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Gauge how far risk limits, derivatives and other risk management instruments change a firm's risk position.

FRM® Part I · Foundations of Risk Management · How Do Firms Manage Financial Risk? · FRM1.FR.2.e

The idea

Risk limits and instruments change a risk position in different ways, and the question is always by how much. Limits constrain it. They include position or notional limits, sensitivity limits such as DV01, VaR limits, concentration limits, and stop-loss limits that trigger review after cumulative losses. A notional limit alone does not control risk, because moving from short to long maturity bonds keeps notional but multiplies DV01. Derivatives change exposure by their delta or duration equivalents. A call with delta 0.6 on 10,000 shares acts like 6,000 shares. A pay-fixed swap shortens a bond portfolio's duration by the fixed leg duration less the floating leg duration, scaled by notional. Diversified VaR for two positions is below the simple sum unless ρ is 1, assuming normally distributed returns. VaRp = √(VaR1² + VaR2² + 2 × ρ × VaR1 × VaR2). A common mistake is to ignore the sign or the offsetting leg.

Formulas

Diversified VaR of two positions

VaR = √(VaR1² + VaR2² + 2 × ρ × VaR1 × VaR2)

VaR1, VaR2
the stand-alone VaRs of the two positions
ρ
the correlation between their returns

Below the simple sum unless ρ = 1, but it can still breach a limit. Assumes normal returns with the same horizon and confidence level for both.

Dollar value of a basis point

DV01 = V × D × 0.0001

V
the market value of the position
D
its modified duration

The change in value for a one basis point move in yield. A notional limit ignores D, so swapping into longer bonds at the same notional multiplies DV01.

A practice question

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

A stop-loss limit on a trading desk is best described as:

  1. A cap on the notional size of positions the desk may holdWhy not: This is a position or notional limit, which restricts size rather than realised losses.
  2. A cap on the share of the portfolio that may be placed with one counterpartyWhy not: This is a concentration limit, which controls diversification rather than losses.
  3. A ceiling on the estimated 99% one-day loss of the deskWhy not: This is a VaR limit, which is a forward-looking measure rather than a trigger on realised loss.
  4. A cumulative loss threshold that triggers review or closing of positions - correct

Stop-loss limits act on realised or marked losses: when cumulative losses reach the threshold, the desk must stop, reduce or seek approval to continue.

A harder one

A trader holds 10,000 call options on a stock, each on one share, with delta 0.6, and is short 3,000 shares. The stock price is USD 50. Using delta, what is the approximate profit or loss if the stock rises 2%?

  1. A gain of about USD 7,000Why not: Treats the options as 10,000 shares (delta of 1) and ignores the actual delta of 0.6.
  2. A gain of about USD 9,000Why not: Adds the short position as if it were long: 6,000 + 3,000 shares.
  3. A gain of about USD 3,000 - correct
  4. A gain of about USD 6,000Why not: Counts only the options' delta exposure and forgets the short stock position.

Net delta = 0.6 x 10,000 - 3,000 = 3,000 shares. Gain = 3,000 x 50 x 0.02 = USD 3,000. The short stock cancels half of the options' delta exposure.

The hardest tier

A desk has a one-day 99% VaR limit of USD 5 million. Its two positions have stand-alone VaRs of USD 3 million and USD 4 million, and a correlation of 0.5 between their returns. What is the diversified VaR, and does it comply with the limit?

  1. USD 7.00 million, so it breaches the limitWhy not: Adds the stand-alone VaRs, which assumes perfect correlation and ignores diversification.
  2. USD 5.00 million, so it is exactly at the limitWhy not: Assumes zero correlation, which gives sqrt(9 + 16) and drops the covariance term.
  3. USD 6.08 million, so it breaches the limit - correct
  4. USD 3.61 million, so it is within the limitWhy not: Subtracts the covariance term instead of adding it, so the sign of the correlation effect is wrong.

VaR = sqrt(3² + 4² + 2 x 0.5 x 3 x 4) = sqrt(9 + 16 + 12) = sqrt(37) = 6.08, above the USD 5 million limit. Diversification helps, but not enough.

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