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VaR (Value at Risk)

VaR is a quantile of an estimated loss distribution, defined for a stated scope, horizon and confidence level. It is not maximum loss and does not describe the severity of outcomes beyond the threshold.

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Who this entry is for — Anyone who needs to read or compare a loss measure without mistaking it for a guaranteed limit. VaR is useful only when its scope, unit, horizon, level, date and method are explicit.

For a defined loss variable L, Value at Risk at level α is a quantile of its estimated distribution:

VaRα(L) = inf {x : P(L ≤ x) ≥ α}

A one-day 95% VaR of EUR 18,000 means that, under the selected data, valuations and model, at least 95% of the estimated distribution does not exceed that loss. Outcomes above EUR 18,000 remain possible; VaR does not measure their size and does not state maximum loss.

In plain terms — VaR marks a threshold in the assumed distribution. It does not cap the tail.

VaR and Expected Shortfall read different parts of the tail Illustrative diagram: the actual shape depends on data, model, horizon and confidence level VaR and Expected Shortfall read different parts of the tail Illustrative diagram: the actual shape depends on data, model, horizon and confidence level VaR: quantile ES: tail average losses beyond the quantile loss magnitude → frequency / density 01 Distribution It is a model or an empiricalestimate of losses, not an… 02 VaR threshold It marks a quantile for achosen horizon and level; it i… 03 Tail Outcomes beyond the thresholdremain possible; VaR alone doe… 04 Expected Shortfall In this continuous diagram, itis average tail loss under the… Cyclepedia · source-checked visual explainer
VaR and Expected Shortfall read different parts of the same estimated distribution.

The coordinates a VaR needs

A VaR figure is incomplete without at least:

Field Question
Scope Which position, strategy or portfolio?
Variable and unit Monetary loss, percentage or another measure?
Horizon One day, ten days or another interval?
Level 95%, 99% or another quantile?
Date and positions Which portfolio and valuation time?
Method Which data, distributions, dependencies and revaluation rules?

Changing one of these coordinates can change the result. Parameters set by a rule for a particular regulated entity are not defaults for every portfolio.

Three estimation families

Method What it does Limitation to disclose
Historical simulation Applies observed past factor changes to the current portfolio and constructs hypothetical P&L The window may not contain the relevant regime
Parametric Estimates a distribution through selected parameters and structure The result depends on the distribution and approximations; “parametric” does not always mean “normal”
Monte Carlo Simulates many paths or shocks from a specified model and revalues the portfolio Dependencies, dynamics, data and pricing remain modelling choices

The method should reflect payoffs, non-linearity, liquidity, data quality and intended use. Comparing methods can expose sensitivity, but it does not make one of them automatically correct.

Backtesting and exceptions

An exception occurs when the observed loss, defined consistently with the test, exceeds forecast VaR. Its count must be read together with confidence level, horizon, sample size, temporal dependence, P&L definition and model changes.

For example, the Basel market-risk framework uses a particular regulatory interpretation over 250 observations: 0–4, 5–9 and at least 10 exceptions for one-day 99% VaR fall into different zones. Those thresholds are designed for that sample and context. They do not support a universal rule such as “more than three exceptions per month means the model is broken”.

Backtesting is a diagnostic, not a guarantee. An unexpected result calls for investigation of data, valuation, assumptions, implementation, use and market conditions, connecting it to model risk.

What VaR does not measure by itself

  • the severity of losses beyond the quantile: that is the role of Expected Shortfall;
  • a worst case or a guaranteed loss limit;
  • out-of-sample shocks, dependency breaks or regime changes;
  • liquidity, funding, counterparty and operational risk unless modelled;
  • decision quality or capital adequacy for every possible use.

VaR is not subadditive for every distribution and portfolio. Diversification benefits embedded in the number therefore need to be understood in light of the model and compared with stress testing and scenario analysis.

Typical mistake — Presenting “VaR 100,000” as maximum loss. Without coordinates it is incomplete; with them it remains a conditional estimate.

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