Metrics

Assumptions

• All metrics are computed on an AEP (aggregate) basis — total loss per scenario. Metrics can equally be computed on an OEP basis; we use AEP throughout unless stated otherwise.
• All calculations assume 100% participation — the reinsurer takes the full share of every contract. Scaling for partial participation is straightforward (multiply by the participation fraction) and is covered in the Engineering chapter.

The EP curve gives us the full loss distribution. But decision-makers don’t stare at curves — they ask specific questions and need specific numbers:

• How much should we reserve for a typical year? Premium must at least cover expected losses, or the portfolio bleeds 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.

Some of the most important metrics that answer these questions are EL, VaR, and TVaR. They are not the only risk metrics — standard deviation, attachment probability, and others play important roles — but they appear in nearly every reinsurance pricing, reserving, and capital decision.

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.” Formal definitions in the risk management 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 — they just approach it from different directions:

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%	$\text{VaR}_{0.995}$

The conversions are:

$$\text{Return period} = \frac{1}{1 - \alpha} \qquad \text{Exceedance probability} = 1 - \alpha \qquad \alpha = 1 - \frac{1}{\text{Return period}}$$

We use $\alpha$ in all formal definitions — it is the universal convention in risk measure literature, and what you will encounter in textbooks and regulatory documents. But we always state the return period or exceedance probability alongside, because that is the language practitioners use.

Watch for Ambiguity

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)? In reinsurance, it almost always means the latter — the 10% exceedance probability. But the ambiguity is real. 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.

Expected Loss (EL)

Intuition

The expected loss (EL) is the average scenario loss — the single most basic measure of risk. It is the loss that the model suggests you would experience on average across all simulated scenarios.

Formal definition

$$\text{EL} = \frac{1}{N} \sum_{s=1}^{N} L_s$$

where $N$ is the number of scenarios and $L_s$ is the total loss in scenario $s$.

Business interpretation

EL is the burning cost (industry shorthand for the long-run average loss). It 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.

What this means in code

helios_re/expected_loss.py

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EL is necessary but not sufficient

Two portfolios can have the same EL but very different risk profiles. A portfolio with 20 scenarios of $50M each has EL = $50M. A portfolio with 19 scenarios of $0 and one scenario of $1,000M also has EL = $50M. The second is far more dangerous — the entire risk is concentrated in a single catastrophic scenario, and if that scenario occurs, the loss is 20× the mean. The average tells you nothing about how concentrated the risk is — you need tail metrics to distinguish them.

EL = 44.6M

Value at Risk (VaR)

Intuition

EL cannot distinguish a steady portfolio from a catastrophic one. Value at Risk (VaR) addresses this by putting a number on the tail — the extreme right end of the loss distribution, where the rare but severe outcomes live. VaR at confidence $\alpha$ is the loss at exceedance probability $(1-\alpha)$ on the EP curve — the loss reached or exceeded in only $(1-\alpha)$ of scenarios. At $\alpha = 0.90$, that is the 1-in-10 loss.

Formal definition

For $N$ equiprobable scenarios sorted in descending order, the $r$-th largest loss has exceedance probability $r/N$. VaR at confidence $\alpha$ is the loss at exceedance probability $(1-\alpha)$:

$$\text{VaR}_\alpha = L_{(k)} \quad \text{where } k = \lfloor (1-\alpha) \cdot N \rfloor$$

where $L_{(r)}$ denotes the $r$-th largest loss. The top $k$ scenarios — those at or above VaR — form the tail used for TVaR.

Note

For continuous distributions, this is equivalent to the $\alpha$-quantile of the CDF: $\inf\{x : P(L \leq x) \geq \alpha\}$. For discrete scenario sets, different quantile conventions can differ by one rank position — we use the EP curve convention throughout.

In Practice

The descending-sorted losses are already an EP curve. VaR reads off a single position — and the top $k$ values (including VaR itself) give you TVaR (next section). In code with 0-based arrays: VaR = losses[k - 1].

Business interpretation

VaR tells you: “This loss is reached or exceeded in only $(1-\alpha)$ of scenarios.” Regulators use VaR to set minimum capital requirements — Solvency II requires capital to cover the 1-in-200 year loss ($\text{VaR}_{0.995}$).

What this means in code

helios_re/var.py

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Key Insight

VaR tells you the threshold but nothing about what happens beyond it. VaR(90%) = $97.0M marks the boundary of the worst 10% of scenarios — but it does not say whether those exceedances are $98M or $500M. This is VaR’s fundamental limitation and the reason TVaR is increasingly preferred for capital thinking.

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

VaR(90%) = 97.0M Tail scenarios (above VaR)

In Practice

Throughout this chapter, we use $\alpha = 0.90$ — the 1-in-10 year loss. This is appropriate for a 20-scenario dataset where $k = \lfloor 20 \times 0.10 \rfloor = 2$ tail scenarios — enough to illustrate the concepts. In production, regulatory frameworks target much longer return periods: Solvency II uses the 1-in-200 year VaR ($\alpha = 0.995$), the Swiss Solvency Test uses TVaR at the 1-in-100 return period ($\alpha = 0.99$). These require catastrophe models with 10,000+ scenarios for stable tail estimates.

Tail Value at Risk (TVaR)

Intuition

VaR tells you where the tail starts. Tail Value at Risk (TVaR) tells you how bad it gets once you’re in it. At confidence level $\alpha$, TVaR is the average loss across the worst $(1-\alpha)$ of scenarios — the expected severity of the tail.

TVaR is also known as Conditional Tail Expectation (CTE) and Expected Shortfall (ES) — three names for the same quantity.

Formal definition

$$\text{TVaR}_\alpha = \frac{1}{k} \sum_{r=1}^{k} L_{(r)}$$

where $k = \lfloor N \cdot (1 - \alpha) \rfloor$ is the number of tail scenarios and $L_{(r)}$ is the $r$-th largest loss (rank $r$ in the descending EP curve). In words: take the worst $k$ scenarios and average their losses. The connection to $\alpha$ is through $k$: a higher confidence level means fewer but more extreme tail scenarios.

Business interpretation

TVaR is the expected loss conditional on being in the tail. It answers the CUO’s question: “If we have a bad year — one of the worst $(1-\alpha)$ — how bad is it on average?”

What this means in code

helios_re/tvar.py

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Key Insight

TVaR is preferred over VaR for capital thinking. VaR answers “what is the threshold?” TVaR answers “how bad is it when things go wrong?” For a reinsurer managing tail risk, the second question is more relevant. Two portfolios with the same VaR can have very different TVaRs — the one with the heavier tail is more dangerous and needs more capital.

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

VaR(90%) = 97.0M TVaR(90%) = 104.5M TVaR = mean of tail scenarios

VaR vs. TVaR: why it matters

Sub-additivity

A coherent risk measure must satisfy:

$$\rho(X + Y) \leq \rho(X) + \rho(Y)$$

This says that the risk of a combined portfolio should not exceed the sum of individual risks — diversification should help, not hurt. TVaR always satisfies this property. VaR does not.

Here is a concrete example. Consider two portfolios, each with 10 equiprobable scenarios:

Scenario	$L_1$	$L_2$	$L_1 + L_2$
1	0	10	10
2	0	1	1
3	0	0	0
4	0	0	0
5	0	0	0
6	0	0	0
7	0	0	0
8	0	0	0
9	1	0	1
10	9	0	9

At $\alpha = 0.80$, $k = \lfloor 0.2 \times 10 \rfloor = 2$ tail scenarios. Sort each portfolio descending — VaR is the $k$-th largest:

$$L_1 \text{ descending: } [9, \mathbf{1}, 0, 0, 0, 0, 0, 0, 0, 0] \quad\Rightarrow\quad \text{VaR}_{0.80}(L_1) = 1$$ $$L_2 \text{ descending: } [10, \mathbf{1}, 0, 0, 0, 0, 0, 0, 0, 0] \quad\Rightarrow\quad \text{VaR}_{0.80}(L_2) = 1$$ $$\text{VaR}_{0.80}(L_1) + \text{VaR}_{0.80}(L_2) = 1 + 1 = 2$$

Now the combined portfolio, sorted descending:

$$L_1 + L_2 \text{ descending: } [10, \mathbf{9}, 1, 1, 0, 0, 0, 0, 0, 0] \quad\Rightarrow\quad \text{VaR}_{0.80}(L_1 + L_2) = 9$$

Combining the portfolios made VaR worse: $9 > 2$. Diversification increased measured risk — a pathological result.

TVaR does not have this problem. With $k = 2$ tail scenarios:

$$\text{TVaR}_{0.80}(L_1) = \frac{9 + 1}{2} = 5 \qquad \text{TVaR}_{0.80}(L_2) = \frac{10 + 1}{2} = 5.5$$ $$\text{TVaR}_{0.80}(L_1 + L_2) = \frac{10 + 9}{2} = 9.5 \leq 10.5 = \text{TVaR}_{0.80}(L_1) + \text{TVaR}_{0.80}(L_2) \quad\checkmark$$

Comparison

Property	VaR	TVaR
What it measures	Threshold of the tail	Mean severity of the tail
Tail sensitivity	None — ignores everything beyond the threshold	Captures average severity, but not the full shape of the tail (just as EL cannot capture the full distribution)
Sub-additivity	Not guaranteed — diversification can increase VaR	Always satisfied

Additional metrics

EL, VaR, and TVaR are the core trio, but reinsurance analytics uses several other metrics for risk profiling, contract pricing, and portfolio construction.

Attachment probability

The probability that a loss exceeds the attachment point of a contract:

$$P_{\text{attach}} = \frac{1}{N} \sum_{s=1}^{N} \mathbf{1}[L_s > A]$$

This tells a reinsurer how often a contract will be triggered. A layer with 30% attachment probability means the reinsurer expects to pay a claim roughly 3 years out of 10. Attachment probability is a key input to pricing — it determines the frequency component of the expected loss for a layer.

Exhaustion probability

The probability that a loss reaches or exceeds the attachment point plus the limit (the contract is fully exhausted):

$$P_{\text{exhaust}} = \frac{1}{N} \sum_{s=1}^{N} \mathbf{1}[L_s \geq A + \ell]$$

The ratio of exhaustion probability to attachment probability reveals how “binary” a layer is. If a layer almost always exhausts when it attaches, the cedent is buying an all-or-nothing contract — the reinsurer pays the full limit or nothing. This shapes how much capital the reinsurer must hold against the layer.

Standard deviation

A measure of volatility around the mean:

$$\sigma = \sqrt{\frac{1}{N} \sum_{s=1}^{N} (L_s - \text{EL})^2}$$

We use the population formula ($1/N$, not $1/(N-1)$) because the $N$ scenarios represent the entire simulated probability space, not a sample drawn from it.

Standard deviation captures how dispersed losses are around EL. It appears in simpler pricing models (e.g., the standard deviation loading principle adds a multiple of $\sigma$ to EL to compute premium). It is less informative about tail behavior than TVaR — two distributions with the same $\sigma$ can have very different tails.

Window TVaR

Standard TVaR captures the average severity of the entire tail beyond a single threshold. Window TVaR refines this by measuring the average loss within a bounded interval of the loss distribution — typically between two return periods or confidence levels.

Given two confidence levels $\alpha_{\text{lo}} < \alpha_{\text{hi}}$, let $k_{\text{lo}} = \lfloor (1-\alpha_{\text{lo}}) \cdot N \rfloor$ and $k_{\text{hi}} = \lfloor (1-\alpha_{\text{hi}}) \cdot N \rfloor$. The $k_{\text{lo}}$ worst scenarios form the broader tail; the $k_{\text{hi}}$ worst form the narrower, more extreme tail. Window TVaR is the mean of the scenarios between these two boundaries:

$$\text{Window TVaR}(\alpha_{\text{lo}}, \alpha_{\text{hi}}) = \frac{1}{k_{\text{lo}} - k_{\text{hi}}} \sum_{r=k_{\text{hi}}+1}^{k_{\text{lo}}} L_{(r)}$$

For example, Window TVaR(80%, 90%) averages the losses between the 1-in-5 and 1-in-10 return periods — the scenarios in the broader tail that are not in the extreme tail.

Window TVaR is especially useful for:

• Pricing excess-of-loss layers — a layer sits between an attachment and exhaustion point, which corresponds to a specific band of the return period spectrum. Window TVaR characterizes the expected loss contribution from that band.
• Differentiating frequency-driven from severity-driven risk — a low-layer window TVaR is influenced by attritional losses, while a high-layer window TVaR captures catastrophic severity.
• Capital allocation — attributing risk capital to specific portions of the portfolio loss distribution rather than treating the entire tail as one block.

In Practice

The window boundaries map directly to EP curve ranks. Window TVaR(80%, 90%) with $N=20$ averages ranks 3–4 (the scenarios between the 1-in-5 and 1-in-10 return periods).

helios_re/window_tvar.py

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Portfolio-level metrics

When contracts combine into a portfolio, new questions arise: How much does a single contract contribute to overall tail risk? How should capital be allocated across contracts? Metrics like Marginal TVaR and contribution-to-TVaR answer these questions. The portfolio aggregation section develops these in detail.

Metrics explorer

helios_re/metrics_summary.py

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

EL = 44.6M VaR(90%) = 97.0M TVaR(90%) = 104.5M

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We now have the tools to measure risk. The next section shows how contracts transform these distributions — turning a subject loss distribution into a ceded loss distribution.