Integrated Catastrophe Pricing Case

From Catastrophe Model Output to an Indicated Premium

A catastrophe pricing indication is a chain of linked calculations. Location-level exposure and event severity determine event losses; the event occurrence assumptions determine annual loss distributions; and selected loss and tail measures feed a risk load and premium. A result is only as useful as the assumptions carried consistently through that chain.

This hypothetical case assumes each listed event can occur at most once in a year, event occurrences are independent, and each event has a fixed covered loss if it occurs. These simplifying assumptions make the annual distributions transparent. A model with dependent events, multiple occurrences, or uncertain event severity requires a different probability calculation.

Definition: For annual event losses $L_1,\ldots,L_n$, the occurrence annual loss is the largest event loss in the year, $M=\max_j I_jL_j$, where $I_j=1$ if event $j$ occurs and $0$ otherwise. The aggregate annual loss is $S=\sum_j I_jL_j$, the sum of all event losses in the year. OEP describes the exceedance distribution of $M$; AEP describes that of $S$.

The distinction matters whenever more than one event can occur in a year. OEP records the largest event, while AEP adds the losses from all events. Neither distribution can be substituted for the other without changing the measure being priced.

Worked Case: Turning Location Inputs into Event Losses

Worked Example: Building an Event Loss Table

A hypothetical insurer covers three commercial locations. The listed damage ratios are assumed to be covered loss divided by insured value, after applicable policy terms. A coastal storm has annual occurrence probability 0.10, and an inland wind event has probability 0.05. Assume the two event occurrences are independent and each event can occur no more than once per year. Calculate each event's portfolio loss and the expected annual loss.

Location	Insured value	Coastal damage ratio	Inland damage ratio
A	$1,000,000	0.10	0.00
B	$2,000,000	0.10	0.20
C	$2,000,000	0.00	0.20

Method and event losses. For each location and event, multiply insured value by the applicable damage ratio. Location A's coastal loss is $1{,}000{,}000(0.10)=\$100{,}000$, and its inland loss is zero. Location B's losses are $2{,}000{,}000(0.10)=\$200{,}000$ and $2{,}000{,}000(0.20)=\$400{,}000$. Location C's losses are zero and $2{,}000{,}000(0.20)=\$400{,}000$.

The coastal event loss is $100{,}000+200{,}000+0=\$300{,}000$. The inland event loss is $0+400{,}000+400{,}000=\$800{,}000$.

Expected annual loss. By linearity of expectation, the expected annual loss (AAL) is the sum of each event loss multiplied by its annual occurrence probability:

\operatorname{AAL} =0.10(300{,}000)+0.05(800{,}000) =30{,}000+40{,}000 =\$70{,}000.

This calculation does not require independence; linearity of expectation applies even if event occurrences are dependent. Independence will be needed to calculate the joint annual distribution in the next step. The common mistake is to add the event losses and call the result expected annual loss. The sum, $1,100,000, is the loss if both events occur, not the expected loss.

Constructing OEP and AEP Distributions

For the case above, let $I_1$ and $I_2$ indicate coastal and inland event occurrence. The annual aggregate loss is $S=300{,}000I_1+800{,}000I_2$; the occurrence loss is $M=\max(300{,}000I_1,800{,}000I_2)$. Independence permits multiplying the occurrence and nonoccurrence probabilities to obtain each joint outcome.

Result: If nonnegative event losses $L_j$ occur independently with probabilities $p_j$, then for $x<0$, $P(M\le x)=0$. For $x\ge 0$, $$ P(M\le x)=\prod_{j:L_j>x}(1-p_j). $$ For every event with loss greater than $x$, that event must not occur for the maximum to be at most $x$. Events with loss at most $x$ impose no restriction.

The domain condition is essential: the product formula applies only for $x\ge0$. Since $M$ is nonnegative, its cumulative probability is zero below zero. For AEP, enumerate possible combinations of event occurrences and sum their losses; the maximum-loss product formula does not give the aggregate distribution.

Worked Example: Calculating OEP and AEP Tail Measures

Continue the hypothetical portfolio. Calculate the probability of each annual loss under OEP and AEP, then find the 95th percentile for both measures and the probability that AEP exceeds $800,000.

Step 1: Joint outcomes. The probability of neither event is $0.90(0.95)=0.855$. Coastal only has probability $0.10(0.95)=0.095$. Inland only has probability $0.90(0.05)=0.045$. Both have probability $0.10(0.05)=0.005$. These probabilities sum to $0.855+0.095+0.045+0.005=1.000$.

Step 2: AEP distribution. The four outcomes produce aggregate losses of $0, $300,000, $800,000, and $1,100,000, respectively. Thus the cumulative probabilities at these ordered losses are $0.855$, $0.855+0.095=0.950$, $0.995$, and $1.000$.

Step 3: OEP distribution. The corresponding maximum event losses are $0, $300,000, $800,000, and $800,000. Therefore the OEP probability at $800,000 is $0.045+0.005=0.050$, while the other probabilities are 0.855 at zero and 0.095 at $300,000.

Step 4: Percentile and exceedance probability. Define the 95th percentile as the smallest loss $x$ for which the cumulative probability is at least 0.95. For AEP, the cumulative probability reaches exactly 0.950 at $300,000, so the 95th-percentile AEP is $300,000. OEP has the same cumulative probability at $300,000, so its 95th percentile is also $300,000. AEP exceeds $800,000 only when both events occur, giving probability 0.005. At exactly $800,000, AEP does not exceed the threshold.

Trap avoided. A common error is to jump from the occurrence of the $800,000 event to an $800,000 95th-percentile AEP. But the cumulative AEP probability at $300,000 is already exactly 0.95. Under the stated smallest-loss definition, the percentile is $300,000, not $800,000. Also distinguish $P(S>800{,}000)=0.005$ from $P(S\ge800{,}000)=0.050$.

Why the Tail Average Uses the Upper Five Percent

A percentile identifies a point on the annual loss distribution, but it does not describe the severity of losses beyond that point. A tail conditional mean supplements the percentile by averaging the upper tail. For a discrete distribution, probability mass at the percentile may need to be split; simply averaging all outcomes at or above the percentile can include more than the intended tail probability.

Definition: The value-at-risk at percentile $\alpha$, denoted $\operatorname{VaR}_{\alpha}$, is the smallest $x$ such that $P(S\le x)\ge\alpha$. The tail conditional mean is $$ \operatorname{TVaR}_{\alpha} =\frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{u}\,du. $$ This definition averages the worst $1-\alpha$ fraction of the distribution, using only the fraction of any probability mass needed at the boundary.

In the case's AEP distribution, the 95th percentile is $300,000, and the probability strictly above it is $0.045+0.005=0.050$. That is exactly the upper five percent, so no probability mass at $300,000 is required in the tail average. The tail average is $[0.045(800{,}000)+0.005(1{,}100{,}000)]/0.05$.

\operatorname{TVaR}_{0.95} =\frac{36{,}000+5{,}500}{0.05} =\$830{,}000.

A short justification for the discrete-tail calculation is to regard each outcome's probability as a portion of the unit interval ordered from lowest to highest loss. The interval from percentile $\alpha$ to 1 has length $1-\alpha$. Each loss value contributes its loss multiplied by the length of its interval inside that upper tail; dividing the resulting weighted sum by $1-\alpha$ gives the tail average. If the probability strictly above the percentile is less than $1-\alpha$, include only the needed fraction of the probability mass at the percentile.

From Expected Loss and Tail Measure to Premium

A pricing indication must identify how modelled losses, risk compensation, expenses, and profit are combined. There is no unique risk-load formula: the actuary selects a measure and calibration appropriate to the purpose, available evidence, and constraints. For this hypothetical case, use five percent of the excess of 95% TVaR over AAL as the risk load. This is a stated pricing choice, not a universal rule.

Worked Example: Calculating the Indicated Catastrophe Premium

Using the hypothetical portfolio's AAL of $70,000 and 95% TVaR of $830,000, calculate the indicated premium. Assume a fixed expense provision of $12,000, variable expenses equal to 20% of premium, and a profit provision equal to 5% of premium. The risk load is 5% of $\operatorname{TVaR}_{0.95}-\operatorname{AAL}$.

Step 1: Risk load. The excess tail measure over expected loss is $830{,}000-70{,}000=\$760{,}000$. The selected risk load is $0.05(760{,}000)=\$38{,}000$.

Step 2: Amount needed before percentage provisions. AAL, risk load, and fixed expense together are $70{,}000+38{,}000+12{,}000=\$120{,}000$.

Step 3: Gross up for variable provisions. If $P$ is premium, the variable expense and profit provisions total $0.20P+0.05P=0.25P$. The premium must satisfy $P=120{,}000+0.25P$, so $0.75P=120{,}000$, giving $P=\$160{,}000$.

Check. At a $160,000 premium, variable expenses are $32,000 and profit is $8,000. Total provision is $70{,}000+38{,}000+12{,}000+32{,}000+8{,}000=\$160{,}000$, as required.

Trap avoided. Adding 25% of $120,000 to the base would give $150,000, but that incorrectly calculates percentage provisions on the pre-provision amount rather than on premium. The correct equation solves for the premium that itself generates those provisions.

Practical Checks Before Relying on the Indication

The arithmetic is only one part of a catastrophe indication. An actuary should verify that exposure values and damage ratios are on compatible bases, that event losses reflect the intended coverage terms, and that annual event probabilities and dependence assumptions match the model being used. A portfolio event loss should also be checked against the sum of its location losses, as in the first calculation.

The independent-event assumption has a specific role here: it allowed the joint probabilities to be calculated as products. If event occurrences are dependent, the four joint probabilities cannot be inferred from the two marginal probabilities alone. Likewise, if an event loss is random conditional on occurrence, a fixed-loss event table is insufficient to determine the full OEP and AEP distributions. Model uncertainty and sensitivity to assumptions may warrant separate analysis rather than being hidden inside a single indicated premium.

This case illustrates an end-to-end catastrophe pricing calculation aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. The central discipline is to preserve the meaning of each quantity: location loss, event loss, maximum event loss, aggregate annual loss, tail measure, risk load, and gross premium are different objects.

Key takeaway: Build the indication in order: calculate covered event losses from location inputs; construct the appropriate OEP or AEP distribution under stated assumptions; calculate the selected loss and tail measures; then solve the premium equation with expenses and profit applied on their specified bases.

Check Your Understanding

Work through each problem using the stated assumptions. Probabilities are annual, and all dollar losses are hypothetical.

Two independent events have fixed losses of $250,000 and $600,000, with occurrence probabilities 0.08 and 0.04. Calculate AAL. (Answer: $44,000)
For the two events in Question 1, calculate the probability that AEP is at least $600,000 and the probability that it exceeds $600,000. (Answer: 0.0400 and 0.0032, respectively)
For Question 1, find the 95th-percentile AEP using the smallest-loss cumulative-probability definition. (Answer: $250,000)
A hypothetical portfolio has AAL $90,000 and $\operatorname{TVaR}_{0.95}$ of $690,000. The selected risk load is 4% of the difference between TVaR and AAL. Calculate the risk load. (Answer: $24,000)
Using Question 4's risk load, add fixed expenses of $6,000, variable expenses of 15% of premium, and profit of 5% of premium. Calculate the indicated premium. (Answer: $150,000)

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