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Work out CVA under stressed conditions together with the resulting stress loss on CVA.

FRM® Part II · Credit Risk Measurement and Management · The Evolution of Stress Testing Counterparty Exposures · FRM2.CR.21.f

The idea

To compute stressed CVA, rebuild the CVA formula with scenario inputs. Marginal PD(t) is the probability of default between t−1 and t. DF(t) is the discount factor and EE(t) the expected exposure. CVA = LGD × Σ DF(t) × EE(t) × marginal PD(t). A spread converts to a hazard rate through the credit triangle, written Hazard. Hazard = spread ÷ LGD. Survival to t is then e raised to the power of minus the hazard rate times t. Marginal PD is the survival to t−1 minus the survival to t. Stress the spread and, if the scenario says so, the exposure profile and LGD, then recompute. The stress loss on CVA is written SLC, and any hedge gain reduces it. SLC = stressed CVA − base CVA. Typical errors are using the spread directly as a probability, leaving LGD out of the credit triangle, omitting discounting, using cumulative instead of marginal PD, and reporting the stressed level of CVA as the loss. A stress on recovery also changes the hazard rate.

Formulas

Unilateral CVA

CVA = LGD × Σ DF(t) × EE(t) × PD(t)

LGD
the counterparty's loss given default
DF(t)
the discount factor to date t
EE(t)
the expected exposure at date t
PD(t)
the marginal probability of default between t − 1 and t

Use marginal, not cumulative, PD and do not forget discounting. With a hazard rate λ, marginal PD is e^(−λ × (t − 1)) − e^(−λ × t).

Hazard rate from the credit triangle

hazard = spread ÷ LGD

spread
the counterparty's credit spread, as a decimal
LGD
loss given default, one minus the recovery rate

Do not use the spread directly as a probability or as the hazard rate. A recovery stress changes LGD here as well as in the CVA formula.

Stress loss on CVA

SLC = stressed CVA − base CVA − hedge gain

stressed CVA
CVA recomputed with scenario spreads, exposures and LGD
base CVA
CVA with current inputs
hedge gain
the gain on CVA hedges, such as CDS, in the scenario

The loss is the increase, not the stressed level. Subtract hedge gains from the increase, never from the stressed CVA.

A practice question

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

Which expression approximates unilateral CVA?

  1. LGD x the sum of discount factor x expected exposure x survival probabilityWhy not: Weights exposure by survival instead of by default in each interval.
  2. LGD x the sum of discount factor x expected negative exposure x the institution's own marginal PDWhy not: Describes DVA rather than CVA.
  3. LGD x the sum over time buckets of discount factor x expected exposure x marginal default probability - correct
  4. LGD x peak exposure x cumulative PD, with no discountingWhy not: Uses peak exposure instead of EE and omits discounting, overstating CVA.

CVA prices the expected loss from the counterparty's default: LGD x DF x EE x probability of default in each interval, summed over time.

A harder one

A counterparty has expected exposure of USD 20 million at the 1-year date and none at other dates. The 1-year discount factor is 0.95 and LGD is 60%. Its stressed 1-year spread is 800 bp. Use hazard rate = spread / LGD, PD to 1 year = 1 - exp(-hazard rate), and CVA = LGD x DF x EE x PD. What is the stressed CVA, closest to?

  1. USD 1.42 million - correct
  2. USD 0.91 millionWhy not: Uses the 8% spread directly as the default probability.
  3. USD 0.88 millionWhy not: Takes the hazard rate equal to the spread, leaving LGD out of the credit triangle.
  4. USD 1.50 million, with no discount factor appliedWhy not: Omits discounting: uses LGD x EE x PD without the 0.95 discount factor.

Hazard = 0.08 / 0.6 = 0.1333; PD = 1 - exp(-0.1333) = 12.48%. CVA = 0.6 x 0.95 x 20 x 0.1248 = 1.42.

The hardest tier

A counterparty's expected exposures at years 1, 2 and 3 are USD 12, 15 and 10 million. Base case: spread 150 bp, LGD 60%. Stress: spread 600 bp, LGD 75% (lower recovery), and all expected exposures multiplied by 1.5. Discount factors are exp(-0.03t); hazard rate = spread / LGD; marginal PD for year t = exp(-L(t-1)) - exp(-Lt). What is the stress loss on CVA, closest to?

  1. USD 2.81 millionWhy not: Reports the stressed CVA level rather than the increase over base CVA.
  2. USD 1.32 millionWhy not: Stresses the spread only, ignoring the LGD and exposure shocks.
  3. USD 2.30 million - correct
  4. USD 2.91 millionWhy not: Uses the stressed LGD as the loss rate but the base LGD in the credit triangle, so the hazard rate is not updated for lower recovery.

Base CVA = 0.505. Stressed CVA with spread 600 bp, LGD 75%, exposures x 1.5 = 2.809. Stress loss = 2.303. A recovery stress changes both the loss rate and the hazard rate.

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