Value at Risk Backtesting Calculator
A VaR backtesting calculator counts realized P&L observations that are strictly worse than their paired positive VaR limits. It reports observed versus expected exception coverage and the Kupiec likelihood-ratio statistic without converting a p-value into a model verdict or compliance decision.
Enter realized P&L and paired VaR limits
Enter chronological rows as signed P&L followed by a positive VaR loss limit. Use one currency, horizon and clean-or-dirty convention throughout.
Both columns must use this same currency.
Expected exception probability equals one minus this confidence.
Enter 20 to 5,000 rows. Example: -250, 200 means realized P&L −250 against a positive VaR loss limit of 200.
Entered exception coverage
Entered Market Risk Diagnostics 1.0.0.
| Row | Realized P&L | Positive VaR limit | Exception threshold | Result |
|---|
How the VaR coverage backtest works
Observed rate = Exceptions ÷ N
LRuc = −2 × [log L(expected rate) − log L(observed rate)]
Each chronological P&L observation is compared with the negative of its same-row positive VaR limit. Equality is not counted; only a realized P&L strictly below that threshold is an exception. Limits may vary by row, but all rows must share one horizon, confidence, currency and P&L convention.
The Kupiec unconditional-coverage statistic compares the Bernoulli log likelihood under the entered expected exception rate with the likelihood under the observed rate. The page shows the one-degree-of-freedom asymptotic chi-square p-value as a calculation output, not as a green/red model decision.
Worked example from the audited fixture
The audited fixture contains 40 chronological USD rows, each with a positive VaR limit of USD 200 at 95% confidence.
- Realized P&L values of −250, −320 and −220 are strictly below −USD 200, so there are three exceptions. The expected count is 40 × 5% = 2.
- The observed exception rate is 7.5%. LRuc is approximately 0.459340365 and the asymptotic p-value is approximately 0.497932416; version 1.0.0 assigns no verdict.
Reproduce it: select “Load audited example” above to use the immutable Batch 34 values.
How to interpret the result
- The exception table is the primary audit evidence. Confirm every P&L is compared with the VaR estimate produced before that outcome was known.
- A p-value is not the probability that the model is correct. Coverage tests can have low power and can fail to identify an inaccurate model.
- Correct unconditional frequency does not establish independence: exceptions could still arrive in harmful clusters.
Assumptions and limits
- This page does not estimate VaR; it evaluates user-entered VaR limits against realized P&L.
- The model cannot verify timestamps, look-ahead bias, horizon alignment, position consistency or clean-versus-dirty P&L treatment.
- The asymptotic reference can be weak for small samples or extreme exception counts.
- Conditional coverage, independence, Basel traffic-light zones and loss-function scoring are excluded.
- No regulatory compliance conclusion, model approval, grade, signal or recommendation is generated.
Value at Risk vs Expected Shortfall vs VaR backtesting
These tools share a signed-P&L vocabulary but should not be substituted for one another. VaR locates a threshold, Expected Shortfall summarizes tail severity, and backtesting checks whether previously produced limits had the entered exception frequency.
| Measure | Evidence unit | Question answered | Main boundary |
|---|---|---|---|
| Historical VaR | One signed P&L sample | Lower-tail threshold | Not a maximum-loss guarantee. |
| Expected Shortfall | Same signed P&L sample | Weighted lower-tail mean | Not a future expected-loss forecast. |
| VaR backtesting | Chronological paired P&L and VaR limits | Exception coverage | Unconditional frequency only. |
Frequently asked questions
- Version 1.0.0 counts an exception when realized signed P&L is strictly below the negative of its same-row positive VaR loss limit.
- No. A P&L exactly equal to the negative VaR limit is not strictly worse and is therefore not counted.
- It is the entered row count multiplied by one minus the selected confidence, such as two expected exceptions across 40 rows at 95%.
- It compares Bernoulli log likelihood under the entered expected exception rate with the likelihood under the observed exception rate.
- It proves nothing by itself. It is a one-degree-of-freedom reference calculation and not the probability that the VaR model is correct.
- No. Version 1.0.0 tests unconditional frequency only and does not test clustering or conditional coverage.
- No. It cannot detect look-ahead bias, horizon mismatches, changing positions, mixed currencies or inconsistent clean-versus-dirty P&L.
- No. It assigns no pass/fail verdict, regulatory zone, model grade, compliance conclusion, signal or recommendation.
Sources and methodology
- Federal Reserve Bank of New York — Methods for Evaluating VaR Estimates — Primary discussion of binomial and interval-forecast VaR evaluation methods and their limitations.
- Federal Reserve Bank of New York — Regulatory Evaluation of VaR Models — Primary discussion of Kupiec unconditional coverage and the limited power of hypothesis tests.
Continue the VaR evidence workflow
Verify the records behind your entered P&L
Before interpreting a loss-tail statistic, confirm that the account statement or platform history uses the currency, observation horizon, open-position treatment and trading-cost convention you selected. Do not mix gross and net outcomes or values converted at different rates.
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Review available statements, history exports and instrument specifications for the account used.
Check XM termsFXOpen
Verify statement currency, costs and position-history conventions before entering values.
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