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Explain the concept of CVA, the adjustment made to a derivative's value for credit risk.

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

The idea

Credit valuation adjustment (CVA) is the difference between the value of a derivative portfolio with no counterparty default and its value allowing for default. So the value allowing for default is the risk-free value minus CVA. Let LGD be one minus the recovery rate, EE the expected positive part of the portfolio value at each date, and PD the marginal default probability in each interval. Summing over the exposure dates gives the market price of the expected loss, written as CVA = LGD × Σ (discounted EE × marginal PD). Because exposure is max(V, 0), it behaves like a call option struck at zero, which is why CVA is positive even when the position is currently negative. Default probabilities are risk-neutral, implied from CDS spreads. The usual mistakes are using potential future exposure, cumulative instead of marginal probabilities, or recovery instead of LGD.

Formulas

Value allowing for counterparty default

Risky value = V − CVA

V
the risk-free value of the portfolio
CVA
the credit valuation adjustment

CVA is a deduction. It stays positive even when V is negative today, because the value can turn positive later.

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

Exposure independent of default. Use expected, not potential future, exposure; marginal, not cumulative, probabilities; and LGD, not recovery.

A practice question

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

Credit valuation adjustment (CVA) on a derivative portfolio is best defined as:

  1. the exposure the bank would have if the counterparty defaulted today, measured at current market values.Why not: Confuses CVA with current exposure (replacement cost), which is an amount at risk, not a value adjustment.
  2. the gap between the default-free portfolio value and its value allowing for counterparty default. - correct
  3. the maximum loss at a high confidence level over the life of the trades, measured as a percentile.Why not: Describes potential future exposure, a percentile measure, not the expected-loss price adjustment.
  4. the adjustment to the portfolio value that reflects the bank's own default risk and funding position.Why not: Describes DVA, the benefit from own default risk, rather than CVA.

CVA = risk-free value minus risky value. It is the market price of the expected credit loss on the counterparty, not an exposure or percentile measure.

A harder one

A swap portfolio has a risk-free present value of USD 4.2 million to a bank. The bank estimates the CVA on the counterparty at USD 0.35 million. What is the value of the portfolio taking counterparty default into account?

  1. USD 4.2 millionWhy not: Ignores the adjustment; the risky value is not equal to the risk-free value when counterparty default is possible.
  2. USD 4.55 millionWhy not: Adds the CVA to the risk-free value; CVA is a deduction because it reflects a possible credit loss.
  3. USD 0.35 millionWhy not: Reports the adjustment itself rather than the adjusted value of the portfolio.
  4. USD 3.85 million - correct

Risky value = risk-free value - CVA = 4.2 - 0.35 = USD 3.85 million.

The hardest tier

A bank holds a large uncollateralised derivative position with a counterparty whose CDS spread widens from 100 bp to 200 bp, while the position's expected exposure and the bank's own spread are unchanged. What is the effect on the bank's reported profit and loss?

  1. A gain, because a wider spread raises the yield the bank earns on the exposure.Why not: Confuses the bank's position with a bond holding; the bank has no extra yield, only a larger expected credit loss.
  2. A gain, because the higher spread reduces the exposure at default through a lower recovery rate.Why not: Recovery rate does not reduce exposure; a lower recovery raises loss given default and CVA rises further.
  3. A mark-to-market loss, because the CVA increases even though no default has occurred. - correct
  4. No effect until the counterparty actually defaults, since the risk-free value of the contract is unchanged.Why not: Treats CVA as realised only at default; under fair value it is recognised through market-implied default probabilities.

CVA is a market price using spread-implied default probabilities. Wider spreads raise the CVA, which is booked as a loss unless hedged, for example with bought CDS protection.

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