Calculating Damage From Vulnerability Curves

From Hazard Intensity to Expected Damage

A vulnerability curve connects a hazard intensity at an exposed location to a damage ratio. Once the intensity is assigned and the curve has been evaluated, the ratio can be applied to the location’s exposure value to estimate expected physical damage. The calculation is simple in form, but it is easy to confuse a ratio with a dollar amount, or an expected amount with the damage that will actually occur.

This tutorial focuses on calculating damage from specified intensities and combining the results across locations. It is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. The examples treat each curve value as a mean damage ratio conditional on the stated intensity; they do not model the variability of damage around that mean.

Definition: For location $i$, let $V_i$ be its exposure value, $h_i$ its specified hazard intensity, and $r_i(h_i)$ the mean damage ratio obtained from its vulnerability curve at that intensity. The expected physical damage at that location, conditional on the intensity, is $$ E[D_i\mid h_i]=V_i r_i(h_i). $$ A damage ratio is dimensionless: it expresses expected damage as a proportion of the exposure value. For example, a ratio of $0.12$ applied to an exposure of $\$500{,}000$ gives expected damage of $\$60{,}000$.

The exposure value must match the basis used for the curve and the damage measure. In these examples, $V_i$ is the value of the property being damaged, and the curve gives physical damage as a fraction of that value. This calculation does not by itself determine an insurer’s payment: deductibles, limits, coverage terms, and other financial adjustments are separate from physical damage.

Evaluating a Curve Between Its Points

A vulnerability curve is often provided as a set of intensity-ratio pairs. When an intensity lies between two adjacent points, linear interpolation estimates the ratio between them. This tutorial uses linear interpolation only where it is stated. If an intensity falls outside the supplied range, do not extend the line automatically: use the model’s specified boundary or extrapolation rule, if one is provided.

Linear interpolation: Suppose a curve has points $(h_1,r_1)$ and $(h_2,r_2)$, where $h_1\leq h\leq h_2$. Under the assumption of a straight line between those points, the interpolated mean damage ratio is $$ r(h)=r_1+\frac{h-h_1}{h_2-h_1}(r_2-r_1). $$ The fraction $(h-h_1)/(h_2-h_1)$ is the proportion of the distance from the lower intensity to the upper intensity.

To justify the formula, a straight line between $(h_1,r_1)$ and $(h_2,r_2)$ has slope $(r_2-r_1)/(h_2-h_1)$. Moving from $h_1$ to $h$ changes the ratio by the slope multiplied by the intensity change $h-h_1$. Adding that change to $r_1$ gives the interpolation formula. At $h=h_1$, it returns $r_1$; at $h=h_2$, it returns $r_2$, as required.

Worked Example: Interpolating for One Property

A hypothetical warehouse has an exposure value of $\$800{,}000$. Its vulnerability curve has points $(0.50,0.10)$ and $(0.80,0.28)$, and the assigned intensity is $0.65$. Assume linear interpolation between these points. Calculate the interpolated mean damage ratio and expected physical damage.

Method. First find the fraction of the intensity interval covered: $$ \frac{0.65-0.50}{0.80-0.50} =\frac{0.15}{0.30} =0.50. $$

The ratio increases by $0.28-0.10=0.18$ over the full interval. At halfway, the increase is $0.50(0.18)=0.09$. Thus, $$ r(0.65)=0.10+0.09=0.19. $$

Convert the ratio to expected dollars: $$ E[D\mid h=0.65]=\$800{,}000(0.19)=\$152{,}000. $$ Check: The intensity is halfway between the two curve points, so the interpolated ratio should be halfway between $0.10$ and $0.28$. Their midpoint is $0.19$, confirming the result. The common mistake avoided here is treating $0.19$ as a dollar amount rather than multiplying it by the exposure value.

Combining Locations in a Portfolio

For a portfolio with specified intensities at each location, calculate each location’s ratio using its own curve and intensity, convert that ratio to expected damage using that location’s exposure, and then sum the expected dollar amounts. The portfolio expected damage ratio is the total expected damage divided by total exposure value.

E[D_{\mathrm{total}}]=\sum_i V_i r_i(h_i), \qquad r_{\mathrm{portfolio}}= \frac{\sum_i V_i r_i(h_i)}{\sum_i V_i}.

The portfolio ratio is therefore an exposure-weighted average of the location ratios, not generally their simple average. To see why, the total expected dollars at location $i$ are $V_i r_i(h_i)$. Dividing their sum by total exposure gives $$ \frac{\sum_i V_i r_i(h_i)}{\sum_i V_i} =\sum_i\left(\frac{V_i}{\sum_j V_j}\right)r_i(h_i). $$ The weights $V_i/\sum_j V_j$ add to one, and larger exposures receive larger weights.

Summing the location-level conditional expectations gives the portfolio conditional expected damage by linearity of expectation. This calculation does not require the damage outcomes at different locations to be independent. It does require that each location’s intensity and applicable curve have been identified consistently.

Worked Example: Aggregating Three Interpolated Damage Estimates

A hypothetical portfolio has three properties. Each curve is to be interpolated linearly between its stated points.

Property A: value $\$700{,}000$, intensity $0.60$, curve points $(0.40,0.10)$ and $(0.80,0.22)$.
Property B: value $\$1{,}300{,}000$, intensity $0.30$, curve points $(0.10,0.02)$ and $(0.50,0.10)$.
Property C: value $\$2{,}000{,}000$, intensity $0.70$, curve points $(0.50,0.15)$ and $(0.90,0.25)$.

Calculate each ratio and expected damage, then calculate the portfolio expected damage and portfolio damage ratio.

Property A. Its intensity fraction is $$ \frac{0.60-0.40}{0.80-0.40}=\frac{0.20}{0.40}=0.50. $$ The ratio increases by $0.22-0.10=0.12$, so $$ r_A=0.10+0.50(0.12)=0.16, \qquad E[D_A]=\$700{,}000(0.16)=\$112{,}000. $$

Property B. Its intensity fraction is $$ \frac{0.30-0.10}{0.50-0.10}=\frac{0.20}{0.40}=0.50. $$ The ratio increases by $0.10-0.02=0.08$, so $$ r_B=0.02+0.50(0.08)=0.06, \qquad E[D_B]=\$1{,}300{,}000(0.06)=\$78{,}000. $$

Property C. Its intensity fraction is $$ \frac{0.70-0.50}{0.90-0.50}=\frac{0.20}{0.40}=0.50. $$ The ratio increases by $0.25-0.15=0.10$, so $$ r_C=0.15+0.50(0.10)=0.20, \qquad E[D_C]=\$2{,}000{,}000(0.20)=\$400{,}000. $$

The total exposure is $\$700{,}000+\$1{,}300{,}000+\$2{,}000{,}000=\$4{,}000{,}000$. The portfolio expected damage is $$ \$112{,}000+\$78{,}000+\$400{,}000=\$590{,}000. $$ Thus the portfolio expected damage ratio is $$ \frac{\$590{,}000}{\$4{,}000{,}000}=0.1475=14.75\%. $$ Check: The exposure weights are $0.175$, $0.325$, and $0.500$. Their weighted ratio is $0.175(0.16)+0.325(0.06)+0.500(0.20)=0.1475$, matching the ratio from total dollars. The common mistake avoided is averaging the three ratios equally even though the properties have different values.

Worked Example: Different Curves at Different Locations

Two hypothetical sites experience different assigned intensities and have different vulnerability curves. Site 1 has value $\$1{,}100{,}000$, intensity $0.45$, and curve points $(0.20,0.04)$ and $(0.50,0.16)$. Site 2 has value $\$600{,}000$, intensity $0.65$, and curve points $(0.30,0.10)$ and $(0.70,0.30)$. Interpolate linearly and calculate total expected damage and the portfolio ratio.

Site 1. The intensity fraction is $$ \frac{0.45-0.20}{0.50-0.20}=\frac{0.25}{0.30}=\frac{5}{6}. $$ The ratio increase across the interval is $0.16-0.04=0.12$. Therefore, $$ r_1=0.04+\frac{5}{6}(0.12)=0.04+0.10=0.14, \qquad E[D_1]=\$1{,}100{,}000(0.14)=\$154{,}000. $$

Site 2. The intensity fraction is $$ \frac{0.65-0.30}{0.70-0.30}=\frac{0.35}{0.40}=0.875. $$ The ratio increase across the interval is $0.30-0.10=0.20$. Therefore, $$ r_2=0.10+0.875(0.20)=0.275, \qquad E[D_2]=\$600{,}000(0.275)=\$165{,}000. $$

Total expected damage is $\$154{,}000+\$165{,}000=\$319{,}000$. Total exposure is $\$1{,}100{,}000+\$600{,}000=\$1{,}700{,}000$, so the portfolio ratio is $$ \frac{\$319{,}000}{\$1{,}700{,}000}=0.187647\ldots\approx 18.76\%. $$ The displayed percentage is rounded to two decimal places. Check: The dollar amounts sum to $\$319{,}000$, and multiplying the unrounded portfolio ratio by $\$1{,}700{,}000$ returns $\$319{,}000$. This method avoids applying one site’s curve to another site merely because both belong to the same portfolio.

Expected Damage Is Not a Guaranteed Event Outcome

A vulnerability curve’s mean damage ratio represents an expected damage fraction conditional on the modeled intensity. Consequently, $V_i r_i(h_i)$ is an expected amount, not a promise that the property will sustain exactly that dollar amount. For example, a calculated expected damage of $\$152{,}000$ does not mean that every event at that intensity produces $\$152{,}000$ of damage. A particular realized outcome can be above or below the mean.

This distinction also applies to portfolios. The sum of location-level expected damages is the portfolio expected damage under the stated intensities and assumptions; it need not equal damage observed in one realized event. Modeling the distribution or variability around the curve’s mean requires additional assumptions beyond this calculation.

Common Calculation Traps

Using an unweighted mean of ratios. When exposure values differ, the portfolio ratio is weighted by exposure value. Calculate expected dollars by location first, or use the weighted-average formula.
Skipping interpolation. Using the nearest curve point instead of evaluating the stated intensity can materially change the result. Show the intensity fraction and the ratio increment.
Applying the ratio to the wrong value. A damage ratio is a proportion. Multiply it by the relevant exposure value to obtain dollars.
Extrapolating without a rule. The interpolation formula applies between the two specified points. An intensity outside that interval requires the model’s stated treatment; the linear formula should not be extended by assumption.
Calling the estimate a realized loss. A conditional mean damage estimate is not a claim that the same amount will occur in a particular event.

Key takeaway: For each location, identify the applicable curve and intensity, calculate the mean damage ratio, and multiply by the location’s exposure value. Sum expected dollars across locations; divide by total exposure only if a portfolio damage ratio is required.

Check Your Understanding

For each hypothetical problem, assume linear interpolation only between the supplied curve points. Report dollar amounts to the cent where needed; ratios in the answers are rounded to four decimal places unless shown as percentages.

A property valued at $\$1{,}200{,}000$ has curve points $(0.50,0.08)$ and $(0.90,0.20)$. Its assigned intensity is $0.75$. Find the interpolated ratio and expected damage. (Answer: ratio $0.1550$; expected damage $\$186{,}000$)
Two buildings share a curve with points $(0.20,0.04)$, $(0.60,0.16)$, and $(0.80,0.24)$. Building A is valued at $\$500{,}000$ with intensity $0.50$; Building B is valued at $\$1{,}500{,}000$ with intensity $0.70$. Calculate total expected damage and the portfolio ratio. (Answer: $\$365{,}000$; ratio $0.1825$, or $18.25\%$)
A portfolio has a $\$400{,}000$ exposure at intensity $0.40$ and a $\$900{,}000$ exposure at intensity $0.80$. Both use a curve with points $(0.20,0.05)$, $(0.60,0.17)$, and $(0.80,0.29)$. Find expected damage at each location, total expected damage, and the portfolio ratio. (Answer: $\$44{,}000$ and $\$261{,}000$; total $\$305{,}000$; ratio $0.2346$, or $23.46\%$)
A hypothetical site has value $\$2{,}000{,}000$, intensity $0.50$, and curve points $(0.30,0.06)$ and $(0.70,0.22)$. The realized event damage is $\$350{,}000$. Calculate expected damage and the realized amount’s dollar difference from that expectation. (Answer: expected damage $\$280{,}000$; realized damage is $\$70{,}000$ above the expectation)
Three sites have exposure values $\$250{,}000$, $\$750{,}000$, and $\$1{,}000{,}000$, with evaluated mean damage ratios $0.08$, $0.16$, and $0.22$, respectively. Calculate total expected damage and the portfolio ratio. What ratio would an incorrect simple average produce? (Answer: total expected damage $\$360{,}000$; portfolio ratio $0.1800$; simple average $0.1533$, which is not the portfolio ratio)

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