Hazard Intensity Measures in Catastrophe Models

What a Hazard Intensity Measure Describes

A catastrophe event is not equally severe at every insured location. A windstorm may produce different wind speeds across a region, an earthquake may cause different levels of shaking, and a flood may leave different depths of water at different properties. A catastrophe model therefore needs a way to describe the hazard experienced at a particular location, not merely a label for the event.

A hazard intensity measure is a defined physical quantity used to describe the severity of a hazard at a specified place and, where relevant, over a specified time interval or measurement convention. It is the link between an event and the physical conditions that may cause damage. An intensity measure is not itself a damage estimate or a dollar loss.

Definition: Let $I_j(x)$ denote the intensity measure at location $x$ for event $j$. Its meaning depends on the hazard and the exact convention used: for example, a specified wind-speed statistic, a measure of earthquake ground motion, or water depth relative to a stated elevation. A valid intensity value must be interpreted together with its units, measurement definition, location, and event.

The choice of measure matters. “Wind speed” is incomplete unless the model specifies matters such as whether it is a sustained speed or a gust, the averaging duration, and the units. Earthquake intensity may be represented by peak ground acceleration (PGA) or by a response-spectrum measure such as spectral acceleration at a stated period. Flood depth must have a reference elevation: depth above ground is not necessarily depth above a building’s first floor.

These measures are not interchangeable just because they describe the same event. A wind gust and a sustained wind speed have different definitions; PGA and spectral acceleration describe different aspects of ground motion; and a flood depth without its reference level may be ambiguous. The model’s damage relationship must use the same intensity definition and units as the intensity data.

From Event to Conditional Damage

Intensity is evaluated at an insured location, then interpreted alongside the characteristics of the exposed property. A stated intensity-to-damage relationship may provide a damage ratio, meaning the fraction of the property value expected to be damaged under the specified conditions. For this tutorial, treat that relationship as supplied model information; its construction and calibration are separate questions.

Let $V$ be the exposed value and $R$ the damage ratio. In a simple model, physical damage $D$ is $V R$. If the relationship assigns a deterministic ratio $r(i)$ to intensity $i$, then damage conditional on that intensity is $V r(i)$. If damage remains uncertain even after intensity is known, the appropriate quantity is the conditional expected ratio.

E[D\mid I=i] = E[VR\mid I=i] = V E[R\mid I=i].

This formula assumes that $V$ is fixed and that physical damage is represented as exposed value multiplied by a damage ratio. The last equality follows because a fixed $V$ can be taken outside a conditional expectation. If the ratio is a deterministic function $r(i)$, then $E[R\mid I=i]=r(i)$, giving $E[D\mid I=i]=V r(i)$. This is a conditional damage estimate for a specified intensity; it is not, by itself, an annual expected loss. Turning event-conditioned estimates into an annual quantity requires additional information about event occurrence.

Key distinction: An event describes what happened; an intensity measure describes the hazard level at a location; a damage ratio describes a modeled physical response to that intensity; and a dollar damage estimate applies that ratio to exposed value. Confusing these stages can lead to a well-computed number with the wrong interpretation.

Worked Examples

Worked Example: Wind Speed and Conditional Damage

A hypothetical commercial building has an exposed value of $500,000. For a particular wind event, the model reports a maximum one-minute sustained wind speed of 82 mph at the building. The supplied hypothetical damage schedule gives a ratio of 0.10 at 80 mph and 0.30 at 100 mph, with linear interpolation between those two values. Find the estimated damage conditional on this event intensity.

Method and assumptions. Treat the reported 82 mph as the same one-minute sustained wind measure used by the schedule. Assume the schedule applies to this building and that exposed value is fixed. No probability distribution, interest rate, or mortality assumption is needed: intensity and the schedule are specified deterministically, and this is a property-damage calculation.

First determine the fraction of the interval from 80 to 100 mph:

\frac{82-80}{100-80}=\frac{2}{20}=0.10.

The damage ratio rises by $0.30-0.10=0.20$ across the interval, so the interpolated ratio is:

$$ 0.10+(0.10)(0.20)=0.12. $$

Apply this ratio to the exposed value:

D=500{,}000(0.12)=\$60{,}000.

Check. The intensity is 10% of the way from 80 to 100 mph, so the ratio should be 10% of the way from 0.10 to 0.30: 0.12. The result, $60,000, is 12% of $500,000. The common mistake is to use 82 as though it were the damage ratio or to apply a schedule for gust speed to a sustained-speed input. Neither is valid; the schedule and intensity must use matching definitions.

Worked Example: Converting an Earthquake Intensity Unit

A hypothetical insured warehouse has an exposed value of $1,200,000. The event intensity at the site is PGA of $0.24g$. A supplied hypothetical schedule, defined for PGA in units of $g$, assigns a damage ratio of 0.06 for $0.10g$ to less than $0.20g$, 0.18 for $0.20g$ to less than $0.30g$, and 0.35 for $0.30g$ or greater. Convert the intensity to cm/s², then calculate estimated damage.

Method and assumptions. Use $1g=980.665\text{ cm/s}^2$. The schedule applies to the warehouse, the reported PGA is a site-level value, and the damage ratio is deterministic once the intensity band is selected. No probability distribution, interest rate, or mortality assumption is used.

Convert the reported value:

0.24g \times 980.665\frac{\text{cm/s}^2}{g} =235.3596\text{ cm/s}^2.

The conversion does not change the physical intensity. In $g$ units, $0.20 \leq 0.24 < 0.30$, so the applicable damage ratio is 0.18. Estimated damage is:

D=1{,}200{,}000(0.18)=\$216{,}000.

Check. $216,000 is 18% of $1,200,000, as required. The common mistake is to compare 235.3596 cm/s² directly with thresholds expressed in $g$, which would select the wrong band. Convert the intensity or the thresholds first, and keep the units consistent. The converted intensity is approximately 235.4 cm/s²; the damage ratio remains the one for $0.24g$.

Worked Example: Expected Damage Under Uncertain Flood Depth

A hypothetical property has an exposed value of $300,000. Conditional on a specified flood event affecting the site, the modeled flood depth $H$, measured above the property’s first-floor elevation, has this discrete distribution: $P(H=0)=0.20$, $P(H=0.5\text{ ft})=0.30$, $P(H=1.5\text{ ft})=0.40$, and $P(H=2.5\text{ ft})=0.10$. A supplied hypothetical damage schedule assigns ratios 0, 0.02, 0.12, and 0.30 to those four depths, respectively. Calculate expected damage conditional on the event.

Method and assumptions. Use the stated discrete distribution of first-floor flood depth and the stated deterministic damage ratio at each depth. The probabilities describe depth conditional on this event and sum to 1.00. No interest rate or mortality assumption is relevant.

First calculate the expected damage ratio by weighting each ratio by its probability:

E[R\mid\text{event}] =(0.20)(0)+(0.30)(0.02)+(0.40)(0.12)+(0.10)(0.30).

The four contributions are $0$, $0.006$, $0.048$, and $0.030$, so:

E[R\mid\text{event}]=0.084.

Multiply by the exposed value:

E[D\mid\text{event}] =300{,}000(0.084)=\$25{,}200.

Check. The weighted damage ratio is 8.4%, between the smallest and largest ratios in the schedule. Applying it to $300,000 gives $25,200. The common mistake is to calculate damage at the average depth and assume it equals expected damage. In general, $E[R(H)]$ need not equal $R(E[H])$; the weighted calculation is correct for the supplied discrete model. This is conditional expected damage, not an annual estimate.

Interpreting an Intensity-Based Estimate

The examples use simple supplied schedules to isolate the role of intensity. Real model outputs can depend on more than a single reported number. The intensity measure may be a vector of values, such as ground motion at several periods, or a spatially varying quantity. A building’s construction, occupancy, condition, and other attributes may also matter to the relationship between hazard and damage. Consequently, a physical intensity should not be interpreted as a universal damage percentage.

When reading or using a catastrophe-model result, check these points before performing arithmetic:

Hazard and metric: Is the measure appropriate to the peril, and is it a gust, sustained speed, PGA, spectral measure, or another defined quantity?
Units: Are the reported value and all thresholds in the same units?
Reference convention: For example, is flood depth measured above ground, a floor, or another elevation?
Location and event: Does the value represent the insured location for the event being analyzed?
Conditional meaning: Is the estimate damage conditional on a specified intensity or event, or has event frequency also been incorporated?

A frequent exam trap is to treat a plausible-looking intensity as self-explanatory. A value of “0.25” could represent a fraction of $g$, a different ground-motion measure, or a value in some other scale. Likewise, a depth of 1 foot has no complete physical interpretation until its reference elevation is stated. Always write the measure and units alongside the number before selecting a damage relationship.

Key takeaway: A hazard intensity measure gives a defined physical description of severity at a location. To estimate conditional damage, match that definition and its units to the supplied intensity-to-damage relationship, then apply the resulting expected damage ratio to exposed value. Keep this conditional estimate distinct from annual loss.

Check Your Understanding

For each problem, state the relevant intensity interpretation before calculating. Monetary answers are rounded to the nearest cent where applicable.

A hypothetical office has exposed value $750,000. Its event-level one-minute sustained wind is 80 mph. A supplied schedule is linear from ratio 0.04 at 70 mph to ratio 0.16 at 90 mph. Find the interpolated ratio and conditional damage. (Answer: damage ratio 0.10; damage $75,000.00)
A hypothetical clinic has value $900,000 and experiences PGA $0.18g$. Convert using $1g=980.665\text{ cm/s}^2$. A schedule defined in $g$ assigns ratio 0.05 for $0.10g$ to less than $0.20g$, and 0.15 for $0.20g$ to less than $0.30g$. Find the converted intensity and estimated damage. (Answer: approximately 176.5 cm/s²; damage $45,000.00)
A hypothetical home has value $250,000. Conditional on a specified flood event, first-floor depths of 0, 1, and 2 ft have probabilities 0.25, 0.50, and 0.25. The corresponding damage ratios are 0, 0.08, and 0.24. Find conditional expected damage. (Answer: expected ratio 0.10; expected damage $25,000.00)
A model reports a flood depth of 0.8 ft above ground, while a hypothetical property’s first floor is 1.1 ft above ground. Assuming the same ground datum and ignoring other water entry paths, find the water depth relative to the first floor. If the supplied schedule assigns zero damage for nonpositive first-floor depth, state the scheduled damage ratio. (Answer: depth is $-0.3$ ft relative to the first floor; scheduled ratio 0)
A hypothetical property has value $400,000. Conditional on one event, intensity is in a lower band with probability 0.60 and an upper band with probability 0.40. The supplied damage ratios are 0.03 and 0.18, respectively. Calculate conditional expected damage, and explain why multiplying by an annual event rate would be a separate step. (Answer: expected ratio 0.09; conditional expected damage $36,000.00; the event rate is not part of the stated conditional distribution)

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