Why Catastrophe Results Need More Than One Measure
A catastrophe model can summarize a portfolio in several ways, and each summary answers a different management question. Average annual loss (AAL) describes expected annual loss over the modeled distribution. A loss quantile describes a point in the tail. An exceedance curve shows how the probability of exceeding a loss level changes as that level rises. None of these measures, on its own, identifies every exposure that might warrant attention.
A decision-maker should first establish what a result measures, which probability is attached to it, and whether it refers to one event or the full year. Then the measures can be used together: for example, to distinguish an exposure with frequent moderate losses from one with a lower expected loss but a more severe tail. The hypothetical results below illustrate this interpretation.
Read the Curve Before Interpreting the Number
An occurrence exceedance probability (OEP) curve concerns the largest individual event in a year. An aggregate exceedance probability (AEP) curve concerns the sum of the year’s losses. If losses are nonnegative, the annual aggregate can exceed a threshold because of one large event or because several smaller event losses add up. Consequently, at the same threshold, AEP can exceed OEP. The two probabilities should not be treated as interchangeable.
A return period is conventionally the reciprocal of an annual exceedance probability. It is a convenient way to describe the frequency implied by that probability, not a schedule for when a loss will occur. A “1-in-100-year” loss level does not mean that a loss cannot occur twice in a short period. Under a stationary annual model with exceedance probability \(q\), the return period is \(1/q\) years.
| Loss threshold (hypothetical) | OEP | OEP return period | AEP | AEP return period |
|---|---|---|---|---|
| $10 million | 0.12 | 8.33 years | 0.20 | 5 years |
| $18 million | 0.04 | 25 years | 0.07 | 14.29 years |
| $24 million | 0.01 | 100 years | 0.02 | 50 years |
These are illustrative model results, not observed industry data. At the $18 million threshold, for example, the modeled annual chance that at least one event exceeds the threshold is 0.04, while the chance that the year’s total exceeds it is 0.07. The difference can signal that multiple events contribute to annual accumulation. It does not, by itself, identify which locations or event combinations produce that accumulation.
Worked Example: Interpreting Two Curves at One Threshold
A hypothetical portfolio report gives an OEP of 0.025 and an AEP of 0.04 at a loss threshold of $16 million. Interpret both probabilities, calculate their return periods, and state which probability is appropriate for a question about the largest single event.
Method. Keep the event-level and annual-total definitions separate, then take the reciprocal of each annual exceedance probability.
The OEP of 0.025 means the probability that the largest single event loss in a year exceeds $16 million is 0.025. Its return period is \(1/0.025=40\) years.
The AEP of 0.04 means the probability that total annual loss exceeds $16 million is 0.04. Its return period is \(1/0.04=25\) years.
The largest-single-event question calls for OEP, so the relevant return period is 40 years. AEP is relevant to the annual total. Here AEP is larger than OEP, which is consistent with annual loss sometimes exceeding the threshold through the combined effect of multiple events.
Check. Both probabilities are annual probabilities between zero and one, and their reciprocals are 40 and 25 years. The 25-year return period is not a 25-year guarantee or waiting time.
Average Annual Loss and Exceedance Curves
AAL is an expectation, not a loss threshold at a chosen probability level. For a nonnegative annual aggregate loss, it can be calculated from the distribution of \(L\), or from the area under its AEP curve when the curve is expressed as \(P(L>x)\) against loss \(x\).
The identity follows by writing each realized loss as the length of the interval below it: for any \(L\ge 0\), \(L=\int_0^\infty \mathbf{1}_{\{L>x\}}\,dx\). Taking expectations on both sides gives \(E[L]=\int_0^\infty E[\mathbf{1}_{\{L>x\}}]\,dx\). Since the expectation of the indicator is \(P(L>x)\), the result follows. This tail-integral formula applies to nonnegative losses; the integral must be finite for AAL to be finite.
On a plotted curve, the integral is an area. With a discrete loss distribution, the curve is a step function, so its area can be calculated interval by interval. AAL can also be found by multiplying each possible annual loss by its probability and summing.
Worked Example: Calculating AAL From a Discrete Distribution
A hypothetical portfolio has the following annual aggregate loss distribution. Losses are in millions of dollars: \(L=0\) with probability 0.60, \(L=4\) with probability 0.25, \(L=10\) with probability 0.10, and \(L=25\) with probability 0.05. Calculate AAL from the distribution and verify it using the AEP curve.
Method 1: expected loss. Multiply each annual loss by its probability and add the products:
The AAL is therefore $3.25 million.
Method 2: area under the AEP curve. For \(0\le x<4\), \(P(L>x)=0.25+0.10+0.05=0.40\). For \(4\le x<10\), \(P(L>x)=0.10+0.05=0.15\). For \(10\le x<25\), \(P(L>x)=0.05\). For \(x\ge25\), it is zero. Thus the area is:
The two methods agree, verifying the calculation. Common mistake avoided: multiplying each threshold by the exceedance probability at that threshold is not the step-curve area. For example, \(4(0.40)+10(0.15)+25(0.05)=4.35\), which is not the AAL. The correct calculation uses each interval’s width and the exceedance probability within that interval.
Quantiles Describe Tail Levels, Not Expected Cost
The quantile \(\operatorname{VaR}_p(L)\) is the smallest loss level at which the cumulative probability reaches or exceeds \(p\). For a discrete distribution, the cumulative probability can jump over \(p\), so the quantile is the loss at the jump. In an exceedance curve, the corresponding point is read from the loss level associated with the selected probability, taking care with the distinction between \(P(L>x)\) and \(P(L\le x)\).
A quantile is useful for asking how large annual loss is at a specified probability level. It is not an estimate of the average cost, nor does it describe the severity of outcomes beyond that quantile. AAL and quantiles therefore answer different questions and can rank exposures differently.
Worked Example: Finding Quantiles in a Discrete Loss Distribution
For the same hypothetical annual distribution used above, find the 95th and 99th percentile loss quantiles. Explain why the 95th percentile is not $10 million.
Method. Build the cumulative distribution from the lowest loss upward. The probabilities are 0.60 at $0 million, 0.25 at $4 million, 0.10 at $10 million, and 0.05 at $25 million.
At $0 million, the cumulative probability is 0.60. At $4 million it is \(0.60+0.25=0.85\). At $10 million it is \(0.85+0.10=0.95\). At $25 million it is \(0.95+0.05=1.00\).
By the definition \(\operatorname{VaR}_p(L)=\inf\{x:F_L(x)\ge p\}\), the 95th percentile is $10 million and the 99th percentile is $25 million. In particular, $10 million is the 95th percentile, not a value above it: the cumulative probability reaches exactly 0.95 there. This example shows why one must use the stated quantile convention rather than assume that a percentile always falls strictly between loss outcomes.
Check. The cumulative probability at $4 million is only 0.85, below 0.95, while the cumulative probability at $10 million reaches 0.95. At $10 million the cumulative probability is still below 0.99, so the 99th percentile is $25 million.
Use Multiple Measures to Identify Exposures
A portfolio decision often involves comparing several exposures rather than selecting one summary statistic. AAL can identify where expected loss is concentrated; tail quantiles can identify exposures with severe modeled outcomes. A lower AAL does not imply a lower tail loss, and a high quantile does not by itself show that the exposure has a high expected cost.
For regional or other portfolio segments, first compare their results on a consistent basis: the same modeled year, loss definition, probability level, and currency units. Then relate the differences to the management question. An exposure that is prominent under a tail measure may warrant further investigation even if its AAL is lower. That observation is a prompt for judgment and follow-up, not a rule that a particular action must always be taken.
Worked Example: Choosing a Follow-Up From AAL and Tail Results
A hypothetical insurer compares two regions using stand-alone annual aggregate loss distributions, with losses in millions of dollars. Region East has losses of $0, $5, and $30 million with probabilities 0.80, 0.15, and 0.05. Region West has losses of $0, $12, and $20 million with probabilities 0.90, 0.08, and 0.02. Calculate each region’s AAL, 95th percentile, and 99th percentile, then identify a reasonable portfolio-management interpretation.
Method. Calculate each expectation as a probability-weighted sum, and find each quantile from the cumulative distribution.
For East, the probabilities sum to \(0.80+0.15+0.05=1.00\). Its AAL is:
The cumulative probabilities for East are 0.80 at $0 million, 0.95 at $5 million, and 1.00 at $30 million. Therefore, its 95th percentile is $5 million and its 99th percentile is $30 million.
For West, the probabilities sum to \(0.90+0.08+0.02=1.00\). Its AAL is:
The cumulative probabilities for West are 0.90 at $0 million, 0.98 at $12 million, and 1.00 at $20 million. Therefore, its 95th percentile is $12 million and its 99th percentile is $20 million.
East has the higher AAL ($2.25 million versus $1.36 million) and the higher 99th percentile ($30 million versus $20 million). West has the higher 95th percentile ($12 million versus $5 million). A useful interpretation is that the regions have different risk profiles: East has greater expected loss and a more severe 99th-percentile outcome, while West’s 95th-percentile loss is higher. Management might investigate the exposures and modeled drivers behind each result before deciding where to focus. Common mistake avoided: using only AAL would miss West’s higher 95th-percentile loss.
These are stand-alone regional results. If the decision concerns a combined portfolio quantile, the regional distributions alone are not enough: the dependence between regional annual losses also matters. A combined AAL, by contrast, is the sum of regional AALs when total loss is the sum of the regional losses, by additivity of expectation. Here that AAL would be \(2.25+1.36=3.61\) million dollars.
Common Interpretation Traps
- Substituting OEP for AEP. OEP concerns the largest event; AEP concerns the annual total. Use the curve that matches the question.
- Treating a return period as a timetable. A return period is the reciprocal of an annual exceedance probability, not a promise about how often events will occur in sequence.
- Reading AAL as a likely annual outcome. AAL is an expectation across modeled outcomes. It need not equal any loss that occurs in a particular year.
- Comparing results on inconsistent bases. Check that the loss definition, time basis, and probability level match before ranking exposures.
- Adding regional quantiles to get a portfolio quantile. A combined quantile depends on how regional losses occur together. It cannot generally be calculated by adding stand-alone quantiles.
A disciplined reading of catastrophe results begins with the loss measure and probability definition, then considers AAL and one or more quantiles together. The interpretation should identify what the reported numbers imply and what remains uncertain, rather than treating a single statistic as a complete decision rule. This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Check Your Understanding
For each hypothetical problem, identify the relevant measure and show the calculation before checking the final answer.
- A model reports OEP 0.025 and AEP 0.04 at $16 million. What are the two return periods, and which probability answers a question about the largest single event? (Answer: OEP return period 40 years; AEP return period 25 years; use OEP for the largest single event.)
- An annual loss is $0 million with probability 0.75, $3 million with probability 0.20, and $9 million with probability 0.05. Find AAL and the 90th percentile. (Answer: AAL $1.05 million; 90th percentile $3 million.)
- An AEP curve gives exceedance probability 0.10 at $5 million and 0.025 at $15 million. Find the return period at each threshold and state what each period means. (Answer: 10 years at $5 million and 40 years at $15 million; each is the reciprocal of that threshold’s annual aggregate exceedance probability.)
- An annual loss is $0 million with probability 0.50, $2 million with probability 0.30, and $8 million with probability 0.20. Calculate AAL using the AEP curve’s step intervals. (Answer: \(2(0.50)+6(0.20)=2.20\), so AAL is $2.20 million.)
- Two regions have stand-alone 99th percentile losses of $18 million and $11 million. Can their combined 99th percentile be reported as $29 million from these figures alone? (Answer: No; the dependence between regional losses is also needed.)