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Show how VaR can inform the drafting of investment guidelines and help make the investment process better.

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

The idea

VaR improves the drafting of guidelines by replacing notional or instrument restrictions, such as a cap on derivatives as a percentage of assets, with limits stated in risk. That means a VaR limit or an ex-ante tracking-error limit with a stated confidence level and horizon. This captures leverage, volatility and correlation, allows derivatives that reduce risk, and caps those that raise it. Let z be the normal quantile, for example 2.326 at 99 percent. A one-year VaR limit at that confidence maps to a tracking-error limit TE by dividing by z: TE = VaR limit ÷ z. Used in the process, risk decomposition reveals unintended or unrewarded bets. Removing them cuts risk with no loss of expected active return, and so raises the information ratio: IR = expected active return ÷ tracking error. Exam point: constrain risk rather than size, and remove risk that is not compensated.

Formulas

Tracking-error limit implied by an active VaR limit

TE limit = VaR limit ÷ z

VaR limit
the active VaR limit as a share of assets, over one year
z
the normal quantile, 2.326 at 99%

Normal returns and a one-year horizon to match an annual tracking error. Multiplying by z, or using 1.645 for a 99% limit, gives the wrong limit.

Information ratio of an active portfolio

IR = expected active return ÷ tracking error

expected active return
the expected return over the benchmark
tracking error
the standard deviation of active return

Removing an uncorrelated, unrewarded bet leaves the numerator unchanged and lowers tracking error, found by subtracting variances, not volatilities.

A practice question

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

Why can a guideline that sets a VaR or tracking-error limit be preferable to one based on notional limits, such as a maximum percentage of assets in derivatives?

  1. It guarantees that no loss will ever exceed the limit during the horizon.Why not: VaR gives no guarantee about losses beyond its confidence level.
  2. It removes the need to monitor positions.Why not: A risk limit still has to be monitored through regular VaR reports.
  3. It measures risk in common units across instruments, so it constrains risk rather than size. - correct
  4. It prohibits derivatives and so lowers risk automatically.Why not: A VaR limit permits derivatives that stay within the risk limit.

Notional limits ignore the risk of the instrument. A VaR or tracking-error limit applies to all instruments in the same way and allows hedging.

A harder one

The guidelines cap derivatives at 10 percent of assets in notional terms. A manager puts 10 percent of assets into long-dated bond futures, which raises portfolio duration sharply and increases tracking error well above the intended level. What does this illustrate?

  1. The 10 percent cap works as intended, since 10 percent of assets is small.Why not: Size is not risk; long-dated bond futures carry far more risk per unit of notional than cash.
  2. The cap should be raised to 20 percent so that the exposure is measured more accurately and consistently.Why not: Raising a notional cap changes the size limit but still does not measure the risk.
  3. Futures should be banned, because any derivative position raises risk.Why not: Derivatives can lower risk; a risk limit allows the hedges and caps the bets.
  4. A notional limit does not control risk, which depends on the instrument's volatility. - correct

Notional exposure treats a 10 percent position in long bond futures and in cash equivalents alike. A risk limit captures what matters, volatility and correlation.

The hardest tier

The board wants the ex-ante active VaR of a manager at 99 percent confidence over one year to be no more than 4.0 percent of assets, under normal returns. The ex-ante tracking-error limit to write into the guidelines is closest to:

  1. 2.43 percentWhy not: Divides by 1.645, the 95 percent quantile, instead of 2.326 for 99 percent.
  2. 1.72 percent - correct
  3. 9.31 percentWhy not: Multiplies by the quantile instead of dividing.
  4. 1.05 percentWhy not: Divides by the product of the 95 and 99 percent quantiles.

Active VaR = z x TE, so TE = 4.0 / 2.326 = 1.72 percent. Translating a VaR limit into a tracking-error limit divides by the quantile.

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