Metrics

Assumptions: AEP basis, 100% participation, demo tier

• All metrics are computed on an aggregate exceedance probability (AEP) basis: total loss per trial. Metrics can equally be computed on an occurrence exceedance probability (OEP) basis; we use AEP throughout unless stated otherwise.
• All results are quoted at 100%, before participation, per the site’s contracts-at-100% convention. (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 operation.
• All quoted numbers come from the 20-trial demo tier. The definitions and code are exact at any trial count; the resolution is not.

The exceedance probability (EP) curve 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), Value at Risk (VaR), and Tail Value at Risk (TVaR). 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)	How much loss to expect on average	$\frac{1}{N}\sum_j L_j$
Value at Risk (VaR)	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)	The average loss of the worst $k$ trials	$\frac{1}{k}\sum_{r=1}^{k} L_{(r)}$
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	How often a layer is triggered, and how often it is fully consumed	$P_{\text{attach}}$ and $P_{\text{exhaust}}$
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	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 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 $\alpha$.

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 and exceedance framings.¹ 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.² That is a convention, not a universal: the same measures appear elsewhere, regulatory documents among them, with $\alpha$ written as the exceedance probability instead.³ 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.

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 takes none, because it averages every trial.

Metrics explorer

The explorer plots EL, VaR and TVaR together on SunCoast’s AEP curve, at a confidence level you choose.

caldera_re/metrics_summary.py 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")

▶ Run Reset

Confidence level (α): 90% — 2 tail trials (worst 10%)

EL = 275.6M VaR(90%) = 489.1M TVaR(90%) = 552.6M

Each of the three marks is defined on its own page, beginning with expected loss.

Footnotes

1. Natalia Humphreys, “Exceedance Probability in Catastrophe Modeling”, Casualty Actuarial Society E-Forum, Winter 2021. The return period “is calculated as a reciprocal of the EP”. Verisk, “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”. ↩

2. Mary R. Hardy, “An Introduction to Risk Measures for Actuarial Applications”, 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. ↩

3. Carlo Acerbi and Dirk Tasche, “On the Coherence of Expected Shortfall”, 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”, 2006, writes both measures the same way, at “the certainty level 1 − α” with α = 1%. ↩