# Expected loss (EL)

> Expected loss: the average trial loss, its formal definition, what it means for premium, and the code that computes it.

Source: https://reinsurance.dev/toolkit/metrics/expected-loss/

**[Expected loss (EL)](https://reinsurance.dev/glossary/#el-expected-loss)** is the average trial loss, the most basic measure of risk. It is the loss that the model suggests you would experience on average across all simulated trials. In vendor language the same quantity is the **average annual loss (AAL)**.[^aal] You will meet both names for one number.

Unless a figure says otherwise, the numbers on this page follow the [assumptions of the Metrics overview](https://reinsurance.dev/toolkit/metrics/): the aggregate exceedance probability (AEP) basis, results at 100%, and the [20-trial demo tier](https://reinsurance.dev/toolkit/trials/#demo-scale).

## Definition and implementation

### Formal definition

$$
\text{EL} = \frac{1}{N} \sum_{j=1}^{N} L_j
$$

where $N$ is the number of trials and $L_j$ is the total loss in trial $j$.

### Business interpretation

EL is the floor for any reasonable premium: a reinsurer must charge at least EL (plus expenses and profit margin) or it will lose money over time.

:::caution[Watch Out]
In pricing conversations, EL is sometimes conflated with the **[burning cost](https://reinsurance.dev/glossary/#burning-cost)**. They are not the same object. A burning cost is an experience-rating estimate: the cedent's historical losses to the layer, trended and developed, expressed as a rate on the subject premium.[^burning-cost] Applied to the premium base, that rate yields a loss-cost estimate: the same quantity EL estimates, but from the cedent's loss history rather than from the model's trials. Experience rating is outside this site's scope; on this site, EL always means the trial average defined above.
:::

### What this means in code

Python (`python/caldera_re/expected_loss.py`, runnable on the web page):

```python
from caldera_re.aep import aep
from caldera_re.data import cedents

def expected_loss(curve):
    """Compute expected loss from an EP curve.

    The expected loss is the mean of all trial losses in the curve.
    """
    return curve['loss'].mean()

if __name__ == '__main__':
    curve = aep(cedents.suncoast)
    el = expected_loss(curve)

    print(f"Expected Loss (EL): {el:.1f}M")
    print(f"  = sum of all trial losses / N")
    print(f"  = {curve['loss'].sum():.1f}M / {len(curve)}")
```

*[Interactive figure: Caldera metrics chart. See the web page: https://reinsurance.dev/toolkit/metrics/expected-loss/]*

### EL is necessary but not sufficient

Two loss distributions can have the same EL but very different risk profiles. A loss distribution with 20 trials of \$50M each has EL = \$50M. A distribution with 19 trials of \$0 and one trial of \$1,000M also has EL = \$50M. The second is far more dangerous: the entire risk is concentrated in a single catastrophic trial, and if that trial occurs, the loss is 20× the mean. The average tells you nothing about how concentrated the risk is. You need **[tail metrics](https://reinsurance.dev/toolkit/metrics/value-at-risk/)** to distinguish them.

[^aal]: Verisk, ["Modeling Fundamentals: What Is AAL?"](https://www.verisk.com/blog/modeling-fundamentals--what-is-aal-/), the vendor's own definition: "The mean of this distribution is the average annual loss (AAL), or the expected loss per year, averaged over many years." Natalia Humphreys, ["Exceedance Probability in Catastrophe Modeling"](https://www.casact.org/sites/default/files/2021-03/02_humphreys.pdf), *Casualty Actuarial Society E-Forum*, Winter 2021, writes the same quantity as $\text{AAL} = \sum_i p_i X_i$, the expected annual loss.

[^burning-cost]: Ana J. Mata, ["Burning Cost"](https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat04732.pub2), *Wiley StatsRef: Statistics Reference Online*. Burning cost is "an experience-rating method used … to estimate the expected loss cost." See also IRMI, ["Burning Cost"](https://www.irmi.com/term/insurance-definitions/burning-cost), and the RAA, [Glossary of Reinsurance Terms](https://www.reinsurance.org/RAA/RAA/About-the-RAA/Glossary/Glossary%20of%20Reinsurance%20Terms.aspx): "the ratio of the reinsurance losses incurred to the ceding company's subject premium … based upon historical experience."