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Show how VaR is used to keep watch over risk.

FRM® Part II · Risk Management and Investment Management · VaR and Risk Budgeting in Investment Management · FRM2.IM.6.f

The idea

VaR is used to watch risk by computing, on a regular schedule, the ex-ante VaR or tracking error of current positions. This is done for each manager, each asset class and the total fund, and the results are compared with the risk budget and limits. Component or marginal VaR shows which positions contribute most, so reductions are targeted. Stand-alone VaR ignores correlation and can mislead. Ex-ante measures reflect the positions held now. Ex-post tracking error looks backward and lags changes in strategy, so a manager whose realised tracking error is low may already run a much higher forward risk. Aggregation must use correlations, not sums. VaR also needs supplementing. Delta-normal VaR understates option risks, and back-testing and stress tests check the model. Exam point: monitor ex-ante, by contribution, against limits.

Formulas

Converting a normal VaR to another confidence level and horizon

VaR₂ = VaR₁ × (z₂ ÷ z₁) × √(T₂ ÷ T₁)

VaR₁
the reported VaR
z₁
the normal quantile of the reported confidence level
z₂
the normal quantile of the limit's confidence level
T₁
the reported horizon in days
T₂
the limit's horizon in days

Normal, independent returns and constant positions. Convert both the quantile and the horizon; doing only one, or inverting the quantile ratio, is the usual error.

A practice question

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

Which VaR output best shows a chief investment officer which position or manager to reduce so that total fund risk falls the most?

  1. Component or marginal VaR. - correct
  2. Stand-alone VaR of each position, ignoring the other positions.Why not: Ignores correlation; a hedge with a large stand-alone VaR may actually reduce total risk.
  3. The historical average return of each position.Why not: Returns do not measure risk contribution.
  4. The number of VaR exceptions recorded in the last year.Why not: Shows model performance, not where risk comes from.

Component and marginal VaR capture diversification, so they identify the contributors to total VaR. Stand-alone VaR does not.

A harder one

A manager's ex-post tracking error over the last 36 months is 1.5 percent. A VaR system computing ex-ante tracking error from its current positions gives 4.0 percent. The limit is 3.0 percent. The best interpretation is that:

  1. the manager is within the limit because the realised tracking error is below 3.0 percent.Why not: Relies on history, which lags changes in positions.
  2. the ex-ante figure must be wrong, since an ex-ante estimate should always be below the ex-post figure.Why not: Nothing requires ex-ante to be below ex-post; they measure different things.
  3. positions have probably become more aggressive recently, so the ex-ante figure governs and the limit is exceeded. - correct
  4. the gap arises only because the two figures use different confidence levels.Why not: Tracking errors are standard deviations and do not depend on a confidence level.

Ex-post tracking error reflects past positions; ex-ante is based on current holdings. A large gap signals a change in risk taking, and the forward-looking figure governs monitoring.

The hardest tier

Two managers have stand-alone VaRs of USD 12 million and USD 9 million, and P&L correlation of 0.25. The aggregate VaR is limited to USD 18 million. The first manager raises its risk by 25 percent, to a stand-alone VaR of USD 15 million. The new aggregate VaR is closest to:

  1. USD 19.3 million - correct
  2. USD 24.0 millionWhy not: Adds the stand-alone VaRs, which ignores diversification.
  3. USD 17.5 millionWhy not: Treats the managers as uncorrelated and omits the correlation term.
  4. USD 20.9 millionWhy not: Scales the old aggregate by 1.25, as if both managers raised their risk by 25 percent.

New VaR = sqrt(15² + 9² + 2 x 0.25 x 15 x 9) = sqrt(373.5) = 19.3, above the 18.0 limit. The old aggregate was sqrt(279) = 16.7, within the limit.

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