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Work out CVA, plus its expression as a spread, in the simple setting where wrong-way risk, netting and collateralization are all absent.

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

The idea

In the simple setting, with exposure independent of default and no netting or collateral, CVA = (1 − R) × Σ discounted EE(tᵢ) × [PD(tᵢ) − PD(tᵢ₋₁)]. Default probabilities usually come from a CDS curve, here taken as flat: hazard rate = spread ÷ (1 − R) and cumulative PD = 1 − e^(−spread × t ÷ (1 − R)). The marginal probability of each interval is the difference of cumulative probabilities; using the cumulative figure at each date counts early defaults many times. To express CVA as a running spread, divide by the notional and by the risky annuity (the sum of discount factor × survival probability, per unit notional): spread = CVA ÷ (notional × risky annuity). Exam point: use the risky, not the risk-free, annuity.

Formulas

Credit valuation adjustment

CVA = (1 − R) × Σ DF(tᵢ) × EE(tᵢ) × [PD(tᵢ) − PD(tᵢ₋₁)]

R
the counterparty's recovery rate, so 1 − R is loss given default
DF(tᵢ)
the risk-free discount factor to date tᵢ
EE(tᵢ)
the expected exposure, the mean of max(V, 0), at date tᵢ
PD(tᵢ)
the counterparty's cumulative default probability to tᵢ; the bracket is the marginal probability for the interval
tᵢ
each exposure date, with t₀ = 0

No wrong-way risk, netting or collateral. Using the cumulative PD at each date counts early defaults many times.

Cumulative default probability from a flat CDS spread

PD(t) = 1 − e^(−spread × t ÷ (1 − R))

spread
the flat CDS spread, as a decimal
t
the horizon, in years
R
the recovery rate

Constant hazard rate spread ÷ (1 − R). Forgetting to divide by 1 − R, or dropping the exponential for spread × t, are the traps.

CVA as a running spread

CVA spread = CVA ÷ (Notional × Annuity)

CVA
the upfront CVA to convert (for a new trade in a netting set, its incremental CVA)
Notional
the trade's notional
Annuity
the risky annuity: the sum of discount factor × counterparty survival probability, per unit of notional

Use the risky annuity, since the spread is paid only while the counterparty survives; the risk-free annuity understates it.

A practice question

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

A bank has a single-date exposure to a counterparty. The discounted expected exposure is USD 2.5 million, the marginal probability of default over the relevant interval is 3% and the recovery rate is 40%. Exposure and default are independent and there is no netting or collateral. What is the CVA, closest to?

  1. USD 75,000Why not: Ignores recovery altogether and treats the whole exposure as lost on default.
  2. USD 27,000Why not: Applies loss given default twice, once as (1 - R) and again as an extra 60% factor.
  3. USD 45,000 - correct
  4. USD 30,000Why not: Multiplies by the recovery rate (40%) instead of the loss given default (60%).

CVA = (1 - R) x EE x PD = 0.60 x 2,500,000 x 0.03 = USD 45,000.

A harder one

A bank has exposure to a counterparty on three annual dates. Discounted expected exposures at years 1, 2 and 3 are USD 1.0 million, USD 2.0 million and USD 1.5 million. The counterparty's CDS curve is flat at 200 bp and the recovery rate is 40%. Cumulative default probabilities are 1 - exp(-s t / (1 - R)). Assuming no wrong-way risk, netting or collateral, the CVA is closest to:

  1. USD 85,300 - correct
  2. USD 56,900Why not: Uses the recovery rate in place of loss given default.
  3. USD 182,700Why not: Uses cumulative rather than marginal default probabilities at each date, counting early defaults repeatedly.
  4. USD 142,200Why not: Leaves out the loss given default factor, so the whole exposure is treated as lost.

Cumulative PDs: 3.28%, 6.45%, 9.52%; marginal PDs 3.28%, 3.17%, 3.07%. CVA = 0.6 x (1.0 x 0.0328 + 2.0 x 0.0317 + 1.5 x 0.0307) million = about USD 85,324.

The hardest tier

A bank has a 4-year swap with a counterparty with notional USD 50 million. Expected exposures (undiscounted) at years 1 to 4 are USD 3.0, 4.5, 4.0 and 2.0 million. The continuously compounded discount rate is 3%, the counterparty CDS spread is 150 bp (flat), recovery is 40%, and default probabilities follow 1 - exp(-s t / (1 - R)). The CVA is computed on annual dates with no wrong-way risk, netting or collateral. The risky annuity is the sum over years of the discount factor times the survival probability. The CVA expressed as a running spread is closest to:

  1. 10.3 bp per year - correct
  2. 24.4 bp per yearWhy not: Weights exposures by cumulative instead of marginal default probabilities before converting to a spread.
  3. 6.2 bp per yearWhy not: Multiplies the spread by loss given default a second time after the CVA already contains it.
  4. 9.7 bp per yearWhy not: Divides by the risk-free annuity (sum of discount factors) instead of the risky annuity.

Discount factors 0.9704, 0.9418, 0.9139, 0.8869; marginal PDs 2.469%, 2.408%, 2.349%, 2.291%. CVA = 0.6 x sum(DF x EE x marginal PD) = USD 180,254. Risky annuity = 3.4927. Spread = 180,254 / (50,000,000 x 3.4927) = 10.3 bp.

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