Distribution semantics

The trial worldview gives us a table of losses. The exceedance probability (EP) curve transforms that table into the most important visualization in reinsurance: a chart showing the probability that losses exceed any given threshold.

What an EP curve represents

An EP curve answers the question: “What is the probability that total trial losses exceed $x$?”

Why this question rather than “What is the probability of exactly $x$?” — because in a continuous loss distribution, the probability of any exact value is effectively zero. The useful question is always about exceedance: how likely is it that losses are at least this bad? This framing directly maps to business decisions: “What is the chance we lose more than our risk appetite?” or “How much capital do we need to survive a 1-in-100 trial?”

When each trial represents one simulated year — as is typical, and as the Helios Re dataset assumes — the EP curve answers the more specific question: “What is the probability that annual losses exceed $x$?”

Formally:

$$EP(x) = P(L \geq x)$$

where $L$ is the trial loss random variable. For a trial-based simulation with $N$ equiprobable trials:

$$EP(x) = \frac{1}{N} \sum_{j=1}^{N} \mathbf{1}[L_j \geq x]$$

In plain English: count how many trials have total loss at or above $x$, divide by the total number of trials $N$.

Key Insight

An EP curve is just a survival function: $EP(x) = P(L \geq x)$. For continuous distributions, this equals $1 - F(x)$ (the complement of the CDF). For discrete trial sets, the distinction between $\geq$ and $>$ matters at the exact trial loss values — our convention uses $\geq$ so that $EP(L_{(r)}) = r/N$. The reinsurance industry uses the survival function because the interesting action is in the tail: “What is the probability of reaching this loss?” is more natural than “What is the probability of staying below it?”

Building an EP curve

The algorithm is straightforward:

1. Compute total loss per trial $L_j$ — the sum of all TELT rows where trial_id = $j$
2. Sort the trial losses in descending order
3. Assign rank $r$ to each (1 = largest loss)
4. The exceedance probability of the $r$-th ranked loss is $r/N$

Step 2 assumes the loss-positive sign convention — the worst trials are the largest values, so descending order puts the tail at the top. Under a loss-negative convention, the sort flips to ascending and every rank formula in this chapter is mirrored.

helios_re/aep.py Python

▶ Run Reset

Reading the EP curve

SunCoast subject loss AEP curve (20 trials). Each point is one trial's total loss. Hover for details. The dashed line shows the mean loss across all trials.

From the chart and table above, you can read off key facts about SunCoast’s subject losses — the cedent’s losses before any contract terms are applied:

• There is a 5% chance (1 in 20 years) of a $616.2M+ loss — that is the single worst trial
• There is a 10% chance (1 in 10 years) of losses reaching or exceeding $489.1M
• There is a 50% chance of exceeding $274.6M — half the trials produce losses above this level
• The average loss across all trials is $275.6M

In Practice

In industry, EP curves are typically plotted with the return period (1/EP) on the x-axis (log scale) and loss on the y-axis. This makes the tail behavior visually prominent. A “1-in-100 year loss” means the loss exceeded with probability 1% per year — equivalently, $EP(x) = 0.01$.

OEP vs. AEP

There are two standard EP curves, distinguished by how they summarize occurrences within a trial. To define them, we need occurrence-level losses. Write $L_{j,t,e}$ for the total loss from the occurrence at timestamp $t$ of event $e$ in trial $j$, summed across all geographies and lines of business.

OEP (Occurrence EP)

The OEP curve uses the largest single occurrence per trial. It answers: “What is the probability that the peak single-occurrence loss in a trial exceeds $x$?”

$$L_j^{\text{OEP}} = \max_{(t,e)} L_{j,t,e}$$

helios_re/oep.py Python

▶ Run Reset

AEP (Aggregate EP)

The AEP curve uses the total loss across all occurrences. It answers: “What is the probability that total trial losses exceed $x$?”

$$L_j^{\text{AEP}} = \sum_{(t,e)} L_{j,t,e}$$

helios_re/oep_and_aep.py Python

▶ Run Reset

The AEP is always greater than or equal to the OEP at every return period, because the total loss is always at least as large as the largest occurrence. The difference between them reflects multi-occurrence risk — the accumulation of losses from multiple occurrences in the same trial.

Worth Noting

OEP and AEP are not contract-specific — they describe how you summarize trial losses for the EP curve, regardless of what generated those losses. An occurrence contract still has an AEP (the total ceded loss across all occurrences that triggered it). The distinction is about aggregation level, not contract type.

EP curve explorer

OEP (peak occurrence) AEP (aggregate) Log scale

SunCoast subject loss EP curves (20 trials). Move your cursor over the chart for details. The dashed AEP curve is always at or above the solid OEP curve — the gap reflects multi-occurrence risk.

The EP curve gives us the full loss distribution. The next section extracts specific numbers from that distribution — the risk metrics that drive every business decision in reinsurance.