# Metrics

> Expected loss, Value at Risk, Tail Value at Risk and the layer probabilities: the question each metric answers, the conventions they share, and a page for each.

Source: https://reinsurance.dev/toolkit/metrics/

:::note[Assumptions: AEP basis, 100% participation, demo tier]
- All metrics are computed on an **[aggregate exceedance probability (AEP)](https://reinsurance.dev/glossary/#aep-aggregate-exceedance-probability)** basis: total loss per trial. Metrics can equally be computed on an [occurrence exceedance probability (OEP)](https://reinsurance.dev/glossary/#oep-occurrence-exceedance-probability) basis; we use AEP throughout unless stated otherwise.
- All results are quoted at **100%**, before participation, per the site's [contracts-at-100% convention](https://reinsurance.dev/financial-modelling/composition/#composing-contracts-from-the-blocks). (The metrics on these pages run on subject losses, where the distinction is moot; it matters once contracts enter.) Participation scales every metric linearly (multiply by the participation fraction) and is applied at portfolio roll-up in the Financial Modelling chapter, using the [scaling](https://reinsurance.dev/financial-modelling/terms/scaling/) operation.
- All quoted numbers come from the [20-trial demo tier](https://reinsurance.dev/toolkit/trials/#demo-scale). The definitions and code are exact at any trial count; the resolution is not.
:::

The [exceedance probability (EP) curve](https://reinsurance.dev/toolkit/distributions/) gives us the full loss distribution. But decision-makers do not read curves. They ask specific questions and need specific numbers:

- **How much loss should we expect in a typical year?** Premium must at least cover expected losses, or the portfolio loses money.
- **What is the worst loss we should plan for at a given confidence level?** Capital requirements, risk appetite limits, and retrocession purchasing all need a threshold.
- **When things go badly, how bad do they actually get?** A threshold tells you where the tail starts. It says nothing about the severity once you're in it.

Three metrics answer these questions: **[expected loss (EL)](https://reinsurance.dev/glossary/#el-expected-loss)**, **[Value at Risk (VaR)](https://reinsurance.dev/glossary/#var-value-at-risk)**, and **[Tail Value at Risk (TVaR)](https://reinsurance.dev/glossary/#tvar-tail-value-at-risk)**. They are not the only risk metrics (standard deviation, attachment probability, and others have pages of their own), but these three run through nearly every reinsurance pricing and capital decision.

## The metric pages

| Metric | The question it answers | Definition |
| --- | --- | --- |
| [Expected loss (EL)](https://reinsurance.dev/toolkit/metrics/expected-loss/) | How much loss to expect on average | $\frac{1}{N}\sum_j L_j$ |
| [Value at Risk (VaR)](https://reinsurance.dev/toolkit/metrics/value-at-risk/) | The loss reached or exceeded in only $(1-\alpha)$ of trials | $L_{(k)}$, with $k = \lfloor (1-\alpha) \cdot N \rfloor$ |
| [Tail Value at Risk (TVaR)](https://reinsurance.dev/toolkit/metrics/tail-value-at-risk/) | The average loss of the worst $k$ trials | $\frac{1}{k}\sum_{r=1}^{k} L_{(r)}$ |
| [VaR vs. TVaR](https://reinsurance.dev/toolkit/metrics/var-vs-tvar/) | Why the VaR of a combined distribution can exceed the sum of the standalone VaRs, and the TVaR cannot | Sub-additivity |
| [Attachment and exhaustion probability](https://reinsurance.dev/toolkit/metrics/attachment-and-exhaustion-probability/) | How often a layer is triggered, and how often it is fully consumed | $P_{\text{attach}}$ and $P_{\text{exhaust}}$ |
| [Additional metrics](https://reinsurance.dev/toolkit/metrics/additional-metrics/) | Volatility, one band of the tail, and what one contract holds of a portfolio's tail or adds to it | $\sigma$, window TVaR, contribution to TVaR, marginal TVaR |
| [If your losses are negative](https://reinsurance.dev/toolkit/metrics/negative-losses/) | What changes when losses are stored as negative numbers | The same definitions, mirrored |

Every metric but one is computed from the per-trial losses that an [EP curve](https://reinsurance.dev/toolkit/distributions/) holds; the count of attaching occurrences needs the occurrence rows. VaR, TVaR and window TVaR index into the curve's sort order; expected loss, the standard deviation and the two layer probabilities do not depend on the order. Unless a page says otherwise, the assumptions at the top of this one hold on every page in the table, and so does the [meaning of $\alpha$](https://reinsurance.dev/toolkit/metrics/#confidence-exceedance-and-return-periods).

## Confidence, exceedance, and return periods

In practice, you will hear risk thresholds described as "the 1-in-200-year loss" or "the 0.5% exceedance VaR": the [return period](https://reinsurance.dev/glossary/#return-period) and exceedance framings.[^returnperiod] Formal definitions in the actuarial risk-measure literature use the complementary **confidence level** $\alpha$: the 1-in-200 loss is $\text{VaR}_{0.995}$. All three framings describe the same point on the EP curve:

| Return period | Exceedance probability | Confidence level $\alpha$ | Notation |
| ---: | ---: | ---: | --- |
| 10 years | 10% | 90% | $\text{VaR}_{0.90}$ |
| 100 years | 1% | 99% | $\text{VaR}_{0.99}$ |
| 200 years | 0.5% | 99.5% | $\text{VaR}_{0.995}$ |

Each framing converts into the others exactly:

$$
\begin{aligned}
\text{Return period} &= \frac{1}{1 - \alpha} \\[2pt]
\text{Exceedance probability} &= 1 - \alpha \\[2pt]
\alpha &= 1 - \frac{1}{\text{Return period}}
\end{aligned}
$$

We use $\alpha$ for the confidence level in all formal definitions, following the actuarial risk-measure literature.[^hardy] That is a convention, not a universal: the same measures appear elsewhere, regulatory documents among them, with $\alpha$ written as the exceedance probability instead.[^es-level] We always state the return period or exceedance probability alongside, because that is the language practitioners use, and because the symbol alone does not say which one is meant.

:::caution[Watch Out]
When someone says **"the 10% VaR,"** do they mean 10% confidence ($\alpha = 0.10$, the 10th percentile) or 10% exceedance ($\alpha = 0.90$, the 1-in-10 loss)? Both meanings are in use. Whenever you see a percentage attached to VaR or TVaR, check whether it refers to the confidence level or the exceedance probability. On this site, a bare subscript always means confidence: $\text{VaR}_{0.90}$ is the 1-in-10 loss, and an exceedance probability is written out in words wherever it is meant.
:::

Confidence levels apply to the tail metrics; [expected loss](https://reinsurance.dev/toolkit/metrics/expected-loss/) takes none, because it averages every trial.

## Metrics explorer

The explorer plots EL, VaR and TVaR together on SunCoast's [AEP curve](https://reinsurance.dev/toolkit/distributions/#oep-vs-aep), at a confidence level you choose.

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

```python
from caldera_re.aep import aep
from caldera_re.data import cedents
from caldera_re.expected_loss import expected_loss
from caldera_re.report import heading
from caldera_re.tvar import tvar
from caldera_re.var import var

curve = aep(cedents.suncoast)

el = expected_loss(curve)
v90 = var(curve, 0.90)
tv90 = tvar(curve, 0.90)

heading("Caldera Re Subject Loss Metrics (AEP basis)")
print(f"Expected Loss (EL):     {el:7.1f}M")
print(f"VaR(90%):               {v90:7.1f}M")
print(f"TVaR(90%):              {tv90:7.1f}M")
```

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

Each of the three marks is defined on its own page, beginning with [expected loss](https://reinsurance.dev/toolkit/metrics/expected-loss/).

[^hardy]: Mary R. Hardy, ["An Introduction to Risk Measures for Actuarial Applications"](https://www.casact.org/sites/default/files/database/studynotes_hardy4.pdf), Society of Actuaries study note, 2006 (the copy hosted by the CAS), §§2.2 to 2.3: "VaR is always specified with a given confidence level α – typically α=95% or 99%", and the conditional tail expectation "is defined using some confidence level α" too.

[^es-level]: Carlo Acerbi and Dirk Tasche, ["On the Coherence of Expected Shortfall"](https://arxiv.org/abs/cond-mat/0104295), *Journal of Banking & Finance* 26(7), 2002, §1, define expected shortfall at level $\alpha$ as the "average loss in the worst 100α% cases", so their $\alpha$ is the exceedance probability. Federal Office of Private Insurance, ["Technical document on the Swiss Solvency Test"](https://www.finma.ch/finmaarchiv/bpv/download/e/sst_techdok_061002_e_wo_li_20070118.pdf), 2006, writes both measures the same way, at "the certainty level 1 − α" with α = 1%.

[^returnperiod]: 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. The return period "is calculated as a reciprocal of the EP". Verisk, ["Modeling Fundamentals: Combining Loss Metrics"](https://www.verisk.com/blog/modeling-fundamentals-combining-loss-metrics/), October 2012, reads the same mapping off a ranked table: the 40th largest of 10,000 simulated years is "the 40/10,000 or 0.40% exceedance probability (also known as the 250-year loss when expressed as a return period)", meaning "a 0.40% probability that the portfolio will experience an aggregate loss of 153,560,977 or greater in a given year".