damage measure and exposure basis. For example, the ratio might be repair cost divided by replacement value for a building with specified construction characteristics. The vulnerability function therefore should be matched to relevant property characteristics, coverage of physical components, and the units of the hazard intensity. A function calibrated for one property class or intensity measure should not be applied to another without justification.
Reading Hazard and Vulnerability Inputs
A hazard input for a particular calculation may be a single modeled intensity at a location, or it may be a conditional distribution of possible intensities given an event description. For example, a scenario could produce a modeled peak gust at each insured location. In a different setup, uncertainty in the modeled gust could be represented by several possible intensities and their conditional probabilities. In either case, the intensity must be expressed in the units used by the vulnerability function.
A vulnerability function is often supplied as a table of intensity and mean damage ratio pairs. Between table points, linear interpolation is a common assumption when the problem specifies no other interpolation rule. For intensities outside the table range, do not silently extend the function: use an explicitly supplied extrapolation rule, or state that the provided information is insufficient. A damage ratio also cannot represent a negative proportion of the exposure; if a fitted curve would produce values outside the physically meaningful range, the model’s stated treatment of that range is needed.
| Quantity | Meaning | Example unit |
|---|---|---|
| Hazard intensity \(I\) | Physical severity of the event at the location | Miles per hour of wind, or a specified shaking measure |
| Mean damage ratio \(g(I)\) | Expected fraction of the exposure damaged at intensity \(I\) | Dimensionless fraction |
| Exposure value \(V\) | Value to which the damage ratio is applied | Dollars |
| Expected damage \(Vg(I)\) | Conditional expected physical damage at that intensity | Dollars |
The mean damage ratio summarizes expected damage, not necessarily a single building’s realized condition. Individual structures may experience more or less damage than the conditional mean. If the vulnerability function is already a conditional mean, do not multiply by a separate probability of damage unless the model defines that probability as an additional component. The expected ratio already averages over the damage outcomes represented by the function.
Why Exposure Multiplies the Mean Damage Ratio
Suppose exposure value \(V\) is fixed for the calculation and physical damage is \(D=VR\). Conditional on intensity \(I\), the expected damage follows directly from the definition of conditional expectation and the fact that \(V\) is fixed:
This relation does not require every building to have exactly the mean damage ratio. It requires that the vulnerability function represent the conditional mean ratio for the exposure being valued. If exposure value itself is uncertain, or if it varies with the event, it cannot automatically be taken outside the conditional expectation; the model must specify how those quantities relate.
Worked Example: Interpolating a Site-Level Wind Estimate
A hypothetical wind scenario produces a reference gust of 120 miles per hour. A specified site adjustment factor is 0.90, so the modeled gust at a warehouse is the reference gust multiplied by that factor. The warehouse has an exposure value of $600,000. Its vulnerability table gives mean damage ratios of 0.02 at 80 miles per hour, 0.08 at 100 miles per hour, and 0.25 at 120 miles per hour. Using linear interpolation, calculate conditional expected damage. Assume the site adjustment and vulnerability table apply to this warehouse and that no policy terms are applied.
Step 1: Calculate the site intensity. The site gust is \(120\times0.90=108\) miles per hour. This lies between the 100- and 120-mile-per-hour table points.
Step 2: Interpolate the mean damage ratio. The intensity is 8 miles per hour above 100, out of a 20-mile-per-hour interval. The increase in mean damage ratio across that interval is \(0.25-0.08=0.17\). Therefore,
Step 3: Convert the ratio to expected dollars. The conditional expected damage is \(600{,}000\times0.148=\$88{,}800\).
Check. The interpolated ratio, 0.148, lies between 0.08 and 0.25, as it should. The resulting damage is 14.8% of $600,000. The method is site adjustment, interpolation, then multiplication by exposure. A common mistake is to apply the 120-mile-per-hour vulnerability point directly, even though the site intensity is 108 miles per hour; that would give \(600{,}000\times0.08? \) No: the 120-mile-per-hour ratio is 0.25, so that incorrect calculation is $150,000.
Uncertain Intensity and Nonlinear Vulnerability
When intensity is uncertain conditional on a specified event, calculate expected damage across the possible intensity outcomes. If the possible intensities are \(I_1,\ldots,I_n\), with conditional probabilities \(p_1,\ldots,p_n\), and exposure value \(V\) is fixed, then:
This is an average of the vulnerability-function outputs, weighted by the probabilities of the corresponding intensities. It is generally not equal to evaluating the vulnerability function at the average intensity. Equality is guaranteed for a linear function over the relevant range, but not for a nonlinear one.
Worked Example: Averaging Over Conditional Intensity Outcomes
For a hypothetical storm event, a coastal facility has a $2,000,000 exposure value. Conditional on this event, its site intensity is 90 miles per hour with probability 0.25, 110 miles per hour with probability 0.50, and 130 miles per hour with probability 0.25. The vulnerability table gives mean damage ratios of 0.01 at 80 miles per hour, 0.05 at 100, 0.20 at 120, and 0.50 at 140. Use linear interpolation between points. Find the conditional expected damage.
Step 1: Find the mean damage ratio at each possible intensity. At 90 miles per hour, the ratio is halfway between 0.01 and 0.05: \(0.01+(10/20)(0.05-0.01)=0.01+0.02=0.03\).
At 110 miles per hour, the ratio is halfway between 0.05 and 0.20: \(0.05+(10/20)(0.20-0.05)=0.05+0.075=0.125\).
At 130 miles per hour, the ratio is halfway between 0.20 and 0.50: \(0.20+(10/20)(0.50-0.20)=0.20+0.15=0.35\).
Step 2: Average the ratios using the conditional probabilities.
Step 3: Multiply by exposure. The conditional expected damage is \(2{,}000{,}000\times0.1575=\$315{,}000\).
Check. The probabilities sum to \(0.25+0.50+0.25=1\), and the weighted mean ratio of 0.1575 falls between the smallest and largest possible ratios. The method averages vulnerability outputs, not intensities. A common mistake is to use the average intensity of 110 miles per hour and report \(2{,}000{,}000\times0.125=\$250{,}000\). That is not the required expectation: the vulnerability curve is nonlinear, so the two calculations differ.
Combining Locations for a Specified Event
If a specified event affects several locations, calculate expected damage at each location using that location’s intensity and exposure, then sum the expected amounts. The expected value of a sum equals the sum of the expected values, so this calculation does not require the locations’ damage outcomes to be independent. Dependence matters for the distribution of total damage, but it is not needed to add location-level expected damages.
A location’s intensity and exposure should not be replaced by portfolio averages before applying a nonlinear vulnerability function. In general, differences in intensity, property characteristics, and exposure values mean that the portfolio calculation is a sum of location-specific terms.
Worked Example: Summing Expected Damage Across Three Properties
A hypothetical hail scenario produces the following site intensities for three properties. Assume all three use the same vulnerability function, with mean damage ratios of 0.02 at intensity 80, 0.10 at 100, 0.30 at 120, and 0.50 at 140. Intensities are in the model’s specified units. Property A has a $1,200,000 exposure value and intensity 105; Property B has an $800,000 exposure value and intensity 90; Property C has a $500,000 exposure value and intensity 125. Use linear interpolation to find the event’s total expected physical damage. Assume the listed exposures are fixed and no policy terms are applied.
Property A. Intensity 105 is one-quarter of the way from 100 to 120. The ratio is \(0.10+(5/20)(0.30-0.10)=0.10+0.05=0.15\). Expected damage is \(1{,}200{,}000\times0.15=\$180{,}000\).
Property B. Intensity 90 is halfway from 80 to 100. The ratio is \(0.02+(10/20)(0.10-0.02)=0.02+0.04=0.06\). Expected damage is \(800{,}000\times0.06=\$48{,}000\).
Property C. Intensity 125 is one-quarter of the way from 120 to 140. The ratio is \(0.30+(5/20)(0.50-0.30)=0.30+0.05=0.35\). Expected damage is \(500{,}000\times0.35=\$175{,}000\).
Total and check. Total expected damage is \(\$180{,}000+\$48{,}000+\$175{,}000=\$403{,}000\). The total exposure is \(\$1{,}200{,}000+\$800{,}000+\$500{,}000=\$2{,}500{,}000\), and the implied exposure-weighted ratio is \(\$403{,}000/\$2{,}500{,}000=0.1612\). It is between the smallest and largest location-level ratios, 0.06 and 0.35. The method avoids a common mistake: applying one average intensity to all properties before using the vulnerability function, which discards their different site intensities.
Calculation Discipline and Common Traps
A reliable calculation follows the chain from event to location intensity, from intensity to mean damage ratio, and from ratio to expected dollars. The process also makes assumptions visible: the intensity measure and units, the relevant vulnerability curve, any interpolation rule, the exposure basis, and whether intensity is fixed or uncertain.
Use the intensity at the insured location, in the units required by the vulnerability function. Apply any stated site adjustment before reading the curve.
Use the supplied function or interpolate according to the stated rule. Do not substitute a ratio for a different property class or intensity measure.
Calculate a ratio for each possible intensity and take the probability-weighted mean. Do not generally evaluate the curve at mean intensity.
Multiply the applicable mean ratio by the corresponding exposure value. For multiple locations, sum the location-level expected damage amounts.
- Using the reference intensity instead of the site intensity. A stated location adjustment changes the intensity at which the vulnerability function must be evaluated.
- Confusing a damage ratio with a probability. A mean damage ratio is an expected fraction of value damaged; it is not automatically the probability that damage occurs.
- Applying a vulnerability function at average intensity. With a nonlinear function, the expected ratio is the probability-weighted average of the outputs, not necessarily the output at average intensity.
- Applying the ratio to the wrong value. The exposure basis must match the vulnerability definition. A ratio against replacement value should be applied to the corresponding replacement-value exposure, not an unrelated amount.
- Extrapolating without an assumption. A table does not establish how the function behaves beyond its stated range. Use only a supplied or explicitly justified rule.
Hazard and vulnerability calculations are building blocks for catastrophe pricing: they translate a physical scenario into an expected amount of damage for specified exposures. Their usefulness depends on preserving the distinctions between event intensity, conditional damage ratio, and exposure value. The calculations here are conditional on the specified event or intensity distribution; they do not by themselves measure annual frequency or incorporate contract terms.
Check Your Understanding
Use linear interpolation between listed points unless a question states otherwise. Treat each exposure value as fixed, and do not apply policy terms.
- A hypothetical wind event produces a site intensity of 85 miles per hour. A building is valued at $450,000. Its mean damage ratio is 0.02 at 60 miles per hour and 0.10 at 100 miles per hour. What is the conditional expected damage? (Answer: $31,500)
- Conditional on a specified event, a facility worth $1,250,000 has intensity 80 with probability 0.20, 100 with probability 0.50, and 120 with probability 0.30. The corresponding mean damage ratios are 0.04, 0.12, and 0.36. Calculate conditional expected damage. (Answer: $220,000)
- Two properties use one vulnerability function: mean damage ratios are 0.03 at intensity 80, 0.11 at 100, and 0.31 at 120. Their exposures and site intensities are $700,000 at 90 and $300,000 at 110. Find total expected damage. (Answer: $112,000)
- A hypothetical location has equal conditional probabilities of intensity 80 and 120. Its vulnerability function gives mean damage ratios of 0.02 at 80, 0.06 at 100, and 0.30 at 120. For a $1,000,000 exposure, compare the correct conditional expected damage with the estimate obtained by evaluating the function at mean intensity. (Answer: Correct estimate $160,000; estimate at mean intensity $60,000)
- At three locations in a specified event, conditional expected damages are $42,000, $115,000, and $73,000. What is total conditional expected damage, and is independence among the locations required for this sum? (Answer: $230,000; no)