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Judge how shifts in credit spread and recovery rate assumptions affect CVA.

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

The idea

CVA rises with the counterparty's credit spread because implied default probabilities rise. The relationship is slightly concave. Let R be the recovery rate and t the time in years. From a flat CDS spread, the cumulative default probability by time t is PD = 1 − exp(−spread × t ÷ (1 − R)). Doubling the spread therefore raises CVA by a bit less than double. Recovery needs care. Loss given default, LGD, is one minus R. If default probabilities are held fixed, CVA is proportional to LGD, so a higher recovery lowers it directly. But if the probabilities are calibrated to a given CDS spread, a higher recovery implies a higher hazard rate, spread ÷ (1 − R). That offsets most of the lower LGD, so CVA is only weakly sensitive to recovery. CVA uses risk-neutral probabilities, so a widening of the risk premium raises it even if historical default rates are unchanged.

A practice question

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

All else equal, including exposure and recovery, what happens to a bank's CVA on a counterparty when the counterparty's CDS spread widens?

  1. CVA decreases, because exposure falls when the counterparty is weaker.Why not: Assumes exposure falls with credit quality; with exposure held fixed here there is no such effect.
  2. CVA is unchanged, because CVA depends only on exposure and recovery.Why not: Omits the default probability, which is a direct input to CVA.
  3. CVA decreases, because a wider spread means the market expects a higher recovery.Why not: Reverses the effect and invents a recovery link; a wider spread raises implied default probability.
  4. CVA increases, because the market-implied default probabilities rise. - correct

CVA is increasing in the default probability. A wider spread raises implied PDs, so CVA rises (holding exposure and recovery constant).

A harder one

A bank calibrates default probabilities to a flat 300 bp CDS spread using hazard rate = spread / (1 - R) and has a single exposure date 5 years out. The recovery assumption is raised from 40% to 60%, with the CDS spread unchanged. What is the approximate effect on CVA?

  1. It rises by about 50%, because the implied hazard rate rises with recovery, and the effect of the lower loss given default is ignored.Why not: Counts only the higher implied hazard rate and forgets the lower loss given default.
  2. It is exactly unchanged, because CVA depends only on the spread and the exposure, never on the recovery assumption.Why not: The offset is large but not exact; the exponential in the default probability leaves a small residual change.
  3. It falls slightly, by about 6%, as lower LGD is mostly offset by a higher implied default probability. - correct
  4. It falls by about 33%, in proportion to loss given default, with default probabilities unchanged.Why not: Holds default probability fixed; when calibrating to a spread, a higher recovery implies a higher hazard rate.

CVA = (1-R)(1 - exp(-s T/(1-R))) x EE: R=40% gives 0.13272, R=60% gives 0.12508, a fall of 5.8%. When spread is held fixed, recovery has only a second-order effect.

The hardest tier

A bank values CVA with exposures of USD 3.0 million at year 5 and USD 4.0 million at year 10 (discounted). The counterparty CDS spread is 500 bp (flat) and the bank calibrates default probabilities as 1 - exp(-s t / (1 - R)). The recovery assumption is cut from 40% to 20% with the spread unchanged. By roughly how much does the CVA change?

  1. It rises by about 10%. - correct
  2. It falls by about 10%, because lower recovery implies lower default probability.Why not: Uses the right size but the wrong sign; the lower implied hazard rate does not outweigh the higher loss given default.
  3. It rises by about 33%, in proportion to loss given default.Why not: Holds default probabilities fixed; after recalibration to the spread the implied hazard rate falls as recovery falls.
  4. It is unchanged, because the spread is unchanged and spread already contains recovery.Why not: Spread pins down the product of PD and LGD only approximately; over long horizons the residual change is material.

CVA at R=40%: USD 1,152,509; at R=20%: USD 1,272,456, a rise of 10.4%. The effect is much smaller than the 33% a fixed-PD view suggests, because CVA is roughly spread x exposure x time, but it is not zero.

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