Reading the Statistics in an Event Loss Table
An event loss table (ELT) summarizes modeled events and the losses they produce for a specified portfolio and set of policy terms. Its rows describe event types or catalog events; they are not necessarily separate observed years. The figures most often needed for pricing are an event’s annual occurrence frequency, its conditional event severity, and the resulting rate-weighted expected loss.
These statistics answer different questions. Frequency describes how often an event is expected to occur. Conditional severity describes the mean loss given that the event occurs. Their product gives that event’s expected loss per year. A rate is not automatically a probability: an event type can have an expected frequency greater than one occurrence per year.
The conditional mean should be calculated over the losses represented by the table’s event definition and coverage terms. If the table includes events that affect no insured location, their zero losses belong in the conditional mean when they are part of the modeled event population. Removing those rows changes the statistic to a mean conditional on a narrower event set.
Frequency Is Not the Same as Probability
An annual frequency is an expected count. For example, a frequency of \(0.20\) occurrences per year means that the expected count over ten years is \(10(0.20)=2\); it does not, by itself, say that the probability of at least one occurrence in a particular year is \(0.20\).
If occurrences follow a Poisson process with annual rate \(\lambda\), the probability of at least one occurrence during a period of \(t\) years is \(1-e^{-\lambda t}\). This probability formula requires the Poisson assumption. If instead a model allows at most one occurrence in a year, the annual count is Bernoulli and its expected count equals its occurrence probability. The event-table convention and model assumptions determine which interpretation is appropriate.
Do not confuse the event’s rate with the probability that its loss exceeds a selected threshold, either. That threshold probability depends on the distribution of event losses, not just on the mean conditional severity. Interpreting exceedance curves is a separate calculation.
Why Rate Times Conditional Severity Gives Expected Loss
Suppose event type \(j\) occurs \(N_j\) times in a year and its occurrence losses are \(L_{j1},\ldots,L_{jN_j}\). The total loss from that type is \(S_j=\sum_{k=1}^{N_j}L_{jk}\). Conditional on \(N_j=n\), if each occurrence has the same conditional mean \(m_j\), linearity of expectation gives \(E[S_j\mid N_j=n]=nm_j\). Taking expectations over the count gives the result below.
For multiple event types, total loss is the sum of the type-specific losses. Expectation is additive, so the total expected loss is \(\sum_j\lambda_jm_j\). This does not require event types to be independent; it does require losses to be counted consistently and the table’s rates and severities to refer to the same portfolio, time basis, and loss measure. Dependence matters for the distribution of aggregate losses, but not for this sum of expectations.
Calculating Table Statistics
The following process keeps the units and conditioning clear.
Check whether it is an expected number of occurrences per year, an annual probability, or a rate over another time horizon. Convert it to the intended time basis before multiplying.
Confirm that the loss is conditional on an occurrence and uses the relevant coverage terms, currency, and portfolio definition.
Multiply its annual frequency by its conditional mean severity. The result has units of dollars per year.
Add the event-level expected losses. Do not average event severities without weighting them by frequency.
Worked Example: Summing Event-Level Contributions
A hypothetical property portfolio has the following annual frequencies and conditional mean covered losses in an ELT. Calculate each event type’s rate-weighted expected loss and the total. Report money to the cent.
| Event type | Annual frequency (per year) | Conditional mean loss |
|---|---|---|
| Coastal windstorm | 0.12 | $2,400,000 |
| Inland hailstorm | 0.30 | $600,000 |
| River flood | 0.04 | $7,500,000 |
Coastal windstorm. Multiply its annual frequency by its conditional mean loss: \(0.12\times \$2,400,000=\$288,000.00\) per year.
Inland hailstorm. \(0.30\times \$600,000=\$180,000.00\) per year.
River flood. \(0.04\times \$7,500,000=\$300,000.00\) per year.
Total. Add the three contributions: \(\$288,000.00+\$180,000.00+\$300,000.00=\$768,000.00\) per year. The sum of frequencies is \(0.12+0.30+0.04=0.46\) occurrences per year. As a check, the rate-weighted mean severity across these event types is \(\$768,000/0.46=\$1,669,565.22\); multiplying by \(0.46\) returns \(\$768,000.00\) per year. This pooled severity is frequency-weighted, not the simple average of the three severities.
Method and trap avoided. The method is to multiply within each event type and then sum. Adding the severities, or taking their unweighted average and multiplying by the total frequency, would give the wrong weight to event types with different frequencies.
From a Conditional Loss Distribution to a Table Entry
An ELT may report the conditional mean severity directly, or provide conditional outcomes and probabilities from which it can be calculated. Those probabilities are conditional on the event occurring and should sum to one. Once the mean is found, it is multiplied by the annual occurrence frequency—not by each conditional severity outcome separately without applying its probability.
Worked Example: Conditional Outcomes and a Poisson Event Rate
In a hypothetical model, a named storm type has an annual frequency of \(0.35\) occurrences per year. Conditional on an occurrence, the covered loss is $500,000 with probability \(0.80\) and $2,000,000 with probability \(0.20\). Find the conditional mean severity, the rate-weighted expected loss per year, and the probability of at least one occurrence in a year if occurrences are Poisson. Give the probability to four decimals and money to the cent.
Step 1: Check the conditional probabilities. They sum to \(0.80+0.20=1.00\), so they define the stated conditional loss distribution.
Step 2: Calculate the conditional mean severity. \(0.80(\$500,000)+0.20(\$2,000,000)=\$400,000+\$400,000=\$800,000.00\).
Step 3: Calculate rate-weighted expected loss. \(0.35(\$800,000)=\$280,000.00\) per year.
Step 4: Calculate the annual occurrence probability under the stated assumption. For a Poisson count with annual mean \(0.35\), \(P(N\geq1)=1-e^{-0.35}=1-0.704688\ldots=0.2953\).
The frequency is \(0.35\) occurrences per year, while the probability of at least one occurrence is \(0.2953\). The first is an expected count; the second is a probability. The conditional loss outcomes determine the mean severity, but do not change the occurrence probability.
Method and trap avoided. Use the conditional probabilities to average severity, then multiply by the event frequency. A common error is to report \(0.35\) as the Poisson probability of at least one event; the correct probability is \(1-e^{-0.35}\).
Interpreting Rates Across Time and Event Types
A rate is meaningful only with its time unit. If an event has frequency \(0.06\) per month, its frequency on a twelve-month basis is \(12(0.06)=0.72\) occurrences per year, provided the rate is constant over the months being combined. Its rate-weighted expected loss must use a compatible annual frequency and a severity on the same loss basis.
For a pooled set of event types, the appropriate conditional mean is weighted by event frequency. If the event types have frequencies \(\lambda_j\) and conditional mean severities \(m_j\), the pooled mean per occurrence is \(\left(\sum_j\lambda_jm_j\right)/\left(\sum_j\lambda_j\right)\), when the total frequency is positive. Multiplying this pooled mean by the total frequency recovers the total rate-weighted expected loss. This identity is a check, not a reason to ignore the individual event contributions.
Worked Example: Annualizing a Frequency Before Weighting
A hypothetical ELT reports a frequency of \(0.05\) occurrences per month for a volcanic ash event type and a conditional mean covered loss of $1,200,000 per occurrence. Assume the monthly frequency is constant for all twelve months. Calculate the annual frequency and annual rate-weighted expected loss. Report money to the cent.
Step 1: Convert the time basis. \(0.05\) occurrences per month \(\times 12\) months per year \(=0.60\) occurrences per year.
Step 2: Weight the conditional severity. \(0.60\times \$1,200,000=\$720,000.00\) per year.
Check. Using the monthly basis gives \(0.05\times \$1,200,000=\$60,000.00\) per month, and \(12\times \$60,000.00=\$720,000.00\) per year. The two calculations agree.
Method and trap avoided. Convert the frequency to the requested time unit before interpreting the result. Multiplying a monthly rate by a severity and labeling the result “per year” would overstate the annual contribution by a factor of twelve.
Common Mistakes and Practical Checks
- Calling every rate a probability. A frequency can exceed one per year. State whether a number is a mean count, a probability, or a rate over another horizon.
- Using unconditional and conditional quantities inconsistently. A severity conditional on an event must be paired with that event’s occurrence frequency. Check whether the table’s “loss” is conditional, already frequency-weighted, or defined in another way.
- Taking an unweighted average across event types. Rare severe events and frequent moderate events contribute in proportion to both their frequencies and severities.
- Dropping modeled zero-loss events without noticing. If the desired mean is conditional on all events in a type, removing events with no portfolio loss changes the population being averaged.
- Mixing horizons or currencies. A monthly frequency and annual loss contribution are not interchangeable, and all losses being combined must be on a consistent monetary basis.
Check Your Understanding
For each problem, distinguish the frequency from any probability and show the rate-weighting logic.
- A hypothetical ELT gives an annual frequency of \(0.08\) for a quake sequence and a conditional mean loss of $3,000,000. What is its rate-weighted expected loss per year? (Answer: $240,000.00 per year)
- A hypothetical event type has Poisson annual frequency \(0.50\). What is the probability of at least one occurrence in a year, to four decimals, and what is the expected number of occurrences in three years? (Answer: probability \(0.3935\); expected count \(1.50\))
- A hypothetical wind event occurs at annual frequency \(0.20\). Conditional on occurrence, losses are $1,000,000 with probability \(0.70\) and $4,000,000 with probability \(0.30\). Find conditional mean severity and rate-weighted expected loss per year. (Answer: $1,900,000.00 severity; $380,000.00 per year)
- A hypothetical ELT lists two event types: Type A has annual frequency \(0.15\) and conditional mean loss $800,000; Type B has annual frequency \(0.05\) and conditional mean loss $5,000,000. Find the total rate-weighted expected loss and the frequency-weighted pooled mean severity. (Answer: $370,000.00 per year; $1,850,000.00 per occurrence)
- A hypothetical event type has frequency \(0.04\) per month and conditional mean loss $2,500,000. Assuming a constant monthly rate throughout the year, calculate its annual frequency and rate-weighted expected loss per year. (Answer: \(0.48\) occurrences per year; $1,200,000.00 per year)