FRM Exam Part II · Credit Value Adjustment

Unilateral Credit Value Adjustment (CVA) Explained

Updated 11 October 2026 · Fact-checked

Unilateral CVA is the market value of the expected loss from a counterparty defaulting, seen from your side only. You compute it as LGD × Σ EE(tᵢ) × PD(tᵢ₋₁, tᵢ) × discount factor. Use discounted expected positive exposure, the marginal default probability in each period, and loss given default. For a claim on the counterparty, subtract CVA from the risk-free value.

Understand Unilateral Credit Value Adjustment (CVA)

A derivative with a counterparty has a risk-free value, which assumes both sides always pay. In reality the counterparty can default. If it defaults when the contract is worth a positive amount to you, you lose part of that value. If the contract is worth a negative amount to you, you still owe it. The loss is one-sided.

Credit value adjustment (CVA) is the price of that risk. It is the expected loss from counterparty default on the derivative, discounted to today. The value after adjustment is: risky value = risk-free value − CVA.

Unilateral CVA looks only at the counterparty's default. It treats you as default-free. The bilateral version also adds your own default (DVA), which is a separate topic.

Three inputs drive it. Exposure is how much you would lose if default happened at a date, measured by expected exposure (EE), the expected value of max(V, 0). Default probability is the chance the counterparty defaults in each time interval. Loss given default (LGD) is 1 − recovery rate. Multiply them for each interval and add up.

The standard formula assumes exposure and default are independent. If they are linked, you have wrong-way or right-way risk, and the simple product no longer holds.

Key formulas to remember

Unilateral CVA (discrete): CVA = LGD × Σ [ EE*(tᵢ) × PD(tᵢ₋₁, tᵢ) ]; EE* is discounted expected exposure, EE(tᵢ) × DF(tᵢ). Assumes exposure is independent of default.
Loss given default: LGD = 1 − R; R is the recovery rate. Use the same R for all dates unless told otherwise.
Marginal default probability: PD(tᵢ₋₁, tᵢ) = Q(tᵢ₋₁) − Q(tᵢ); Q(t) is the survival probability to time t. Q(0) = 1.
Survival probability from constant hazard: Q(t) = e^(−λt); λ is the constant default intensity. Approximately λ ≈ credit spread ÷ LGD (credit triangle).
Adjusted value: Risky value = Risk-free value − CVA; Unilateral CVA is a non-negative number that reduces the value of your claim on the counterparty.
Single-period shortcut: CVA ≈ PD × LGD × discounted EE; Use only when the question gives one period or one average exposure.

How to solve Unilateral Credit Value Adjustment (CVA) questions

Follow this order for any CVA calculation question. It keeps each input in the right place.

1Identify the party whose default matters. In unilateral CVA it is only the counterparty.
2Write down LGD. If you are given the recovery rate R, use LGD = 1 − R.
3Get the default probability for each interval. If given survival probabilities, take Q(tᵢ₋₁) − Q(tᵢ). If given a hazard rate, use Q(t) = e^(−λt).
4Take the expected exposure for each date. It must be the positive part only. If given discount factors, discount it.
5Multiply EE × discount factor × marginal PD for each interval.
6Sum across intervals and multiply by LGD.
7Subtract CVA from the risk-free value if the question asks for the adjusted price.
8Sanity check: CVA must be non-negative, and no larger than LGD × peak discounted EE × cumulative PD over the horizon.

Quickest way: Table method for CVA

When to use it: Use it when the question gives two to four time buckets with exposures and default probabilities.

Pull LGD out of the sum. Multiply once at the end.
Compute each bucket as discounted EE × marginal PD.
Add the buckets, then multiply by LGD.
Check the units. Exposure in USD gives CVA in USD.
Eliminate options that are negative, that are larger than LGD × peak discounted EE × cumulative PD over the whole horizon, or that use cumulative instead of marginal PD.

Common mistakes in Unilateral Credit Value Adjustment (CVA)

Using cumulative default probability for every bucket
The table lists cumulative PD and it is quicker to copy.
Fix: Use the marginal PD for each interval, the difference between consecutive cumulative values. Using cumulative values counts the same default several times.
Forgetting to multiply by LGD
Students stop after PD × EE and treat it as the loss.
Fix: PD × EE is expected exposure at default. Multiply by 1 − R to get the loss.
Using expected value of the contract instead of expected positive exposure
Netting positive and negative values looks natural.
Fix: Take the expected value of max(V, 0). A negative value gives you no gain when the counterparty defaults.
Skipping discounting
The question gives discount factors in a separate line.
Fix: CVA is a present value. Multiply each term by its discount factor unless EE is already stated as discounted.
Adding CVA to the risk-free value
Confusing the sign of the adjustment.
Fix: For a claim on a risky counterparty, value falls. Risky value = risk-free value − CVA.
Mixing in your own default
Bilateral CVA and DVA are taught nearby.
Fix: Unilateral CVA ignores your default. Own-default benefit is DVA and appears only in bilateral CVA.

Worked examples

Example 1

A bank has an OTC swap with a counterparty. Recovery rate is 40%. Over two years, discounted expected exposures are USD 2.0 million (year 1) and USD 3.0 million (year 2). Marginal default probabilities are 2% in year 1 and 3% in year 2. Calculate unilateral CVA.

Show the solution

LGD = 1 − 0.40 = 0.60.
Year 1 term: 2.0 × 0.02 = 0.04 million.
Year 2 term: 3.0 × 0.03 = 0.09 million.
Sum = 0.04 + 0.09 = 0.13 million.
CVA = 0.60 × 0.13 = 0.078 million.

Answer: CVA = USD 78,000

Example 2

A derivative has a risk-free value of EUR 5.00 million to a bank. Survival probabilities for the counterparty are 1.00 at t = 0, 0.97 at year 1 and 0.93 at year 2. Discounted EE is EUR 4.0 million for year 1 and EUR 6.0 million for year 2. LGD is 50%. Find the risky value.

Show the solution

Marginal PD year 1 = 1.00 − 0.97 = 0.03.
Marginal PD year 2 = 0.97 − 0.93 = 0.04.
Year 1 term: 4.0 × 0.03 = 0.12.
Year 2 term: 6.0 × 0.04 = 0.24.
Sum = 0.36. CVA = 0.50 × 0.36 = 0.18 million.
Risky value = 5.00 − 0.18 = 4.82 million.

Answer: Risky value = EUR 4.82 million (CVA = EUR 0.18 million)

Exam tips

Check whether the exposure given is already discounted. Questions often hide this in one phrase.
Look for the word cumulative versus marginal in the default probability table.
Read the question for which party's default counts. Unilateral means only the counterparty.
Expect a conceptual twist: the formula assumes no wrong-way risk. If exposure rises as credit quality falls, the simple CVA understates the loss.
Do the arithmetic in millions, then convert units at the end.

Practice questions from Credit Value Adjustment

Unilateral Credit Value Adjustment (CVA): frequently asked questions

What is the CVA formula for FRM Part II?

CVA = LGD × Σ discounted EE(tᵢ) × marginal PD(tᵢ₋₁, tᵢ). It is the discounted expected loss from counterparty default. It assumes exposure and default are independent.

Is CVA a loss or a price adjustment?

It is both. It is the expected loss, and it is also the amount deducted from the risk-free price to get the risky price. Banks charge it to counterparties and often hedge it.

What is the difference between unilateral and bilateral CVA?

Unilateral CVA counts only the counterparty's default and is subtracted from the risk-free value of your claim on the counterparty. Bilateral CVA also includes your own default through DVA: bilateral value = risk-free value − CVA + DVA. DVA is added to value and reduces the net adjustment. It rises when your own credit spread widens, producing an accounting gain.

Why use expected positive exposure and not net value?

On default you lose only when the contract has positive value to you. If it is negative, you still owe it. So exposure is max(V, 0).