FRM Exam Part II · Backtesting VaR

Purpose and Objectives of VaR Backtesting

Updated 11 October 2026 · Fact-checked

VaR backtesting compares a model's daily VaR forecast with the P&L that followed, to check whether losses beyond VaR occur as often as the confidence level implies. Banks use it to validate models, set capital multipliers and find weaknesses. Hypothetical (clean) P&L tests the model; actual (dirty) P&L tests real results.

Understand Purpose and Objectives of VaR Backtesting

A VaR model makes a claim. A 99% one-day VaR says that, if the model is right, the loss on a day should exceed VaR only about 1% of the time. Backtesting checks that claim against what actually happened.

The method is simple. Each day, record the VaR forecast made at the start of the day. At the end of the day, take the P&L and compare. If the loss is larger than VaR, that day is an exception (also called an exceedance or breach). Count exceptions over a window, commonly 250 trading days, and compare the count with the expected number. At 99% over 250 days, you expect about 2.5.

Banks backtest for three reasons. First, validation: to see whether the model is accurate and unbiased. Too many exceptions suggest risk is understated. Too few suggest the model is too conservative and ties up capital. Second, regulation: under Basel, the number of exceptions feeds the capital multiplier through the traffic light approach. Third, diagnosis: exceptions point to causes such as bad data, wrong mapping, stale volatility or missing risk factors.

The choice of P&L matters. Actual (dirty) P&L is the reported trading result. It includes intraday trading, fees, commissions, reserves and carry (interest accrual). It reflects what the bank really made or lost, but it is contaminated by items the VaR model does not try to capture. Clean P&L removes fees, commissions, intraday trading and similar items, leaving the result of holding the opening positions. Hypothetical P&L revalues the previous day's closing portfolio using the next day's market moves, with positions held fixed. In practice, clean and hypothetical P&L are used to mean nearly the same thing. Check how the question defines them.

Hypothetical (clean) P&L is the purest test of the model, because it matches what VaR measures: the risk of a fixed portfolio over one day. Actual P&L is a test of whether the bank's risk estimates are relevant to real outcomes. Supervisors typically look at both. Basel's backtesting framework counts exceptions on both actual and hypothetical P&L, and the more conservative view drives the outcome.

Key formulas to remember

Exception definition: Exception on day t if P&L(t) < −VaR(t); VaR is quoted as a positive loss. A loss larger than VaR is an exception. A loss equal to VaR is not.
Expected number of exceptions: E[N] = T × (1 − c); T is the number of days and c is the VaR confidence level. At 99% over 250 days, E[N] = 2.5.
Exception rate: Exception rate = N ÷ T; Compare with 1 − c. A rate well above 1 − c suggests VaR is too low.
Hypothetical P&L: Hypothetical P&L = value of yesterday's closing positions at today's prices − their value at yesterday's prices; Positions held fixed. No intraday trades, fees or new deals.
Actual P&L link: Actual P&L = hypothetical P&L + fees and commissions + intraday trading result + reserve changes and other items; Conceptual breakdown. The extra items are what make actual P&L dirty.

How to solve Purpose and Objectives of VaR Backtesting questions

Use this order for any question on the purpose of backtesting or the type of P&L.

1Identify what is being asked: why backtest, which P&L to use, or how to count exceptions.
2Note the VaR horizon and confidence level. Work out the expected exception rate as 1 − c.
3Check which P&L series is given. Label it actual (dirty), clean or hypothetical, and note what it includes.
4Compare each day's P&L with that day's VaR forecast, using the VaR set before the P&L occurred. Count a loss larger than VaR as an exception.
5Compare the count with the expected number, T × (1 − c).
6Interpret: too many exceptions means VaR is too low or the model is flawed. Too few means it may be too conservative.
7If P&L is contaminated by fees, intraday trades or reserves, say that exceptions may reflect non-model items, not model error. Name the cleaner series as the better test of the model.

Quickest way: Three-check shortcut

When to use it: Use it for MCQs that ask which P&L suits a purpose or what an exception count implies.

Model accuracy test: pick hypothetical or clean P&L.
Test of real-world results, or regulatory reporting: actual P&L is also relevant.
Expected exceptions = T × (1 − c). Compare quickly and decide whether VaR looks too low, too high or acceptable.
Reject any option that says backtesting uses future VaR forecasts or that fees belong in clean P&L.

Common mistakes in Purpose and Objectives of VaR Backtesting

Saying clean P&L includes fees and commissions.
Students think clean means final reported profit.
Fix: Clean P&L strips out fees, commissions and intraday trading. Actual (dirty) P&L includes them.
Treating hypothetical P&L as the real trading result.
The word P&L suggests something actually booked.
Fix: Hypothetical P&L is a recalculation holding yesterday's closing positions fixed. It is never booked.
Comparing P&L with the VaR calculated after the day ends.
Students forget VaR is a forecast.
Fix: Use the VaR made at the start of the day, before the P&L is known.
Thinking the purpose of backtesting is only to catch too many exceptions.
Capital rules emphasise breaches.
Fix: Too few exceptions also signal a problem: an over-conservative model wastes capital. Backtesting checks calibration in both directions.
Counting a loss equal to VaR as an exception.
Careless reading of the definition.
Fix: An exception needs the loss to exceed VaR. Equal is not a breach.
Blaming the model for every exception on actual P&L.
Students ignore contamination from non-model items.
Fix: Check whether fees, intraday trades or reserves caused the breach. Rerun on clean or hypothetical P&L.

Worked examples

Example 1

A bank reports a 99% one-day VaR and backtests over 250 trading days. It records 7 exceptions on hypothetical P&L. What number of exceptions is expected, and what does the result suggest?

Show the solution

Expected exceptions = 250 × (1 − 0.99) = 250 × 0.01 = 2.5.
Observed exceptions = 7, which is well above 2.5.
Observed exception rate = 7 ÷ 250 = 2.8%, against 1% expected.
Because hypothetical P&L excludes intraday trading and fees, the breaches are not explained by those items.

Answer: Expected 2.5 exceptions. Seven is well above that, so the model likely understates risk. A formal test is needed to judge significance, but hypothetical P&L points to model weakness.

Example 2

On one day a bank's 99% VaR is USD 10 million. Hypothetical P&L is −USD 8 million. Actual P&L is −USD 11 million, after USD 4 million of intraday trading losses. Is this an exception on each basis, and what does it suggest?

Show the solution

Hypothetical: loss USD 8 million < VaR USD 10 million, so no exception.
Actual: loss USD 11 million > VaR USD 10 million, so an exception.
Difference in P&L = −11 − (−8) = −3 million, from intraday trading, fees and other items. The stated intraday loss of USD 4 million implies other items of +USD 1 million.
The model passes on the fixed portfolio. The breach comes from items outside the model's one-day static portfolio view.

Answer: No exception on hypothetical P&L and an exception on actual P&L. The breach is driven by intraday trading and other non-model items, not by a failure of the VaR model for the fixed portfolio.

Exam tips

Know the three P&L labels cold. Questions often ask which series isolates model error: hypothetical or clean.
Always compute T × (1 − c) first. It anchors every interpretation question.
Watch for the wording 'too few exceptions'. It is a valid problem, not a good result.
In case-style questions, trace each breach to a cause: model, data or non-model P&L items.
Remember that VaR is a forecast made before the day. Backtesting never uses hindsight VaR.

Practice questions from Backtesting VaR

Purpose and Objectives of VaR Backtesting in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Purpose and Objectives of VaR Backtesting: frequently asked questions

What is VaR backtesting in FRM Part II?

It is a comparison of daily VaR forecasts with subsequent P&L. You count days where the loss exceeds VaR and check that the count is consistent with the confidence level. It validates the model and supports regulatory capital decisions.

What is the difference between clean P&L and dirty P&L?

Dirty (actual) P&L is the reported trading result, including fees, commissions, intraday trading and reserve changes. Clean P&L removes those items so it reflects only the price moves on the positions held at the start of the day.

Hypothetical vs actual P&L in backtesting: which is better?

Hypothetical P&L is the better test of model accuracy because it holds the portfolio fixed, as VaR assumes. Actual P&L shows whether real results breach VaR, which matters to supervisors. Both are commonly examined.

Why would a bank want few exceptions but not zero?

A well-calibrated 99% VaR should be breached about 1% of the time. Zero or very few exceptions over a long window suggest VaR is too high, so capital and limits are tighter than needed.