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Show how exposure changes under collateralization, and judge what risk the remargining period, threshold and minimum transfer amount introduce.

FRM® Part II · Credit Risk Measurement and Management · Future Value and Exposure · FRM2.CR.19.h

The idea

With a margin agreement the close-out exposure, written Exposure, is floored at zero. Exposure = max(V − C, 0). Here V is the value at close-out and C is the collateral actually held, which reflects the value at the last successful margin call. The gap between them comes from the margin period of risk. It is the remargining interval plus time for the call, any dispute or grace period, and the close-out and replacement of the position, often taken as about ten business days. The threshold is an unsecured amount, so no collateral is called below it. The minimum transfer amount (MTA) means a call smaller than the MTA is not made, so stale uncollateralised value remains. Initial margin held reduces exposure further. A rough guide to the exposure left is threshold plus MTA plus the value move over the margin period. Longer remargining periods raise exposure.

Formulas

Exposure with collateral held

Exposure = max(V − C, 0)

V
the portfolio value at close-out
C
the collateral actually held, set at the last successful margin call

C is set from the value at the last call, less any threshold, and a call below the MTA is not made. Using today's value for C ignores the margin period of risk.

A practice question

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

In a collateralised relationship, the exposure at the time of close-out can be written as:

  1. max(V, 0) - C, which can be negative when the collateral exceeds the positive value of the portfolioWhy not: Applies the floor before deducting collateral, wrongly allowing negative exposure.
  2. max(V - C, 0), where C is set from the value at the close-out date itselfWhy not: Assumes the collateral is perfectly up to date, ignoring the margin period of risk.
  3. V - C with no floor, since collateral makes the exposure symmetricWhy not: Exposure is floored at zero; a collateral surplus does not produce a negative exposure.
  4. max(V - C, 0), with V the value at close-out and C the collateral held from the last margin call - correct

Collateral C is based on the value at the last margin call, while V moves until close-out. The residual exposure is max(V - C, 0).

A harder one

A netting set has a threshold of USD 5 million and a minimum transfer amount of USD 1 million, with no initial margin. At the last margin date the value was 18, and the collateral held was set to V - threshold. At close-out the value is 21.5 (USD million). The exposure is:

  1. USD 8.5 million - correct
  2. USD 3.5 millionWhy not: Assumes the collateral held equals the full value at last call, 21.5 - 18, ignoring that the threshold remains unsecured.
  3. USD 16.5 millionWhy not: Treats the threshold as the collateral held, 21.5 - 5.
  4. USD 21.5 millionWhy not: Ignores the collateral altogether.

Collateral held = 18 - 5 = 13, since the call exceeds the MTA. Exposure = max(21.5 - 13, 0) = 8.5, made up of the unsecured threshold of 5 and the 3.5 move since the last call.

The hardest tier

Daily margining implies a margin period of risk of 10 business days, made up of the remargining interval plus 9 days for the call, dispute and close-out. The bank moves to weekly (5-day) remargining, so the margin period of risk becomes 14 days. The daily standard deviation of the mark-to-market is USD 0.6 million, there is no threshold, and the 99% quantile is 2.326. The new 99% collateralised exposure is closest to:

  1. USD 4.41 millionWhy not: Ignores the change and keeps the 10-day margin period.
  2. USD 3.12 millionWhy not: Uses only the 5-day remargining interval, not the full 14-day period.
  3. USD 6.18 millionWhy not: Scales the original exposure linearly by 14/10 instead of by the square root.
  4. USD 5.22 million - correct

Exposure = 2.326 x 0.6 x sqrt(14) = 5.22, against 2.326 x 0.6 x sqrt(10) = 4.41 before. Exposure grows with the square root of the margin period of risk, not linearly.

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