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Show how the CVA figure changes once collateralization is built in, paying attention to initial margins, thresholds and the margin period of risk.

FRM® Part II · Credit Risk Measurement and Management · CVA · FRM2.CR.20.j

The idea

Collateral reduces exposure to the amount not covered at close-out. Variation margin held at time t reflects the portfolio value one margin period of risk (MPoR) earlier, less any threshold. Write L for that earlier value and VM for the variation margin held. VM = max(L − Threshold, 0). Write V for the value at time t, and IM for the initial margin held. Exposure = max(V − VM − IM, 0). A larger threshold leaves more exposure uncollateralised and raises CVA. A longer MPoR lets more value change occur without margin, raising CVA. With zero threshold and no initial margin, collateralised exposure is roughly proportional to the square root of the MPoR. Initial margin lowers exposure further, and a minimum transfer amount adds small residual exposure. Compute the collateralised expected exposure first, then apply LGD and default probabilities as usual.

Formulas

Variation margin held under a threshold

VM = max(L − Threshold, 0)

L
the portfolio value one margin period of risk earlier
Threshold
the unsecured amount below which no collateral is called

Collateral lags the value by the MPoR. Applying the threshold to today's value ignores that lag.

Exposure with variation and initial margin

Exposure = max(V − VM − IM, 0)

V
the portfolio value at time t
VM
the variation margin held
IM
the initial margin held

Initial margin is deducted, never added. Compute this exposure in each scenario, average it, then apply LGD and default probabilities.

A practice question

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

All else equal, how does lengthening the margin period of risk (MPoR) in a collateral agreement affect the CVA?

  1. It has no effect, because the threshold alone determines the residual exposure after collateral is called.Why not: Ignores that the residual exposure is driven by value changes over the MPoR.
  2. It reduces CVA, because collateral has more time to be delivered.Why not: Treats a longer period as more time to collect; it is the interval of uncovered exposure, which grows.
  3. It increases CVA, because more value change can occur between the last margin call and close-out. - correct
  4. It reduces CVA, because the default probability is applied over a longer horizon and falls over it.Why not: Confuses the default probability horizon with the margin period; the MPoR only widens the uncovered gap.

The MPoR is the time between the last successful margin exchange and the close-out. Exposure after collateral is roughly the value change over that window, so longer MPoR means higher residual exposure and higher CVA.

A harder one

Under a collateral agreement with a threshold of USD 3 million, no initial margin, and no minimum transfer amount, collateral held at time t reflects the portfolio value one MPoR earlier. That earlier value was USD 9 million and the value at time t (when the counterparty defaults) is USD 14 million. What is the exposure at default?

  1. USD 5 millionWhy not: Assumes the collateral equals the earlier value in full, ignoring the threshold.
  2. USD 8 million - correct
  3. USD 11 millionWhy not: Applies the threshold to the current value, ignoring that collateral lags by the margin period of risk.
  4. USD 3 millionWhy not: Takes the threshold as the exposure, forgetting the value change over the MPoR.

Collateral = max(9 - 3, 0) = 6. Exposure = max(14 - 6, 0) = USD 8 million: the threshold (3) plus the move over the MPoR (5).

The hardest tier

A collateralised netting set (zero threshold, no initial margin) has a CVA of USD 80,000 with a 10-day MPoR. Assume the collateralised expected exposure is proportional to the square root of the MPoR (normally distributed value changes) and nothing else changes. If the MPoR lengthens to 20 days, the CVA is closest to:

  1. USD 320,000Why not: Squares the ratio of MPoRs instead of taking its square root.
  2. USD 160,000Why not: Assumes exposure scales linearly with the MPoR rather than with its square root.
  3. USD 56,569Why not: Divides by the square root of 2, reversing the direction of the effect.
  4. USD 113,137 - correct

Exposure ~ sigma x sqrt(MPoR), so the ratio is sqrt(20/10) = 1.414. CVA = 80,000 x 1.414 = USD 113,137.

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