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Summarise the Basel requirements that govern VaR backtesting.

FRM® Part II · Market Risk Measurement and Management · Backtesting VaR · FRM2.MR.4.f

The idea

Under the legacy Basel internal-models VaR backtesting framework, a bank backtests its one-day, 99 percent (one-tailed) VaR against daily trading outcomes over the most recent 250 business days. The capital VaR uses a 10-day horizon, but the backtest does not. The number of exceptions puts the bank in a zone: green for 0 to 4, yellow for 5 to 9, red for 10 or more. The market risk charge is the larger of the previous day's VaR and the multiplier times the average VaR of the previous 60 business days. The multiplier is at least 3. It stays 3.00 in the green zone, rises through 3.40, 3.50, 3.65, 3.75 and 3.85 for 5 to 9 exceptions, and is 4.00 in the red zone, where the increase is automatic and the model is presumed flawed. The multiplier depends on the count of exceptions, not on the size of the losses. Do not count exceptions on the 10-day VaR. The current framework, the FRTB internal models approach, uses expected shortfall for capital, so treat these zones as the legacy VaR regime.

Formulas

Legacy Basel market risk charge from VaR

MRC = max(VaRprev, k × VaRavg)

VaRprev
the previous day's 10-day 99% VaR
k
the multiplier, 3.00 in the green zone, 3.40 to 3.85 in the yellow zone and 4.00 in the red zone
VaRavg
the average 10-day VaR over the previous 60 business days

The multiplier comes from exceptions of the one-day 99% VaR over the last 250 business days. Applying k to yesterday's VaR instead of the average is a common slip.

A practice question

From the question bank, with the reason each wrong option is wrong.

Under the Basel market risk framework, which comparison is used to backtest a bank's internal VaR model?

  1. The 10-day, 99% VaR against 10-day trading outcomes over the most recent 250 business days.Why not: Backtesting uses the one-day VaR, even though the capital charge is based on a 10-day horizon.
  2. The one-day, 95% VaR against daily trading outcomes over the most recent 60 business days.Why not: Uses the wrong confidence level and the 60-day window that applies to averaging VaR for the charge, not to backtesting.
  3. The one-day, 99% VaR against daily trading outcomes over the last 250 business days. - correct
  4. The 10-day, 99% VaR against monthly trading outcomes over the most recent 12 months.Why not: Mismatches horizon and frequency; the framework counts daily exceptions of the one-day VaR.

Basel backtesting counts exceptions of the one-day 99% VaR against daily P&L over the last 250 business days, even though the capital VaR is for 10 days.

A harder one

A bank has 11 exceptions in its last 250 business days. Its risk manager argues that the period was extraordinary. Which statement describes the Basel treatment?

  1. The multiplier is raised to 3.85, the highest yellow-zone value.Why not: 3.85 corresponds to 9 exceptions; 10 or more is the red zone with a 4.00 multiplier.
  2. The supervisor has full discretion and no increase applies unless it finds the model flawed.Why not: Describes discretion in the yellow zone; in the red zone the increase is automatic.
  3. The multiplier increases by 0.5 pending an investigation.Why not: Invents a partial add-on; the red zone add-on is the full step to 4.00.
  4. Red zone: the multiplier automatically rises to 4.00 and the model is presumed flawed. - correct

Ten or more exceptions fall in the red zone. The add-on is automatic (multiplier 4.00 in total), and the model is presumed to be flawed.

The hardest tier

A bank has 9 exceptions in its last 250 business days. Its 10-day 99% VaR yesterday was USD 80 million and its average 10-day VaR over the previous 60 business days is USD 60 million. Using the Basel rule (larger of yesterday's VaR and multiplier times the 60-day average), the general market risk charge is closest to which amount?

  1. USD 308 millionWhy not: Applies the multiplier to yesterday's VaR instead of the 60-day average.
  2. USD 231 million - correct
  3. USD 180 millionWhy not: Uses the green-zone multiplier of 3 rather than the yellow-zone 3.85 for 9 exceptions.
  4. USD 240 millionWhy not: Treats 9 exceptions as the red zone with a 4.00 multiplier.

Nine exceptions give a multiplier of 3.85. Multiplier x average = 3.85 x 60 = 231, which exceeds yesterday's VaR of 80, so the charge is 231.

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