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Explain why credit risk capital is harder to calculate for derivatives than for loans.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.j

The idea

Credit exposure on a loan is essentially known: it is the amount lent, which falls as the loan is repaid, and a loss arises only if the borrower defaults. A derivative is different. Its value depends on future market variables, so the exposure at default is uncertain and has to be simulated. It can also be positive or negative. The bank loses only if the counterparty defaults while the contract is worth a positive amount to the bank, whereas a negative value must still be paid. Netting agreements, collateral and close-out terms change exposure and must be modelled by netting set. Exposure = max(net value − collateral, 0). Exposure evolves over the contract's life. Wrong-way risk arises when exposure tends to be high just when the counterparty is likely to default. Capital therefore needs a joint model of market and credit risk, not the simple loan formulas.

Formulas

Credit exposure to a derivatives counterparty

exposure = max(net value − collateral, 0)

net value
the value to the bank of all trades in the netting set
collateral
collateral held against the netting set

Without enforceable netting, sum the positive trade values separately. A negative value gives zero exposure but still has to be paid.

A practice question

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

Why is credit risk capital harder to calculate for a derivative than for a loan?

  1. The value of a derivative is always positive to the bank, so the exposure never falls below zero.Why not: A derivative can have positive or negative value; only a positive value is exposed to default.
  2. Derivative counterparties cannot default, so there is no credit loss to measure.Why not: Derivative counterparties can and do default; the problem is that the amount at risk is uncertain.
  3. The exposure to a derivative depends on uncertain future market variables, whereas a loan's exposure is the amount lent. - correct
  4. Recovery rates on derivatives are always zero, so loss given default cannot be modelled.Why not: Recovery on derivative claims is not zero and is modelled like other claims; the main complication is the uncertain exposure.

A loan's exposure is known (principal), but a derivative's value depends on market moves, can be an asset or a liability, and must be simulated. This uncertainty is the core difficulty.

A harder one

At a future date, a derivative's value to the bank under four equally likely market scenarios is -3, +2, +5 and +9 (USD million). The expected positive exposure at that date is closest to:

  1. USD 9.00 millionWhy not: Uses the maximum of the scenarios, which is a peak exposure, not an expected one.
  2. USD 5.33 millionWhy not: Averages only the positive scenarios, ignoring that the negative scenario still counts with zero exposure and probability weight.
  3. USD 4.00 million - correct
  4. USD 3.25 millionWhy not: Takes the expected value including the negative scenario, but exposure is floored at zero in each scenario.

EPE = (0 + 2 + 5 + 9) / 4 = USD 4.00 million. The negative scenario contributes zero exposure but still carries a 25% probability weight.

The hardest tier

A counterparty has an average default probability of 3%. If market rates rise (probability 50%), the bank's derivative exposure is USD 8 million and the counterparty's default probability is 5%; if rates fall (probability 50%), the exposure is zero and the default probability is 1%. LGD is 60%. Taking into account the link between exposure and default, the expected loss is closest to:

  1. USD 0.072 millionWhy not: Assumes exposure and default are independent, using average PD of 3% and average exposure of 4, which hides the wrong-way risk.
  2. USD 0.120 million - correct
  3. USD 0.240 millionWhy not: Uses the rising-rates state only and forgets to weight it by its 50% probability.
  4. USD 0.144 millionWhy not: Applies the average 3% PD to the full USD 8 million exposure, overstating exposure in the state where it is zero.

EL = 0.5 x 0.05 x 0.6 x 8 + 0.5 x 0.01 x 0.6 x 0 = USD 0.120 million, versus 0.072 if exposure and default were independent. Wrong-way risk (exposure high when default is likely) raises the loss.

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