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Give the definitions of debt valuation adjustment (DVA) and of credit valuation adjustment (CVA).

FRM® Part II · Credit Risk Measurement and Management · Credit Risk · FRM2.CR.12.b

The idea

Credit valuation adjustment (CVA) is the present value of the expected loss from the counterparty defaulting before the contract ends. Split the life into intervals. Let R be the recovery rate, qi the probability of default in interval i and vi the present value of the expected positive exposure at that time. Summing the expected loss over all the intervals of the contract's life gives the adjustment: CVA = Σ (1 − R) × qi × vi. Debt valuation adjustment (DVA) is the mirror image. It is the present value of the expected loss the counterparty bears if you default, using your own default probabilities and the exposure you create for the counterparty. Writing V for the no-default value of the contract, the value after both adjustments is: Adjusted = V − CVA + DVA. DVA is a gain to you: when your own credit spread widens, your liabilities are worth less and DVA rises. Exam point: keep the signs straight and use the right party's default probability in each term.

Formulas

Credit valuation adjustment

CVA = (1 − R) × Σ qᵢ × vᵢ

R
the recovery rate on the counterparty
Σ
the sum over the intervals of the contract's life
qᵢ
the counterparty's probability of defaulting in interval i
vᵢ
the present value of the expected positive exposure at that time

Assumes exposure and default are independent; wrong-way risk breaks this. Use (1 − R), not R, and interval default probabilities, not cumulative ones. DVA is the same sum using your own default probabilities.

Value of a derivative after both credit adjustments

adjusted value = no-default value − CVA + DVA

no-default value
the value assuming neither party can default
CVA
the credit valuation adjustment
DVA
the debt valuation adjustment

CVA is a cost and DVA a gain. Reversing either sign is the common error.

A practice question

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

In the pricing of an OTC derivative, credit valuation adjustment (CVA) is best defined as:

  1. the present value of the expected loss the counterparty bears if the bank itself defaults, added to the value.Why not: Describes DVA, which depends on the bank's own default, not CVA.
  2. the present value of the expected loss from the counterparty defaulting, deducted from the no-default value of the contract. - correct
  3. the notional amount multiplied by the counterparty's default probability.Why not: Uses the notional instead of the expected exposure and ignores recovery.
  4. the difference between the contract's market value and its collateral, regardless of default probability.Why not: That is an uncollateralised exposure measure; CVA also needs default probabilities and recovery.

CVA is the value of the counterparty-default option the bank has implicitly sold: the sum over time of (1 - R) x default probability x discounted expected positive exposure. It lowers the value of the contract to the bank.

A harder one

A bank estimates CVA over three one-year intervals. The counterparty's default probabilities in each interval are 1.0%, 1.2% and 1.4%, the present values of expected positive exposure are USD 2.0 million, 3.0 million and 2.5 million, and the recovery rate is 40%. The CVA is closest to:

  1. USD 91,000.Why not: Forgets the loss-given-default factor (1 - R) and uses the full exposure.
  2. USD 36,400.Why not: Multiplies by the recovery rate R instead of (1 - R).
  3. USD 54,600. - correct
  4. USD 64,800.Why not: Multiplies the cumulative default probability (3.6%) by the peak exposure and (1 - R) rather than summing interval by interval.

CVA = 0.6 x (0.010 x 2.0 + 0.012 x 3.0 + 0.014 x 2.5) million = 0.6 x 0.091 million = USD 54,600.

The hardest tier

A bank has a bilateral derivative with a counterparty. Counterparty default probabilities are 2% and 3% in the next two years, with present values of expected positive exposure of USD 4 million and 5 million. The bank's own default probabilities are 1% and 1.5%, with present values of expected negative exposure (what the bank would owe) of USD 3 million and 2 million. Recovery is 40% for both. The net adjustment to the no-default value, CVA and DVA together, is closest to:

  1. a reduction of USD 174,000.Why not: Subtracts DVA as if it were a second cost instead of adding it.
  2. a reduction of USD 102,000. - correct
  3. an increase of USD 102,000.Why not: Has the sign reversed: CVA (138,000) exceeds DVA (36,000), so the net is a reduction.
  4. a reduction of USD 138,000.Why not: Includes only CVA and ignores the DVA of 36,000.

CVA = 0.6 x (0.02 x 4 + 0.03 x 5) = 0.138 million. DVA = 0.6 x (0.01 x 3 + 0.015 x 2) = 0.036 million. Net adjustment = -0.138 + 0.036 = -0.102 million.

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