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Account for the general effect of aggregation on exposure, and its effect when transaction values are correlated.

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

The idea

Under a legally enforceable netting agreement, the positive and negative values of the trades are added before the zero floor is applied. Netting therefore changes the exposure on the whole set of trades with one counterparty to Exposure = max(sum of V, 0). Without netting, exposure would be the sum of max(V, 0) taken trade by trade. Because the floor is convex, netted exposure is never greater than the sum of the stand-alone exposures. The size of the benefit depends on correlation. For zero-mean normal values, expected exposure (EE) is about 0.399 times the standard deviation. For two trades, let s1 and s2 be their standard deviations and ρ their correlation. The portfolio standard deviation is then given by Sigma = √(s1² + s2² + 2 × ρ × s1 × s2). With perfect positive correlation, portfolio sigma equals s1 + s2 and there is no benefit. Zero correlation gives a partial benefit, and negative correlation gives the most. Without an enforceable netting agreement the positive exposures simply add. Check legal enforceability before applying any netting credit.

Formulas

Exposure of a netting set

Netted exposure = max(ΣVᵢ, 0)

Vᵢ
the value to the bank of each trade in the enforceable netting set

Only under a legally enforceable netting agreement. Without one, exposure is the sum of each trade's max(Vᵢ, 0).

Expected exposure of a zero-mean normal value

EE = 0.399 × σ

σ
the standard deviation of the value at that date

Mean of zero and normal values only; 0.399 is 1 ÷ √(2π). For a netting set use the portfolio σ, not the sum of the trades' σ.

Standard deviation of two netted trade values

σP = √(σ₁² + σ₂² + 2 × ρ × σ₁ × σ₂)

σ₁, σ₂
the standard deviation of each trade's value
ρ
the correlation between the two values

With ρ = 1, σP is the sum and netting gives no benefit; the lower ρ, the larger the benefit. Getting the sign of ρ wrong is the trap.

A practice question

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

Which statement describes the general effect of aggregating trades under a legally enforceable netting agreement?

  1. Netted exposure can exceed the sum of the stand-alone exposures when the trades in the set are negatively correlatedWhy not: Negative correlation increases the netting benefit; netting can never add exposure.
  2. Netted exposure never exceeds the sum of stand-alone exposures, as values offset before the floor at zero - correct
  3. Netted exposure equals the sum of stand-alone exposures for any trades with the same counterparty, whether or not an agreement is enforceableWhy not: Netting requires a legally enforceable agreement; without it the positive exposures add.
  4. Netted exposure equals that of the largest single trade, since smaller trades are absorbedWhy not: Netting nets all the values; it does not just keep the largest trade.

Exposure = max(sum of V, 0) with netting versus sum of max(V,0) without. Because max(.,0) is convex, the first is never larger than the second.

A harder one

Two trades have zero-mean, normally distributed mark-to-market values at a future date, with standard deviations of USD 10 million and USD 6 million and a correlation of 0.3. Take the EE of a zero-mean normal as its standard deviation divided by 2.5066. The expected exposure of the netting set is closest to which figure?

  1. USD 6.38 millionWhy not: Adds the two stand-alone EEs, (10 + 6)/2.5066, which is the no-netting figure.
  2. USD 4.65 millionWhy not: Sets the correlation to zero, sqrt(100 + 36)/2.5066.
  3. USD 5.23 million - correct
  4. USD 3.99 millionWhy not: Uses a correlation of minus 0.3 in the variance, sqrt(100 + 36 - 36)/2.5066.

Portfolio standard deviation = sqrt(100 + 36 + 2 x 0.3 x 10 x 6) = sqrt(172) = 13.12. EE = 13.12/2.5066 = 5.23, below the 6.38 without netting.

The hardest tier

A netting set of zero-mean normal trades has a standard deviation of USD 13.12 million. A new trade with a standard deviation of USD 8 million and a correlation of minus 0.5 with the existing netting set is added. The change in the netting set's expected exposure (EE of a zero-mean normal = standard deviation/2.5066) is closest to:

  1. A rise of USD 0.66 millionWhy not: Right size but the wrong sign; negative correlation reduces portfolio standard deviation.
  2. A fall of USD 0.66 million - correct
  3. A rise of USD 3.19 millionWhy not: Adds the new trade's stand-alone EE, 8/2.5066, with no allowance for netting.
  4. A rise of USD 2.13 millionWhy not: Uses a correlation of plus 0.5 instead of minus 0.5.

New standard deviation = sqrt(13.12² + 8² + 2 x (-0.5) x 13.12 x 8) = sqrt(172 + 64 - 105) = 11.45. Change in EE = (11.45 - 13.12)/2.5066 = -0.66, so a trade can reduce the netting set's exposure.

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